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Fourier transform

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Solo Hermelin

The document provides a detailed explanation of the Fourier Transform, including its properties such as linearity, symmetry, conjugate functions, scaling, derivatives, convolution, and modulation. It presents mathematical proofs and identities related to the Fourier Transform and includes examples of its applications. Additionally, it includes links to visual aids for understanding convolution and signal processing concepts.

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Fourier Transform SOLO HERMELIN Updated: 22.07.07 Run This http://www.solohermelin.com
Fourier Transform ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF ωω exp:F SOLO Jean Baptiste Joseph Fourier 1768-1830 F (ω) is known as Fourier Integral or Fourier Transform and is in general complex ( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+= Using the identities ( ) ( )t d tj δ π ω ω =∫ +∞ ∞− 2 exp we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1 F= ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ]00 2 1 2 exp 2 expexp 2 exp ++−=−=−=     −= ∫∫ ∫ ∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +∞ ∞− tftfdtfd d tjf d tjdjf d tjF ττδττ π ω τωτ π ω ωττωτ π ω ωω ( ) ( ){ } ( ) ( )∫ +∞ ∞− == π ω ωωω 2 exp: d tjFFtf -1 F ( ) ( ) ( ) ( )[ ]00 2 1 ++−=−∫ +∞ ∞− tftfdtf ττδτ If f (t) is continuous at t, i.e. f (t-0) = f (t+0) This is true if (sufficient not necessary) f (t) and f ’ (t) are piecewise continue in every finite interval1 2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫ +∞ ∞− dttf
Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Linearity1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫ +∞ ∞− F Symmetry2 ( )tF -1 F F ( )ωπ −f2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }tFdttjtFf dt tjtFf d tjFtf t F=−=−⇒=⇒= ∫∫∫ +∞ ∞− +∞ ∞− ↔ +∞ ∞− ωωπ π ωω π ω ωω ω exp2 2 exp 2 exp Proof: Conjugate Functions3 ( )tf * -1 F F ( )ω−* F Proof: ( ) ( ) ( ) ( ) ( ) ( ){ }tf d tjF d tjFtf **** 2 exp 2 exp 1- F=−=−= ∫∫ +∞ ∞− →− +∞ ∞− π ω ωω π ω ωω ωω
Fourier Transform ( ){ } ( ) ( ) ( )       =      −=−= ∫∫ +∞ ∞− = +∞ ∞− a F aa d a jfdttjtaftaf ta ωτ τ ω τω τ 1 expexp:F ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωωωωω ω ωω FjdttjjtfF d d dttjtftfF nn n n −=−−=→−== ∫∫ +∞ ∞− +∞ ∞− FF expexp: SOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Scaling4 Derivatives5 Proof: ( )taf -1 F F       a F a ω1 Proof: Corollary: for a = -1 ( )tf − -1 F F ( )ω−F ( ) ( )tftj n − -1 F F ( )ω ω F d d n n ( )tf td d n n -1 F F ( ) ( )ωω Fj n ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωω π ω ωωω π ω ωωω Fj d tjjFtf td dd tjFFtf nn n n 1-1- FF ==→== ∫∫ +∞ ∞− +∞ ∞− 2 exp 2 exp
Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Convolution6 Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωττωττωτωτ ττωττωττττωτττ τ 212121 212121 expexpexp expexpexp: FFFdjfdduujufjf ddttjtfjfdtdtfftjdtff ut =         −=         −−= −−−−=         −−=         − ∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− F ( ) ( )tftf 21 -1 F F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫ +∞ ∞− −= τττ dtfftftf 2121 :* -1 F F ( ) ( )ωω 21 FF The animations above graphically illustrate the convolution of two rectangle functions (left) and two Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a function of t, the position indicated by the vertical green line. The gray region indicates the product as a function of g (τ) f (t-τ) , so its area as a function of t is precisely the convolution. http://mathworld.wolfram.com/Convolution.html
Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 Proof: ( ) ( ) ( )∫ +∞ ∞− −= dttjtfF ωω exp11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− =−=−= π ω ωω π ω ωω π ω ωω 22 exp 2 exp 2 * 112 * 2 * 12 * 1 d FF d dttjtfFdt d tjFtfdttftf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− −=== π ω ωω π ω ωω π ω ωω 22 exp 2 exp 21122121 d FF d dttjtfFdt d tjFtfdttftf ( ) ( ) ( )∫ +∞ ∞− −= π ω ωω 2 exp * 2 * 2 d tjFtf ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− =−→−= dttjtfFdttjtfF ωωωω expexp 1111 ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− −=−= ωωω π ωωω π dFFdFFdttftf 212121 2 1 2 1
Signal Duration and BandwidthSOLO ( )tf -1 F F ( )ωFRelationships from Parseval’s Formula ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 ( ) ( ) ,2,1,0 2 1 2 22 == ∫∫ ∞+ ∞− ∞+ ∞− nd d Sd dttst m m m ω ω ω π Choose ( ) ( ) ( ) ( )tstjtftf m −== 21 ( ) ( )tftj n − -1 F F ( )ω ω F d d n n and use 5a Choose ( ) ( ) ( ) n n td tsd tftf == 21 and use 5b ( )tf td d n n -1 F F ( ) ( )ωω Fj n ( ) ( ) ,2,1,0 2 1 22 2 == ∫∫ ∞+ ∞− ∞+ ∞− ndSdt td tsd m n n ωωω π ( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0 2 * ==      = ∫∫ +∞ ∞− +∞ ∞− mnd d Sd S j dt td tsd tstj m m n n n n mm ω ω ω ωω π Choosec ( ) ( ) n n td tsd tf =1 ( ) ( ) ( )tstjtf m −=2
Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Modulation9 Shifting: for any a real8 Proof: ( ) ttf 0cos ω -1 F F ( ) ( )[ ]00 2 1 ωωωω −++ FF Proof: ( ) ( )[ ]tjtjt 000 expexp 2 1 cos ωωω −+= ( )atf − -1 F F ( ) ( )ωω ajF −exp ( ) ( )tajtf exp -1 F F ( )aF −ω ( ){ } ( ) ( ) ( ) ( )( ) ( ) ( )ωωττωτω τ Fajdajfdttjatfatf at −=+−=−−=− ∫∫ +∞ ∞− =− +∞ ∞− expexpexp:F ( ) ( ){ } ( ) ( ) ( ) ( ) ( )( ) ( )aFdttajtfdttjtajtftajtf −=−−=−= ∫∫ +∞ ∞− +∞ ∞− ωωω expexpexp:expF use shifting property with a=±ω0
( )atf − -1 F F ( ) ( )ωω ajF −exp Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform (Summary) Linearity1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫ +∞ ∞− F Symmetry2 ( )tF -1 F F ( )ωπ −f2 Conjugate Functions3 ( )tf * -1 F F ( )ω−* F Scaling4 ( )taf -1 F F       a F a ω1 Derivatives5 ( ) ( )tftj n − -1 F F ( )ω ω F d d n n ( )tf td d n n -1 F F ( ) ( )ωω Fj n Convolution6 ( ) ( )tftf 21 -1 F F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫ +∞ ∞− −= τττ dtfftftf 2121 :* -1 F F ( ) ( )ωω 21 FF ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 Shifting: for any a real8 ( ) ( )tajtf exp -1 F F ( )aF −ω Modulation9 ( ) ttf 0 cos ω -1 F F ( ) ( )[ ]00 2 1 ωωωω −++ FF ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− −=−= ωωω π ωωω π dFFdFFdttftf 212121 2 1 2 1
Fourier Transform SOLO ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF ωω exp:F ( ) ( ){ } ( ) ( )∫ +∞ ∞− == π ω ωωω 2 exp d tjFFtf 1- F -1 F F ( ) ( ) ( )∫ +∞ ∞− =− dttjtfF ωω exp * - complex conjugate( ) ( ) ( )∫ +∞ ∞− = dttjtfF ωω exp** ( ) ( ) imaginarytf realtf ( ){ } ( ){ } 0Re 0Im = = tf tf ( ) ( ) ( ) ( )tftf tftf * * −= = ( ) ( ) ( ) ( )ωω ωω * * FF FF −=− =− ( ) realtf ( ) ( )ωω * FF =− ( ) imaginarytf ( ) ( )ωω * FF −=− Therefore Fourier Transform of Real or Imaginary Functions
Fourier Transform SOLO ( ) realtf ( ) ( )ωω * FF =− ( ) imaginarytf ( ) ( )ωω * FF −=− ( ) realtf ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe ( ) imaginarytf ( ) ( ) ( ) ( )       −= −−= ωω ωω FF FF ImIm ReRe ( ) ( ) ( ) ( ){ } ( ) ( )∫ +∞ ∞− −==+= dttjtftfFjFF ωωωω exp:ImRe F ( ){ } ( ) ( ) ( ) ( ) ( )ωωω −==−−=− ∫∫ +∞ ∞− +∞ ∞− Fdttjtfdttjtftf expexpF ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: ( ) realtf ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )      =−−=⇔−−= =−+=⇔−+= ωωω ωωω FjFFtftftftf FFFtftftftf evenodd eveneven Im5.05.0: Re5.05.0: F F ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )      =−−=⇔−−= =−+=⇔−+= ωωω ωωω FFFtftftftf FjFFtftftftf evenodd eveneven Re5.05.0: Im5.05.0: F F ( ) imaginarytf Fourier Transform of Real or Imaginary Functions (continue – 1)
Fourier Transform ω ( )ωFRe ( )ωFIm Real & Even t ( )tfIm ( )tfRe Real & Even SOLO ω ( )ωFRe ( )ωFIm Imaginary & Odd t ( )tfIm ( )tfRe Real & Odd ω ( )ωFRe ( )ωFIm Imag. &Even t ( )tfIm ( )tfRe Imag.& Even ω ( )ωFRe ( )ωFIm Real & Odd t ( )tfIm ( )tfRe Imag. & Odd ( ) realtf ( ) ( ) ( ) ( )[ ]tftftftf even −+== 5.0: ( ) ( ) ( ) ( )[ ]tftftftf even −+== 5.0: ( ) imaginarytf ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −== −+== FF FFtftf even ReRe 5.0FF ( ) ( )ωω * FF =− ( ) ( ) ( ) ( )[ ]tftftftf odd −−== 5.0: ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −−== −−== FjFj FFtftf even ImIm 5.0FF ( ) realtf ( ) ( )ωω * FF =− ( ) ( )ωω * FF −=− ( ) ( ) ( ) ( )[ ]tftftftf odd −−== 5.0: ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −== −+== FjFj FFtftf even ImIm 5.0FF ( ) imaginarytf ( ) ( )ωω * FF −=− ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −−== −−== FF FFtftf even ReRe 5.0FF Fourier Transform of Real or Imaginary Functions (continue – 2)
Fourier Transform SOLO ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( )∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +−+= −+=−== dtttftfjdtttftf dttjttftfdttjtftfF oddevenoddeven oddeven ωω ωωωω sincos sincosexp:F ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: ( ) ( ) ( )tftftf oddeven += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ +∞+∞+∞+∞ −→ ∞− +∞ ∞− =+−=+= 0000 0 cos2coscoscoscoscos dtttfdtttfdfdtttfdtttfdtttf eveneven f eveneven t eveneven even ωωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0coscoscoscoscos 000 0 =+−=+= ∫∫∫∫∫ +∞+∞ − +∞ −→ ∞− +∞ ∞− dtttfdfdtttfdtttfdtttf odd f oddodd t oddodd odd ωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0sinsinsinsinsin 000 0 =+−−=+= ∫∫∫∫∫ +∞+∞+∞ −→ ∞− +∞ ∞− dtttfdfdtttfdtttfdtttf even f eveneven t eveneven even ωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ +∞+∞+∞ − +∞ −→ ∞− +∞ ∞− =+−−=+= 0000 0 sin2sinsinsinsinsin dtttfdtttfdfdtttfdtttfdtttf oddodd f oddodd t oddodd odd ωωττωτωωω τ τ     Therefore ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞+∞+∞ ∞− −=−== 00 sin2cos2exp: dtttfjdtttfdttjtftfF oddeven ωωωω F ( ) ( ) ( )[ ] ( ) ( )∫ +∞ =−+= 0 cos25.0 dtttfFFF eveneven ωωωω ( ) ( ) ( )[ ] ( ) ( )∫ +∞ =−−= 0 sin25.0 dtttfjFFF oddodd ωωωω Odd and Even Parts
Fourier Transform ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ++−= ++=== π ω ωωωω π ω ωωωω π ω ωωωω π ω ωωω 2 cosImsinRe 2 sinImcosRe 2 sincosImRe 2 exp: d tFtFj d tFtF d tjtFjF d tjFFtf -1 F SOLO ( ) 00: <∀= ttfCausal Causal Functions A causal functions is a equal zero for negative t ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: Since and ( ) 0>− ttf we have ( ) ( ) ( ) 022 >== ttftftf oddeven ( ) realtf ( ) ( ){ } ( ) ( ) ( ) ( )[ ]∫ +∞ ∞− −== π ω ωωωωω 2 sinImcosRe d tFtFFtf -1 F ( ) causalrealtf & ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 sinIm4 2 cosRe4 2 sinIm22 2 cosRe22 00 >−== −==== ∫∫ ∫∫ ∞+∞+ +∞ ∞− +∞ ∞− t d tF d tF d tFtf d tFtftf oddeven π ω ωω π ω ωω π ω ωω π ω ωω ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe ( ) ( ) ( ) ( ) ( ) 0sinIm 2 cosRe 2 00 >−== ∫∫ +∞+∞ tdtFdtFtf ωωω π ωωω π
Fourier TransformSOLO ( ) 00: <∀= ttfCausalReal & Causal Functions ( ) ( ) ( ) 022 >== ttftftf oddeven ( ) causalrealtf & ( ) ( ) ( ) ( ) ( ) 0sinIm 2 cosRe 2 00 >−== ∫∫ +∞+∞ tdtFdtFtf ωωω π ωωω π ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ +∞ ∞− −=+= dttjttfFjFF ωωωωω sincosImRe ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫∫ +∞ ∞− +∞+∞ ∞− +∞+∞ ∞− −=      −== dtduttuuFdttdutuuFdtttfF ω π ω π ωω cossinIm 2 cossinIm 2 cosRe 00 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫∫ +∞ ∞− +∞+∞ ∞− +∞+∞ ∞− −=      −=−= dvdtttvvFdttdvtvvFdtttfF ω π ω π ωω sincosRe 2 sincosRe 2 sinIm 00 Therefore ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ −= dtduttuuFF ω π ω cossinIm 2 Re 0 ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ −= dtdvttvvFF ω π ω sincosRe 2 Im 0 But also ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ = dtduttuuFF ω π ω coscosRe 2 Re 0 ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ = dtdvttvvFF ω π ω sinsinRe 2 Im 0 Real & Causal Functions Real & Causal Functions ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe
Fourier Transform SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− +=+= π ω ωω π ω ωω 2 sin 2 cos d tFj d tFtftftf oddeven ( ) ( ) ( ) ( )tf tftf tf eveneven −= −+ = 2 : ( ) ( ) ( ) ( )tf tftf tf oddodd −−= −− = 2 : ( ) ( ) ( ) ( ) ( ) ( )tf d tF tftf tf eveneven −== −+ = ∫ +∞ ∞− π ω ωω 2 cos 2 ( ) ( ) ( ) ( ) ( ) ( )∫ +∞ ∞− −−== −− = tf d tFj tftf tf oddodd π ω ωω 2 sin 2 ( ) ( ) ( )ωωω FjFF ImRe += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +− +=+= π ω ωω π ω ωω π ω ωω π ω ωω 2 sinRe 2 sinIm 2 cosIm 2 cosReImRe d tFj d tF d tFj d tFtfjtftf ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −= π ω ωω π ω ωω 2 sinIm 2 cosReRe d tF d tFtf ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− += π ω ωω π ω ωω 2 sinRe 2 cosImIm d tF d tFtf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −=−=−+−=− π ω ωω π ω ωω 2 sin 2 cos d tFj d tFtftftftftf oddevenoddeven
Fourier TransformSOLO Examples of Fourier Transform
Fourier TransformSOLO Examples of Fourier Transform
Fourier TransformSOLO Examples of Fourier Transform
Fourier TransformSOLO Examples of Fourier Transform
Fourier TransformSOLO Examples of Fourier Transform
Fourier TransformSOLO Examples of Fourier Transform
Fourier Transform ( )       > < =Π 2 1 0 2 1 1 t t t 2 1 2 1 − ( )tΠ t Rectangle 1 ( ) ( )      −<− < > = τ ττ τ τ t tt t t 2/1 2/ 2/1 lim Limiter τ ( )[ ] ( )tt sgnlimlim2 0 =→ ττ 0 ( )tτlim t 2/1 2/1− τ τ− SOLO Special Symbols ( )     > <− =Λ 10 11 t tt t 11− ( )tΛ t Triangle 1 ( )    < > = 00 01 t t tH 0 ( )tH t Heaviside unit step 1 ( )    <− > = 01 01 sgn t t t 0 ( )tsgn t Signum 1 1− 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1td d
Fourier Transform ( ) ( ) ( ) ( )    ≠ =∞ =         > ≤ =                     <− ≤ > =      = →→→ 00 0 0 2/1 lim 2/1 2/ 2/1 limlimlim: 000 t t t t t tt t td d t td d t τ ττ τ ττ τ δ ττττ SOLO Special Symbols 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1td d δ (t) function Since ( )( ) ( )tt sgn 2 1 limlim 0 = → ττ we have also δ (t) function is defined as: ( ) ( )t td d t sgn 2 1 =δ 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1 0 t ( )tδ Area = 1 0→τ ( ) ( )      −<− < > = τ ττ τ τ t tt t t 2/1 2/ 2/1 lim Limiter τ ( )[ ] ( )tt sgnlimlim2 0 = → ττ 0 ( )tτlim t 2/1 2/1− τ τ−
Fourier TransformSOLO Special Symbols Properties of δ (t) function 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1 0 t ( )tδ Area = 1 0→τ ( ) ( )tt −= δδδ (t) is a even function:2 ( ) ( )    ≠ =∞ =         > ≤ = → 00 0 0 2/1 lim 0 t t t t t τ ττ δ τ 1 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]00 2 1 ++−=−=− ∫∫ +∞ ∞− −=+∞ ∞− τττδτδ δδ ffdtttfdtttf uu Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]00 2 1 00 2 1 lim 2 1 lim sgn 2 1 limsgn 2 1 limsgnlim 2 1 lim 0 0 sgn 2 1 ++−= ++−−−−+−+=    −+−+= −−−=−=− →∞ + − − →∞ + − →∞ + −→∞ + − →∞ =+ − →∞ ∫∫ ∫∫∫ ττ ττ ττττδ τ τ δ ff fTfTffTfTftfdtfdTfTf tfdtttftdtfdtttf T T T T T T T T TT T T T t dt d tT T T 4 Fourier Transform ( ){ } ( ) ( ) ( ) ( ) 10exp 2 1 0exp 2 1 exp =++−=−= ∫ +∞ ∞− jjdttjtt ωδδF
Fourier Transform ( )                   −=       + =       =       =       −= = + = → → → → → − → → εεε εε εε επ εεπ ε ε ε π δ ε ε ε ε ε ε ε ε ε x Ln x x J x Ai x x x x x x 2 exp 1 lim 11 lim 1 lim sin 1 lim 4 exp 2 1 lim lim lim 1 2 0 /1 0 0 0 2 0 1 0 220 SOLO Special Symbols δ (t) function The δ (t) function can be defined as the following limit as ε→0 Ai is the Airry function, ( ) ∫ ∞       += 0 3 3 cos 1 dttx t xAi π ( ) ( )[ ]∫ + − −−= π π τττ π dxnjxJn sinexp 2 1 Friedrich Wilhelm Bessel 1784 - 1846 Edmond Nicolas Laguerre 1834 - 1886 Jn (x) is the Bessel function of the first kind, and Ln (x) is the Laguerre polynomial of arbitrary positive order.
Fourier TransformSOLO Special Symbols δ (t) function The δ (t) function can be defined also by the limit n→∞ ( )                   + = →∞ x xn x n 2 1 sin 2 1 sin 2 1 lim π δ ( ) ( ) ( ) ( )tnsincn tnn xnnx n n n →∞ →∞ →∞ = Π= −= lim lim explim 22 πδ( )     > ≤ =Π 2/10 2/11 x x x ( ) ( ) x x xsinc π πsin =
Fourier TransformSOLO δ (t) function ( )       > ≤ = 2 0 2 1 : 02 τ τ τδ π τ t te t tfj Use It’s Fourier Transform is ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )τπ τπ πττ δ τ τ πτ τ ππ ττ ff ff ffj e dtedtetf tffj tffjtfj − − = − ===∆ + − −+ − − ∞+ ∞− − ∫∫ 0 0 2/ 2/0 22/ 2/ 22 sin 2 11 0 0 For any function φ (t), defined at t=0- and t=0+, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]+− − → + → − → + → + − → +∞ ∞− → +=+= +== − + − + ∫∫∫∫ 00 2 11 lim 1 lim 1 lim 1 lim 1 limlim 0 2/ 0 2/ 0 0 0 2/ 2 0 2/ 0 2 0 2/ 2/ 2 00 000 ϕϕϕ τ ϕ τ ϕ τ ϕ τ ϕ τ ϕδ τ τ τ τ τ π τ τ π τ τ τ π τ τ τ tttt dttedttedttedttt tfjtfjtfj ( ) ( )tt δδτ τ = →0 lim
Fourier Transform ( )       > ≤ = 2 0 2 1 : 02 τ τ τδ π τ t te t tfj SOLO δ (t) function ( ) ( )[ ] ( )τπ τπ τ ff ff f − − =∆ 0 0sin ( ) ( )tt δδτ τ = →0 lim ( ) ( )[ ] ( ) 1 sin limlim 0 0 00 = − − =∆ →→ τπ τπ τ τ τ ff ff f ( )[ ] ( )τπ τπ ff ff − − 0 0sin ( ) ∫ +∞ ∞− = fdet tfj π δ 2
Fourier Transform ( )      ∆>− ∆≤− ∆=∆ 2/0 2/ 1 : 0 0 fff fff ffS f SOLO δ (f) function Define: In the time domain we obtain: ( ) ( ) ( ) ( ) tfj ff ff tfjff ff tfjtfj ff e tf tf tj e f fde f fdefSts 0 0 0 0 0 2 2/ 2/ 22/ 2/ 22 sin 2 11 π π ππ π π π ∆ ∆ = ∆ = ∆ == ∆+ ∆− ∆+ ∆− +∞ ∞− ∆∆ ∫∫ For any function Φ (f), defined at f=f0- and f=f 0+ , we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]+− − + + − Φ+Φ=Φ ∆ +Φ ∆ = Φ ∆ +Φ ∆ =Φ ∆ =Φ ∆− →∆ ∆+ →∆ ∆− →∆ ∆+ →∆ ∆+ ∆− →∆ +∞ ∞− ∆ →∆ ∫∫∫∫ 00 2/ 0 2/ 0 2/ 0 2/ 0 2/ 2/ 00 2 11 lim 1 lim 1 lim 1 lim 1 limlim 0 0 0 0 0 0 0 0 0 0 ffff f ff f dff f dff f dff f dfffS f ff f ff f f f ff f ff f f ff ff f f f ( ) ( )0 0 lim fffS f f −=∆ →∆ δ ( ) ( )0 0 :lim fffS f f −=∆ →∆ δ ( ) tfj f f ets 02 0 lim π =∆ →∆ ( ) ( ) ∫ +∞ ∞− −− =− tdeff tffj 02 0 π δ
Fourier Transform ( ) ( )∑−= += N Nn N Tntftf : SOLO fN (f) N-Periodic Extension of a function f (t) Define N- extension of f (t)
Fourier Transform ( ) ( )∑−= += N Nn N Tntt δδ : SOLO δN (f) function Define Let find the Fourier transform of δN (f) ( ) ( ) ( ) ( )[ ] [ ] ( )Tf TNf ee j j ee eee eee e ee etdenTttdetf TfjTfj TNfjTNfj TfjTfjTfj TNfjTNfj Tfj Tfj NTfjTNfj N Nn Tnfj N Nn tfjtfj NN π π δδ ππ ππ πππ ππ π π ππ πππ sin 2 1 2sin 2 2 1 1 2 1 2 2 1 2 2 1 2 2 1 2 2 1222 222             + = − − = = −         − = − − = =+==∆ −       +−      + −       +      +− +− −=−= +∞ ∞− − +∞ ∞− − ∑∑ ∫∫ We can see that ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ,2,1,0 sin 12sin sin 1212sin ±±=∆= + = + +++ =      +∆ kf Tf TNf kTf NkTNf T k f NN π π ππ ππ N-extension of δ (t)
Fourier TransformSOLO δN (f) function (continue – 1) ( ) ( )∑−= += N Nn N Tntt δδ : ( ) ( )[ ] ( ) ∑−= = + =∆ N Nn Tnfj N e Tf TNf f π π π 2 sin 12sin δN (t) is a periodic function with a time period of T . ΔN (f) is a periodic function with a frequency period of f0 = 1/T . ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] Tn n TTnj e dfedff N Nn n n N Nn T T TnfjN Nn T T Tnfj T T N 1sin1 2 00 01 2/1 2/1 22/1 2/1 2 2/1 2/1 ====∆ ∑∑∑ ∫∫ −= ≠← =← −= − −−= + − + −  π π π π π
Fourier TransformSOLO δN (f) function (continue – 2) When N → ∞ the peak goes to infinity and the null-to-null bandwidth goes to zero. This resembles to a delta function. To prove that this is the case let compute: ΔN (f) is a periodic function with a frequency period of f0 = 1/T , with peak amplitude of (2 N+1) and null-to-null bandwidth of 2/ [(2N+1) T]. ( ) ( ) ( ) T dff T T N 1 2/1 2/1 =∆∫ + − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 limlim 2/1 2/1 2 2/1 2/1 Φ=Φ=Φ∆ ∑ ∫∫ −= + − ∞→ + − ∞→ T dffedfff N Nn T T Tnfj N T T N N π
Fourier TransformSOLO δN (f) function (continue – 3) ( ) ( ) ( ) T dff T T N 1 2/1 2/1 =∆∫ + − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 limlim 2/1 2/1 2 2/1 2/1 Φ=Φ=Φ∆ ∑ ∫∫ −= + − ∞→ + − ∞→ T dffedfff N Nn T T Tnfj N T T N N π Therefore ( ) ( ) ( ) ( ) ∑∫ ∞+ −∞= + − ∞→ ∞       +=∆=∆ m T T N N T m f T dfff δ 1 lim: 2/1 2/1
Fourier TransformSOLO δN (f) function (continue – 4) Let compute the convolution between f (t) and δN (f) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tfTntfdTntfdTntfttf N N Nn N Nn N Nn N =+=+−=+−=∗ ∑∑ ∫∫ ∑ −=−= +∞ ∞− +∞ ∞− −= ττδτττδτδ : Therefore ( ) ( ) ( )ttftf NN δ∗= Using this relation the Fourier Transform of fN (t) is given by ( ) ( ) ( ) ( ) ( )[ ] ( )Tf TfN fFffFfF NN π π sin 12sin + =∆= ( ) ( ) ( ) ( )Tntfttftf N Nn NN +=∗= ∑−= δ If N → ∞ then ( ) ( ) ( ) ( ) ( ) ∑∑ ∑ ∞+ −∞= ∞+ −∞= +∞ −∞= ∞∞       −      =      −=       −=∆= mm m T m f T m F TT m ffF T T m f T fFffFfF δδ δ 11 1 ( ) ( ) ∑ ∑ ∑ ∞+ −∞= ∞+ −∞= − +∞ −∞= ∞       =             −      = += m t T m j m n e T m F T T m f T m F T Tntftf π δ 2 1 1 1 F
Fourier TransformSOLO δN (f) function (continue – 4) ( ) ∑ +∞ −∞= ∞       −      = m T m f T m F T fF δ 1 ( ) ( ) ∑∑ +∞ −∞= +∞ −∞= ∞       =+= m t T m j n e T m F T Tntftf π21 f∞ (t) is a periodic function with a time period of T . F∞ (f) is a periodic function with a frequency period of f0 = 1/T . We obtained the Fourier Series description of a periodic function ( ) ( )∫∑ +∞ ∞− +∞ −∞= ∞ =      == tdetf TT m F T aeatf t T m j m m t T m j m ππ 22 11 If we define ( ) ( )     > ≤ = 2/0 2/ 0 Tt Tttf tf ( ) ( )∫ + − − = 2/ 2/ 2 0 T T tfj tdetffF π then ( ) ( )∫∑ + − +∞ −∞= ∞ == 2/ 2/ 22 1 T T t T m j m m t T m j m tdetf T aeatf ππ
Fourier Transform ( ) ππ ≤≤−= xxxf SOLO Simple Fourier Series ( ) ( ) ( ) 0cos 1 cos 1 === ∫∫ + − + − π π π π ππ dxxnxdxxnxfan ( ) ( ) ( ) ( ) ( ) ( ) nn xn n xnx dxxnxdxxnxfb n n 1 0 2 0 1 2 sincos2 sin 1 sin 1 + + − + − − =               +      −= == ∫∫ ππ π π π π π ππ http://en.wikipedia.org/wiki/Fourier_series Square wave equation ( ) ( ) ( ) ( )( )xN N xxxfSN 1sin 1 1 3sin 3 1 sin − − +++=  Sawtooth wave equation http://en.wikipedia.org/wiki/Sawtooth_wave http://en.wikipedia.org/wiki/Square_wave ( ) ( ) ( ) ( ) ( )xn n xxxfS n N sin 1 22sinsin2 1+ − ++−=  http://mathworld.wolfram.com/FourierSeries.html
Fourier Transform ( ) ( ) ∑ ∞ =       = 1 22 sin 2 sin 8 k Triangle k xkk xf π π SOLO Simple Fourier Series Triangular wave equation
Fourier Transform ( ) ( ) ∑ ∞ =       = 1 22 sin 2 sin 8 k Triangle k xkk xf π π SOLO Simple Fourier Series Triangular wave equation http://mathworld.wolfram.com/FourierSeries.html
SignalsSOLO Signal Duration and Bandwidth then ( ) ( )∫ +∞ ∞− − = tdetsfS tfi π2 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 t t∆2 t ( ) 2 ts f f f∆2 ( ) 2 fS ( ) ( ) ( ) 2/1 2 22 :               − =∆ ∫ ∫ ∞+ ∞− +∞ ∞− tdts tdtstt t ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = tdts tdtst t 2 2 : Signal Duration Signal Median ( ) ( ) ( ) 2/1 2 22 2 4 :               − =∆ ∫ ∫ ∞+ ∞− +∞ ∞− fdfS fdfSff f π ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = fdfS fdfSf f 2 2 2 : π Signal Bandwidth Frequency Median Fourier
Signals ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 SOLO Signal Duration and Bandwidth (continue – 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− + +∞ ∞− +∞ ∞− − +∞ ∞− +∞ ∞− − +∞ ∞− =         =         =         = dffSfSdfdesfS dfdesfSdfdefSsdss tfi tfitfi ττ τττττττ π ππ 2 22 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 ( ) ( ) ( )∫ +∞ ∞− == fdefSfi td tsd ts tfi π π 2 2' ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− + +∞ ∞− +∞ ∞− − +∞ ∞− +∞ ∞− − +∞ ∞− =         −=         −=         −= dffSfSfdfdesfSfi dfdesfSfidfdefSfsidss tfi tfitfi 222 22 2'2 '2'2'' πττπ ττπττπτττ π ππ ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSds 22 ττ Parseval Theorem From From ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSfdtts 2222 4' π
Signals ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− − ∞+ ∞− +∞ ∞− +∞ ∞− − ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ===== dffS fd fd fSd fS i dffS fdtdetstfS dffS tdfdefStst dffS tdtstst tdts tdtst t fifi 22 2 2 2 22 2 2 : π ππ SOLO Signal Duration and Bandwidth ( ) ( )∫ +∞ ∞− − = tdetsfS tfi π2 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 Fourier ( ) ( )∫ +∞ ∞− − −= tdetsti fd fSd tfi π π 2 2 ( ) ( )∫ +∞ ∞− = fdefSfi td tsd tfi π π 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− − =         ==== tdts td td tsd tsi tdts tdfdefSfts tdts fdtdetsfSf tdts fdfSfSf fdfS fdfSf f fifi 22 2 2 2 22 2 2222 : ππ ππππ
Signals ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− =≤         dffSfdttstdttsdttstdtts 222222 2 2 4' 4 1 π ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSdts 22 τ SOLO Signal Duration and Bandwidth (continue – 1) 0&0 == ftChange time and frequency scale to get From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttgdttfdttgtf 22 Choose ( ) ( ) ( ) ( ) ( )ts td tsd tgtsttf ':& === ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttsdttstdttstst 22 ''we obtain ( ) ( )∫ +∞ ∞− dttstst 'Integrate by parts ( )    = += →    = = sv dtstsdu dtsdv stu ' ' ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− −−= dttststdttsstdttstst '' 2 0 2  ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −= dttsdttstst 2 2 1 ' ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSfdtts 2222 4' π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− =≤ dffS dffSf dtts dttst dtts dffSf dtts dttst 2 222 2 2 2 222 2 2 44 4 1 ππ assume ( ) 0lim = →∞ tst t
SignalsSOLO Signal Duration and Bandwidth (continue – 2) ( ) ( ) ( ) ( ) ( ) ( )      22 2 222 2 2 4 4 1 ft dffS dffSf dtts dttst ∆ ∞+ ∞− +∞ ∞− ∆ ∞+ ∞− +∞ ∞−                             ≤ ∫ ∫ ∫ ∫ π Finally we obtain ( ) ( )ft ∆∆≤ 2 1 0&0 == ftChange time and frequency scale to get Since Schwarz Inequality: becomes an equality if and only if g (t) = k f (t), then for: ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttgdttfdttgtf 22 ( ) ( ) ( ) ( )tftsteAt td sd tgeAts tt ααα αα 222: 22 −=−=−==⇒= −− we have ( ) ( )ft ∆∆= 2 1
Laplace’s Transform ( ) ( ) ∫∫∫ +∞ ∞− = = +∞ ∞− = = +∞ ∞− === sde j jde j fdet ts f js tj f js tfj π ω π δ πω ω ω πω ω π 2 1 2 1 2: : 2: : 2 ( ) ( ){ } ( ) σσ <== +∫ ∞ − f ts dtetftfsF 0 L SOLO Laplace L-Transform Laplace’s Transform To find the Inverse Laplace’s Transform (L -1 ) we use: ( ) ( ) ( ) ( ) ∫ ∫∫ ∫∫ ∞ ∞+ ∞− − ∞+ ∞− ∞ − ∞+ ∞−         =        = 00 ττττ ττ dsdefdsedefdsesF j j ts j j tss j j ts ( ) ( ) ( )tfdtf =−∫ +∞ ∞− ττδτ ( ) ( )∫ ∞+ ∞− = j j ts dsesF j tf π2 1 For a signal f (t) we define the Laplace’s Transform (L) Pierre-Simon Laplace 1749-1827 ( ) ( )∫ ∞ −= 0 2 ττδτπ dtfj ( )tfjπ2=
Laplace’s TransformSOLO Laplace L-Transform (continue – 1) The Inverse Laplace’s Transform (L -1 ) is given by: ( ) ( )∫ ∞+ ∞− = j j ts dsesF j tf π2 1 Using Jordan’s Lemma (see “Complex Variables” presentation or the end of this one) Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. x yΓ R we can write ( ) ( ){ } ( ) ( )∫∫ ∞+ ∞− + + === j j tsts f f dsesF j dsesF j sFtf σ σ ππ 2 1 2 11-L ( ) ( ){ } ( ) ( ) ( )∫∫∫ =+== ∞+ ∞− dsesF j dsesF j dsesF j sFtf ts C ts j j ts πππ 2 1 2 1 2 1 0    1-L If the F (s) has no poles for σ > σf+, according to Cauchy’s Theorem we can use a closed infinite region to the left of σf+, to obtain
Laplace’s TransformSOLO Properties of Laplace L-Transform s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 1 ( ) { } if M i ii zsFc σmaxRe 1 >∑= Linearity ( )∑= M i ii tfc 1 3 ( ) ( ) ( ) ( ) ( ) ( )+−+−+− −−−− 000 1121 nnnn ffsfssFs Differentiation ( ) n n td tfd 4 ( ) ( )∫∞− → + + t t df ss sF ξξ 0 lim 1Integration ( )∫∞− t df ξξ 5 ( ) s sFReal Definite Integration ( )∫ t df 0 ξξ ( )∫∫ t ddf 0 0 ξλλ ξ ( ) 2 s sF 2       a s F a 1Scaling ( )taf
Laplace’s TransformSOLO Properties of Laplace L-Transform (continue – 1) s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 6 ( ) n n sd sFdMuliplicity by tn ( ) ( )tftn − 7 ( )∫ ∞ 0 dssFDivision by t ( ) t tf 8 ( )sFe sλTime shifting ( ) ( )λλ ±± tutf 9 ( )asF Complex Translations ( )tfe ta± 10 ( ) ( )sHsF ⋅ Convolution t - plane ( ) ( ) ( ) ( )∫ ∞ −⋅=∗ 0 τττ dthfthtf 11 ( ) ( ) ( ) ( )∫ ∞+ ∞− −=∗ j j dsHF j sHsF j σ σ τττ ππ 2 1 2 1Convolution s - plane ( ) ( )thtf ⋅
Laplace’s TransformSOLO Properties of Laplace L-Transform (continue – 2) s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 12 Initial Value Theorem ( ) ( )sFstf st ∞→→ =+ limlim 0 13 Final Value Theorem ( ) ( )sFstf st 0 limlim →∞→ = 14 Parseval’s Theorem ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞ ∞ ∞+ ∞− ∞ −=−= −=− j j j j ts j j ts dssGsF j dsdtetgsF j dttgdsesF j dttgtf σ σ σ σ σ σ ππ π 2 1 2 1 2 1 0 00
( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t ( ) ( ){ } ( ) σσ <== +∫ ∞ − f ts dtetftfsF 0 L SOLO Sampling and z-Transform ( ) ( ){ } ( ) σδδ < − ==       −== − ∞ = − ∞ = ∑∑ 0 1 1 00 sT n sTn n T e eTnttsS LL ( ) ( ){ } ( ) ( ) ( ) ( ) ( ){ } ( ) ( )       << − = =       − == − ∞+ ∞− −− ∞ = − ∞ = +∫ ∑∑ 0 00 ** 1 1 2 1 σσσξξ π δ δ ξ σ σ ξ f j j tsT n sTn n d e F j ttf eTnfTntTnf tfsF L L L ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )              − = − − = − = ∑∫ ∑∫ ∑ −− − −− Γ −− −− Γ −− ∞ = − ts e ofPoles tsts F ofPoles tsts n nsT e F Resd e F j e F Resd e F j eTnf sF ξ ξξ ξ ξξ ξ ξ ξ π ξ ξ ξ π 1 1 0 * 112 1 112 1 2 1 Poles of ( ) Ts e ξ−− −1 1 Poles of ( )ξF planes T nsn π ξ 2 += ωj ωσ j+ 0=s Laplace Transforms The signal f (t) is sampled at a time period T. 1Γ 2 Γ ∞→R ∞→R Poles of ( ) Ts e ξ−− −1 1 Poles of ( )ξF planeξ T nsn π ξ 2 += ωj ωσ j+ 0=s Z Transform
Fourier Transform ( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t SOLO Sampling and z-Transform (continue – 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∑ ∞+ −∞= ∞+ −∞= −−→ ∞+ −∞= −− +→ += − −−       += −       + −=       +             − −− −= − −= −− −− nn Tse n ts T n js T n js e ofPoles ts T n jsF TeT T n jsF T n jsF e T n js e F RessF ts n ts π π π π ξ ξ ξ ξπ ξ π ξ ξ ξ ξ 21 2 lim 2 1 2 lim 1 1 2 2 1 1 * Poles of ( )ξF ωj σ 0=s T π2 T π2 T π2 Poles of ( )ξ* F plane js ωσ += The signal f (t) is sampled at a time period T. The poles of are given by( )ts e ξ−− −1 1 ( ) ( ) T n jsnjTsee n njTs π ξπξπξ 2 21 2 +=⇒=−−⇒==−− ( ) ∑ +∞ −∞=       += n T n jsF T sF π21*
Fourier TransformSOLO F F-1 frequency-B/2 B/2 B F F-1 -B/2 B/2 B 1/Ts-1/Ts frequency Sample Sampling a function at an interval Ts (in time domain) Anti-aliasing filters is used to enforce band-limited assumption. causes it to be replicated at 1/ Ts intervals in the other (frequency) domain. Sampling and z-Transform (continue – 2)
Fourier Transform ( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t SOLO Sampling and z-Transform (continue – 3) 0=z planez Poles of ( )zF C The signal f (t) is sampled at a time period T. The z-Transform is defined as: ( ){ } ( ) ( ) ( ) ( ) ( ) ( )         − −=== ∑ ∑ = − → ∞ = − = iF iF i iF Ts FofPoles T F n n ze ze F zTnf zFsFtf ξξ ξ ξ ξξ ξξξ 1 0 * 1 lim:Z ( ) ( )      < >≥ = ∫ − 00 0 2 1 1 n RzndzzzF jTnf fC C n π
Fourier TransformSOLO Sampling and z-Transform (continue – 4) ( ) ( ) ( )∑∑ ∞ = − +∞ −∞= =      += 0 * 21 n nsT n eTnf T n jsF T sF πWe found The δ (t) function we have: ( ) 1=∫ +∞ ∞− dttδ ( ) ( ) ( )τδτ fdtttf =−∫ +∞ ∞− The following series is a periodic function: ( ) ( )∑ −= n Tnttd δ: therefore it can be developed in a Fourier series: ( ) ( ) ∑∑       −=−= n n n T tn jCTnttd πδ 2exp: where: ( ) T dt T tn jt T C T T n 1 2exp 1 2/ 2/ =      = ∫ + − πδ Therefore we obtain the following identity: ( )∑∑ −=      − nn TntT T tn j δπ2exp Second Way
Fourier Transform ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF νπνπ 2exp:2 F ( ) ( ) ( )∑∑ ∞ = − +∞ −∞= =      += 0 * 21 n nsT n eTnf T n jsF T sF π ( ) ( ){ } ( ) ( )∫ +∞ ∞− == ννπνπνπ dtjFFtf 2exp2:2-1 F SOLO Sampling and z-Transform (continue – 5) We found Using the definition of the Fourier Transform and it’s inverse: we obtain ( ) ( ) ( )∫ +∞ ∞− = ννπνπ dTnjFTnf 2exp2 ( ) ( ) ( ) ( ) ( ) ( )∑∫∑ ∞ = +∞ ∞− ∞ = −=−= 0 111 0 * exp2exp2exp nn n sTndTnjFsTTnfsF ννπνπ ( ) ( ) ( )[ ]∫ ∑ +∞ ∞− +∞ −∞= −−== 111 * 2exp22 νννπνπνπ dTnjFjsF n ( ) ( ) ∑∫ ∑ +∞ −∞= +∞ ∞− +∞ −∞=             −=      −−== nn T n F T d T n T FjsF νπνννδνπνπ 2 11 22 111 * We recovered (with –n instead of n) ( ) ∑ +∞ −∞=       += n T n jsF T sF π21* Second Way (continue) Making use of the identity: with 1/T instead of T and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑       −−=−− nn T n T Tnj 11 1 2exp ννδννπ ( )∑∑ −=      − nn TntT T tn j δπ2exp
Claude Elwood Shannon 1916 – 2001 http://en.wikipedia.org/wiki/Claude_E._Shannon Fourier TransformSOLO Henry Nyquist 1889 - 1976 http://en.wikipedia.org/wiki/Harry_Nyquist Nyquist-Shannon Sampling Theorem The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this band- limiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English). The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication"). http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
Fourier TransformSOLO Nyquist-Shannon Sampling Theorem (continue – 1) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. • Start with a band limited signal s (t) ( ) 2 0 fB fforfS >≡ • Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz. ( ) ∑ ∞+ −∞=             −= k sT kfjSfS 1 2* π fjs π2= ( ) ( ) ( )      −= ∑ +∞ −∞=n sTnttsts δ* ( )             −= ∑ ∞+ −∞=k sT jksSsS π2 * L-1 L F F-1
Fourier Transform 2 1 2 B T B s −< SOLO Nyquist-Shannon Sampling Theorem (continue – 2) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. B B Ts =      > 2 2 1 (Nyquist Sampling Rate) • Complex signal band-limited to B/2 Hz requires B complex samples/second, or 2 B real samples/seconds (twice the highest frequency) • Start with a band-limited signal f (t) ( ) 2 0 fB fforfF >≡ • Sample f (t) at a time period Ts, replicates spectrum every 1/Ts Hz. Nyquist-Shannon Sampling Theorem:
Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) • Start with a band limited signal s (t) ( ) 2 0 fB fforfS >≡ • Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz. ( )             −= ∑ ∞+ −∞=k sT kfSfS 1 * ( ) ( ) ( ) ( ) ( )∑ ∑ ∞+ −∞= +∞ −∞= −=       −= n ss n s TntTns Tnttsts δ δ* ( ) ( )∫ +∞ ∞− − = tdetsfS tfj π2 ( ) ( )∫ +∞ ∞− = fdefSts tfj π2F F-1 Continuous Fourier Transform F F-1 Discretization of a Continuous Signal ( ) ( )∫ +∞ ∞− == fdefSTnts sTnfj s π2 ( ) ( ) ( )∑∑ ∞+ −∞=       − = ∞+ −∞= − == n n f f j s T f n Tnfj sDTFT s s s s eTnseTnsfS π π 2 1 2 : DTFT provides an approximation of the continuous-time Fourier transform. Discrete Time Fourier Transform (DTFT) Define
Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-1) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. Discretization of a Continuous Signal ( ) ( )∫ +∞ ∞− == fdefSTnts sTnfj s π2 DTFT-1 DTF T Discrete Time Fourier Transform (DTFT) ( ) ( ) ( )∑∑ ∞+ −∞=       − = ∞+ −∞= − == n n f f j s T f n Tnfj sDTFT s s s s eTnseTnsfS π π 2 1 2 : We can see that ( ) ( ) ( ) ( )∑∑ ∞+ −∞= −      −∞+ −∞=       + − ===+ n DTFT nkj n f f j s n n f fkf j ssDTFT fSeeTnseTnsfkfS ss s  1 2 22 π ππ The Discrete Time Fourier Transform SDTFT (fs) is periodic with period fs. Let compute ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∑ ∑ ∑ ∫∫ ∑∫ ∞+ −∞= ∞+ −∞= =← ≠← + − −      ∞+ −∞= + − −     + − ∞+ −∞= −     + −       = − − = − = == n s sn nm nm ss f fs nm f f j s n f f nm f f j s f f n nm f f j s f f m f f j DTFT Tms Tnm nm fTns f nm j e Tns fdeTnsdfeTnsdfefS s s s s s s s s s s s s 1sin 2 1 0 2/ 2/ 2 2/ 2/ 22/ 2/ 22/ 2/ 2    π π π π πππ ( ) ( )∑ +∞ −∞= − = n Tnfj sDTFT s eTnsfS π2 : ( ) ( ) ( ) ( ) ∫ + − = s s s T T nTfj DTFTss dfefSTTns 2/1 2/1 2π
Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-2) Normalization of the frequency DTFT-1 DTFT ( ) ( )∑ +∞ −∞= − = n Tnfj sDTFT s eTnsfS π2 : ( ) ( ) ( ) ( ) ∫ + − = s s s T T nTfj DTFTss dfefSTTns 2/1 2/1 2π ( ) ( )[ ] [ ]2/1,2/1 2/1,2/1 : * * +−∈ +−∈ = f TTf Tff ss s ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π DTFT-1 DTFT ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π Example ( ) 1,,1,002 −== − NneAns nfj π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )( )1* 0 0 * * ** ** *2 *21 0 *2* 0 0 0 00 00 0 0 0 *sin *sin 1 1 −−− −− −− −−− −−− −− −−− = −− − − = − − = − − == ∑ Nffj ffj Nffj ffjffj NffjNffj ffj NffjN n nffj DTFT e ff Nff A e e ee ee A e e AeAfS π π π ππ ππ π π π π π |SDTFT(f*)| Normalized Frequency
Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-3) ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π DTFT-1 DTFT ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π Example ( )    ≥= = = − 22&8,,00 21,,10,902 nn ne ns nfj  π ( )    ≥= = = − 27&4,,00 26,,10,302 nn ne ns nfj  π Frequency Resolution Increases with Observation Time N Ts DTFT DTFT
Fourier Transform ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π SOLO The Discrete Fourier Transform (DFT) Assume a periodic sequence, sampled at a time period Ts, such that s (n Ts) = s [(n+kN) Ts] The Discrete Fourier Transform (DFT) requires an input function that is discrete and whose non-zero values have a limited (finite) duration. Unlike the Discrete-time Fourier transform (DTFT), it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. Its inverse transform cannot reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 1) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: where is a primitive N'th root of unity and is periodic N j eW π2 : − = n Nm N j n N j Nmn N j Nmn WeeeW =                =        = −− + − +  1 222 πππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( )[ ] ( )[ ]                 N N N s s s s s s W NNNNNNN NNNNNNN NN NN NN S DFT DFT DFT DFT DFT TNs TNs Ts Ts Ts WWWWW WWWWW WWWWW WWWWW WWWWW NS NS S S S                     ⋅− ⋅− ⋅ ⋅ ⋅                       =                     − − −−−−−−− −−−−−−− −− −− −− 1 2 2 1 0 1 2 2 1 0 1121211101 1222221202 1222221202 1121211101 1020201000 [ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 2) nNmn WW =+ [ ] [ ] N H NN I N WW 1 = N j eW π2 − = 1 2 * − == WeW N j π [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       = −−−−−−− −−−−−−− −− −− −− 1121211101 1222221202 1222221202 1121211101 1020201000 NNNNNNN NNNNNNN NN NN NN N WWWWW WWWWW WWWWW WWWWW WWWWW W       [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       == −+−−+−−−−−− −+−−+−−−−−− +−+−−− +−+−−− +−+−−− 1112121110 2122222120 2122222120 1112121110 0102020100 * NNNNNNN NNNNNNN NN NN NN T N H N WWWWW WWWWW WWWWW WWWWW WWWWW WW       Let multiply those two matrices [ ] [ ]( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( )       = ≠= − − = − − == +++++= − − − −− = − +−−−− ∑ mkN mk W W W W W WWWWWWWWWW mk mk N mk NmkN j jmk mNNkmjjkmkmk mk H NN 0 1 1 1 1 1 1 0 111100 ,  Where IN is the NxN identity matrix
Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 3) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we defined the Discrete Fourier Transform: [ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix We found that [ ] [ ] N H NN I N WW 1 = Where IN is the NxN identity matrix Therefore the Inverse Discrete Fourier Transform (IDFT) is [ ] N H NN SW N s 1 = ( ) ( ) ( )∑∑ − = − = − == 1 0 21 0 11 N n nk N j DFT N k nk DFTs ekS N WkS N Tns π D.F.T. I.D.F.T.
Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 4) Second way to find the Inverse Discrete Fourier Transform (IDFT). Let compute: ( ) ( ) ( ) ( ) ( ) ∑ ∑∑∑∑ − = − = −−− = − = −−− = + == 1 0 1 0 21 0 1 0 21 0 2 N n N k rnk N j s N k N n rnk N j s N k rk N j DFT eTnseTnsekS πππ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( )    ≠− =− =     −+    − −+−                 − −       − − =     −+    − −+−     − − =     −+    −− −+−− = − − = −       − = −− −− −− −− − = −− ∑ Nmrn NmrnN rn N jrn N rnjrn rn N rn N rn rn N rn N jrn N rnjrn rn N rn rn N jrn N rnjrn e e e e e rn N j rnj rn N j N rn N j N k rnk N j 0 cossin cossin sin sin cossin cossin sin sin 2 sin 2 cos1 2sin2cos1 1 1 1 1 2 2 2 2 1 0 2 ππ ππ π π π π ππ ππ π π ππ ππ π π π π π ( ) ( )[ ] ,2,1,0 1 0 2 ±±=+=∑ − = + mTmNrsNekS s N k rk N j DFT π
Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 1) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: where is a primitive N'th root of unity and is periodic N j eW π2 : − = n Nm N j n N j Nmn N j Nmn WeeeW =                =        = −− + − +  1 222 πππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ]                     ⋅− ⋅− ⋅ ⋅ ⋅                     =                     − − −− −− −− −− s s s s s NN NN NN NN DFT DFT DFT DFT DFT TNs TNs Ts Ts Ts WWWWW WWWWW WWWWW WWWWW WWWWW NS NS S S S 1 2 2 1 0 1 2 2 1 0 12210 23320 23420 12210 00000        
Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 5) The DFT ant Inverse DFT (IDFT) are given by ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFTs ekS N Tns π ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π IDFT DFT with the periodic properties ( )[ ] ( ) ,2,1,0 ±±= =+ m TnsTmNns ss ( ) ( ) ,2,1,0 ±±= =+ m kSNmkS DFTDFT The sequence s (0), s (Ts),…,s [(N-1) Ts] can be interpreted to be a sequence of finite length, given for r = 0, 1,…,N-1, and zero otherwise or a periodic sequence, defined for all r.
Fourier Transform ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 6) The DFT ant Inverse DFT (IDFT) are given by ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFTs ekS N Tns π IDFT DFT ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π IDTFT DTFT The DTFT ant Inverse DTFT (IDTFT) where given by We can see that DFT is a sampled version of DTFT by tacking: ( ) ( )[ ] [ ]2/1,2/1 2/1,2/1 1,,1,0 * * +−∈ +−∈ −==⇒== f TTf Nk TN k f N k fTf ss s s  ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π
Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue –7) We can see that DFT is a sampled version of DTFT : ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π By changing f0 from 0.25 to 0.275 we move |SDTFT (f)| to the right, and since the sampling points didn’t change, we obtain different |SDFT (k)| values.
Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 8) We can see that DFT is a sampled version of DTFT : ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π Increase sampling density from N=20 to N=60.
SOLO Properties of The Discrete Fourier Transform (DFT) (continue – 9) ( )mns − ( ) mk N j DFT ekS π2 − Linearity1 ( ) ( )nsns 2211 αα + Shift of a Sequence2 3 4 5 Periodic Convolution 6 7 Conjugate 8 9 IDFT DFT ( ) ( )∑ − = − = 1 0 2 : N n nk N j DFT enskS π ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFT ekS N ns π ( ) ( )kSkS DFTDFT 2211 αα + ( ) ( )nsns 21 , Periodic Sequence (Period N) ( ) ( )kSkS DFTDFT 21 , DFT (Period N) ( ) nl N j ens π2 − ( )lkSDFT − ( ) ( )∑ − = −⋅ 1 0 21 N m mnsms ( ) ( )kSkS DFTDFT 21 ⋅ ( ) ( )nsns 21 ⋅ ( ) ( )∑ − = −⋅ 1 0 21 1 N l DFTDFT lkSlS N ( )ns∗ ( )kSDFT − ∗ ( )ns −∗ ( )kSDFT ∗ Real & Imaginary ( )[ ]nsRe ( )[ ]nsImj ( ) ( ) ( )[ ] 2/kSkSkS DFTDFTeven −+= ∗ ( ) ( ) ( )[ ] 2/kSkSkS DFTDFTodd −−= ∗
SOLO Properties of The Discrete Fourier Transform (DFT) (continue – 10) ( ) ( ) ( )[ ] 2/: nsnsnseven −+= ∗ ( )kSDFTReEven Part10 11 12 Symmetric Proprties (only when s (n) is real) Parseval’s Formula IDFT DFT ( ) ( )∑ − = − = 1 0 2 : N n nk N j DFT enskS π ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFT ekS N ns π ( ) ( )nsns 21 , Periodic Sequence (Period N) ( ) ( )kSkS DFTDFT 21 , DFT (Period N) ( )lkSDFT − ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( ) ( ) ( ) ( )         −−∠=∠ −= −−= −= −= ∗ kSkS kSkS kSmkSm kSkS kSkS DFTDFT DFTDFT DFTDFT DFTDFT DFTDFT II ReRe Odd Part ( ) ( ) ( )[ ] 2/: nsnsnsodd −−= ∗
Fourier TransformSOLO Fast Fourier Transform (FFT) John Wilder Tukey 1915 – 2000 http://en.wikipedia.org/wiki/John_Tukey James W. Cooley 1926 - http://www.ieee.org/portal/pages/about/awards/bios/2002kilby.html The Cooley-Tukey algorithm, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of smaller DFTs of sizes N1 and N2, recursively, in order to reduce the computation time to O(N log N) for highly-composite N (smooth numbers). FFTs became popular after J. W. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm (first invented by Gauss) and describing how to perform it conveniently on a computer
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm The radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley-Tukey algorithm, although highly optimized Cooley-Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage. ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: 1,1, 22/1 2 * 2 +==−====→= −−− − ππ ππ jNj evenN NN j N j eWeWWeWeW Suppose N is a power of 2; i.e. N=2L (L is integer). Since N is a even integer, let compute SDFT (k) by separate s (nTs) into two (N/2)-point sequences consisting of the even-numbered points (n=2r) and odd numbered points (n=2r+1). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∑ − = − = − = + − = ++= ++= 12/ 0 2 12/ 0 2 12/ 0 12 12/ 0 2 122 122 N n kr N k N N n kr N N n kr N N n kr NDFT WrsWWrs WrsWrskS
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 1) 2/ 2/ 222 2 N N j N j N WeeW ==        = −− ππ We divided the N-point DFT into two N/2-points DFTs. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      kH N n kr N k N kG N n kr N N n kr N k N N n kr NDFT WrsWWrs WrsWWrskS ∑∑ ∑∑ − = − = − = − = ++= ++= 12/ 0 2/ 12/ 0 2/ 12/ 0 2 12/ 0 2 122 122 Since
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 2) We divided the N-point DFT into two N/2-points DFTs. Reduction of an 8-points FFT to two 4-points FFTs A 2-points FFT Reduction of an 4-points FFT to two 2-points FFTs
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 3) Flow Diagram for an 8-points FFT
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 2) ( ) ( )kkj kN N j Nk N eeW 1 2 2/ −==        = − − π π We divided the N-point DFT into two N/2-points DFTs. ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ∑∑ − = − − = +         ++=++= 12/ 0 1 2/ 12/ 0 2/ 2/2/ N n kn N Nk N N n Nnk N kn NDFT WWNnsnsWNnsWnskS k  Since N/2 is an even integer (N=2L ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )       tgofFFTN N n nl N WW N N n nl N ng DFT WngWNnsnslkS NN L 2/ 12/ 0 2/ 2 12/ 0 2 2/ 2 2/2 ∑∑ − = = = − = =++== ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )       thofFFTN N n nl N WW N N n nl N nh n NDFT WnhWWNnsnslkS NN L 2/ 12/ 0 2/ 2 12/ 0 2 2/ 2 2/12 ∑∑ − = = = − = =+−=+=
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 3) We divided the N-point DFT into two N/2-points DFTs. Reduction of an 8-points FFT to two 4-points FFTs Reduction of an 4-points FFT to two 2-points FFTs A 2-points FFT (Butterfly)
Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 4) Flow Diagram for an 8-points FFT
Fourier Transform ( ) ( ) 1,,1,0: 1 0 2 −== ∑ − = − NkeTnskS N n nk N j sDFT  π 8 64 24 64 8 16 256 64 256 24 32 1024 160 1024 64 64 4096 384 4096 160 128 16384 896 16384 384 SOLO Fast Fourier Transform (FFT) Arithmetic Operations for a Radix FFT versus DFT For N = 2L we have L stages of Radix FFT and: For N-point DFT we have: For each row we have N complex additions and N complex multiplications, therefore for the N rows we have Number of complex additions DFT = Number of complex multiplications DFT = NxN=N2 Number of complex additions FFT =N L=N log2 N Number of complex additions FFT =N/2 (multiplications per stage) x L -1 =N/2 log2 (N/2) Operation Complex additions Complex multiplications DFT DFTFFT FFT N=2L Approximate number of Complex Arithmetic Operations Required for 2L-point DFT and FFT computations
SOLO Complex Variables Laurent’s Series (1843) Power Series If f (z) is analytic inside and on the boundary of the ring shaped region R bounded by two concentric circles C1 and C2 with center at z0 and respective radii r1 and r2 (r1 > r2), then for all z in R: Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r P1 P0 z'( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 2 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 1 0 = − = ∫ + nzd zz zf i a C nn π Proof: Since z is inside R we have R1 <|z-z0|=r < R2 , and |z’-z0|= R1 on C1 and R2 on C2. Start with the Cauchy’s Integral Formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫∫∫∫ − − − =→ − + − + − + − = 212 0 1 1 01 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0 CCC P P P PC dz zz zf dz zz zf zfdzdz zz zf dz zz zf dz zz zf dz zz zf zf   
SOLO Complex Variables Laurent’s Series (continue - 1) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r >R2, |z’-z0|=R1. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For I integral:                     − − − − − +        − − ++ − − + − = − − − − = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n zs C n za C za C Rzzzazzzaza zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−⋅++−⋅+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0000100 0 0 1 0 0 02 00 2 0 2 01 2 00 2 '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1              π πππ ( ) ∫ −1 ' ' ' 2 1 C zd zz zf iπ We have: ( ) ( ) n n n C n n n R r rR MR dR rRR Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ ππ where |f (z)|<M in R and r/R1< 1, therefore: 0 ∞→ → n nR
SOLO Complex Variables Laurent’s Series (continue - 2) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r > R2, |z’-z0|=R2. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For II integral:                     − − − − − +        − − ++ − − + − = − − − − = − − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1'' 1 1 ' 1 11 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n za C n za CC Rzzzazzza zzzz zdzfzz i zzzz zdzf izzzz zdzf i zdzf i n n − +− +− − − +−+−− +−++−= −− − + −        − ++ −        − += − +−− ∫ ∫∫∫ 1 001 1 001 0 0 1 0 1 00 2 0 0 0 01 0 01 00 ' ''' 2 1 1 ' '' 2 11 ' '' 2 1 '' 2 1              π πππ ( ) ∫ −C zd zz zf i ' ' ' 2 1 π We have: ( ) ( ) n n n C n n n r R rR RM dR rRr MR zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫− 2 2 2 2 0 2 2 2 0 0 2' ''' 2 1 0 π θ ππ where |f (z)|<M in R and R2/r< 1, therefore: 0 ∞→ − → n nR Return to Table of Contents
Z2 Transform C1 x y R C2r2 z0 z r z' C r1 SOLO Z-Transform Two Sided ( ) ( )∑ ∞ −∞= − = n n zTnfzF Example 1 ( ) Tn aTnf = ( ) ( )∫ − = C n dzzzF j Tnf 1 2 1 π ( )          << − =      =      >< − =      =      == ∑ ∑ ∑ ∑∑ −∞= ∞+ = +∞ =∞+ −∞= ∞+ −∞= − 1 0 0 0 /1 / 0 1 1 n k T T Tk TT n T n T T n T n n T n nTn naz az az a z a z z a nza z az a z a zazF
Z2 TransformSOLO Z-Transform Two Sided Example 2 −+ −+ << << gg ff r z r rr ξ ξ ξξ −+ << gg rzr −−++ << gfgf rrzrr ( ) ( ){ } ( )∫       = − C d z GF j TngTnf ξ ξ ξξ π 1 2 1 Z −−++ −− ++ << <<< ><< gfgf fg gf rrzrr nrrz nrzr 0&/ 0&/ ξ ξ ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∑ ∑ ∫∑       =      =             == − <<∞+ −∞= − − +∞= −∞= − << − +∞= −∞= − −+ < − > + C r z r C n n n n n rr C n n n n d z GF j d z TngF j zTngdzF j zTngTnfTngTnf gg n f n f ξ ξ ξξ π ξ ξ ξξ π ξξ π ξ ξ 11 1 2 1 2 1 2 1 00 Z
Z2 TransformSOLO Z-Transform Two Sided Example 2 (continue – 1) { }             >< − =             −− = −− <> − =             − − = −− = ∫ ∫ → << → << zban ba z ba z b z b z a aResd b z b z a a j zban z ba z ba Resd z baj ba TT TT TT C T T T T b z b b z T T T T C TT TTTTa b z a TT TnTn T T T T T T &0 1111 1 2 1 &0 1 1 1 11 1 1 1 11 2 1 ξ ξ ξ ξ ξ ξ ξ ξ ξ ξπ ξξ ξ ξ ξ ξπ ξ ξ ξ ξ Z ( ) ( ){ } ( )∫       = − C d z GF j TngTnf ξ ξ ξξ π 1 2 1 Z −−++ −− ++ << <<< ><< gfgf fg gf rrzrr nrrz nrzr 0/ 0/ ξ ξ
Z TransformSOLO Properties of Z-Transform Functions Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0 1 ( ) −+ <<∑= ii ff M i ii rzrzFc minmax 1 Linearity ( )∑= M i ii kfc 1 2 ( ) ( ) ,2,10 ==−− kkfmkf ( )zFz m− Shifting ( )mkf − ( ) ( ) ( ) ∑= −−− −+ m k kmm zkfzFz 1 ( )mkf + ( ) ( ) ( ) ∑= − − m k kmm zkfzFz 1 ( )1+kf ( ) ( )0fzFz − 3 Scaling ( )kfak ( ) ( ) ( ) −+ ∞ = −−− <<= ∑ ff k k razrazakfzaF 0 11
Z TransformSOLO Properties of Z-Transform Functions (continue – 1) 4 Periodic Sequence ( )kf ( ) ( ) −+ << − 111 1 ffN N rzrzF z z N = number of units in a period Rf1- ,+ = radiuses of convergence in F(1) (z) F(1) (z) = Z -Transform of the first period 5 Multiplication by k ( )kfk ( ) −+ <<− ff rzr zd zFd z 6 Convolution ( ) ( ) ( ) ( )∑ ∞ = −=∗ 0 : m mkhmfkhkf ( ) ( ) ( ) ( )−−++ <<⋅ hfhf rrzrrzHzF ,min,max 7 Initial Value ( ) ( )zFf z ∞→ = lim0 8 Final Value ( ) ( ) ( ) ( ) existsfifzFzkf zk ∞−= →∞→ 1limlim 1 Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0
Z TransformSOLO Properties of Z-Transform Functions (continue – 2) 9 Complex Conjugate ( )kf * ( ) −+ << ff rzrzF ** 10 Product ( ) ( )khkf ⋅ ( ) ( ) −−++ − <=<∫ hfhf C rrzrr z zd zHzF j ,1, 2 1 1 π 12 Correlation ( ) ( ) ( ) ( ) ( ) ( ) 1,1, 2 1 11 0 ≥<=<=−⋅=⊗ −−++ −− ∞ = ∫∑ krrzrr z zd zzHzF j kmhmfkhkf hfhf C k m π 11 Parceval’s Theorem ( ) ( ) ( ) ( ) −−++ − ∞ = <=<=⋅ ∫∑ hfhf Ck rrzrr z zd zHzF j khkf ,1, 2 1 1 0 π Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0
Z TransformSOLO Table of Z-Transform Functions Z - Domain k - Domain ( )kf ( ) ( ) f k k RzzkfzF >= ∑ ∞ = − 0 1 ( )mkf + ( ) ( ) ( ) ( )[ ]110 11 −−−−− +−− mfzfzfzFz mm 2 ( )mkf − ( )zFz m− 3 ( ) ( ) ( )kfkfkf −+=∆ 1: ( ) ( ) ( )01 fzzFz −−4 ( ) ( ) ( ) ( )kfkfkfkf ++−+=∆ 122:2 ( ) ( ) ( ) ( ) ( )1021 2 fzfzzzFz −−−−5 ( )kf3 ∆ ( ) ( ) ( ) ( ) ( ) ( ) ( )2130331 23 fzfzzfzzzzFz −−−+−−−6
L2 Transform ( ) [ [ 0>= − aetf ta SOLO Laplace Transform Two Sided ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∞+ <−< ∞− ∞+ ∞− ∞+ ∞− −− +∞ ∞− − ∞+ << ∞− +∞ ∞− − −+ <−>+ −== == j j j j ts ts j j tts gg tftf dsGF j ddtetgF j dtetgdeF j dtetgtftgtf ξ ξ ξ ξ ξ ξ ξ ξ σ σσσσ σ σ σ ξ σ σσσ σ ξ ξξξ π ξξ π ξξ π 2 1 2 1 2 1 00 2 L Hence ( ) ( ){ } ( ) ( ) ( ) ( ) −− ++ ∞+ ∞− +∞ ∞− − <<−< −<<> −== ∫∫ ff gf j j ts tfor tfor dsGF j dtetgtftgtf σσσσ σσσσ ξξξ π ξ ξ σ σ ξ ξ 0 0 2 1 2L Example 1 { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  srealasreala tastas tastaststata asasas e as e dtedtedteee =<−=> ∞ +− ∞− −−+∞ +− ∞− −− +∞ ∞− −−− + + −− = +− + −− =+== ∫∫∫ σσ 11 0 0 0 0 2 L { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsreala as tasreal as e f f ta σσ σσ L aa− ( )sreal=σ ( )simag=ω ( ) ta etf − = t 1
L2 Transform ( ) 0>= − aetf ta SOLO Laplace Transform Two Sided Example 2 { } ab d bsa ee fg j j tbta =<<−=−       −− −      − −= −− ∞+ ∞− −− ∫ σσσσσ ξ ξξ ξ σ σ ξ ξ 11 2L { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsreala as tasreal as e f f ta σσ σσ L ( ) 0>= − betg tb Find the two sided Laplace transform of f (t) g (t) { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsrealb bs tbsreal bs e f f tb σσ σσ L { } ba d bsa ee gf j j tbta +=−<<<−       +− −      + = ++ ∞+ ∞− −− ∫ σσσσσ ξ ξξ ξ σ σ ξ ξ 11 2L ( )basbsa Res a +− −=      −−− −= = 111 ξξξ ( )basbsa Res a ++ −=      +−+ = −= 111 ξξξ C1 σ ω b−σ a 0<t 0 0 =∫<t C2 σ ω b+σa− 0>t 0 0 =∫>t
SOLO References A. Papoulis, “The Fourier Integral and its Applications”, McGraw Hill, 1962 R.N. Bracewell, “The Fourier Transform and its Applications”, McGraw Hill, 1965, 1978 J.W. Goodman,“Introduction to Fourier Optics”, McGraw Hill, 1968 H. Stark, Ed. “Applications of Optical Fourier Transform”, Academic Press, 1982 A. Papoulis, “Systems and Transforms with Applications in Optics”, McGraw Hill, 1968 Fourier Transform Athanasios Papoulis 1921-2002 Ronald N. Bracewell 1921 - Joseph W. Goodman William Ayer Professor, Emeritus Packard 352 Department of Electrical Engineering Stanford University Stanford, CA 94305 Email: goodman@ee.stanford.edu
January 6, 2015 98 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
Raymond Paley 1907 - 1933 Norbert Wiener 1894 - 1964 Paley – Wiener Condition A necessary and Sufficient condition for a square-integrable function A (ω) ≥ 0 to be the Fourier spectrum of a causal function is the convergence of the integral: ( ) ∞< +∫ +∞ ∞− ω ω ω d A 2 1 ln SOLO
The Mellin Transform ( ) ( )∫ ∞ − = 0 1 : s M exfsF SOLO Hjalmar Mellin 1854 - 1933 Putting: tdexdex tt −− −=→= ( )11 −−− = sts ex ( ) ( )∫ +∞ ∞− −− = tdeefsF tst M We can see that the Mellin Transform of the function f (t) is identical to the Bilateral Laplace Transform of f (e-t ).
SOLO Example ( ) ∫ ∞ 0 sin dk k kr Let compute: x y R ε A B C D E F G H Rx =Rx −= For this use the integral: 0=∫ABCDEFGHA zi dz z e Since z = 0 is outside the region of integration 0=+++= ∫∫∫∫∫ − − BCDEF ziR xi GHA zi R xi ABCDEFGHA zi dz z e dx x e dz z e dx x e dz z e ε ε ∫∫∫∫∫∫ ∞∞ ∞→ → − ∞→ → ∞→ → − −∞→ → === − =+ 00 0000 sin 2 sin 2 sin lim2limlimlim dk k rk idx x x idx x x idx x ee dx x e dx x e R R R xixi R R xi RR xi R ε ε ε ε ε ε ε ε πθθθε ε ππ ε ε π θ θ ε ε ε ε θ θθ idideidei e e dz z e i ii eii i eiez GHA zi −==== ∫∫∫∫ →→ = → 00 1 0 0 00 limlimlim  ( ) 01 2 2 0 /2 /2sin 0 sin 00 ∞→ −− ≥ − = →−=≤=≤= ∫∫∫∫∫ R RRReRii i eRieRz BCDEF zi e R dedededeRi eR e dz z e i ii π θθθθ π πθ πθθ π θ ππ θ θ θ θθ Therefore: 0 sin 2 0 =−= ∫∫ ∞ πidk k rk idz z e ABCDEFGHA zi ( ) 2 sin 0 π =∫ ∞ dk k kr Complex Variables
SOLO Complex Variables Cauchy’s Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside and on a simple closed curve C, then: ( ) ( ) ( )yxviyxuzf ,, +=Since is analytic and has continuous first order derivative ( ) y u i y v x v i x u zd fd zf iyzxz ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ == == ' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Cauchy - Riemann ( ) ( ) ( ) ( ) ( ) 0 00 =      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ −= ++−=++= ∫∫∫∫ ∫∫∫∫ RR dydx y v x u idydx y u x v dyudxvidyvdxudyidxviudzzf CCCC    q.e.d. Augustin Louis Cauchy )1789-1857(
SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane, and m is a positive constant. ( ) 0lim =∫Γ →∞ zdzFe zmi R x y Γ R Proof: ( ) 0lim =∫Γ →∞ zdzFe zmi R using: q.e.d. ( ) ( )∫∫ = Γ = π θθ θ θ θ 0 deRieRFezdzFe iieRmi eRz zmi i i ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫ − − − − − − =≤= =≤ 2/ 0 sin 1 0 sin 1 0 sin 0 sincos 00 2 π θ π θ π θθ π θθθθ π θθ π θθ θθθ θθθ θθ dRe R M dRe R M dReRFe deRieRFedeRieRFedeRieRFe Rm k Rm k iRm iiRmRmiiieRmiiieRmi ii 2/0/2sin πθπθθ ≤≤≥ for π2/π 1 θsin πθ /2 θ ( ) ( )Rm k Rm k Rm k iieRmi e R M de R M de R M deRieRFe i −− − − − −=≤≤ ∫∫∫ 1 222 2/ 0 /2 1 2/ 0 sin 1 0 π π π θ π θθ θθθ θ ( ) ( ) 01 2 limlim 0 =−≤ − →∞→∞ ∫ Rm kR iieRmi R e R M deRieRFe i π θθ θ θ Marie Ennemond Camille Jordan 1838 - 1922
SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim >=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the upper part of z plane. 1 ( ) 00lim <=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the down part of z plane. 2 ( ) 00lim >=∫Γ →∞ mzdzFe zm R x y Γ R where Γ is the semicircle, in the right part of z plane. 3 ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. 4 x yΓ R
SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type (Bromwwich-Wagner) ( )∫ ∞+ ∞− jc jc ts sdsFe iπ2 1 The contour from c - i ∞ to c + i ∞ is called Bromwich Contour Thomas Bromwich 1875 - 1929 x y 0< Γt R c x y 0>Γt R c ( ) ( ) ( ) ( ) ( ) ( ) ( )    < > ==         +== ∫ ∫∫∫ Γ ∞+ ∞− →∞ ∞+ ∞− 0 0 2 1 lim 2 1 2 1 tzFeRes tzFeRes zdzF i sdsFesdsFe i sdsFe i tf tz planezRight tz planezLeft ts ic ic ts R ic ic ts π ππ where Γ is the semicircle, in the right part of z plane, for t < 0. where Γ is the semicircle, in the left part of z plane, for t > 0. This integral is also the Inverse Laplace Transform.

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Fourier transform

  • 1. Fourier Transform SOLO HERMELIN Updated: 22.07.07 Run This http://www.solohermelin.com
  • 2. Fourier Transform ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF ωω exp:F SOLO Jean Baptiste Joseph Fourier 1768-1830 F (ω) is known as Fourier Integral or Fourier Transform and is in general complex ( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+= Using the identities ( ) ( )t d tj δ π ω ω =∫ +∞ ∞− 2 exp we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1 F= ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ]00 2 1 2 exp 2 expexp 2 exp ++−=−=−=     −= ∫∫ ∫ ∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +∞ ∞− tftfdtfd d tjf d tjdjf d tjF ττδττ π ω τωτ π ω ωττωτ π ω ωω ( ) ( ){ } ( ) ( )∫ +∞ ∞− == π ω ωωω 2 exp: d tjFFtf -1 F ( ) ( ) ( ) ( )[ ]00 2 1 ++−=−∫ +∞ ∞− tftfdtf ττδτ If f (t) is continuous at t, i.e. f (t-0) = f (t+0) This is true if (sufficient not necessary) f (t) and f ’ (t) are piecewise continue in every finite interval1 2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫ +∞ ∞− dttf
  • 3. Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Linearity1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫ +∞ ∞− F Symmetry2 ( )tF -1 F F ( )ωπ −f2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }tFdttjtFf dt tjtFf d tjFtf t F=−=−⇒=⇒= ∫∫∫ +∞ ∞− +∞ ∞− ↔ +∞ ∞− ωωπ π ωω π ω ωω ω exp2 2 exp 2 exp Proof: Conjugate Functions3 ( )tf * -1 F F ( )ω−* F Proof: ( ) ( ) ( ) ( ) ( ) ( ){ }tf d tjF d tjFtf **** 2 exp 2 exp 1- F=−=−= ∫∫ +∞ ∞− →− +∞ ∞− π ω ωω π ω ωω ωω
  • 4. Fourier Transform ( ){ } ( ) ( ) ( )       =      −=−= ∫∫ +∞ ∞− = +∞ ∞− a F aa d a jfdttjtaftaf ta ωτ τ ω τω τ 1 expexp:F ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωωωωω ω ωω FjdttjjtfF d d dttjtftfF nn n n −=−−=→−== ∫∫ +∞ ∞− +∞ ∞− FF expexp: SOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Scaling4 Derivatives5 Proof: ( )taf -1 F F       a F a ω1 Proof: Corollary: for a = -1 ( )tf − -1 F F ( )ω−F ( ) ( )tftj n − -1 F F ( )ω ω F d d n n ( )tf td d n n -1 F F ( ) ( )ωω Fj n ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωω π ω ωωω π ω ωωω Fj d tjjFtf td dd tjFFtf nn n n 1-1- FF ==→== ∫∫ +∞ ∞− +∞ ∞− 2 exp 2 exp
  • 5. Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Convolution6 Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωττωττωτωτ ττωττωττττωτττ τ 212121 212121 expexpexp expexpexp: FFFdjfdduujufjf ddttjtfjfdtdtfftjdtff ut =         −=         −−= −−−−=         −−=         − ∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− F ( ) ( )tftf 21 -1 F F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫ +∞ ∞− −= τττ dtfftftf 2121 :* -1 F F ( ) ( )ωω 21 FF The animations above graphically illustrate the convolution of two rectangle functions (left) and two Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a function of t, the position indicated by the vertical green line. The gray region indicates the product as a function of g (τ) f (t-τ) , so its area as a function of t is precisely the convolution. http://mathworld.wolfram.com/Convolution.html
  • 6. Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 Proof: ( ) ( ) ( )∫ +∞ ∞− −= dttjtfF ωω exp11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− =−=−= π ω ωω π ω ωω π ω ωω 22 exp 2 exp 2 * 112 * 2 * 12 * 1 d FF d dttjtfFdt d tjFtfdttftf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− −=== π ω ωω π ω ωω π ω ωω 22 exp 2 exp 21122121 d FF d dttjtfFdt d tjFtfdttftf ( ) ( ) ( )∫ +∞ ∞− −= π ω ωω 2 exp * 2 * 2 d tjFtf ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− =−→−= dttjtfFdttjtfF ωωωω expexp 1111 ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− −=−= ωωω π ωωω π dFFdFFdttftf 212121 2 1 2 1
  • 7. Signal Duration and BandwidthSOLO ( )tf -1 F F ( )ωFRelationships from Parseval’s Formula ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 ( ) ( ) ,2,1,0 2 1 2 22 == ∫∫ ∞+ ∞− ∞+ ∞− nd d Sd dttst m m m ω ω ω π Choose ( ) ( ) ( ) ( )tstjtftf m −== 21 ( ) ( )tftj n − -1 F F ( )ω ω F d d n n and use 5a Choose ( ) ( ) ( ) n n td tsd tftf == 21 and use 5b ( )tf td d n n -1 F F ( ) ( )ωω Fj n ( ) ( ) ,2,1,0 2 1 22 2 == ∫∫ ∞+ ∞− ∞+ ∞− ndSdt td tsd m n n ωωω π ( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0 2 * ==      = ∫∫ +∞ ∞− +∞ ∞− mnd d Sd S j dt td tsd tstj m m n n n n mm ω ω ω ωω π Choosec ( ) ( ) n n td tsd tf =1 ( ) ( ) ( )tstjtf m −=2
  • 8. Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform Modulation9 Shifting: for any a real8 Proof: ( ) ttf 0cos ω -1 F F ( ) ( )[ ]00 2 1 ωωωω −++ FF Proof: ( ) ( )[ ]tjtjt 000 expexp 2 1 cos ωωω −+= ( )atf − -1 F F ( ) ( )ωω ajF −exp ( ) ( )tajtf exp -1 F F ( )aF −ω ( ){ } ( ) ( ) ( ) ( )( ) ( ) ( )ωωττωτω τ Fajdajfdttjatfatf at −=+−=−−=− ∫∫ +∞ ∞− =− +∞ ∞− expexpexp:F ( ) ( ){ } ( ) ( ) ( ) ( ) ( )( ) ( )aFdttajtfdttjtajtftajtf −=−−=−= ∫∫ +∞ ∞− +∞ ∞− ωωω expexpexp:expF use shifting property with a=±ω0
  • 9. ( )atf − -1 F F ( ) ( )ωω ajF −exp Fourier TransformSOLO ( )tf -1 F F ( )ωFProperties of Fourier Transform (Summary) Linearity1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫ +∞ ∞− F Symmetry2 ( )tF -1 F F ( )ωπ −f2 Conjugate Functions3 ( )tf * -1 F F ( )ω−* F Scaling4 ( )taf -1 F F       a F a ω1 Derivatives5 ( ) ( )tftj n − -1 F F ( )ω ω F d d n n ( )tf td d n n -1 F F ( ) ( )ωω Fj n Convolution6 ( ) ( )tftf 21 -1 F F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫ +∞ ∞− −= τττ dtfftftf 2121 :* -1 F F ( ) ( )ωω 21 FF ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− = ωωω π dFFdttftf 2 * 12 * 1 2 1 Parseval’s Formula7 Shifting: for any a real8 ( ) ( )tajtf exp -1 F F ( )aF −ω Modulation9 ( ) ttf 0 cos ω -1 F F ( ) ( )[ ]00 2 1 ωωωω −++ FF ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− −=−= ωωω π ωωω π dFFdFFdttftf 212121 2 1 2 1
  • 10. Fourier Transform SOLO ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF ωω exp:F ( ) ( ){ } ( ) ( )∫ +∞ ∞− == π ω ωωω 2 exp d tjFFtf 1- F -1 F F ( ) ( ) ( )∫ +∞ ∞− =− dttjtfF ωω exp * - complex conjugate( ) ( ) ( )∫ +∞ ∞− = dttjtfF ωω exp** ( ) ( ) imaginarytf realtf ( ){ } ( ){ } 0Re 0Im = = tf tf ( ) ( ) ( ) ( )tftf tftf * * −= = ( ) ( ) ( ) ( )ωω ωω * * FF FF −=− =− ( ) realtf ( ) ( )ωω * FF =− ( ) imaginarytf ( ) ( )ωω * FF −=− Therefore Fourier Transform of Real or Imaginary Functions
  • 11. Fourier Transform SOLO ( ) realtf ( ) ( )ωω * FF =− ( ) imaginarytf ( ) ( )ωω * FF −=− ( ) realtf ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe ( ) imaginarytf ( ) ( ) ( ) ( )       −= −−= ωω ωω FF FF ImIm ReRe ( ) ( ) ( ) ( ){ } ( ) ( )∫ +∞ ∞− −==+= dttjtftfFjFF ωωωω exp:ImRe F ( ){ } ( ) ( ) ( ) ( ) ( )ωωω −==−−=− ∫∫ +∞ ∞− +∞ ∞− Fdttjtfdttjtftf expexpF ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: ( ) realtf ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )      =−−=⇔−−= =−+=⇔−+= ωωω ωωω FjFFtftftftf FFFtftftftf evenodd eveneven Im5.05.0: Re5.05.0: F F ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )      =−−=⇔−−= =−+=⇔−+= ωωω ωωω FFFtftftftf FjFFtftftftf evenodd eveneven Re5.05.0: Im5.05.0: F F ( ) imaginarytf Fourier Transform of Real or Imaginary Functions (continue – 1)
  • 12. Fourier Transform ω ( )ωFRe ( )ωFIm Real & Even t ( )tfIm ( )tfRe Real & Even SOLO ω ( )ωFRe ( )ωFIm Imaginary & Odd t ( )tfIm ( )tfRe Real & Odd ω ( )ωFRe ( )ωFIm Imag. &Even t ( )tfIm ( )tfRe Imag.& Even ω ( )ωFRe ( )ωFIm Real & Odd t ( )tfIm ( )tfRe Imag. & Odd ( ) realtf ( ) ( ) ( ) ( )[ ]tftftftf even −+== 5.0: ( ) ( ) ( ) ( )[ ]tftftftf even −+== 5.0: ( ) imaginarytf ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −== −+== FF FFtftf even ReRe 5.0FF ( ) ( )ωω * FF =− ( ) ( ) ( ) ( )[ ]tftftftf odd −−== 5.0: ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −−== −−== FjFj FFtftf even ImIm 5.0FF ( ) realtf ( ) ( )ωω * FF =− ( ) ( )ωω * FF −=− ( ) ( ) ( ) ( )[ ]tftftftf odd −−== 5.0: ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −== −+== FjFj FFtftf even ImIm 5.0FF ( ) imaginarytf ( ) ( )ωω * FF −=− ( ){ } ( ){ } ( ) ( )[ ] ( ) ( )ωω ωω −−== −−== FF FFtftf even ReRe 5.0FF Fourier Transform of Real or Imaginary Functions (continue – 2)
  • 13. Fourier Transform SOLO ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( )∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +−+= −+=−== dtttftfjdtttftf dttjttftfdttjtftfF oddevenoddeven oddeven ωω ωωωω sincos sincosexp:F ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: ( ) ( ) ( )tftftf oddeven += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ +∞+∞+∞+∞ −→ ∞− +∞ ∞− =+−=+= 0000 0 cos2coscoscoscoscos dtttfdtttfdfdtttfdtttfdtttf eveneven f eveneven t eveneven even ωωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0coscoscoscoscos 000 0 =+−=+= ∫∫∫∫∫ +∞+∞ − +∞ −→ ∞− +∞ ∞− dtttfdfdtttfdtttfdtttf odd f oddodd t oddodd odd ωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0sinsinsinsinsin 000 0 =+−−=+= ∫∫∫∫∫ +∞+∞+∞ −→ ∞− +∞ ∞− dtttfdfdtttfdtttfdtttf even f eveneven t eveneven even ωττωτωωω τ τ     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫∫∫∫ +∞+∞+∞ − +∞ −→ ∞− +∞ ∞− =+−−=+= 0000 0 sin2sinsinsinsinsin dtttfdtttfdfdtttfdtttfdtttf oddodd f oddodd t oddodd odd ωωττωτωωω τ τ     Therefore ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ +∞+∞+∞ ∞− −=−== 00 sin2cos2exp: dtttfjdtttfdttjtftfF oddeven ωωωω F ( ) ( ) ( )[ ] ( ) ( )∫ +∞ =−+= 0 cos25.0 dtttfFFF eveneven ωωωω ( ) ( ) ( )[ ] ( ) ( )∫ +∞ =−−= 0 sin25.0 dtttfjFFF oddodd ωωωω Odd and Even Parts
  • 14. Fourier Transform ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ++−= ++=== π ω ωωωω π ω ωωωω π ω ωωωω π ω ωωω 2 cosImsinRe 2 sinImcosRe 2 sincosImRe 2 exp: d tFtFj d tFtF d tjtFjF d tjFFtf -1 F SOLO ( ) 00: <∀= ttfCausal Causal Functions A causal functions is a equal zero for negative t ( ) ( ) ( )[ ] ( )tftftftf eveneven −=−+= 5.0: ( ) ( ) ( )[ ] ( )tftftftf oddodd −−=−−= 5.0: Since and ( ) 0>− ttf we have ( ) ( ) ( ) 022 >== ttftftf oddeven ( ) realtf ( ) ( ){ } ( ) ( ) ( ) ( )[ ]∫ +∞ ∞− −== π ω ωωωωω 2 sinImcosRe d tFtFFtf -1 F ( ) causalrealtf & ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 sinIm4 2 cosRe4 2 sinIm22 2 cosRe22 00 >−== −==== ∫∫ ∫∫ ∞+∞+ +∞ ∞− +∞ ∞− t d tF d tF d tFtf d tFtftf oddeven π ω ωω π ω ωω π ω ωω π ω ωω ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe ( ) ( ) ( ) ( ) ( ) 0sinIm 2 cosRe 2 00 >−== ∫∫ +∞+∞ tdtFdtFtf ωωω π ωωω π
  • 15. Fourier TransformSOLO ( ) 00: <∀= ttfCausalReal & Causal Functions ( ) ( ) ( ) 022 >== ttftftf oddeven ( ) causalrealtf & ( ) ( ) ( ) ( ) ( ) 0sinIm 2 cosRe 2 00 >−== ∫∫ +∞+∞ tdtFdtFtf ωωω π ωωω π ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ +∞ ∞− −=+= dttjttfFjFF ωωωωω sincosImRe ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫∫ +∞ ∞− +∞+∞ ∞− +∞+∞ ∞− −=      −== dtduttuuFdttdutuuFdtttfF ω π ω π ωω cossinIm 2 cossinIm 2 cosRe 00 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫∫ +∞ ∞− +∞+∞ ∞− +∞+∞ ∞− −=      −=−= dvdtttvvFdttdvtvvFdtttfF ω π ω π ωω sincosRe 2 sincosRe 2 sinIm 00 Therefore ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ −= dtduttuuFF ω π ω cossinIm 2 Re 0 ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ −= dtdvttvvFF ω π ω sincosRe 2 Im 0 But also ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ = dtduttuuFF ω π ω coscosRe 2 Re 0 ( ) ( ) ( ) ( )∫ ∫ +∞ ∞− +∞ = dtdvttvvFF ω π ω sinsinRe 2 Im 0 Real & Causal Functions Real & Causal Functions ( ) ( ) ( ) ( )      −−= −= ωω ωω FF FF ImIm ReRe
  • 16. Fourier Transform SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− +=+= π ω ωω π ω ωω 2 sin 2 cos d tFj d tFtftftf oddeven ( ) ( ) ( ) ( )tf tftf tf eveneven −= −+ = 2 : ( ) ( ) ( ) ( )tf tftf tf oddodd −−= −− = 2 : ( ) ( ) ( ) ( ) ( ) ( )tf d tF tftf tf eveneven −== −+ = ∫ +∞ ∞− π ω ωω 2 cos 2 ( ) ( ) ( ) ( ) ( ) ( )∫ +∞ ∞− −−== −− = tf d tFj tftf tf oddodd π ω ωω 2 sin 2 ( ) ( ) ( )ωωω FjFF ImRe += ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− +− +=+= π ω ωω π ω ωω π ω ωω π ω ωω 2 sinRe 2 sinIm 2 cosIm 2 cosReImRe d tFj d tF d tFj d tFtfjtftf ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −= π ω ωω π ω ωω 2 sinIm 2 cosReRe d tF d tFtf ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− += π ω ωω π ω ωω 2 sinRe 2 cosImIm d tF d tFtf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −=−=−+−=− π ω ωω π ω ωω 2 sin 2 cos d tFj d tFtftftftftf oddevenoddeven
  • 23. Fourier Transform ( )       > < =Π 2 1 0 2 1 1 t t t 2 1 2 1 − ( )tΠ t Rectangle 1 ( ) ( )      −<− < > = τ ττ τ τ t tt t t 2/1 2/ 2/1 lim Limiter τ ( )[ ] ( )tt sgnlimlim2 0 =→ ττ 0 ( )tτlim t 2/1 2/1− τ τ− SOLO Special Symbols ( )     > <− =Λ 10 11 t tt t 11− ( )tΛ t Triangle 1 ( )    < > = 00 01 t t tH 0 ( )tH t Heaviside unit step 1 ( )    <− > = 01 01 sgn t t t 0 ( )tsgn t Signum 1 1− 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1td d
  • 24. Fourier Transform ( ) ( ) ( ) ( )    ≠ =∞ =         > ≤ =                     <− ≤ > =      = →→→ 00 0 0 2/1 lim 2/1 2/ 2/1 limlimlim: 000 t t t t t tt t td d t td d t τ ττ τ ττ τ δ ττττ SOLO Special Symbols 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1td d δ (t) function Since ( )( ) ( )tt sgn 2 1 limlim 0 = → ττ we have also δ (t) function is defined as: ( ) ( )t td d t sgn 2 1 =δ 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1 0 t ( )tδ Area = 1 0→τ ( ) ( )      −<− < > = τ ττ τ τ t tt t t 2/1 2/ 2/1 lim Limiter τ ( )[ ] ( )tt sgnlimlim2 0 = → ττ 0 ( )tτlim t 2/1 2/1− τ τ−
  • 25. Fourier TransformSOLO Special Symbols Properties of δ (t) function 0 ( )t td d τlim t ( )τ2/1 ττ− Area = 1 0 t ( )tδ Area = 1 0→τ ( ) ( )tt −= δδδ (t) is a even function:2 ( ) ( )    ≠ =∞ =         > ≤ = → 00 0 0 2/1 lim 0 t t t t t τ ττ δ τ 1 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]00 2 1 ++−=−=− ∫∫ +∞ ∞− −=+∞ ∞− τττδτδ δδ ffdtttfdtttf uu Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]00 2 1 00 2 1 lim 2 1 lim sgn 2 1 limsgn 2 1 limsgnlim 2 1 lim 0 0 sgn 2 1 ++−= ++−−−−+−+=    −+−+= −−−=−=− →∞ + − − →∞ + − →∞ + −→∞ + − →∞ =+ − →∞ ∫∫ ∫∫∫ ττ ττ ττττδ τ τ δ ff fTfTffTfTftfdtfdTfTf tfdtttftdtfdtttf T T T T T T T T TT T T T t dt d tT T T 4 Fourier Transform ( ){ } ( ) ( ) ( ) ( ) 10exp 2 1 0exp 2 1 exp =++−=−= ∫ +∞ ∞− jjdttjtt ωδδF
  • 26. Fourier Transform ( )                   −=       + =       =       =       −= = + = → → → → → − → → εεε εε εε επ εεπ ε ε ε π δ ε ε ε ε ε ε ε ε ε x Ln x x J x Ai x x x x x x 2 exp 1 lim 11 lim 1 lim sin 1 lim 4 exp 2 1 lim lim lim 1 2 0 /1 0 0 0 2 0 1 0 220 SOLO Special Symbols δ (t) function The δ (t) function can be defined as the following limit as ε→0 Ai is the Airry function, ( ) ∫ ∞       += 0 3 3 cos 1 dttx t xAi π ( ) ( )[ ]∫ + − −−= π π τττ π dxnjxJn sinexp 2 1 Friedrich Wilhelm Bessel 1784 - 1846 Edmond Nicolas Laguerre 1834 - 1886 Jn (x) is the Bessel function of the first kind, and Ln (x) is the Laguerre polynomial of arbitrary positive order.
  • 27. Fourier TransformSOLO Special Symbols δ (t) function The δ (t) function can be defined also by the limit n→∞ ( )                   + = →∞ x xn x n 2 1 sin 2 1 sin 2 1 lim π δ ( ) ( ) ( ) ( )tnsincn tnn xnnx n n n →∞ →∞ →∞ = Π= −= lim lim explim 22 πδ( )     > ≤ =Π 2/10 2/11 x x x ( ) ( ) x x xsinc π πsin =
  • 28. Fourier TransformSOLO δ (t) function ( )       > ≤ = 2 0 2 1 : 02 τ τ τδ π τ t te t tfj Use It’s Fourier Transform is ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )τπ τπ πττ δ τ τ πτ τ ππ ττ ff ff ffj e dtedtetf tffj tffjtfj − − = − ===∆ + − −+ − − ∞+ ∞− − ∫∫ 0 0 2/ 2/0 22/ 2/ 22 sin 2 11 0 0 For any function φ (t), defined at t=0- and t=0+, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]+− − → + → − → + → + − → +∞ ∞− → +=+= +== − + − + ∫∫∫∫ 00 2 11 lim 1 lim 1 lim 1 lim 1 limlim 0 2/ 0 2/ 0 0 0 2/ 2 0 2/ 0 2 0 2/ 2/ 2 00 000 ϕϕϕ τ ϕ τ ϕ τ ϕ τ ϕ τ ϕδ τ τ τ τ τ π τ τ π τ τ τ π τ τ τ tttt dttedttedttedttt tfjtfjtfj ( ) ( )tt δδτ τ = →0 lim
  • 29. Fourier Transform ( )       > ≤ = 2 0 2 1 : 02 τ τ τδ π τ t te t tfj SOLO δ (t) function ( ) ( )[ ] ( )τπ τπ τ ff ff f − − =∆ 0 0sin ( ) ( )tt δδτ τ = →0 lim ( ) ( )[ ] ( ) 1 sin limlim 0 0 00 = − − =∆ →→ τπ τπ τ τ τ ff ff f ( )[ ] ( )τπ τπ ff ff − − 0 0sin ( ) ∫ +∞ ∞− = fdet tfj π δ 2
  • 30. Fourier Transform ( )      ∆>− ∆≤− ∆=∆ 2/0 2/ 1 : 0 0 fff fff ffS f SOLO δ (f) function Define: In the time domain we obtain: ( ) ( ) ( ) ( ) tfj ff ff tfjff ff tfjtfj ff e tf tf tj e f fde f fdefSts 0 0 0 0 0 2 2/ 2/ 22/ 2/ 22 sin 2 11 π π ππ π π π ∆ ∆ = ∆ = ∆ == ∆+ ∆− ∆+ ∆− +∞ ∞− ∆∆ ∫∫ For any function Φ (f), defined at f=f0- and f=f 0+ , we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]+− − + + − Φ+Φ=Φ ∆ +Φ ∆ = Φ ∆ +Φ ∆ =Φ ∆ =Φ ∆− →∆ ∆+ →∆ ∆− →∆ ∆+ →∆ ∆+ ∆− →∆ +∞ ∞− ∆ →∆ ∫∫∫∫ 00 2/ 0 2/ 0 2/ 0 2/ 0 2/ 2/ 00 2 11 lim 1 lim 1 lim 1 lim 1 limlim 0 0 0 0 0 0 0 0 0 0 ffff f ff f dff f dff f dff f dfffS f ff f ff f f f ff f ff f f ff ff f f f ( ) ( )0 0 lim fffS f f −=∆ →∆ δ ( ) ( )0 0 :lim fffS f f −=∆ →∆ δ ( ) tfj f f ets 02 0 lim π =∆ →∆ ( ) ( ) ∫ +∞ ∞− −− =− tdeff tffj 02 0 π δ
  • 31. Fourier Transform ( ) ( )∑−= += N Nn N Tntftf : SOLO fN (f) N-Periodic Extension of a function f (t) Define N- extension of f (t)
  • 32. Fourier Transform ( ) ( )∑−= += N Nn N Tntt δδ : SOLO δN (f) function Define Let find the Fourier transform of δN (f) ( ) ( ) ( ) ( )[ ] [ ] ( )Tf TNf ee j j ee eee eee e ee etdenTttdetf TfjTfj TNfjTNfj TfjTfjTfj TNfjTNfj Tfj Tfj NTfjTNfj N Nn Tnfj N Nn tfjtfj NN π π δδ ππ ππ πππ ππ π π ππ πππ sin 2 1 2sin 2 2 1 1 2 1 2 2 1 2 2 1 2 2 1 2 2 1222 222             + = − − = = −         − = − − = =+==∆ −       +−      + −       +      +− +− −=−= +∞ ∞− − +∞ ∞− − ∑∑ ∫∫ We can see that ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ,2,1,0 sin 12sin sin 1212sin ±±=∆= + = + +++ =      +∆ kf Tf TNf kTf NkTNf T k f NN π π ππ ππ N-extension of δ (t)
  • 33. Fourier TransformSOLO δN (f) function (continue – 1) ( ) ( )∑−= += N Nn N Tntt δδ : ( ) ( )[ ] ( ) ∑−= = + =∆ N Nn Tnfj N e Tf TNf f π π π 2 sin 12sin δN (t) is a periodic function with a time period of T . ΔN (f) is a periodic function with a frequency period of f0 = 1/T . ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] Tn n TTnj e dfedff N Nn n n N Nn T T TnfjN Nn T T Tnfj T T N 1sin1 2 00 01 2/1 2/1 22/1 2/1 2 2/1 2/1 ====∆ ∑∑∑ ∫∫ −= ≠← =← −= − −−= + − + −  π π π π π
  • 34. Fourier TransformSOLO δN (f) function (continue – 2) When N → ∞ the peak goes to infinity and the null-to-null bandwidth goes to zero. This resembles to a delta function. To prove that this is the case let compute: ΔN (f) is a periodic function with a frequency period of f0 = 1/T , with peak amplitude of (2 N+1) and null-to-null bandwidth of 2/ [(2N+1) T]. ( ) ( ) ( ) T dff T T N 1 2/1 2/1 =∆∫ + − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 limlim 2/1 2/1 2 2/1 2/1 Φ=Φ=Φ∆ ∑ ∫∫ −= + − ∞→ + − ∞→ T dffedfff N Nn T T Tnfj N T T N N π
  • 35. Fourier TransformSOLO δN (f) function (continue – 3) ( ) ( ) ( ) T dff T T N 1 2/1 2/1 =∆∫ + − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 limlim 2/1 2/1 2 2/1 2/1 Φ=Φ=Φ∆ ∑ ∫∫ −= + − ∞→ + − ∞→ T dffedfff N Nn T T Tnfj N T T N N π Therefore ( ) ( ) ( ) ( ) ∑∫ ∞+ −∞= + − ∞→ ∞       +=∆=∆ m T T N N T m f T dfff δ 1 lim: 2/1 2/1
  • 36. Fourier TransformSOLO δN (f) function (continue – 4) Let compute the convolution between f (t) and δN (f) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tfTntfdTntfdTntfttf N N Nn N Nn N Nn N =+=+−=+−=∗ ∑∑ ∫∫ ∑ −=−= +∞ ∞− +∞ ∞− −= ττδτττδτδ : Therefore ( ) ( ) ( )ttftf NN δ∗= Using this relation the Fourier Transform of fN (t) is given by ( ) ( ) ( ) ( ) ( )[ ] ( )Tf TfN fFffFfF NN π π sin 12sin + =∆= ( ) ( ) ( ) ( )Tntfttftf N Nn NN +=∗= ∑−= δ If N → ∞ then ( ) ( ) ( ) ( ) ( ) ∑∑ ∑ ∞+ −∞= ∞+ −∞= +∞ −∞= ∞∞       −      =      −=       −=∆= mm m T m f T m F TT m ffF T T m f T fFffFfF δδ δ 11 1 ( ) ( ) ∑ ∑ ∑ ∞+ −∞= ∞+ −∞= − +∞ −∞= ∞       =             −      = += m t T m j m n e T m F T T m f T m F T Tntftf π δ 2 1 1 1 F
  • 37. Fourier TransformSOLO δN (f) function (continue – 4) ( ) ∑ +∞ −∞= ∞       −      = m T m f T m F T fF δ 1 ( ) ( ) ∑∑ +∞ −∞= +∞ −∞= ∞       =+= m t T m j n e T m F T Tntftf π21 f∞ (t) is a periodic function with a time period of T . F∞ (f) is a periodic function with a frequency period of f0 = 1/T . We obtained the Fourier Series description of a periodic function ( ) ( )∫∑ +∞ ∞− +∞ −∞= ∞ =      == tdetf TT m F T aeatf t T m j m m t T m j m ππ 22 11 If we define ( ) ( )     > ≤ = 2/0 2/ 0 Tt Tttf tf ( ) ( )∫ + − − = 2/ 2/ 2 0 T T tfj tdetffF π then ( ) ( )∫∑ + − +∞ −∞= ∞ == 2/ 2/ 22 1 T T t T m j m m t T m j m tdetf T aeatf ππ
  • 38. Fourier Transform ( ) ππ ≤≤−= xxxf SOLO Simple Fourier Series ( ) ( ) ( ) 0cos 1 cos 1 === ∫∫ + − + − π π π π ππ dxxnxdxxnxfan ( ) ( ) ( ) ( ) ( ) ( ) nn xn n xnx dxxnxdxxnxfb n n 1 0 2 0 1 2 sincos2 sin 1 sin 1 + + − + − − =               +      −= == ∫∫ ππ π π π π π ππ http://en.wikipedia.org/wiki/Fourier_series Square wave equation ( ) ( ) ( ) ( )( )xN N xxxfSN 1sin 1 1 3sin 3 1 sin − − +++=  Sawtooth wave equation http://en.wikipedia.org/wiki/Sawtooth_wave http://en.wikipedia.org/wiki/Square_wave ( ) ( ) ( ) ( ) ( )xn n xxxfS n N sin 1 22sinsin2 1+ − ++−=  http://mathworld.wolfram.com/FourierSeries.html
  • 39. Fourier Transform ( ) ( ) ∑ ∞ =       = 1 22 sin 2 sin 8 k Triangle k xkk xf π π SOLO Simple Fourier Series Triangular wave equation
  • 40. Fourier Transform ( ) ( ) ∑ ∞ =       = 1 22 sin 2 sin 8 k Triangle k xkk xf π π SOLO Simple Fourier Series Triangular wave equation http://mathworld.wolfram.com/FourierSeries.html
  • 41. SignalsSOLO Signal Duration and Bandwidth then ( ) ( )∫ +∞ ∞− − = tdetsfS tfi π2 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 t t∆2 t ( ) 2 ts f f f∆2 ( ) 2 fS ( ) ( ) ( ) 2/1 2 22 :               − =∆ ∫ ∫ ∞+ ∞− +∞ ∞− tdts tdtstt t ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = tdts tdtst t 2 2 : Signal Duration Signal Median ( ) ( ) ( ) 2/1 2 22 2 4 :               − =∆ ∫ ∫ ∞+ ∞− +∞ ∞− fdfS fdfSff f π ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = fdfS fdfSf f 2 2 2 : π Signal Bandwidth Frequency Median Fourier
  • 42. Signals ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 SOLO Signal Duration and Bandwidth (continue – 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− + +∞ ∞− +∞ ∞− − +∞ ∞− +∞ ∞− − +∞ ∞− =         =         =         = dffSfSdfdesfS dfdesfSdfdefSsdss tfi tfitfi ττ τττττττ π ππ 2 22 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 ( ) ( ) ( )∫ +∞ ∞− == fdefSfi td tsd ts tfi π π 2 2' ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− + +∞ ∞− +∞ ∞− − +∞ ∞− +∞ ∞− − +∞ ∞− =         −=         −=         −= dffSfSfdfdesfSfi dfdesfSfidfdefSfsidss tfi tfitfi 222 22 2'2 '2'2'' πττπ ττπττπτττ π ππ ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSds 22 ττ Parseval Theorem From From ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSfdtts 2222 4' π
  • 43. Signals ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− − ∞+ ∞− +∞ ∞− +∞ ∞− − ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ===== dffS fd fd fSd fS i dffS fdtdetstfS dffS tdfdefStst dffS tdtstst tdts tdtst t fifi 22 2 2 2 22 2 2 : π ππ SOLO Signal Duration and Bandwidth ( ) ( )∫ +∞ ∞− − = tdetsfS tfi π2 ( ) ( )∫ +∞ ∞− = fdefSts tfi π2 Fourier ( ) ( )∫ +∞ ∞− − −= tdetsti fd fSd tfi π π 2 2 ( ) ( )∫ +∞ ∞− = fdefSfi td tsd tfi π π 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− − =         ==== tdts td td tsd tsi tdts tdfdefSfts tdts fdtdetsfSf tdts fdfSfSf fdfS fdfSf f fifi 22 2 2 2 22 2 2222 : ππ ππππ
  • 44. Signals ( ) ( ) ( ) ( ) ( )∫∫∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− +∞ ∞− =≤         dffSfdttstdttsdttstdtts 222222 2 2 4' 4 1 π ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSdts 22 τ SOLO Signal Duration and Bandwidth (continue – 1) 0&0 == ftChange time and frequency scale to get From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttgdttfdttgtf 22 Choose ( ) ( ) ( ) ( ) ( )ts td tsd tgtsttf ':& === ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttsdttstdttstst 22 ''we obtain ( ) ( )∫ +∞ ∞− dttstst 'Integrate by parts ( )    = += →    = = sv dtstsdu dtsdv stu ' ' ( ) ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− −−= dttststdttsstdttstst '' 2 0 2  ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− −= dttsdttstst 2 2 1 ' ( ) ( )∫∫ +∞ ∞− +∞ ∞− = dffSfdtts 2222 4' π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− =≤ dffS dffSf dtts dttst dtts dffSf dtts dttst 2 222 2 2 2 222 2 2 44 4 1 ππ assume ( ) 0lim = →∞ tst t
  • 45. SignalsSOLO Signal Duration and Bandwidth (continue – 2) ( ) ( ) ( ) ( ) ( ) ( )      22 2 222 2 2 4 4 1 ft dffS dffSf dtts dttst ∆ ∞+ ∞− +∞ ∞− ∆ ∞+ ∞− +∞ ∞−                             ≤ ∫ ∫ ∫ ∫ π Finally we obtain ( ) ( )ft ∆∆≤ 2 1 0&0 == ftChange time and frequency scale to get Since Schwarz Inequality: becomes an equality if and only if g (t) = k f (t), then for: ( ) ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− ≤ dttgdttfdttgtf 22 ( ) ( ) ( ) ( )tftsteAt td sd tgeAts tt ααα αα 222: 22 −=−=−==⇒= −− we have ( ) ( )ft ∆∆= 2 1
  • 46. Laplace’s Transform ( ) ( ) ∫∫∫ +∞ ∞− = = +∞ ∞− = = +∞ ∞− === sde j jde j fdet ts f js tj f js tfj π ω π δ πω ω ω πω ω π 2 1 2 1 2: : 2: : 2 ( ) ( ){ } ( ) σσ <== +∫ ∞ − f ts dtetftfsF 0 L SOLO Laplace L-Transform Laplace’s Transform To find the Inverse Laplace’s Transform (L -1 ) we use: ( ) ( ) ( ) ( ) ∫ ∫∫ ∫∫ ∞ ∞+ ∞− − ∞+ ∞− ∞ − ∞+ ∞−         =        = 00 ττττ ττ dsdefdsedefdsesF j j ts j j tss j j ts ( ) ( ) ( )tfdtf =−∫ +∞ ∞− ττδτ ( ) ( )∫ ∞+ ∞− = j j ts dsesF j tf π2 1 For a signal f (t) we define the Laplace’s Transform (L) Pierre-Simon Laplace 1749-1827 ( ) ( )∫ ∞ −= 0 2 ττδτπ dtfj ( )tfjπ2=
  • 47. Laplace’s TransformSOLO Laplace L-Transform (continue – 1) The Inverse Laplace’s Transform (L -1 ) is given by: ( ) ( )∫ ∞+ ∞− = j j ts dsesF j tf π2 1 Using Jordan’s Lemma (see “Complex Variables” presentation or the end of this one) Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. x yΓ R we can write ( ) ( ){ } ( ) ( )∫∫ ∞+ ∞− + + === j j tsts f f dsesF j dsesF j sFtf σ σ ππ 2 1 2 11-L ( ) ( ){ } ( ) ( ) ( )∫∫∫ =+== ∞+ ∞− dsesF j dsesF j dsesF j sFtf ts C ts j j ts πππ 2 1 2 1 2 1 0    1-L If the F (s) has no poles for σ > σf+, according to Cauchy’s Theorem we can use a closed infinite region to the left of σf+, to obtain
  • 48. Laplace’s TransformSOLO Properties of Laplace L-Transform s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 1 ( ) { } if M i ii zsFc σmaxRe 1 >∑= Linearity ( )∑= M i ii tfc 1 3 ( ) ( ) ( ) ( ) ( ) ( )+−+−+− −−−− 000 1121 nnnn ffsfssFs Differentiation ( ) n n td tfd 4 ( ) ( )∫∞− → + + t t df ss sF ξξ 0 lim 1Integration ( )∫∞− t df ξξ 5 ( ) s sFReal Definite Integration ( )∫ t df 0 ξξ ( )∫∫ t ddf 0 0 ξλλ ξ ( ) 2 s sF 2       a s F a 1Scaling ( )taf
  • 49. Laplace’s TransformSOLO Properties of Laplace L-Transform (continue – 1) s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 6 ( ) n n sd sFdMuliplicity by tn ( ) ( )tftn − 7 ( )∫ ∞ 0 dssFDivision by t ( ) t tf 8 ( )sFe sλTime shifting ( ) ( )λλ ±± tutf 9 ( )asF Complex Translations ( )tfe ta± 10 ( ) ( )sHsF ⋅ Convolution t - plane ( ) ( ) ( ) ( )∫ ∞ −⋅=∗ 0 τττ dthfthtf 11 ( ) ( ) ( ) ( )∫ ∞+ ∞− −=∗ j j dsHF j sHsF j σ σ τττ ππ 2 1 2 1Convolution s - plane ( ) ( )thtf ⋅
  • 50. Laplace’s TransformSOLO Properties of Laplace L-Transform (continue – 2) s - Domaint - Domain ( )tf ( ) ( ) { } + >= ∫ ∞ − f st sdtetfsF σRe 0 12 Initial Value Theorem ( ) ( )sFstf st ∞→→ =+ limlim 0 13 Final Value Theorem ( ) ( )sFstf st 0 limlim →∞→ = 14 Parseval’s Theorem ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞ ∞ ∞+ ∞− ∞ −=−= −=− j j j j ts j j ts dssGsF j dsdtetgsF j dttgdsesF j dttgtf σ σ σ σ σ σ ππ π 2 1 2 1 2 1 0 00
  • 51. ( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t ( ) ( ){ } ( ) σσ <== +∫ ∞ − f ts dtetftfsF 0 L SOLO Sampling and z-Transform ( ) ( ){ } ( ) σδδ < − ==       −== − ∞ = − ∞ = ∑∑ 0 1 1 00 sT n sTn n T e eTnttsS LL ( ) ( ){ } ( ) ( ) ( ) ( ) ( ){ } ( ) ( )       << − = =       − == − ∞+ ∞− −− ∞ = − ∞ = +∫ ∑∑ 0 00 ** 1 1 2 1 σσσξξ π δ δ ξ σ σ ξ f j j tsT n sTn n d e F j ttf eTnfTntTnf tfsF L L L ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )              − = − − = − = ∑∫ ∑∫ ∑ −− − −− Γ −− −− Γ −− ∞ = − ts e ofPoles tsts F ofPoles tsts n nsT e F Resd e F j e F Resd e F j eTnf sF ξ ξξ ξ ξξ ξ ξ ξ π ξ ξ ξ π 1 1 0 * 112 1 112 1 2 1 Poles of ( ) Ts e ξ−− −1 1 Poles of ( )ξF planes T nsn π ξ 2 += ωj ωσ j+ 0=s Laplace Transforms The signal f (t) is sampled at a time period T. 1Γ 2 Γ ∞→R ∞→R Poles of ( ) Ts e ξ−− −1 1 Poles of ( )ξF planeξ T nsn π ξ 2 += ωj ωσ j+ 0=s Z Transform
  • 52. Fourier Transform ( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t SOLO Sampling and z-Transform (continue – 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∑ ∞+ −∞= ∞+ −∞= −−→ ∞+ −∞= −− +→ += − −−       += −       + −=       +             − −− −= − −= −− −− nn Tse n ts T n js T n js e ofPoles ts T n jsF TeT T n jsF T n jsF e T n js e F RessF ts n ts π π π π ξ ξ ξ ξπ ξ π ξ ξ ξ ξ 21 2 lim 2 1 2 lim 1 1 2 2 1 1 * Poles of ( )ξF ωj σ 0=s T π2 T π2 T π2 Poles of ( )ξ* F plane js ωσ += The signal f (t) is sampled at a time period T. The poles of are given by( )ts e ξ−− −1 1 ( ) ( ) T n jsnjTsee n njTs π ξπξπξ 2 21 2 +=⇒=−−⇒==−− ( ) ∑ +∞ −∞=       += n T n jsF T sF π21*
  • 53. Fourier TransformSOLO F F-1 frequency-B/2 B/2 B F F-1 -B/2 B/2 B 1/Ts-1/Ts frequency Sample Sampling a function at an interval Ts (in time domain) Anti-aliasing filters is used to enforce band-limited assumption. causes it to be replicated at 1/ Ts intervals in the other (frequency) domain. Sampling and z-Transform (continue – 2)
  • 54. Fourier Transform ( )tf ( ) ( )∑ ∞ = −= 0n T Tntt δδ ( ) ( ) ( ) ( ) ( )∑ ∞ = −== 0 * n T TntTnfttftf δδ ( )tf * ( )tf T t SOLO Sampling and z-Transform (continue – 3) 0=z planez Poles of ( )zF C The signal f (t) is sampled at a time period T. The z-Transform is defined as: ( ){ } ( ) ( ) ( ) ( ) ( ) ( )         − −=== ∑ ∑ = − → ∞ = − = iF iF i iF Ts FofPoles T F n n ze ze F zTnf zFsFtf ξξ ξ ξ ξξ ξξξ 1 0 * 1 lim:Z ( ) ( )      < >≥ = ∫ − 00 0 2 1 1 n RzndzzzF jTnf fC C n π
  • 55. Fourier TransformSOLO Sampling and z-Transform (continue – 4) ( ) ( ) ( )∑∑ ∞ = − +∞ −∞= =      += 0 * 21 n nsT n eTnf T n jsF T sF πWe found The δ (t) function we have: ( ) 1=∫ +∞ ∞− dttδ ( ) ( ) ( )τδτ fdtttf =−∫ +∞ ∞− The following series is a periodic function: ( ) ( )∑ −= n Tnttd δ: therefore it can be developed in a Fourier series: ( ) ( ) ∑∑       −=−= n n n T tn jCTnttd πδ 2exp: where: ( ) T dt T tn jt T C T T n 1 2exp 1 2/ 2/ =      = ∫ + − πδ Therefore we obtain the following identity: ( )∑∑ −=      − nn TntT T tn j δπ2exp Second Way
  • 56. Fourier Transform ( ) ( ){ } ( ) ( )∫ +∞ ∞− −== dttjtftfF νπνπ 2exp:2 F ( ) ( ) ( )∑∑ ∞ = − +∞ −∞= =      += 0 * 21 n nsT n eTnf T n jsF T sF π ( ) ( ){ } ( ) ( )∫ +∞ ∞− == ννπνπνπ dtjFFtf 2exp2:2-1 F SOLO Sampling and z-Transform (continue – 5) We found Using the definition of the Fourier Transform and it’s inverse: we obtain ( ) ( ) ( )∫ +∞ ∞− = ννπνπ dTnjFTnf 2exp2 ( ) ( ) ( ) ( ) ( ) ( )∑∫∑ ∞ = +∞ ∞− ∞ = −=−= 0 111 0 * exp2exp2exp nn n sTndTnjFsTTnfsF ννπνπ ( ) ( ) ( )[ ]∫ ∑ +∞ ∞− +∞ −∞= −−== 111 * 2exp22 νννπνπνπ dTnjFjsF n ( ) ( ) ∑∫ ∑ +∞ −∞= +∞ ∞− +∞ −∞=             −=      −−== nn T n F T d T n T FjsF νπνννδνπνπ 2 11 22 111 * We recovered (with –n instead of n) ( ) ∑ +∞ −∞=       += n T n jsF T sF π21* Second Way (continue) Making use of the identity: with 1/T instead of T and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑       −−=−− nn T n T Tnj 11 1 2exp ννδννπ ( )∑∑ −=      − nn TntT T tn j δπ2exp
  • 57. Claude Elwood Shannon 1916 – 2001 http://en.wikipedia.org/wiki/Claude_E._Shannon Fourier TransformSOLO Henry Nyquist 1889 - 1976 http://en.wikipedia.org/wiki/Harry_Nyquist Nyquist-Shannon Sampling Theorem The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this band- limiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English). The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication"). http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
  • 58. Fourier TransformSOLO Nyquist-Shannon Sampling Theorem (continue – 1) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. • Start with a band limited signal s (t) ( ) 2 0 fB fforfS >≡ • Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz. ( ) ∑ ∞+ −∞=             −= k sT kfjSfS 1 2* π fjs π2= ( ) ( ) ( )      −= ∑ +∞ −∞=n sTnttsts δ* ( )             −= ∑ ∞+ −∞=k sT jksSsS π2 * L-1 L F F-1
  • 59. Fourier Transform 2 1 2 B T B s −< SOLO Nyquist-Shannon Sampling Theorem (continue – 2) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. B B Ts =      > 2 2 1 (Nyquist Sampling Rate) • Complex signal band-limited to B/2 Hz requires B complex samples/second, or 2 B real samples/seconds (twice the highest frequency) • Start with a band-limited signal f (t) ( ) 2 0 fB fforfF >≡ • Sample f (t) at a time period Ts, replicates spectrum every 1/Ts Hz. Nyquist-Shannon Sampling Theorem:
  • 60. Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) • Start with a band limited signal s (t) ( ) 2 0 fB fforfS >≡ • Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz. ( )             −= ∑ ∞+ −∞=k sT kfSfS 1 * ( ) ( ) ( ) ( ) ( )∑ ∑ ∞+ −∞= +∞ −∞= −=       −= n ss n s TntTns Tnttsts δ δ* ( ) ( )∫ +∞ ∞− − = tdetsfS tfj π2 ( ) ( )∫ +∞ ∞− = fdefSts tfj π2F F-1 Continuous Fourier Transform F F-1 Discretization of a Continuous Signal ( ) ( )∫ +∞ ∞− == fdefSTnts sTnfj s π2 ( ) ( ) ( )∑∑ ∞+ −∞=       − = ∞+ −∞= − == n n f f j s T f n Tnfj sDTFT s s s s eTnseTnsfS π π 2 1 2 : DTFT provides an approximation of the continuous-time Fourier transform. Discrete Time Fourier Transform (DTFT) Define
  • 61. Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-1) • Signal can be recovered if Fourier spectrum of the sampling signal do not overlap. Discretization of a Continuous Signal ( ) ( )∫ +∞ ∞− == fdefSTnts sTnfj s π2 DTFT-1 DTF T Discrete Time Fourier Transform (DTFT) ( ) ( ) ( )∑∑ ∞+ −∞=       − = ∞+ −∞= − == n n f f j s T f n Tnfj sDTFT s s s s eTnseTnsfS π π 2 1 2 : We can see that ( ) ( ) ( ) ( )∑∑ ∞+ −∞= −      −∞+ −∞=       + − ===+ n DTFT nkj n f f j s n n f fkf j ssDTFT fSeeTnseTnsfkfS ss s  1 2 22 π ππ The Discrete Time Fourier Transform SDTFT (fs) is periodic with period fs. Let compute ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∑ ∑ ∑ ∫∫ ∑∫ ∞+ −∞= ∞+ −∞= =← ≠← + − −      ∞+ −∞= + − −     + − ∞+ −∞= −     + −       = − − = − = == n s sn nm nm ss f fs nm f f j s n f f nm f f j s f f n nm f f j s f f m f f j DTFT Tms Tnm nm fTns f nm j e Tns fdeTnsdfeTnsdfefS s s s s s s s s s s s s 1sin 2 1 0 2/ 2/ 2 2/ 2/ 22/ 2/ 22/ 2/ 2    π π π π πππ ( ) ( )∑ +∞ −∞= − = n Tnfj sDTFT s eTnsfS π2 : ( ) ( ) ( ) ( ) ∫ + − = s s s T T nTfj DTFTss dfefSTTns 2/1 2/1 2π
  • 62. Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-2) Normalization of the frequency DTFT-1 DTFT ( ) ( )∑ +∞ −∞= − = n Tnfj sDTFT s eTnsfS π2 : ( ) ( ) ( ) ( ) ∫ + − = s s s T T nTfj DTFTss dfefSTTns 2/1 2/1 2π ( ) ( )[ ] [ ]2/1,2/1 2/1,2/1 : * * +−∈ +−∈ = f TTf Tff ss s ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π DTFT-1 DTFT ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π Example ( ) 1,,1,002 −== − NneAns nfj π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )( )1* 0 0 * * ** ** *2 *21 0 *2* 0 0 0 00 00 0 0 0 *sin *sin 1 1 −−− −− −− −−− −−− −− −−− = −− − − = − − = − − == ∑ Nffj ffj Nffj ffjffj NffjNffj ffj NffjN n nffj DTFT e ff Nff A e e ee ee A e e AeAfS π π π ππ ππ π π π π π |SDTFT(f*)| Normalized Frequency
  • 63. Fourier TransformSOLO The Discrete Time Fourier Transform (DTFT) (continue-3) ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π DTFT-1 DTFT ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π Example ( )    ≥= = = − 22&8,,00 21,,10,902 nn ne ns nfj  π ( )    ≥= = = − 27&4,,00 26,,10,302 nn ne ns nfj  π Frequency Resolution Increases with Observation Time N Ts DTFT DTFT
  • 64. Fourier Transform ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π SOLO The Discrete Fourier Transform (DFT) Assume a periodic sequence, sampled at a time period Ts, such that s (n Ts) = s [(n+kN) Ts] The Discrete Fourier Transform (DFT) requires an input function that is discrete and whose non-zero values have a limited (finite) duration. Unlike the Discrete-time Fourier transform (DTFT), it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. Its inverse transform cannot reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
  • 65. Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 1) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: where is a primitive N'th root of unity and is periodic N j eW π2 : − = n Nm N j n N j Nmn N j Nmn WeeeW =                =        = −− + − +  1 222 πππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( )[ ] ( )[ ]                 N N N s s s s s s W NNNNNNN NNNNNNN NN NN NN S DFT DFT DFT DFT DFT TNs TNs Ts Ts Ts WWWWW WWWWW WWWWW WWWWW WWWWW NS NS S S S                     ⋅− ⋅− ⋅ ⋅ ⋅                       =                     − − −−−−−−− −−−−−−− −− −− −− 1 2 2 1 0 1 2 2 1 0 1121211101 1222221202 1222221202 1121211101 1020201000 [ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
  • 66. Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 2) nNmn WW =+ [ ] [ ] N H NN I N WW 1 = N j eW π2 − = 1 2 * − == WeW N j π [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       = −−−−−−− −−−−−−− −− −− −− 1121211101 1222221202 1222221202 1121211101 1020201000 NNNNNNN NNNNNNN NN NN NN N WWWWW WWWWW WWWWW WWWWW WWWWW W       [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       == −+−−+−−−−−− −+−−+−−−−−− +−+−−− +−+−−− +−+−−− 1112121110 2122222120 2122222120 1112121110 0102020100 * NNNNNNN NNNNNNN NN NN NN T N H N WWWWW WWWWW WWWWW WWWWW WWWWW WW       Let multiply those two matrices [ ] [ ]( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( )       = ≠= − − = − − == +++++= − − − −− = − +−−−− ∑ mkN mk W W W W W WWWWWWWWWW mk mk N mk NmkN j jmk mNNkmjjkmkmk mk H NN 0 1 1 1 1 1 1 0 111100 ,  Where IN is the NxN identity matrix
  • 67. Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 3) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we defined the Discrete Fourier Transform: [ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix We found that [ ] [ ] N H NN I N WW 1 = Where IN is the NxN identity matrix Therefore the Inverse Discrete Fourier Transform (IDFT) is [ ] N H NN SW N s 1 = ( ) ( ) ( )∑∑ − = − = − == 1 0 21 0 11 N n nk N j DFT N k nk DFTs ekS N WkS N Tns π D.F.T. I.D.F.T.
  • 68. Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 4) Second way to find the Inverse Discrete Fourier Transform (IDFT). Let compute: ( ) ( ) ( ) ( ) ( ) ∑ ∑∑∑∑ − = − = −−− = − = −−− = + == 1 0 1 0 21 0 1 0 21 0 2 N n N k rnk N j s N k N n rnk N j s N k rk N j DFT eTnseTnsekS πππ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( )    ≠− =− =     −+    − −+−                 − −       − − =     −+    − −+−     − − =     −+    −− −+−− = − − = −       − = −− −− −− −− − = −− ∑ Nmrn NmrnN rn N jrn N rnjrn rn N rn N rn rn N rn N jrn N rnjrn rn N rn rn N jrn N rnjrn e e e e e rn N j rnj rn N j N rn N j N k rnk N j 0 cossin cossin sin sin cossin cossin sin sin 2 sin 2 cos1 2sin2cos1 1 1 1 1 2 2 2 2 1 0 2 ππ ππ π π π π ππ ππ π π ππ ππ π π π π π ( ) ( )[ ] ,2,1,0 1 0 2 ±±=+=∑ − = + mTmNrsNekS s N k rk N j DFT π
  • 69. Fourier Transform ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 1) For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: where is a primitive N'th root of unity and is periodic N j eW π2 : − = n Nm N j n N j Nmn N j Nmn WeeeW =                =        = −− + − +  1 222 πππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ]                     ⋅− ⋅− ⋅ ⋅ ⋅                     =                     − − −− −− −− −− s s s s s NN NN NN NN DFT DFT DFT DFT DFT TNs TNs Ts Ts Ts WWWWW WWWWW WWWWW WWWWW WWWWW NS NS S S S 1 2 2 1 0 1 2 2 1 0 12210 23320 23420 12210 00000        
  • 70. Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 5) The DFT ant Inverse DFT (IDFT) are given by ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFTs ekS N Tns π ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π IDFT DFT with the periodic properties ( )[ ] ( ) ,2,1,0 ±±= =+ m TnsTmNns ss ( ) ( ) ,2,1,0 ±±= =+ m kSNmkS DFTDFT The sequence s (0), s (Ts),…,s [(N-1) Ts] can be interpreted to be a sequence of finite length, given for r = 0, 1,…,N-1, and zero otherwise or a periodic sequence, defined for all r.
  • 71. Fourier Transform ( ) ( )∑ − = − = 1 0 2 : N n nk N j sDFT eTnskS π SOLO The Discrete Fourier Transform (DFT) (continue – 6) The DFT ant Inverse DFT (IDFT) are given by ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFTs ekS N Tns π IDFT DFT ( ) ( )∑ +∞ −∞= − = n nfj DTFT ensfS *2* : π ( ) ( )∫ + − = 2/1 2/1 *2 ** dfefSns nfj DTFT π IDTFT DTFT The DTFT ant Inverse DTFT (IDTFT) where given by We can see that DFT is a sampled version of DTFT by tacking: ( ) ( )[ ] [ ]2/1,2/1 2/1,2/1 1,,1,0 * * +−∈ +−∈ −==⇒== f TTf Nk TN k f N k fTf ss s s  ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π
  • 72. Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue –7) We can see that DFT is a sampled version of DTFT : ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π By changing f0 from 0.25 to 0.275 we move |SDTFT (f)| to the right, and since the sampling points didn’t change, we obtain different |SDFT (k)| values.
  • 73. Fourier TransformSOLO The Discrete Fourier Transform (DFT) (continue – 8) We can see that DFT is a sampled version of DTFT : ( ) ( ) ( ) 1,,1,0: 1 0 2 −=== = − = − ∑ NkfSeTnskS sTN k fDTFT N n nk N j sDFT  π Increase sampling density from N=20 to N=60.
  • 74. SOLO Properties of The Discrete Fourier Transform (DFT) (continue – 9) ( )mns − ( ) mk N j DFT ekS π2 − Linearity1 ( ) ( )nsns 2211 αα + Shift of a Sequence2 3 4 5 Periodic Convolution 6 7 Conjugate 8 9 IDFT DFT ( ) ( )∑ − = − = 1 0 2 : N n nk N j DFT enskS π ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFT ekS N ns π ( ) ( )kSkS DFTDFT 2211 αα + ( ) ( )nsns 21 , Periodic Sequence (Period N) ( ) ( )kSkS DFTDFT 21 , DFT (Period N) ( ) nl N j ens π2 − ( )lkSDFT − ( ) ( )∑ − = −⋅ 1 0 21 N m mnsms ( ) ( )kSkS DFTDFT 21 ⋅ ( ) ( )nsns 21 ⋅ ( ) ( )∑ − = −⋅ 1 0 21 1 N l DFTDFT lkSlS N ( )ns∗ ( )kSDFT − ∗ ( )ns −∗ ( )kSDFT ∗ Real & Imaginary ( )[ ]nsRe ( )[ ]nsImj ( ) ( ) ( )[ ] 2/kSkSkS DFTDFTeven −+= ∗ ( ) ( ) ( )[ ] 2/kSkSkS DFTDFTodd −−= ∗
  • 75. SOLO Properties of The Discrete Fourier Transform (DFT) (continue – 10) ( ) ( ) ( )[ ] 2/: nsnsnseven −+= ∗ ( )kSDFTReEven Part10 11 12 Symmetric Proprties (only when s (n) is real) Parseval’s Formula IDFT DFT ( ) ( )∑ − = − = 1 0 2 : N n nk N j DFT enskS π ( ) ( )∑ − = + = 1 0 2 1 N k nk N j DFT ekS N ns π ( ) ( )nsns 21 , Periodic Sequence (Period N) ( ) ( )kSkS DFTDFT 21 , DFT (Period N) ( )lkSDFT − ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( ) ( ) ( ) ( )         −−∠=∠ −= −−= −= −= ∗ kSkS kSkS kSmkSm kSkS kSkS DFTDFT DFTDFT DFTDFT DFTDFT DFTDFT II ReRe Odd Part ( ) ( ) ( )[ ] 2/: nsnsnsodd −−= ∗
  • 76. Fourier TransformSOLO Fast Fourier Transform (FFT) John Wilder Tukey 1915 – 2000 http://en.wikipedia.org/wiki/John_Tukey James W. Cooley 1926 - http://www.ieee.org/portal/pages/about/awards/bios/2002kilby.html The Cooley-Tukey algorithm, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of smaller DFTs of sizes N1 and N2, recursively, in order to reduce the computation time to O(N log N) for highly-composite N (smooth numbers). FFTs became popular after J. W. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm (first invented by Gauss) and describing how to perform it conveniently on a computer
  • 77. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm The radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley-Tukey algorithm, although highly optimized Cooley-Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage. ( ) ( ) ( )∑∑ − = − = − == 1 0 1 0 2 : N n nk s N n nk N j sDFT WTnseTnskS π For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform: 1,1, 22/1 2 * 2 +==−====→= −−− − ππ ππ jNj evenN NN j N j eWeWWeWeW Suppose N is a power of 2; i.e. N=2L (L is integer). Since N is a even integer, let compute SDFT (k) by separate s (nTs) into two (N/2)-point sequences consisting of the even-numbered points (n=2r) and odd numbered points (n=2r+1). ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∑ − = − = − = + − = ++= ++= 12/ 0 2 12/ 0 2 12/ 0 12 12/ 0 2 122 122 N n kr N k N N n kr N N n kr N N n kr NDFT WrsWWrs WrsWrskS
  • 78. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 1) 2/ 2/ 222 2 N N j N j N WeeW ==        = −− ππ We divided the N-point DFT into two N/2-points DFTs. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      kH N n kr N k N kG N n kr N N n kr N k N N n kr NDFT WrsWWrs WrsWWrskS ∑∑ ∑∑ − = − = − = − = ++= ++= 12/ 0 2/ 12/ 0 2/ 12/ 0 2 12/ 0 2 122 122 Since
  • 79. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 2) We divided the N-point DFT into two N/2-points DFTs. Reduction of an 8-points FFT to two 4-points FFTs A 2-points FFT Reduction of an 4-points FFT to two 2-points FFTs
  • 80. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 3) Flow Diagram for an 8-points FFT
  • 81. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 2) ( ) ( )kkj kN N j Nk N eeW 1 2 2/ −==        = − − π π We divided the N-point DFT into two N/2-points DFTs. ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ∑∑ − = − − = +         ++=++= 12/ 0 1 2/ 12/ 0 2/ 2/2/ N n kn N Nk N N n Nnk N kn NDFT WWNnsnsWNnsWnskS k  Since N/2 is an even integer (N=2L ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )       tgofFFTN N n nl N WW N N n nl N ng DFT WngWNnsnslkS NN L 2/ 12/ 0 2/ 2 12/ 0 2 2/ 2 2/2 ∑∑ − = = = − = =++== ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )       thofFFTN N n nl N WW N N n nl N nh n NDFT WnhWWNnsnslkS NN L 2/ 12/ 0 2/ 2 12/ 0 2 2/ 2 2/12 ∑∑ − = = = − = =+−=+=
  • 82. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 3) We divided the N-point DFT into two N/2-points DFTs. Reduction of an 8-points FFT to two 4-points FFTs Reduction of an 4-points FFT to two 2-points FFTs A 2-points FFT (Butterfly)
  • 83. Fourier TransformSOLO Fast Fourier Transform (FFT) The radix-2 DIT Algorithm (continue – 4) Flow Diagram for an 8-points FFT
  • 84. Fourier Transform ( ) ( ) 1,,1,0: 1 0 2 −== ∑ − = − NkeTnskS N n nk N j sDFT  π 8 64 24 64 8 16 256 64 256 24 32 1024 160 1024 64 64 4096 384 4096 160 128 16384 896 16384 384 SOLO Fast Fourier Transform (FFT) Arithmetic Operations for a Radix FFT versus DFT For N = 2L we have L stages of Radix FFT and: For N-point DFT we have: For each row we have N complex additions and N complex multiplications, therefore for the N rows we have Number of complex additions DFT = Number of complex multiplications DFT = NxN=N2 Number of complex additions FFT =N L=N log2 N Number of complex additions FFT =N/2 (multiplications per stage) x L -1 =N/2 log2 (N/2) Operation Complex additions Complex multiplications DFT DFTFFT FFT N=2L Approximate number of Complex Arithmetic Operations Required for 2L-point DFT and FFT computations
  • 85. SOLO Complex Variables Laurent’s Series (1843) Power Series If f (z) is analytic inside and on the boundary of the ring shaped region R bounded by two concentric circles C1 and C2 with center at z0 and respective radii r1 and r2 (r1 > r2), then for all z in R: Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r P1 P0 z'( ) ( ) ( )∑∑ ∞ = − ∞ = − +−= 1 00 0 n n n n n n zz a zzazf ( ) ( ) ,2,1,0' ' ' 2 1 2 1 0 = − = ∫ +−− nzd zz zf i a C nn π ( ) ( ) ,2,1,0' ' ' 2 1 1 1 0 = − = ∫ + nzd zz zf i a C nn π Proof: Since z is inside R we have R1 <|z-z0|=r < R2 , and |z’-z0|= R1 on C1 and R2 on C2. Start with the Cauchy’s Integral Formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫∫∫∫ − − − =→ − + − + − + − = 212 0 1 1 01 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0 CCC P P P PC dz zz zf dz zz zf zfdzdz zz zf dz zz zf dz zz zf dz zz zf zf   
  • 86. SOLO Complex Variables Laurent’s Series (continue - 1) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r >R2, |z’-z0|=R1. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For I integral:                     − − − − − +        − − ++ − − + − = − − − − = − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1 '' 1 ' 1 ' 1 1 ' 1 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n zs C n za C za C Rzzzazzzaza zzzz zdzf i zz zz zz zdzf i zz zz zdzf izz zdzf i n n +−⋅++−⋅+= −− − + −         − ++−         − + − = ∫ ∫∫∫ − 0000100 0 0 1 0 0 02 00 2 0 2 01 2 00 2 '' '' 2 ' '' 2 1 ' '' 2 1 ' '' 2 1              π πππ ( ) ∫ −1 ' ' ' 2 1 C zd zz zf iπ We have: ( ) ( ) n n n C n n n R r rR MR dR rRR Mr zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫ 11 1 2 0 1 110 0 2'' '' 2 0 π θ ππ where |f (z)|<M in R and r/R1< 1, therefore: 0 ∞→ → n nR
  • 87. SOLO Complex Variables Laurent’s Series (continue - 2) Power Series Pierre Alphonse Laurent 1813 - 1854 C1 x y R C2R2 R1 z0 z z' r Proof (continue – 1): Since z and z’ are inside R we have R1 >|z-z0|=r > R2, |z’-z0|=R2. From Cauchy’s Integral Formula: ( ) ( ) ( ) ∫∫ − − − = 21 ' ' ' ' ' ' CC dz zz zf dz zz zf zf Use the identity: α α ααα α − +++++≡ − − 1 1 1 1 12 n n  For II integral:                     − − − − − +        − − ++ − − + − = − − − − = − − − nn zz zz zz zzzz zz zz zz zz zz zzzzzz 0 0 0 0 1 0 0 0 0 0 0 00 ' ' 1 1'' 1 1 ' 1 11 ' 1  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n R C n n n za C n za CC Rzzzazzza zzzz zdzfzz i zzzz zdzf izzzz zdzf i zdzf i n n − +− +− − − +−+−− +−++−= −− − + −        − ++ −        − += − +−− ∫ ∫∫∫ 1 001 1 001 0 0 1 0 1 00 2 0 0 0 01 0 01 00 ' ''' 2 1 1 ' '' 2 11 ' '' 2 1 '' 2 1              π πππ ( ) ∫ −C zd zz zf i ' ' ' 2 1 π We have: ( ) ( ) n n n C n n n r R rR RM dR rRr MR zzzz zdzfzz R       − = − ≤ −− − ≤ ∫∫− 2 2 2 2 0 2 2 2 0 0 2' ''' 2 1 0 π θ ππ where |f (z)|<M in R and R2/r< 1, therefore: 0 ∞→ − → n nR Return to Table of Contents
  • 88. Z2 Transform C1 x y R C2r2 z0 z r z' C r1 SOLO Z-Transform Two Sided ( ) ( )∑ ∞ −∞= − = n n zTnfzF Example 1 ( ) Tn aTnf = ( ) ( )∫ − = C n dzzzF j Tnf 1 2 1 π ( )          << − =      =      >< − =      =      == ∑ ∑ ∑ ∑∑ −∞= ∞+ = +∞ =∞+ −∞= ∞+ −∞= − 1 0 0 0 /1 / 0 1 1 n k T T Tk TT n T n T T n T n n T n nTn naz az az a z a z z a nza z az a z a zazF
  • 89. Z2 TransformSOLO Z-Transform Two Sided Example 2 −+ −+ << << gg ff r z r rr ξ ξ ξξ −+ << gg rzr −−++ << gfgf rrzrr ( ) ( ){ } ( )∫       = − C d z GF j TngTnf ξ ξ ξξ π 1 2 1 Z −−++ −− ++ << <<< ><< gfgf fg gf rrzrr nrrz nrzr 0&/ 0&/ ξ ξ ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∑ ∑ ∫∑       =      =             == − <<∞+ −∞= − − +∞= −∞= − << − +∞= −∞= − −+ < − > + C r z r C n n n n n rr C n n n n d z GF j d z TngF j zTngdzF j zTngTnfTngTnf gg n f n f ξ ξ ξξ π ξ ξ ξξ π ξξ π ξ ξ 11 1 2 1 2 1 2 1 00 Z
  • 90. Z2 TransformSOLO Z-Transform Two Sided Example 2 (continue – 1) { }             >< − =             −− = −− <> − =             − − = −− = ∫ ∫ → << → << zban ba z ba z b z b z a aResd b z b z a a j zban z ba z ba Resd z baj ba TT TT TT C T T T T b z b b z T T T T C TT TTTTa b z a TT TnTn T T T T T T &0 1111 1 2 1 &0 1 1 1 11 1 1 1 11 2 1 ξ ξ ξ ξ ξ ξ ξ ξ ξ ξπ ξξ ξ ξ ξ ξπ ξ ξ ξ ξ Z ( ) ( ){ } ( )∫       = − C d z GF j TngTnf ξ ξ ξξ π 1 2 1 Z −−++ −− ++ << <<< ><< gfgf fg gf rrzrr nrrz nrzr 0/ 0/ ξ ξ
  • 91. Z TransformSOLO Properties of Z-Transform Functions Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0 1 ( ) −+ <<∑= ii ff M i ii rzrzFc minmax 1 Linearity ( )∑= M i ii kfc 1 2 ( ) ( ) ,2,10 ==−− kkfmkf ( )zFz m− Shifting ( )mkf − ( ) ( ) ( ) ∑= −−− −+ m k kmm zkfzFz 1 ( )mkf + ( ) ( ) ( ) ∑= − − m k kmm zkfzFz 1 ( )1+kf ( ) ( )0fzFz − 3 Scaling ( )kfak ( ) ( ) ( ) −+ ∞ = −−− <<= ∑ ff k k razrazakfzaF 0 11
  • 92. Z TransformSOLO Properties of Z-Transform Functions (continue – 1) 4 Periodic Sequence ( )kf ( ) ( ) −+ << − 111 1 ffN N rzrzF z z N = number of units in a period Rf1- ,+ = radiuses of convergence in F(1) (z) F(1) (z) = Z -Transform of the first period 5 Multiplication by k ( )kfk ( ) −+ <<− ff rzr zd zFd z 6 Convolution ( ) ( ) ( ) ( )∑ ∞ = −=∗ 0 : m mkhmfkhkf ( ) ( ) ( ) ( )−−++ <<⋅ hfhf rrzrrzHzF ,min,max 7 Initial Value ( ) ( )zFf z ∞→ = lim0 8 Final Value ( ) ( ) ( ) ( ) existsfifzFzkf zk ∞−= →∞→ 1limlim 1 Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0
  • 93. Z TransformSOLO Properties of Z-Transform Functions (continue – 2) 9 Complex Conjugate ( )kf * ( ) −+ << ff rzrzF ** 10 Product ( ) ( )khkf ⋅ ( ) ( ) −−++ − <=<∫ hfhf C rrzrr z zd zHzF j ,1, 2 1 1 π 12 Correlation ( ) ( ) ( ) ( ) ( ) ( ) 1,1, 2 1 11 0 ≥<=<=−⋅=⊗ −−++ −− ∞ = ∫∑ krrzrr z zd zzHzF j kmhmfkhkf hfhf C k m π 11 Parceval’s Theorem ( ) ( ) ( ) ( ) −−++ − ∞ = <=<=⋅ ∫∑ hfhf Ck rrzrr z zd zHzF j khkf ,1, 2 1 1 0 π Z - Domaink - Domain ( )kf ( ) ( ) −+ ∞ = − <<= ∑ ff k k rzrzkfzF 0
  • 94. Z TransformSOLO Table of Z-Transform Functions Z - Domain k - Domain ( )kf ( ) ( ) f k k RzzkfzF >= ∑ ∞ = − 0 1 ( )mkf + ( ) ( ) ( ) ( )[ ]110 11 −−−−− +−− mfzfzfzFz mm 2 ( )mkf − ( )zFz m− 3 ( ) ( ) ( )kfkfkf −+=∆ 1: ( ) ( ) ( )01 fzzFz −−4 ( ) ( ) ( ) ( )kfkfkfkf ++−+=∆ 122:2 ( ) ( ) ( ) ( ) ( )1021 2 fzfzzzFz −−−−5 ( )kf3 ∆ ( ) ( ) ( ) ( ) ( ) ( ) ( )2130331 23 fzfzzfzzzzFz −−−+−−−6
  • 95. L2 Transform ( ) [ [ 0>= − aetf ta SOLO Laplace Transform Two Sided ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫ ∫ ∫∫ ∞+ <−< ∞− ∞+ ∞− ∞+ ∞− −− +∞ ∞− − ∞+ << ∞− +∞ ∞− − −+ <−>+ −== == j j j j ts ts j j tts gg tftf dsGF j ddtetgF j dtetgdeF j dtetgtftgtf ξ ξ ξ ξ ξ ξ ξ ξ σ σσσσ σ σ σ ξ σ σσσ σ ξ ξξξ π ξξ π ξξ π 2 1 2 1 2 1 00 2 L Hence ( ) ( ){ } ( ) ( ) ( ) ( ) −− ++ ∞+ ∞− +∞ ∞− − <<−< −<<> −== ∫∫ ff gf j j ts tfor tfor dsGF j dtetgtftgtf σσσσ σσσσ ξξξ π ξ ξ σ σ ξ ξ 0 0 2 1 2L Example 1 { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  srealasreala tastas tastaststata asasas e as e dtedtedteee =<−=> ∞ +− ∞− −−+∞ +− ∞− −− +∞ ∞− −−− + + −− = +− + −− =+== ∫∫∫ σσ 11 0 0 0 0 2 L { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsreala as tasreal as e f f ta σσ σσ L aa− ( )sreal=σ ( )simag=ω ( ) ta etf − = t 1
  • 96. L2 Transform ( ) 0>= − aetf ta SOLO Laplace Transform Two Sided Example 2 { } ab d bsa ee fg j j tbta =<<−=−       −− −      − −= −− ∞+ ∞− −− ∫ σσσσσ ξ ξξ ξ σ σ ξ ξ 11 2L { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsreala as tasreal as e f f ta σσ σσ L ( ) 0>= − betg tb Find the two sided Laplace transform of f (t) g (t) { } ( ) ( )      >=<=− + + <=<= − − = + − − 0& 1 0& 1 2 tsrealb bs tbsreal bs e f f tb σσ σσ L { } ba d bsa ee gf j j tbta +=−<<<−       +− −      + = ++ ∞+ ∞− −− ∫ σσσσσ ξ ξξ ξ σ σ ξ ξ 11 2L ( )basbsa Res a +− −=      −−− −= = 111 ξξξ ( )basbsa Res a ++ −=      +−+ = −= 111 ξξξ C1 σ ω b−σ a 0<t 0 0 =∫<t C2 σ ω b+σa− 0>t 0 0 =∫>t
  • 97. SOLO References A. Papoulis, “The Fourier Integral and its Applications”, McGraw Hill, 1962 R.N. Bracewell, “The Fourier Transform and its Applications”, McGraw Hill, 1965, 1978 J.W. Goodman,“Introduction to Fourier Optics”, McGraw Hill, 1968 H. Stark, Ed. “Applications of Optical Fourier Transform”, Academic Press, 1982 A. Papoulis, “Systems and Transforms with Applications in Optics”, McGraw Hill, 1968 Fourier Transform Athanasios Papoulis 1921-2002 Ronald N. Bracewell 1921 - Joseph W. Goodman William Ayer Professor, Emeritus Packard 352 Department of Electrical Engineering Stanford University Stanford, CA 94305 Email: [email protected]
  • 98. January 6, 2015 98 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
  • 99. Raymond Paley 1907 - 1933 Norbert Wiener 1894 - 1964 Paley – Wiener Condition A necessary and Sufficient condition for a square-integrable function A (ω) ≥ 0 to be the Fourier spectrum of a causal function is the convergence of the integral: ( ) ∞< +∫ +∞ ∞− ω ω ω d A 2 1 ln SOLO
  • 100. The Mellin Transform ( ) ( )∫ ∞ − = 0 1 : s M exfsF SOLO Hjalmar Mellin 1854 - 1933 Putting: tdexdex tt −− −=→= ( )11 −−− = sts ex ( ) ( )∫ +∞ ∞− −− = tdeefsF tst M We can see that the Mellin Transform of the function f (t) is identical to the Bilateral Laplace Transform of f (e-t ).
  • 101. SOLO Example ( ) ∫ ∞ 0 sin dk k kr Let compute: x y R ε A B C D E F G H Rx =Rx −= For this use the integral: 0=∫ABCDEFGHA zi dz z e Since z = 0 is outside the region of integration 0=+++= ∫∫∫∫∫ − − BCDEF ziR xi GHA zi R xi ABCDEFGHA zi dz z e dx x e dz z e dx x e dz z e ε ε ∫∫∫∫∫∫ ∞∞ ∞→ → − ∞→ → ∞→ → − −∞→ → === − =+ 00 0000 sin 2 sin 2 sin lim2limlimlim dk k rk idx x x idx x x idx x ee dx x e dx x e R R R xixi R R xi RR xi R ε ε ε ε ε ε ε ε πθθθε ε ππ ε ε π θ θ ε ε ε ε θ θθ idideidei e e dz z e i ii eii i eiez GHA zi −==== ∫∫∫∫ →→ = → 00 1 0 0 00 limlimlim  ( ) 01 2 2 0 /2 /2sin 0 sin 00 ∞→ −− ≥ − = →−=≤=≤= ∫∫∫∫∫ R RRReRii i eRieRz BCDEF zi e R dedededeRi eR e dz z e i ii π θθθθ π πθ πθθ π θ ππ θ θ θ θθ Therefore: 0 sin 2 0 =−= ∫∫ ∞ πidk k rk idz z e ABCDEFGHA zi ( ) 2 sin 0 π =∫ ∞ dk k kr Complex Variables
  • 102. SOLO Complex Variables Cauchy’s Theorem C x y R Proof: ( ) 0=∫C dzzf If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside and on a simple closed curve C, then: ( ) ( ) ( )yxviyxuzf ,, +=Since is analytic and has continuous first order derivative ( ) y u i y v x v i x u zd fd zf iyzxz ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ == == ' y u x v y v x u ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ & Cauchy - Riemann ( ) ( ) ( ) ( ) ( ) 0 00 =      ∂ ∂ − ∂ ∂ +      ∂ ∂ − ∂ ∂ −= ++−=++= ∫∫∫∫ ∫∫∫∫ RR dydx y v x u idydx y u x v dyudxvidyvdxudyidxviudzzf CCCC    q.e.d. Augustin Louis Cauchy )1789-1857(
  • 103. SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then where Γ is the semicircle arc of radius R, center at origin, in the upper part of z plane, and m is a positive constant. ( ) 0lim =∫Γ →∞ zdzFe zmi R x y Γ R Proof: ( ) 0lim =∫Γ →∞ zdzFe zmi R using: q.e.d. ( ) ( )∫∫ = Γ = π θθ θ θ θ 0 deRieRFezdzFe iieRmi eRz zmi i i ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫ − − − − − − =≤= =≤ 2/ 0 sin 1 0 sin 1 0 sin 0 sincos 00 2 π θ π θ π θθ π θθθθ π θθ π θθ θθθ θθθ θθ dRe R M dRe R M dReRFe deRieRFedeRieRFedeRieRFe Rm k Rm k iRm iiRmRmiiieRmiiieRmi ii 2/0/2sin πθπθθ ≤≤≥ for π2/π 1 θsin πθ /2 θ ( ) ( )Rm k Rm k Rm k iieRmi e R M de R M de R M deRieRFe i −− − − − −=≤≤ ∫∫∫ 1 222 2/ 0 /2 1 2/ 0 sin 1 0 π π π θ π θθ θθθ θ ( ) ( ) 01 2 limlim 0 =−≤ − →∞→∞ ∫ Rm kR iieRmi R e R M deRieRFe i π θθ θ θ Marie Ennemond Camille Jordan 1838 - 1922
  • 104. SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Jordan’s Lemma Generalization If |F (z)| ≤ M/Rk for z = R e iθ where k > 0 and M are constants, then for Γ a semicircle arc of radius R, and center at origin: ( ) 00lim >=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the upper part of z plane. 1 ( ) 00lim <=∫Γ →∞ mzdzFe zmi R x y Γ R where Γ is the semicircle, in the down part of z plane. 2 ( ) 00lim >=∫Γ →∞ mzdzFe zm R x y Γ R where Γ is the semicircle, in the right part of z plane. 3 ( ) 00lim <=∫Γ →∞ mzdzFe zm R where Γ is the semicircle, in the left part of z plane. 4 x yΓ R
  • 105. SOLO Complex Variables The Residue Theorem, Evaluations of Integral and Series Evaluation of Integrals Integral of the Type (Bromwwich-Wagner) ( )∫ ∞+ ∞− jc jc ts sdsFe iπ2 1 The contour from c - i ∞ to c + i ∞ is called Bromwich Contour Thomas Bromwich 1875 - 1929 x y 0< Γt R c x y 0>Γt R c ( ) ( ) ( ) ( ) ( ) ( ) ( )    < > ==         +== ∫ ∫∫∫ Γ ∞+ ∞− →∞ ∞+ ∞− 0 0 2 1 lim 2 1 2 1 tzFeRes tzFeRes zdzF i sdsFesdsFe i sdsFe i tf tz planezRight tz planezLeft ts ic ic ts R ic ic ts π ππ where Γ is the semicircle, in the right part of z plane, for t < 0. where Γ is the semicircle, in the left part of z plane, for t > 0. This integral is also the Inverse Laplace Transform.

Editor's Notes

  • #18: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #19: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #20: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #21: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #22: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #23: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #24: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #25: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #26: R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978 Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
  • #27: http://mathworld.wolfram.com/DeltaFunction.html http://en.wikipedia.org/wiki/Dirac_delta_function
  • #28: http://mathworld.wolfram.com/DeltaFunction.html
  • #42: Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley &amp; Sons, 1992, pp.72-74 François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75 Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138 Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
  • #43: Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley &amp; Sons, 1992, pp.72-74 François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75 Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138 Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
  • #44: Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley &amp; Sons, 1992, pp.72-74 François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75 Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138
  • #45: Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley &amp; Sons, 1992, pp.72-74 François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75 Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138 Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
  • #46: Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley &amp; Sons, 1992, pp.72-74 François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75 Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138 Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
  • #49: Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 385-386
  • #50: Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 385-386
  • #51: Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 385-386
  • #61: http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989
  • #62: http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989
  • #63: http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989 M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004
  • #64: M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004 http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989
  • #65: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #66: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #67: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #68: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #69: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #70: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #71: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #72: http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #73: M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004 http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #74: M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004 http://en.wikipedia.org/wiki/Discrete_Fourier_transform Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
  • #75: Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 525 http://en.wikipedia.org/wiki/Discrete_Fourier_transform
  • #76: Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 525 http://en.wikipedia.org/wiki/Discrete_Fourier_transform
  • #77: http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #78: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #79: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #80: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #81: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #82: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #83: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
  • #84: Elliot, D.F., Rao, K.R., “Fast Transforms: Algorithms, Analysis. Applications”, Academic Press, 1982, Ch.4, pp. 58 - 98 Oppenheim, A.V., Schafer, R.W., “Discrete-Time Signal Processing”, Prentice Hall, 1989, Ch.9, pp. 581 - 661 http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm
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  • #93: Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 450-451
  • #94: Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 450-451
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  • #101: Bracewell, R.N.,”The Fourier Transform and its Applications”,2nd Ed., McGraw-Hill, 1965, 1978, pp.254-257
  • #102: M.R. Spiegel, “Complex Variables”, Schaum’s Outline Series, 1964, pg.184
  • #104: Spiegel, M.R., “Complex Variables with an introduction to Conformal Mapping and its applications”, Schaum’s Outline Series, McGraw-Hill, 1964, pp.176 and 182-183 A.A. Hauser, “Complex Variables with Physical Applications”, Simon &amp; Schuster Tech Outlines, 1971, pp.239-240 A. Angot, “Complements de Mathematiques”
  • #105: F.B. Hildebrand, “Advanced Calculus for Applications”, 2nd Ed., Prentice Hall, 1976, Ch.10, “Functions of a Complex Variable”, pp.590
  • #106: F.B. Hildebrand, “Advanced Calculus for Applications”, 2nd Ed., Prentice Hall, 1976, Ch.10, “Functions of a Complex Variable”, pp.590