![Example
1-1
1
t
f(t)
dtetfjF tj
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−
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1
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−
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j
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sin2
-10 -5 0 5 10
-1
0
1
2
3
F()
-10 -5 0 5 10
0
1
2
3
|F()|
-10 -5 0 5 10
0
2
4arg[F()]](https://image.slidesharecdn.com/nss-fourier-200826040929/85/Nss-fourier-33-320.jpg)
![Example
dtetfjF tj
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−−
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t
f(t)
e−t
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+−
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+
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j
1
-10 -5 0 5 10
0
0.5
1
|F(j)|
-10 -5 0 5 10
-2
0
2
arg[F(j)]
=2](https://image.slidesharecdn.com/nss-fourier-200826040929/85/Nss-fourier-35-320.jpg)
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2
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Nss fourier
- 1. Significance of Fourier Series and Fourier Transform Dr.R.Subasri Professor Kongu Engineering College Perundurai Courtesy: Referred and collected from various web sources and arranged
- 2. Any periodic function f(t) can be expressed as a weighted sum (infinite) of sine and cosine functions of increasing frequency: Our building block: Add enough of them to get any signal f(x) you want! )+ xAsin( Fourier Series
- 3. A Sum of Sinusoids
- 4. • Decompose a periodic input signal into primitive periodic components. T nt b T nt a a tf n n n n + += = = 2 sin 2 cos 2 )( 11 0 DC Part Even Part Odd Part T is a period of all the above signals Let 0=2/T )sin()cos( 2 )( 0 1 0 1 0 tnbtna a tf n n n n ++= = = DC part is the average value of the given continuous time signal fundamental angular frequency. the n-th harmonic of the periodic function
- 5. The integrations can be performed from 0 to 2 ( ) dfa = 2 00 2 1 ( ) ,,ndncosfan 21 1 2 0 == ( ) ,,ndnsinfbn 21 1 2 0 ==
- 6. Waveform Symmetry • Even Functions • Odd Functions )()( tftf −= )()( tftf −−=
- 7. Even Functions f() The value of the function would be the same when we walk equal distances along the X-axis in opposite directions. ( ) ( ) ff =− Mathematically speaking -
- 8. Odd Functions The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions. ( ) ( ) ff −=− Mathematically speaking - f()
- 9. Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function. 10 0 10 5 0 5
- 10. Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function. 10 0 10 5 0 5
- 11. The Fourier series of an even function ( )f is expressed in terms of a cosine series. ( ) = += 1 0 cos n n naaf The Fourier series of an odd function ( )f is expressed in terms of a sine series. ( ) = = 1 sin n n nbf
- 12. Example 1. Find the Fourier series of the following periodic function. 0 f ( ) 2 3 4 5 A -A ( ) −= = 2 0 whenA whenAf ( ) ( ) ff =+ 2
- 13. ( ) ( ) ( ) 0 2 1 2 1 2 1 2 0 2 0 2 0 0 = −+= += = dAdA dfdf dfa
- 14. ( ) ( ) 0 11 1 1 2 0 2 0 2 0 = −+ = −+= = n nsin A n nsin A dncosAdncosA dncosfan
- 15. ( ) ( ) ncosncoscosncos n A n ncos A n ncos A dnsinAdnsinA dnsinfbn −++−= + −= −+= = 20 11 1 1 2 0 2 0 2 0
- 18. Therefore, the corresponding Fourier series is ++++ 7sin 7 1 5sin 5 1 3sin 3 1 sin 4A In writing the Fourier series we may not be able to consider infinite number of terms for practical reasons. The question therefore, is – how many terms to consider?
- 19. When we consider 4 terms as shown in the previous slide, the function looks like the following. 1.5 1 0.5 0 0.5 1 1.5 f ( )
- 20. When we consider 6 terms, the function looks like the following. 1.5 1 0.5 0 0.5 1 1.5 f ( )
- 21. When we consider 8 terms, the function looks like the following. 1.5 1 0.5 0 0.5 1 1.5 f ( )
- 22. When we consider 12 terms, the function looks like the following. 1.5 1 0.5 0 0.5 1 1.5 f ( )
- 26. ++++ 7sin 7 1 5sin 5 1 3sin 3 1 sin 4A Therefore, the corresponding Fourier series is Frequency Spectrum
- 27. Spectral representation The frequency representation of periodic and aperiodic signals indicates how their power or energy is allocated to different frequencies. Such a distribution over frequency is called the spectrum of the signal. For a periodic signal the spectrum is discrete function of frequency and povides information as to how the power of the signal is distributed over the different frequencies present in the signal. We thus learn not only what frequency components are present in the signal but also the strength of these frequency components On the other hand, the spectrum of an aperiodic signal is a continuous function of frequency.
- 28. Application of Fourier analysis The frequency representation of signals and systems is extremely important in signal processing and in communications. It explains filtering, modulation of messages in a communication system, the meaning of bandwidth, and how to design filters. Likewise, the frequency representation turns out to be essential in the sampling of analog signals the bridge between analog and digital signal processing.
- 29. Fourier Series vs. Fourier Integral −= = n tjn nectf 0 )( Fourier Series: Fourier Integral: dtetf T c T T tjn Tn − − = 2/ 2/ 0 )( 1 dtetfjF tj − − = )()( = − dejFtf tj )( 2 1 )( Period Function Discrete Spectra Non-Period Function Continuous Spectra
- 30. Fourier Transform Pair dtetfjF tj − − = )()( = − dejFtf tj )( 2 1 )( Synthesis Analysis Fourier Transform: Inverse Fourier Transform:
- 31. dtetfjF tj − − = )()( Continuous Spectra )()()( += jjFjFjF IR )( |)(| = j ejF FR(j) FI(j) () Magnitude Phase
- 32. Example 1-1 1 t f(t) dtetfjF tj − − = )()( dte tj − − = 1 1 1 1 1 − − − = tj e j )( − − = jj ee j = sin2
- 33. Example 1-1 1 t f(t) dtetfjF tj − − = )()( dte tj − − = 1 1 1 1 1 − − − = tj e j )( − − = jj ee j = sin2 -10 -5 0 5 10 -1 0 1 2 3 F() -10 -5 0 5 10 0 1 2 3 |F()| -10 -5 0 5 10 0 2 4arg[F()]
- 34. Example dtetfjF tj − − = )()( dtee tjt −− = 0 t f(t) e−t dte tj +− = 0 )( + = j 1
- 35. Example dtetfjF tj − − = )()( dtee tjt −− = 0 t f(t) e−t dte tj +− = 0 )( + = j 1 -10 -5 0 5 10 0 0.5 1 |F(j)| -10 -5 0 5 10 -2 0 2 arg[F(j)] =2
- 36. Parseval’s Theorem: Energy Preserving (Fourier Transform) = − − djFdttf 22 |)(| 2 1 |)(| dtetftf tj − − = )(*)](*[F −− = − djFjF )]([*)( 2 1 dttftfdttf − − = )(*)(|)(| 2 * )( = − dtetf tj )(* −= jF = − djF 2 |)(| 2 1
- 37. Parseval’s Theorem: Power Preserving (Fourier Series) 2 2 0 )( 1 = k k T cdttf T 222 0 2 0 2 1 )( 1 kk T baadttf T ++=
- 38. C1- In one period, the number of discontinuous points is finite. C2- In one period, the number of maximum and minimum points is finite. C3- In one period, the function is absolutely integrable.
- 39. Existence of the Fourier Transform − dttf |)(| Sufficient Condition: f(t) is absolutely integrable, i.e.,
- 45. Thank you