![CT Sinusoidal Signal
𝒙 𝒕 = 𝑨𝒄𝒐𝒔 𝝎𝒕 + 𝝓
Periodic with 3 characteristics: 𝑨, 𝝎, 𝝓.
𝑨 : amplitude
𝝓 : phase [radians; degrees]
𝝎 = 𝟐𝝅𝒇 : angular frequency [rad/s]
𝒇 = 𝟏/𝑻 : frequency [𝐇𝐳 = 𝟏/𝒔]
𝑻 = 𝟏/𝒇 : Period [s]
10/26/2022
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![DT Real Exponential Signal
𝒙[𝒏] = 𝑩 𝒆𝒂𝒏
𝑩 and 𝒂 are real
𝒂 is positive or negative => two special cases:
1. Decaying exponential, 𝒂 < 𝟎
2. Growing exponential, 𝒂 > 𝟎
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![Complex Exponential Signal
CT Complex Exponential Signal
𝒙(𝒕) = 𝑨𝒆𝝀𝒕
𝑨 and 𝝀 are complex constants (e.g. 𝒆𝒋𝝎𝒕)
DT Complex Exponential Signal
𝒙[𝒏] = 𝑪𝒆𝒃𝒏
𝑪 and 𝒃 are complex constants (e.g. 𝒆𝒋𝛀𝒏)
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![Complex Sinusoidal Signals
A special case of a complex exponential
𝒙(𝒕) = 𝑨𝒆𝒋𝝎𝒕
𝑨 is complex and 𝝀 = 𝒋𝝎 (𝝎 is real)
𝒙[𝒏] = 𝑪𝒆𝒋𝛀𝒏
𝑪 is complex and 𝒃 = 𝒋𝛀 (𝛀 is real)
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![Relationship Between
CT Complex Exponentials & Real Sinusoids
Euler’s formula: 𝒆𝒋𝝎𝒕
= 𝒄𝒐𝒔 𝝎𝒕 + 𝒋𝒔𝒊𝒏(𝝎𝒕)
Complex sinusoid as the sum of two real sinusoids
𝑨𝒆𝒋𝜽
𝒆𝒋𝝎𝒕
= 𝑨 𝒄𝒐𝒔 𝝎𝒕 + 𝜽 + 𝒋𝑨 𝒔𝒊𝒏(𝝎𝒕 + 𝜽)
Real sinusoid as the sum of two complex sinusoids:
𝑨𝒄𝒐𝒔 𝝎𝒕 + 𝜽 =
𝑨
𝟐
[𝒆𝒋(𝝎𝒕+𝜽)
+𝒆−𝒋(𝝎𝒕+𝜽)
]
𝑨𝒔𝒊𝒏 𝝎𝒕 + 𝜽 =
𝑨
𝟐𝒋
[𝒆𝒋(𝝎𝒕+𝜽)
−𝒆−𝒋(𝝎𝒕+𝜽)
]
10/26/2022
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![Relationship Between
DT Complex Exponentials & Real Sinusoids
Euler’s formula: 𝒆𝒋𝛀𝒏
= 𝒄𝒐𝒔 𝛀𝒏 + 𝒋𝒔𝒊𝒏(𝛀𝒏)
Complex sinusoid as the sum of two real sinusoids
𝑪𝒆𝒋𝜽
𝒆𝒋𝛀𝒏
= 𝑪 𝒄𝒐𝒔 𝛀𝒏 + 𝜽 + 𝒋𝑪 𝒔𝒊𝒏(𝛀𝒏 + 𝜽)
Real sinusoid as the sum of two complex sinusoids:
𝑪𝒄𝒐𝒔 𝛀𝒏 + 𝜽 =
𝑪
𝟐
[𝒆𝒋(𝛀𝒏+𝜽)
+𝒆−𝒋(𝛀𝒏+𝜽)
]
𝑪 𝒔𝒊𝒏 𝛀𝒏 + 𝜽 =
𝑪
𝟐𝒋
[𝒆𝒋(𝛀𝒏+𝜽)
−𝒆−𝒋(𝛀𝒏+𝜽)
]
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![Unit Step Signal
CT unit step signal u(t):
𝒖 𝒕 =
𝟏 𝒕 ≥ 𝟎
𝟎 𝒕 < 𝟎
DT unit step signal u[n]:
𝒖[𝒏] =
𝟏 𝒏 ≥ 𝟎
𝟎 𝒏 < 𝟎
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chapter two of system and signals course.pdf
- 1. Signals and Systems – S&S Arab International University (AIU) Faculty of Informatics and Communication Engineering Elementary Signals 1
- 2. Outlines Elementary signals Sinusoidal signal Exponential signal • Relation between exponential and sinusoidal signals Unit Step Rectangular Pulse Signum Unit Impulse 10/26/2022 S&S 2
- 3. Elementary signals Elementary signals are useful to: Represent more complicated signals. Model many physical signals in nature. Simplify the analysis and design of linear systems. 10/26/2022 S&S 3
- 4. CT Sinusoidal Signal 𝒙 𝒕 = 𝑨𝒄𝒐𝒔 𝝎𝒕 + 𝝓 Periodic with 3 characteristics: 𝑨, 𝝎, 𝝓. 𝑨 : amplitude 𝝓 : phase [radians; degrees] 𝝎 = 𝟐𝝅𝒇 : angular frequency [rad/s] 𝒇 = 𝟏/𝑻 : frequency [𝐇𝐳 = 𝟏/𝒔] 𝑻 = 𝟏/𝒇 : Period [s] 10/26/2022 S&S 4
- 6. Frequency is the rate of change with respect to time. 10/26/2022 S&S 6
- 7. Units of period and frequency 10/26/2022 S&S 7
- 9. 10/26/2022 S&S 9 Varying Sine Waves 𝑥(𝑡) = 𝐴 sin(2𝑓𝑡 + 𝜙)
- 10. Phase describes the position of the waveform relative to time 0. 10/26/2022 S&S 10
- 11. DT Sinusoidal Signal 𝒙 𝒏 = 𝑨𝒄𝒐𝒔 𝛀𝒏 + 𝝓 Not all DT sinusoidal signals are periodic. To be periodic: 𝛀 𝟐𝝅 = 𝒎 𝑵 must be a rational number 𝑵 is the number of samples contained in a single cycle of 𝒙 𝒏 . 10/26/2022 S&S 11
- 12. DT Sinusoidal Signal 𝐀 = 𝟏, 𝝓 = 𝟎, 𝑵 = 𝟏𝟐 10/26/2022 S&S 12
- 13. DT Sinusoidal Signal 𝑥 𝑛 = 𝑐𝑜𝑠 5π𝑛 Ω = 5𝜋 → Ω 2𝜋 = 5 2 = 𝑚 𝑁 𝑁 = 2𝑚 5 𝑚 = 5, 10, 15, ⋯ → 𝑁 = 2, 4, 6, ⋯ 10/26/2022 S&S 13
- 14. 10/26/2022 S&S 14 Effect of Increasing the Frequency of 𝐜𝐨𝐬 𝛀𝒏
- 15. 10/26/2022 S&S 15 Effect of Increasing the Frequency of 𝐜𝐨𝐬 𝛀𝒏
- 16. CT Real Exponential Signal 𝒙 𝒕 = 𝑩 𝒆𝒂𝒕 𝑩 and 𝒂 are real 𝑩 is the amplitude of signal at 𝒕 = 𝟎 𝒂 is positive or negative => two special cases: 1. Decaying exponential, 𝒂 < 𝟎 2. Growing exponential, 𝒂 > 𝟎 If 𝒂 = 𝟎 => 𝒙 𝒕 = 𝑩 DC value 10/26/2022 S&S 16
- 17. CT - Exponential Signal 10/26/2022 S&S 17 𝒂 < 𝟎 𝒂 > 𝟎
- 18. DT Real Exponential Signal 𝒙[𝒏] = 𝑩 𝒆𝒂𝒏 𝑩 and 𝒂 are real 𝒂 is positive or negative => two special cases: 1. Decaying exponential, 𝒂 < 𝟎 2. Growing exponential, 𝒂 > 𝟎 10/26/2022 S&S 18
- 19. DT - Exponential Signal 10/26/2022 S&S 19 𝒂 < 𝟎 𝒂 > 𝟎
- 20. Complex Exponential Signal CT Complex Exponential Signal 𝒙(𝒕) = 𝑨𝒆𝝀𝒕 𝑨 and 𝝀 are complex constants (e.g. 𝒆𝒋𝝎𝒕) DT Complex Exponential Signal 𝒙[𝒏] = 𝑪𝒆𝒃𝒏 𝑪 and 𝒃 are complex constants (e.g. 𝒆𝒋𝛀𝒏) 10/26/2022 S&S 20
- 21. Complex Sinusoidal Signals A special case of a complex exponential 𝒙(𝒕) = 𝑨𝒆𝒋𝝎𝒕 𝑨 is complex and 𝝀 = 𝒋𝝎 (𝝎 is real) 𝒙[𝒏] = 𝑪𝒆𝒋𝛀𝒏 𝑪 is complex and 𝒃 = 𝒋𝛀 (𝛀 is real) 10/26/2022 S&S 21
- 22. CT Complex Sinusoidal Signals Expressing 𝐴 in polar form |𝐴|𝑒𝑗𝜃 and using Euler’s formula, we can rewrite 𝑥 𝑡 as 𝑥 𝑡 = 𝑨 𝐜𝐨𝐬 𝝎𝒕 + 𝜽 + 𝑗|𝑨|𝐬𝐢𝐧(𝝎𝒕 + 𝜽) 𝓡𝒆 𝒙 𝒕 𝕴𝒎{𝒙 𝒕 } 𝑥 𝑡 is periodic with fundamental period 𝑻 = 𝟐𝝅 𝝎 10/26/2022 S&S 22
- 26. Relationship Between CT Complex Exponentials & Real Sinusoids Euler’s formula: 𝒆𝒋𝝎𝒕 = 𝒄𝒐𝒔 𝝎𝒕 + 𝒋𝒔𝒊𝒏(𝝎𝒕) Complex sinusoid as the sum of two real sinusoids 𝑨𝒆𝒋𝜽 𝒆𝒋𝝎𝒕 = 𝑨 𝒄𝒐𝒔 𝝎𝒕 + 𝜽 + 𝒋𝑨 𝒔𝒊𝒏(𝝎𝒕 + 𝜽) Real sinusoid as the sum of two complex sinusoids: 𝑨𝒄𝒐𝒔 𝝎𝒕 + 𝜽 = 𝑨 𝟐 [𝒆𝒋(𝝎𝒕+𝜽) +𝒆−𝒋(𝝎𝒕+𝜽) ] 𝑨𝒔𝒊𝒏 𝝎𝒕 + 𝜽 = 𝑨 𝟐𝒋 [𝒆𝒋(𝝎𝒕+𝜽) −𝒆−𝒋(𝝎𝒕+𝜽) ] 10/26/2022 S&S 26
- 27. Relationship Between DT Complex Exponentials & Real Sinusoids Euler’s formula: 𝒆𝒋𝛀𝒏 = 𝒄𝒐𝒔 𝛀𝒏 + 𝒋𝒔𝒊𝒏(𝛀𝒏) Complex sinusoid as the sum of two real sinusoids 𝑪𝒆𝒋𝜽 𝒆𝒋𝛀𝒏 = 𝑪 𝒄𝒐𝒔 𝛀𝒏 + 𝜽 + 𝒋𝑪 𝒔𝒊𝒏(𝛀𝒏 + 𝜽) Real sinusoid as the sum of two complex sinusoids: 𝑪𝒄𝒐𝒔 𝛀𝒏 + 𝜽 = 𝑪 𝟐 [𝒆𝒋(𝛀𝒏+𝜽) +𝒆−𝒋(𝛀𝒏+𝜽) ] 𝑪 𝒔𝒊𝒏 𝛀𝒏 + 𝜽 = 𝑪 𝟐𝒋 [𝒆𝒋(𝛀𝒏+𝜽) −𝒆−𝒋(𝛀𝒏+𝜽) ] 10/26/2022 S&S 27
- 28. 𝒆𝒋𝛀𝒏 = 𝒆𝒋 𝝅 𝟒 𝒏 = 𝒄𝒐𝒔 𝝅 𝟒 𝒏 + 𝒋𝒔𝒊𝒏 𝝅 𝟒 𝒏 𝒏 = 𝟎, 𝟏, 𝟐, … , 𝟕 10/26/2022 S&S 28 Relationship Between DT Complex Exponentials & Real Sinusoids
- 29. Unit Step Signal CT unit step signal u(t): 𝒖 𝒕 = 𝟏 𝒕 ≥ 𝟎 𝟎 𝒕 < 𝟎 DT unit step signal u[n]: 𝒖[𝒏] = 𝟏 𝒏 ≥ 𝟎 𝟎 𝒏 < 𝟎 10/26/2022 S&S 29
- 31. Remember: Fundamental period of complex exponential and sinusoidal signals. • 𝑒𝑗𝜔0𝑡 and cos 𝜔0𝑡 + 𝜃 are periodic for any value of 𝜔0. 𝑇0 = 2𝜋 𝜔0 is the fundamental period. • 𝑒𝑗Ω0𝑛 and cos Ω0𝑛 are periodic only if Ω0 2𝜋 = 𝑚 𝑁 is a rational number. 𝑁0 is the smallest 𝑁. • cos2 𝜃 = 1 2 + 1 2 cos 2𝜃 ; sin2 𝜃 = 1 2 − 1 2 cos 2𝜃 10/26/2022 S&S 31
- 32. Ex0: Determine if each function 𝒙 given below is periodic; and if it is, find its fundamental period. a) 𝑥 𝑡 = cos 2𝜋𝑡 + 𝜋 3 b) 𝑥 𝑛 = 𝑒𝑗 𝜋 4 𝑛 c) 𝑥 𝑡 = sin2 𝑡 d) 𝑥 𝑛 = cos2 ( 𝜋 8 n) e) 𝑥 𝑛 = cos(2n) 10/26/2022 S&S 32
- 33. Ex1: Determine whether the following signals are energy signals, power signals, or neither. a ) 𝑥 𝑡 = 𝑒−𝑎𝑡 𝑢 𝑡 , 𝑎 > 0 b ) 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔0𝑡 + 𝜃) 10/26/2022 S&S 33
- 34. End of Lecture 2 10/26/2022 S&S 34