Signals and systems-4
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The document discusses Fourier representations of signals and linear time-invariant (LTI) systems. It begins by introducing Fourier series (FS) and Fourier transforms (FT) for continuous and discrete time periodic and non-periodic signals. It then provides examples of using Fourier series to represent continuous time periodic signals, including deriving FS coefficients and determining time domain signals from given FS coefficients. Key aspects covered are the FS representation as a weighted sum of complex sinusoids and interpretation of Fourier coefficients in the frequency domain.
![Fourier Representations of Signals and LTI
Systems
Time Property Periodic Non periodic
Continuous
(t)
Fourier Series
(FS)
Fourier Transform
(FT)
Discrete
[n]
Discrete Time Fourier Series
(DTFS)
Discrete Time Fourier Transform
(DTFT)
Prof: Sarun Soman, MIT, Manipal 2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Signals and systems-4
- 1. Signals and Systems-IV Prof: Sarun Soman Manipal Institute of Technology Manipal
- 2. Fourier Representations of Signals and LTI Systems Time Property Periodic Non periodic Continuous (t) Fourier Series (FS) Fourier Transform (FT) Discrete [n] Discrete Time Fourier Series (DTFS) Discrete Time Fourier Transform (DTFT) Prof: Sarun Soman, MIT, Manipal 2
- 3. Continuous Time Periodic Signals: Fourier Series FS of a signal x(t) ݔ ݐ = ܺ[݇]݁ఠబ௧ ஶ ୀିஶ )ݐ(ݔ fundamental period is T, fundamental frequency ߱ = ଶగ ் A signal is represented as weighted superposition of complex sinusoids. Representing signal as superposition of complex sinusoids provides an insightful characterization of signal. The weight associated with a sinusoid of a given frequency represents the contribution of that sinusoid to the overall signal. Prof: Sarun Soman, MIT, Manipal 3
- 4. Jean Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) Prof: Sarun Soman, MIT, Manipal 4
- 5. Continuous Time Periodic Signals: Fourier Series Prof: Sarun Soman, MIT, Manipal 5
- 6. Continuous Time Periodic Signals: Fourier Series ܺ ݇ − Fourier Coefficient ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧݀ݐ ் Fourier series coefficients are known as a frequency –domain representation of .)ݐ(ݔ Eg. Determine the FS representation of the signal. ݔ ݐ = 3 cos గ ଶ ݐ + గ ସ using the method of inspection. Prof: Sarun Soman, MIT, Manipal 6
- 7. Example ܶ = 4, ߱ = ߨ 2 FS representation of a signal x(t) ݔ ݐ = ܺ[݇]݁ఠబ௧ ஶ ୀିஶ ݔ ݐ = ܺ[݇]݁ గ ଶ ௧ ஶ ୀିஶ (1) Using Euler’s formula to expand given .)ݐ(ݔ ݔ ݐ = 3 ݁ గ ଶ௧ା గ ସ + ݁ ି గ ଶ௧ା గ ସ 2 )ݐ(ݔ = 3 2 ݁ గ ସ݁ గ ଶ ௧ + 3 2 ݁ି గ ସ݁ି గ ଶ ௧ (2) Equating each term in eqn (2) to the terms in eqn (1) X k = 3 2 eି୨ ସ, k = 1 3 2 e୨ ସ, k = −1 0, otherwise Prof: Sarun Soman, MIT, Manipal 7
- 8. Example All the power of the signal is concentrated at two frequencies ࣓ = ࣊ and ࣓ = − ࣊ . Determine the FS coefficients for the signal )ݐ(ݔ Ans: ܶ = 2, ߱ = ߨ Magnitude & Phase Spectra t -2 0 2 4 6-1 x(t) ݁ିଶ௧ Prof: Sarun Soman, MIT, Manipal 8
- 9. Example ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧݀ݐ ் ܺ ݇ = 1 2 න ݁ିଶ௧݁ିగ௧݀ݐ ଶ = 1 2 න ݁ି(ଶାగ)௧݀ݐ ଶ ܺ ݇ = −1 2(2 + ݆݇ߨ) ݁ି(ଶାగ)௧ | ଶ = 1 4 + ݆2݇ߨ 1 − ݁ିସ݁ିଶగ ݁ିଶగ = 1 = 1 − ݁ିସ 4 + ݆݇2ߨ Find the time domain signal whose FS coefficients are ܺ ݇ = ݆ߜ ݇ − 1 − ݆ߜ ݇ + 1 + ߜ ݇ − 3 + ߜ ݇ + 3 , ߱ = ߨ Ans: FS of a signal x(t) ݔ ݐ = ܺ[݇]݁ఠబ௧ ஶ ୀିஶ ݔ ݐ = ܺ[݇]݁గ௧ ஶ ୀିஶ = ݆݁(ଵ)గ௧ − ݆݁(ିଵ)గ௧ + ݁(ଷ)గ௧ + ݁(ିଷ)గ௧ Prof: Sarun Soman, MIT, Manipal 9
- 10. Example = ݆(2݆ sin ߨ)ݐ + 2 cos 3ߨݐ = − ܖܑܛ ࢚࣊ + ܛܗ܋ ࢚࣊ Find the FS coefficient of periodic signal )ݐ(ݔ as shown in Fig. Ans: ܶ = 6, ߱ = ߨ 3 ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧݀ݐ ் ଶ ି ் ଶ ܺ ݇ = 1 6 න ݁)ݐ(ݔି గ ଷ ௧ ݀ݐ ଷ ିଷ = 1 6 න 1 ݁ି గ ଷ௧ ݀ݐ + න (−1) ଶ ଵ ିଵ ିଶ ݁ି గ ଷ ௧ ݀ݐ = 1 6 ݁ି గ ଷ ௧ −݆݇ ߨ 3 |ିଶ ିଵ + ݁ି గ ଷ ௧ ݆݇ ߨ 3 |ଵ ଶ 0 2 4-2 -4 t x(t) Prof: Sarun Soman, MIT, Manipal 10
- 11. Example = 1 6 ݁ି గ ଷ(ିଶ) − ݁ି గ ଷ(ିଵ) ݆݇ ߨ 3 + ݁ି గ ଷ(ଶ) − ݁ି గ ଷ(ଵ) ݆݇ ߨ 3 = 1 6 ݁ ଶగ ଷ + ݁ି ଶగ ଷ ݆݇ ߨ 3 − ݁ గ ଷ + ݁ି గ ଷ ݆݇ ߨ 3 = 1 ݆2ߨ݇ 2 ܿݏ 2ߨ݇ 3 − 2 cos ߨ݇ 3 , ݇ ≠ 0 For ݇ = 0 ܺ 0 = 1 6 න ݐ݀)ݐ(ݔ ଷ ିଷ = 1 6 න 1 ݀ݐ + න (−1) ଶ ଵ ିଵ ିଶ ݀ݐ = 1 6 −1 + 2 − 1 = 0 The DC component is zero. Prof: Sarun Soman, MIT, Manipal 11
- 12. Example Find the FS coefficient of the signal .)ݐ(ݔ Ans: ܶ = 2, ߱ = ߨ ݔ ݐ = ൜ 1 + ,ݐ −1 < ݐ < 0 1 − ,ݐ 0 < ݐ < 1 ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧ ݀ݐ ் ଶ ି ் ଶ ܺ ݇ = 1 2 න ݁)ݐ(ݔିగ௧݀ݐ ଵ ିଵ = 1 2 ቈන 1 + ݐ ݁ିగ௧ ݀ݐ ିଵ + න 1 + ݐ ݁ିగ௧݀ݐ ଵ = 1 2 ቈන 1 ݁ିగ௧ ݀ݐ ିଵ + න ݐ ݁ିగ௧ ݀ݐ ିଵ + න 1 ݁ିగ௧ ݀ݐ ଵ + න ݐ ݁ିగ௧ ݀ݐ ଵ 10-1-2 2 t x(t) Prof: Sarun Soman, MIT, Manipal 12
- 13. Example ܺ ݇ = 1 ߨଶ݇ଶ 1 − −1 , ݇ ≠ 0 For ݇ = 0 ܺ 0 = 1 2 ቈන (1 ିଵ + ݐ)݀ݐ + න 1 − ݐ ݀ݐ ଵ = 1 2 ܵ݅݊ܿ function ܿ݊݅ݏ ݑ = sin ߨݑ ߨݑ The functional form ୱ୧୬ గ௨ గ௨ often occurs in Fourier Analysis Prof: Sarun Soman, MIT, Manipal 13
- 14. Continuous Time Periodic Signals: Fourier Series – The maximum of the function is unity at ݑ = 0. – The zero crossing occur at integer values of .ݑ – Mainlobe- portion of the function b/w the zero crossings at ݑ = ±1. – Sidelobes- The smaller ripples outside the mainlobe. – The magnitude dies off as ଵ ௨ . Prof: Sarun Soman, MIT, Manipal 14
- 15. Continuous Time Periodic Signals: Fourier Series Determine the FS representation of the square wave depicted in Fig. Ans: The period is T , ߱ = ଶగ ் The signal has even symmetry, integrate over the range − ் ଶ ݐ ் ଶ ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧݀ݐ ் ଶ ି ் ଶ ܺ ݇ = 1 ܶ න (1)݁ିఠబ௧݀ݐ ் ଶ ି ் ଶ ܺ ݇ = 1 ܶ න (1) ்ೞ ି்ೞ ݁ିఠబ௧݀ݐ ܺ ݇ = −1 ܶ݇߱ ݁ିఠబ௧|ି்ೞ ்ೞ ܺ ݇ = −1 ܶ݇߱ ݁ିఠబ்ೞ − ݁ఠబ்ೞ ܺ ݇ = 2 ܶ݇߱ ݁ఠబ்ೞ − ݁ିఠబ்ೞ ݆2 ܺ ݇ = 2 ܶ݇߱ sin ݇߱ܶ௦ , ݇ ≠ 0 Prof: Sarun Soman, MIT, Manipal 15
- 16. Example For ݇ = 0 ܺ 0 = 1 ܶ න ݀ݐ ்ೞ ି்ೞ = 2ܶ௦ ܶ ܺ ݇ = 2 ܶ݇߱ sin ݇߱ܶ௦ ߱ = 2ߨ ܶ ܺ ݇ = sin ߨ݇ 2ܶ௦ ܶ ߨ݇ ܺ ݇ = 2ܶ௦ ܶ sin ߨ݇ 2ܶ௦ ܶ ߨ݇ 2ܶ௦ ܶ ܺ ݇ = 2ܶ௦ ܶ ܿ݊݅ݏ ݇ 2ܶ௦ ܶ 2ܶ௦ ܶ = 1 8 = 12.5% 2ܶ௦ ܶ = 1 2 = 50% Prof: Sarun Soman, MIT, Manipal 16
- 17. Example Use the defining equation for the FS coefficients to evaluate the FS representation for the following signals. ݔ ݐ = sin 3ߨݐ + cos 4ߨݐ Ans: ܶଵ = 2 3 , ܶଶ = 1 2 )ݐ(ݔ will be periodic with T=2sec. Fundamental frequency ߱ = ߨ ݔ ݐ ݔ ݐ = ܺ[݇]݁ఠబ௧ ஶ ୀିஶ ܺ ݇ = 1 2 , ݇ = ±4 1 ݆2 , ݇ = 3 −1 ݆2 , ݇ = −3 Prof: Sarun Soman, MIT, Manipal 17
- 18. 0 x(t) t 2 1 3 2 3 4 3 − 8 3 -2 − 2 3 − 4 3 Example Find X[k] Ans: m x(t) 0 2δ(t) 1 −ߜ ݐ − 1 3 − ߜ ݐ + 2 3 2 ߜ ݐ − 2 3 + ߜ ݐ + 4 3 3 −ߜ ݐ − 1 − ߜ ݐ + 2 4 ߜ ݐ − 4 3 + ߜ ݐ + 8 3 1 Prof: Sarun Soman, MIT, Manipal 18
- 19. Example m x(t) -1 −ߜ ݐ + 1 3 − ߜ ݐ − 2 3 -2 ߜ ݐ + 2 3 + ߜ ݐ − 4 3 -3 −ߜ ݐ + 1 − ߜ ݐ − 2 -4 ߜ ݐ + 4 3 + ߜ ݐ − 8 3 -1 0 x(t) t 2 − 1 3 2 3 4 3 − 8 3 -2 − 2 3 − 4 3 1-1 0 x(t) t 2 − 1 3 1 3 4 3 -2 − 4 3 ܺ ݇ = 1 ܶ න ݁)ݐ(ݔିఠబ௧ ݀ݐ ் ଶ ି ் ଶ ܺ ݇ = 3 4 න ݁)ݐ(ݔି ଷగ ଶ ௧ ݀ݐ ଶ ଷ ି ଶ ଷ Prof: Sarun Soman, MIT, Manipal 19
- 20. Example ܺ ݇ = 3 4 න 2δ t − δ t − 1 3 − δ t + 1 3 ݁ି ଷగ ଶ ௧ ݀ݐ ଶ ଷ ି ଶ ଷ Using sifting property ܺ ݇ = 3 4 2 − ݁ି గ ଶ − ݁ గ ଶ ܺ ݇ = 6 4 − 6 4 cos ݇ ߨ 2 Prof: Sarun Soman, MIT, Manipal 20
- 21. Discrete Time Periodic Signals: The Discrete Time Fourier Series DTFS representation of a periodic signal with fundamental frequency Ω = ଶగ ே ݔ ݊ = ܺ[݇]݁Ωబ ேିଵ ୀ Where ܺ ݇ = 1 ܰ ]݊[ݔ ேିଵ ୀ ݁ିΩబ Prof: Sarun Soman, MIT, Manipal 21
- 22. Discrete Time Periodic Signals: The Discrete Time Fourier Series ]݊[ݔand ܺ ݇ are exactly characterized by a finite set of N numbers. DTFS is the only Fourier representation that can be numerically evaluated and manipulated in a computer. ݔ ݊ is ‘N’ periodic in ‘n’ ܺ[݇] is ‘N’ periodic in ‘k’ Prof: Sarun Soman, MIT, Manipal 22
- 23. Example Find the frequency domain representation of the signal depicted in Fig. Ans: ܰ = 5, Ω = 2ߨ 5 ܺ ݇ = 1 ܰ ]݊[ݔ ேିଵ ୀ ݁ିΩబ The signal has odd symmetry, sum over n=-2 to 2 ܺ ݇ = 1 5 ]݊[ݔ ଶ ୀିଶ ݁ି ଶగ ହ = 1 5 ൜0 + 1 2 ݁ ଶగ ହ + 1 − 1 2 ݁ି ଶగ ହ + 0ൠ = 1 5 1 + ݆ sin 2ߨ݇ 5 ● 1 ●● -2 0 2 -4 y[n] n4 -6 ● ●6● 1 2ൗ Prof: Sarun Soman, MIT, Manipal 23
- 24. Example X[k] will be periodic with period ‘N’. Values of X[k] for k=-2 to 2. Calculator in radians mode ܺ −2 = 1 5 1 − ݆ sin 4ߨ 5 = 0.232݁ି.ହଷଵ ܺ −1 = 1 5 1 − ݆ sin 2ߨ 5 = 0.276݁ି. ܺ 0 = 1 5 ܺ 1 = 1 5 1 + ݆ sin 2ߨ 5 = 0.276݁. ܺ 2 = 1 5 1 + ݆ sin 4ߨ 5 = 0.232݁.ହଷଵ Mag & phase plot. Prof: Sarun Soman, MIT, Manipal 24
- 25. Example Use the defining equation for the DTFS coefficients to evaluate the DTFS representation for the following signals. ݔ ݊ = cos 6ߨ 17 ݊ + ߨ 3 ݔ ݊ = ܺ[݇]݁Ωబ ேିଵ ୀ ܰ = 17, Ω = 2ߨ 17 ݔ ݊ = 1 2 ݁ గ ଵା గ ଷ + ݁ ି గ ଵା గ ଷ ݔ ݊ = 1 2 ݁ గ ଷ݁(ଷ) ଶగ ଵ + 1 2 ݁ି గ ଷ݁(ିଷ) ଶగ ଵ ܺ[݇] = 1 2 ݁ గ ଷ, ݇ = 3 1 2 ݁ି గ ଷ, ݇ = −3 0, ݇ ݊ ݁ݏ݅ݓݎ݄݁ݐ = {−8, −7, … , 8} Prof: Sarun Soman, MIT, Manipal 25
- 26. Example ݔ ݊ = 2 sin 4ߨ 19 ݊ + cos 10ߨ 19 ݊ + 1 Ans: ܰ = 19, Ω = 2ߨ 19 = 1 ݆ ݁ ସగ ଵଽ − ݁ି ସగ ଵଽ + 1 2 ݁ ଵగ ଵଽ + ݁ି ଵగ ଵଽ + 1 = −݆݁ ଶ ଶగ ଵଽ + ݆݁ ିଶ ଶగ ଵଽ + 1 2 ݁ ହ ଶగ ଵଽ + 1 2 ݁ ିହ ଶగ ଵଽ + 1݁ ଶగ ଵଽ ݔ ݊ = ܺ[݇]݁Ωబ ேିଵ ୀ ܺ ݇ = 1 2 , ݇ = ±5 ݆, ݇ = −2 1, ݇ = 0 −݆, ݇ = 2 0, ݇ ݊ ݁ݏ݅ݓݎ݄݁ݐ = {−9,8, . . , 9} Prof: Sarun Soman, MIT, Manipal 26
- 27. Example ݔ ݊ = [ −1 (ߜ ݊ − 2݉ ஶ ୀିஶ + ߜ ݊ + 3݉ )] Ans ܰ = 12, Ω = ߨ 6 ܺ ݇ = 1 ܰ ]݊[ݔ ேିଵ ୀ ݁ିΩబ m x[n] 0 2ߜ[݊] 1 −ߜ ݊ − 2 − ߜ[݊ + 3] 2 ߜ ݊ − 4 + ߜ[݊ + 6] 3 −ߜ ݊ − 6 − ߜ[݊ + 9] 4 ߜ ݊ − 8 + ߜ[݊ + 12] 5 −ߜ ݊ − 10 − ߜ[݊ + 15] 6 ߜ ݊ − 12 + ߜ[݊ + 18] m x[n] -1 −ߜ ݊ + 2 − ߜ[݊ − 3] -2 ߜ ݊ + 4 + ߜ[݊ − 6] -3 −ߜ ݊ + 6 − ߜ[݊ − 9] -4 ߜ ݊ + 8 + ߜ[݊ − 12] -5 −ߜ ݊ + 10 − ߜ[݊ − 15] -6 ߜ ݊ + 2 + ߜ[݊ − 18] Prof: Sarun Soman, MIT, Manipal 27
- 28. Example ܺ ݇ = 1 12 ]݊[ݔ ୀିହ ݁ି గ ݔ ݊ = 0, ݂݊ ݎ = ±5,6 Prof: Sarun Soman, MIT, Manipal 28
- 29. Example ݔ ݊ = cos ݊ߨ 30 + 2 sin ݊ߨ 90 Ans: Ωଵ = ߨ݊ 30 = 2ߨ݊ 60 ܰଵ = 60, ܰଶ = 180 ܰଵ ܰଶ = 1 3 ݔ ݊ will be periodic with period N=180, Ω = ଶగ ଵ଼ ݔ ݊ = 1 2 ݁ గ ଷ + ݁ି గ ଷ + 1 ݆ ݁ గ ଽ − ݁ି గ ଽ ݔ ݊ = ܺ[݇]݁Ωబ ேିଵ ୀ ݔ ݊ = 1 2 ݁(ଷ) ଶగ ଵ଼ + ݁(ିଷ) ଶగ ଵ଼ − ݆ ݁(ଵ) ଶగ ଵ଼ − ݁(ିଵ) ଶగ ଵ଼ ܺ ݇ = ݆, ݇ = −1 −݆, ݇ = 1 1 2 , ݇ = ±3 0, ݊ ݁ݏ݅ݓݎ݄݁ݐ − 89 ≤ ݇ ≤ 90 Prof: Sarun Soman, MIT, Manipal 29
- 30. Example Inverse DTFS: used to determine the time domain signal x[n] from DTFS coefficients X[k]. Ans: ܰ = 9, Ω = 2ߨ 9 Take ݇ = −4 4 ݐ Find ܺ[݇]from the plot. ݔ ݊ = ܺ[݇]݁Ωబ ேିଵ ୀ ݔ ݊ = ܺ[݇]݁ ଶగ ଽ ସ ୀିସ ࡷ ࢄ[] -4 0 -3 ݁ ଶగ ଷ -2 2݁ గ ଷ -1 0 0 ݁గ 1 0 2 2݁ି గ ଷ 3 ݁ି ଶగ ଷ 4 0 Prof: Sarun Soman, MIT, Manipal 30
- 31. Example ݔ ݊ = ݁ ଶగ ଷ ݁(ିଷ) ଶగ ଽ + 2݁ గ ଷ݁(ିଶ) ଶగ ଽ + ݁గ݁() ଶగ ଽ + 2݁ି గ ଷ݁(ଶ) ଶగ ଽ + ݁ି ଶగ ଷ ݁(ଷ) ଶగ ଽ ݔ ݊ = 2 cos 6ߨ݊ 9 − 2ߨ 3 + 4 sin 4ߨ݊ 9 − ߨ 3 − 1 Find x[n]. Ans: ܰ = 12, Ω = 2ߨ 12 Prof: Sarun Soman, MIT, Manipal 31
- 32. Example ݔ ݊ = ܺ[݇]݁ ଶగ ଵହ ସ ୀିସ From table expression for ܺ[݇] ܺ ݇ = ݁ି గ ݔ ݊ = ݁ି గ ݁ ଶగ ଵହ ସ ୀିସ ݔ ݊ = ݁ గ ଶ ଵହ ି ଵ ସ ୀିସ Let ݈ = ݇ + 4 ݔ ݊ = ݁ గ ଶ ଵହ ି ଵ ିସ ଼ ୀ ࢄ[] -4 ݁ ସగ -3 ݁ ଷగ -2 ݁ ଶగ -1 ݁ గ 0 1 1 ݁ି గ 2 ݁ି ଶగ 3 ݁ି ଷగ 4 ݁ି ସగ Prof: Sarun Soman, MIT, Manipal 32
- 33. Example ݔ ݊ = ݁ ିସగ ଶ ଵଶ ି ଵ ݁ గ ଶ ଵଶ ି ଵ ଼ ୀ ݔ ݊ = ݁ ିସగ ଶ ଵଶି ଵ 1 − ݁ ଽగ ଶ ଵଶ ି ଵ 1 − ݁ గ ଶ ଵଶି ଵ Prof: Sarun Soman, MIT, Manipal 33
- 34. Example = ݁ ିସగ ଶ ଵଶି ଵ ݁ ଽ ଶగ ଶ ଵଶି ଵ ݁ ି ଽ ଶగ ଶ ଵଶି ଵ − ݁ ଽ ଶగ ଶ ଵଶି ଵ ݁ గ ଶ ଶ ଵଶ ି ଵ ݁ ି గ ଶ ଶ ଵଶ ି ଵ − ݁ గ ଶ ଶ ଵଶ ି ଵ = ݁ ି ଽ ଶగ ଶ ଵଶି ଵ − ݁ ଽ ଶగ ଶ ଵଶି ଵ ݁ ି గ ଶ ଶ ଵଶି ଵ − ݁ గ ଶ ଶ ଵଶି ଵ = sin 9 2 ߨ 2݊ 12 − 1 6 sin ߨ 2 2݊ 12 − 1 6 Prof: Sarun Soman, MIT, Manipal 34