Signals and systems-5
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Discussion on continuous and discrete time Fourier Transform.Solved examples using properties of Fourier transform. Introduction to Z transform
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Signals and systems-5
- 1. Signals and Systems-V Prof: Sarun Soman Manipal Institute of Technology Manipal
- 2. Non-periodic Signals: Fourier-Transform Representations No restrictions on the period of the sinusoids used to represent non-periodic signal. Frequencies can take a continuum of values. For CT non periodic signal the range is from −∞ to ∞ For DT non periodic signal the range is from −ߨ to ߨ CTFT ݔ ݐ = 1 2ߨ න ܺ(݆߱)݁ఠ௧݀߱ ஶ ିஶ (1) DTFT ݔ ݊ = 1 2ߨ න ܺ(݆Ω)݁Ω݀Ω గ ିగ (2) Prof: Sarun Soman, MIT, Manipal 2
- 3. Continuous Time Non-periodic Signals: The Fourier Transform CTFT is used to represent a continuous time non-periodic signal as a superposition of complex sinusoids. ݔ ݐ = 1 2ߨ න ܺ(݆߱)݁ఠ௧݀߱ ஶ ିஶ Where ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧݀ݐ ஶ ିஶ ܺ ݆߱ is the frequency domain representation of )ݐ(ݔ The weight on each sinusoid is ఠ ଶగ Prof: Sarun Soman, MIT, Manipal 3
- 4. Continuous Time Non-periodic Signals: The Fourier Transform CTFT is used to analyze the characteristics of CT systems and the interaction b/w CT signals and systems. Eq(1) and (2) may not converge for all functions of x(t) Dirichlet conditions for non periodic signal x(t) is absolutely integrable න )ݐ(ݔ ݀ݐ < ∞ ஶ ିஶ x(t) has a finite number of maxima, minima and discontinuities in any finite interval. The size of each discontinuity is finite Eg. Unit step function is not absolutely integrable Prof: Sarun Soman, MIT, Manipal 4
- 5. Example 1.Find the FT of ݔ ݐ = ݁ଶ௧.)ݐ−(ݑ Ans: ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧݀ݐ ஶ ିஶ ܺ ݆߱ = න ݁ଶ௧ ିஶ ݁ିఠ௧݀ݐ = ݁ ଶିఠ ௧ 2 − ݆߱ |ିஶ = 1 2 − ݆߱ 2.ݔ ݐ = ݁ି ௧ Ans: ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧ ݀ݐ ஶ ିஶ ܺ ݆߱ = න ݁௧ ିஶ ݁ିఠ௧ ݀ݐ + න ݁ି௧ ݁ିఠ௧ ݀ݐ ஶ = ݁ ఠାଵ ௧ ݆߱ + 1 |ିஶ + ݁ି ఠାଵ ௧ −(݆߱ + 1) | ஶ = 1 1 + ݆߱ + 1 ݆߱ + 1 = 2 1 + ݆߱ Prof: Sarun Soman, MIT, Manipal 5
- 6. Example Find the FT of ݔ ݐ Ans: Rectangular pulse is absolutely integrable provided ܶ < ∞ ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧ ݀ݐ ஶ ିஶ ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧݀ݐ ் ି் = − 1 ݆߱ ݁ିఠ௧݀|ݐି் ் = ݁ఠ் − ݁ିఠ் ݆߱ = 2 sin ߱ܶ ߱ , ߱ ≠ 0 For ߱ = 0 lim ఠ→ 2 sin ߱ܶ ߱ = 2ܶ Zero crossing points ߱ܶ = ±݉ߨ ߱ = ± ݉ߨ ܶ , ݉ = ±1, ±2, ±3 … . . Prof: Sarun Soman, MIT, Manipal 6
- 7. Example Ans: ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧ ݀ݐ ஶ ିஶ ܺ ݆߱ = න (1) ିଶ ݁ିఠ௧݀ݐ + න (−1) ଶ ݁ିఠ௧݀ݐ ܺ ݆߱ = ݁ିఠ௧ −݆߱ |ିଶ + ݁ିఠ௧ ݆߱ | ଶ = ݁ଶఠ − 1 ݆߱ + ݁ିଶఠ − 1 ݆߱ = ݆ 2 ߱ + 2 cos 2߱ ݆߱ Find FT t x(t) 2-2 1 Prof: Sarun Soman, MIT, Manipal 7
- 8. Example ݔ ݐ = ߜ()ݐ Draw the spectrum Ans: ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧ ݀ݐ ஶ ିஶ ܺ ݆߱ = න ߜ()ݐ ݁ିఠ௧ ݀ݐ ஶ ିஶ Using sifting property ܺ ݆߱ = 1 Inverse CTFT Determine the time domain signal ܺ ݆߱ = ݁ିଶఠ)߱(ݑ Ans: ݔ ݐ = 1 2ߨ න ܺ(݆߱)݁ఠ௧ ݀߱ ஶ ିஶ ݔ ݐ = 1 2ߨ න ݁ିଶఠ݁ఠ௧݀߱ ஶ = 1 2ߨ ݁(ିଶା௧)ఠ (−2 + ݆)ݐ | ஶ = 1 2ߨ ݁(ିଶା௧)ஶ − 1 −2 + ݆ݐ ܺ ݆߱ ߱ 0 Prof: Sarun Soman, MIT, Manipal 8
- 9. Example = 1 2ߨ(2 − ݆)ݐ Find inverse CTFT ࢄ ࣓ = ൝ ܛܗ܋ ࣓ , ࣓ < ࣊ , ࢚ࢎࢋ࢙࢘࢝ࢋ Ans: )ݐ(ݔ = 1 2ߨ න ܺ(݆߱)݁ఠ௧ ݀ ஶ ିஶ ߱ = 1 2ߨ න ݁ଶఠ + ݁ିଶఠ 2 గ ସ ି గ ସ ݁ఠ௧݀߱ = 1 2ߨ න 1 2 ݁ ଶା௧ ఠ గ ସ ି గ ସ ݀߱ + 1 2ߨ න 1 2 ݁(௧ିଶ)ఠ݀߱ గ ସ ି గ ସ = 1 2ߨ ݁ ଶା௧ ఠ 2(ݐ + 2) |ି గ ସ గ ସ + 1 2ߨ ݁ ௧ିଶ ఠ 2(ݐ − 2) |ି గ ସ గ ସ Prof: Sarun Soman, MIT, Manipal 9
- 10. Example = 1 2ߨ sin ߨ 4 ݐ + 2 (ݐ + 2) + sin ߨ 4 ݐ − 2 (ݐ − 2) ݔ ݐ = 1 2ߨ sin ߨ 4 ݐ + 2 (ݐ + 2) + sin ߨ 4 ݐ − 2 (ݐ − 2) 1 8 , ݐ = ±2 Find the time domain signal corresponding to the frequency spectrum. Ans: Prof: Sarun Soman, MIT, Manipal 10
- 11. Example ܺ ݆߱ = ൜ ݁ିଶఠ , ߱ < 2 0, ݁ݏ݅ݓݎ݄݁ݐ )ݐ(ݔ = 1 2ߨ න ܺ(݆߱)݁ఠ௧ ݀ ஶ ିஶ ߱ )ݐ(ݔ = 1 2ߨ න ݁ିଶఠ ݁ఠ௧ ݀ ଶ ିଶ ߱ = 1 2ߨ ݁ ௧ିଶ ఠ (ݐ − 2) |ିଶ ଶ = 1 ߨ(ݐ − 2) sin 2(ݐ − 2) ݔ ݐ = 1 ߨ(ݐ − 2) sin 2(ݐ − 2) , ݐ ≠ 2 2 ߨ , ݐ = 2 Find the time domain signal corresponding to the spectrum. Ans: )ݐ(ݔ = 1 2ߨ න ܺ(݆߱)݁ఠ௧݀ ஶ ିஶ ߱ )ݐ(ݔ = 1 2ߨ න ݁ఠ௧ ݀ ௐ ିௐ ߱ = 1 ݆ߨݐ ݁ௐ௧ − ݁ିௐ௧ 2 Prof: Sarun Soman, MIT, Manipal 11
- 12. Example = sin ܹݐ ߨݐ )ݐ(ݔ = ܹ ߨ sin ܹݐ ܹݐ , ݐ ≠ 0 For ݐ = 0 lim ௧→ sin ܹݐ ߨݐ ݔ ݐ = ܹ ߨ Zero crossing points ܹݐ = ±݉ߨ, ݉ = ±1,2,3 … . ݐ = ± ݉ߨ ܹ Prof: Sarun Soman, MIT, Manipal 12
- 13. Properties of Fourier Transform Linearity Linearity property is the basis of the partial fraction method for determining inverse FT. Eg. Find )ݐ(ݔ ܺ ݆߱ = −݆߱ (݆߱)ଶ+3݆߱ + 2 ܽݔ ݐ + ܾݔ ݐ ܽܺ ݆߱ + ܾܻ(݆߱) Prof: Sarun Soman, MIT, Manipal 13
- 14. Example = ܿଵ ݆߱ + 1 + ܿଶ ݆߱ + 2 ܿଵ = 1, ܿଶ = −2 ܺ ݆߱ = 1 ݆߱ + 1 − 2 ݆߱ + 2 Using the transformation table 1 1 ݆߱ + 1 + −2 1 ݆߱ + 2 ↔ 1 ݁ି௧ݑ ݐ + (−2)݁ିଶ௧)ݐ(ݑ ݔ ݐ = ݁ି௧ݑ ݐ − 2݁ିଶ௧)ݐ(ݑ Symmetry Property: Real and Imaginary Signals. If )ݐ(ݔ is real and even ܺ(݆߱) is real If )ݐ(ݔ is real and odd ܺ(݆߱) is imaginary Time Shift properties ݐ(ݔ − ݐ) ↔ ݁ିఠబ௧ܺ(݆߱) • Shift in time domain leaves the magnitude spectrum unchanged • Introduces a phase shift that is linear function of frequency(݁ିఠబ௧). ݁ି௧)ݐ(ݑ ↔ 1 ݆߱ + ܽ Prof: Sarun Soman, MIT, Manipal 14
- 15. Properties of Fourier Transform Differentiation Property Differentiation in time ݀ ݀ݐ )ݐ(ݔ ↔ ݆߱ܺ(݆߱) • Differentiation in time domain corresponds to multiplying by j߱ in frequency domain. • This operation accentuates high frequency components. Eg. ݁ି௧)ݐ(ݑ ↔ 1 ݆߱ + ܽ ݀ ݀ݐ ݁ି௧)ݐ(ݑ ↔ (݆߱) 1 ݆߱ + ܽ Differentiation in Frequency −݆ݐ )ݐ(ݔ ↔ ݀ ݀߱ ܺ(݆߱) Eg. Use differentiation property to find FT of ݔ ݐ = ݁ݐି௧)ݐ(ݑ Ans: Using differentiation property −݆ݐ )ݐ(ݔ ↔ ݀ ݀߱ ܺ(݆߱) ݐ )ݐ(ݔ ↔ 1 −݆ ݀ ݀߱ ܺ(݆߱) ݁ݐି௧ ↔ ݆ ݀ ݀߱ 1 ݆߱ + ܽ Prof: Sarun Soman, MIT, Manipal 15
- 16. Properties of Fourier Transform ݁ݐି௧ ↔ 1 ݆߱ + ܽ ଶ Integration න ݔ ߬ ݀߬ = 1 ݆߱ ܺ ݆߱ + ߨܺ(݆0)ߜ(߱) ௧ ିஶ • De emphasizing high frequency components. Eg. FT of unit step using integration property Ans: ݑ ݐ = න ߜ ߬ ݀߬ ௧ ିஶ ߜ()ݐ ↔ 1 Using integration property න ߜ ߬ ݀߬ ௧ ିஶ ↔ 1 ݆߱ 1 + ߨߜ ߱ Convolution property ݔ ݐ ∗ ݄()ݐ ↔ ܺ ݆߱ )݆߱(ܪ Eg. Let the input to a system with impulse response ݄ ݐ = 2݁ିଶ௧ )ݐ(ݑ be ݔ ݐ = 3݁ି௧ ݑ ݐ . Prof: Sarun Soman, MIT, Manipal 16
- 17. Properties of Fourier Transform Ans: 2݁ିଶ௧)ݐ(ݑ ↔ 2 ݆߱ + 2 3݁ି௧ ↔ 3 ݆߱ + 1 Using convolution property ݕ ݐ = ݔ ݐ ∗ ݄()ݐ ܻ ݆߱ = ܺ ݆߱ )݆߱(ܪ ܻ ݆߱ = 6 (݆߱ + 2)(݆߱ + 1) = ܿଵ ݆߱ + 2 + ܿଶ ݆߱ + 1 ܿଵ = −6, ܿଶ = 6 ܻ ݆߱ = −6 ݆߱ + 2 + 6 ݆߱ + 1 ݕ ݐ = −6݁ିଶ௧ ݑ ݐ + 6݁ି௧ )ݐ(ݑ Modulation property ݔ ݐ ݖ ݐ ↔ 1 2ߨ ܺ ݆߱ ∗ ܼ(݆߱) Prof: Sarun Soman, MIT, Manipal 17
- 18. Properties of Fourier Transform • Slope of the linear phase term is equal to the time shift (ݐ). Eg. ݔ ݐ = ݁ି௧ାଶ ݐ(ݑ − 2) Ans: ݁ି௧ ݑ ݐ ↔ 1 ݆߱ + 1 ݁ି௧ାଶ ݐ(ݑ − 2) ↔ ݁ିఠ(ଶ) 1 ݆߱ + 1 Frequency Shift Properties ݁ఊ௧ )ݐ(ݔ ↔ ܺ(݆(߱ − ߛ)) • A frequency shift corresponds to multiplication in time domain by a complex sinusoid whose frequency is equal to the shift. Eg. ݔ ݐ ↔ 2 ߱ sin(߱ߨ) ݁ଵ௧)ݐ(ݔ ↔ 2 ߱ − 10 sin(ߨ(߱ − 10)) Scaling Property )ݐ(ݔ ↔ ܺ(݆߱) )ݐܽ(ݔ ↔ 1 ܽ ܺ ݆ ߱ ܽ Scaling the signal in time domain introduces inverse scaling in frequency domain representation & an amplitude scaling. Prof: Sarun Soman, MIT, Manipal 18
- 19. Properties of Fourier Transform Parseval’s Theorem Parseval’s theorem states that energy or power in time domain representation is equal to the energy or power in frequency domain. න )ݐ(ݔ ଶ݀ݐ ஶ ିஶ = 1 2ߨ න ܺ(݆߱) ଶ݀߱ ஶ ିஶ Duality property There is a consistent symmetry b/w the time and Frequency domain representation of signals. A rectangular pulse in either time or frequency domain corresponds to a sinc function in either frequency or time. Prof: Sarun Soman, MIT, Manipal 19
- 20. Properties of Fourier Transform We may interchange time and frequency This interchangeability property is termed duality. Prof: Sarun Soman, MIT, Manipal 20
- 21. Properties of Fourier Transform ݂()ݐ ி் ܨ ݆߱ )ݐ݆(ܨ ி் 2ߨ݂(−߱) Using duality property find the duality property of ‘1’ Ans: ߜ()ݐ ி் 1 1 ி் 2ߨߜ −߱ Find the FT of ݔ ݐ = ଵ ଵା௧ Ans: ݁ି௧ݑ ݐ ி் 1 ݆߱ + 1 Replace ߱ by ݐ 1 ݆ݐ + 1 Prof: Sarun Soman, MIT, Manipal 21
- 22. Properties of Fourier Transform Using duality ݁ି௧ݑ ݐ ி் 1 ݆߱ + 1 1 ݆ݐ + 1 ி் 2ߨ݁ఠ)߱−(ݑ Prof: Sarun Soman, MIT, Manipal 22
- 23. Discrete Time Non-periodic Signals: The Discrete Time Fourier Transform DTFT is used to represent a discrete-time -periodic signal as a superposition of complex sinusoids. DTFT would involve a continuum of frequencies on the interval−ߨ < Ω < ߨ ݔ ݊ = 1 2ߨ න ܺ(݁Ω )݁Ω ݀ గ ିగ Ω Where ܺ ݁Ω = ݁]݊[ݔିΩ ஶ ୀିஶ ܺ ݁Ω is termed as the frequency domain representation of ]݊[ݔ Prof: Sarun Soman, MIT, Manipal 23
- 24. Example Find the DTFT of the exponential sequence ݔ ݊ = ଵ ସ ݊[ݑ + 4] Ans: ܺ ݁Ω = ݁]݊[ݔିΩ ஶ ୀିஶ = 1 4 ݁ିΩ ஶ ୀିସ Let ݊ + 4 = ݈ = 1 4 ିସ ݁ିΩ(ିସ) ஶ ୀ = 1 4 ିସ ݁Ωସ 1 4 ݁ିΩ ஶ ୀ = 256݁ସΩ 1 1 − 1 4 ݁ିΩ Evaluate the DTFT of signal x[n] shown in Fig. Find the expression for magnitude and phase spectra. 0 1 2 3 -1-2-3 n ]݊[ݔ 1 -1 Prof: Sarun Soman, MIT, Manipal 24
- 25. Example ܺ ݁Ω = ݁]݊[ݔିΩ ஶ ୀିஶ = ݔ −3 ݁ଷΩ + ݔ −2 ݁ଶΩ + ݔ 2 ݁ିଶΩ + ݔ 3 ݁ିଷΩ = ݁ଷΩ + ݁ଶΩ + ݁ିଶΩ − ݁ିଷΩ = 2݆ sin 3Ω + 2 cos 2Ω ܺ ݁Ω = 2 ܿݏଶ 2Ω + ݊݅ݏଶ 2Ω < ܺ ݁Ω = ݊ܽݐିଵ sin 3Ω cos 2Ω ݔ ݊ = ܽ , ܽ < 1 Ans: ܺ ݁Ω = ݁]݊[ݔିΩ ஶ ୀିஶ = (ܽ݁ିΩ)+ (ܽ݁Ω)ି ିஶ ୀିଵ ஶ ୀ = 1 1 − ܽ݁ିΩ + ܻ ܻ = (ܽ݁Ω )ି ିஶ ୀିଵ Let ݊ = −݉ Prof: Sarun Soman, MIT, Manipal 25
- 26. Example ܻ = (ܽ݁Ω ) ஶ ୀଵ ܻ = (ܽ݁Ω) ஶ ୀ − 1 = 1 1 − ܽ݁Ω − 1 = ܽ݁Ω 1 − ܽ݁Ω ܺ ݁Ω = 1 1 − ܽ݁ିΩ + ܽ݁Ω 1 − ܽ݁Ω = 1 − ܽଶ 1 + ܽଶ − 2ܽ cos Ω Obtain the DTFT of rectangular pulse ݔ ݊ = ൜ 1, ݊ ≤ ܯ 0, ݊ > ܯ Ans: ܺ ݁Ω = ݁]݊[ݔିΩ ஶ ୀିஶ Prof: Sarun Soman, MIT, Manipal 26
- 27. Example ܺ ݁Ω = 1݁ିΩ ெ ୀିெ Let ݈ = ݊ + ܯ ܺ ݁Ω = ݁ିΩ(ିெ) ଶெ ୀ = ݁Ωெ ݁ିΩ ଶெ ୀ = ݁Ωெ 1 − ݁ିΩ ଶெାଵ 1 − ݁ିΩ = ݁Ωெ ݁ି Ω ଶ ଶெାଵ ݁ Ω ଶ ଶெାଵ − ݁ି Ω ଶ ଶெାଵ 1 − ݁ିΩ = ݁Ωெ ݁ି Ω ଶ ଶெାଵ ݁ି Ω ଶ ݁ Ω ଶ ଶெାଵ − ݁ି Ω ଶ ଶெାଵ ݁ Ω ଶ − ݁ି Ω ଶ = sin Ω 2ܯ + 1 2 sin Ω 2 , Ω ≠ 0,2ߨ … Ω=0 Prof: Sarun Soman, MIT, Manipal 27
- 28. Example ܺ ݁Ω = lim Ω↔ cos Ω 2ܯ + 1 2 ∗ 2ܯ + 1 2 cos Ω 2 ∗ 1 2 = 2ܯ + 1 Find the DTFT of the signal ݔ ݊ = cos ߨ݊ 5 + ݆ sin ߨ݊ 5 ; ݊ ≤ 10 Ans: ݔ ݊ = ݁ గ ହ ܺ ݁Ω = ݁ గ ହ ଵ ୀିଵ ݁ିΩ Let ݊ + 10 = ݉ = ݁ ି గ ହିΩ షభబ ଶ ୀ Prof: Sarun Soman, MIT, Manipal 28
- 29. Example = ݁ ିଵ గ ହ ିΩ 1 − ݁ ଶଵ గ ହ ିΩ 1 − ݁ గ ହ ିΩ = sin 21 2 ߨ 5 − Ω sin 1 2 ߨ 5 − Ω Prof: Sarun Soman, MIT, Manipal 29
- 30. Inverse DTFT Find the inverse DTFT using partial fraction expansion. ܺ ݁Ω = 3 − 1 4 ݁ିΩ 1 − 1 16 ݁ିଶΩ Ans: ܺ ݁Ω = ܣ 1 − 1 4 ݁ିΩ + ܤ 1 + 1 4 ݁ିΩ ܣ = 3 − 1 4 ݁ିΩ 1 + 1 4 ݁ିΩ |షೕΩୀସ ܣ = 1 ܤ = 3 − 1 4 ݁ିΩ 1 − 1 4 ݁ିΩ |షೕΩୀିସ ܤ = 2 ܺ ݁Ω = 1 1 − 1 4 ݁ିΩ + 2 1 + 1 4 ݁ିΩ ݔ ݊ = 1 4 ݑ ݊ + 2 −1 4 ]݊[ݑ Prof: Sarun Soman, MIT, Manipal 30
- 31. z transform • DTFT- complex sinusoidal representation of a DT signal • ݖ transform – Representation in terms of complex exponential signals. • ݖ transform is the discrete time counterpart to Laplace transform Why ݖ transform? • More general classification of DT signal. • A broader characterization of DT LTI systems & its interaction with signals. Prof: Sarun Soman, MIT, Manipal 31
- 32. Z transform Eg. DTFT exists only if impulse response is absolutely summable. DTFT exists only for stable LTI systems. ݖ transform of the impulse response exists for unstable LTI systems and signals. ݖ transform of the impulse response is the transfer function of the system. ݖ = ݁ݎΩ ݎ − ݉ܽ݃݊݅,݁݀ݑݐ Ω − ݈ܽ݊݃݁ ݔ ݊ = ݖ complex exponential signal. Prof: Sarun Soman, MIT, Manipal 32
- 33. Z transform ݔ ݊ = ݎ cos Ω݊ + ݆ݎ sin Ω݊ If ݎ = 1, ]݊[ݔ is a complex sinusoid. Applying ]݊[ݔ to an LTI system ݕ ݊ = ݄ ݊ ∗ ]݊[ݔ = ݄ ݇ ݊[ݔ − ݇] ஶ ୀିஶ ݔ ݊ = ݖ ݕ ݊ = ݄[݇]ݖି ஶ ୀିஶ Prof: Sarun Soman, MIT, Manipal 33
- 34. z transform = ݖ ݄[݇]ݖି ஶ ୀିஶ Transfer function ܪ ݖ = ݄[݇]ݖି ஶ ୀିஶ ݖ transform of ]݊[ݔ ܺ ݖ = ݖ]݊[ݔି ஶ ୀିஶ (1) Convergence • ݖ transform exist when eqn(1) converges. • Necessary condition is absolute summability. ݖ]݊[ݔି ஶ ୀିஶ < ∞ (2) ݖ = ݁ݎΩ ݖି = ݎି Equation (2) can be written as ݎ]݊[ݔି ஶ ୀିஶ < ∞ Prof: Sarun Soman, MIT, Manipal 34
- 35. z transform • The range ′′ݎ for which eq(2) converges is termed as Region of Convergence(ROC) • ݎ]݊[ݔି is absolutely summable even though ]݊[ݔ is not. • Ability to work with signals that doesn't have a DTFT is a significant advantage offered by the ݖ transform. Z-plane. Prof: Sarun Soman, MIT, Manipal 35
- 36. transform ࢠ transform of a causal exponential signal Determine the ݖ transform of the signal ݔ ݊ = ߙ .]݊[ݑ Depict the ROC and the location of poles and zeros of ܺ()ݖ in the ݖ plane. Ans: ܺ ݖ = ݖ]݊[ݔି ஶ ୀିஶ ܺ ݖ = ߙݖ]݊[ݑି ஶ ୀିஶ = ߙ ݖ ஶ ୀ The sum converges only if ߙ ݖ < 1 ݖ > ߙ ܺ ݖ = 1 1 − ߙݖିଵ , ݖ > ߙ ܺ()ݖin pole-zero form = ݖ ݖ − ߙ , ݖ > ߙ Pole zero plot and ROC Prof: Sarun Soman, MIT, Manipal 36
- 37. ݖ transform ࢠ transform of non-causal exponential signal Determine the ݖ transform of the signal ݕ ݊ = −ߙݑ −݊ − 1 .Depict the ROC and the locations of poles and zeros of ܺ ݖ in the ݖ plane. Ans: ܻ ݖ = ݖ]݊[ݕି ஶ ୀିஶ = − ߙ ିଵ ିஶ ݖି Let ݇ = −݊ ܻ ݖ = − ݖ ߙ ஶ ୀଵ = − ݖ ߙ ஶ ୀ − 1 = 1 − ݖ ߙ ஶ ୀ The sum converges, provided ௭ ఈ < 1 ݖ < ߙ = 1 − 1 1 − ߙݖିଵ , ݖ < ߙ Prof: Sarun Soman, MIT, Manipal 37
- 38. transform = 1 − ߙݖିଵ − 1 1 − ߙݖିଵ = −ߙݖିଵ 1 − ߙݖିଵ = − ݖ ߙ − ݖ = ݖ ݖ − ߙ , ݖ < ߙ ROC plot ݖ transform is same but ROC is different z transform of a two sided signal Determine the z-transform of ݔ ݊ = −ݑ −݊ − 1 + ଵ ଶ .]݊[ݑ Depict the ROC and the locations of poles and zeros of ܺ()ݖ in the plane. ܺ ݖ = 1 2 ݖ]݊[ݑି ஶ ୀିஶ − ݊−[ݑ − 1]ݖି = 1 2ݖ − 1 ݖ ିଵ ୀିஶ ஶ ୀ = 1 2ݖ + 1 − ݖ ஶ ୀ ஶ ୀ Both the sum converges when ݖ > 1 2 ܽ݊݀ ݖ < 1 Prof: Sarun Soman, MIT, Manipal 38
- 39. ݖ transform ܺ ݖ = 1 1 − 1 2 ݖିଵ + 1 − 1 1 − ݖ , 1 2 < ݖ < 1 Pole zero form ܺ ݖ = ݖ ݖ − 1 2 + ݖ ݖ − 1 ܺ ݖ = ݖଶ − ݖ + ݖଶ − 1 2 ݖ ݖ − 1 2 ݖ − 1 ܺ ݖ = ݖ 2ݖ − 3 2 ݖ − 1 2 ݖ − 1 , 1 2 < ݖ < 1 Find the z transform and ROC ݔ ݊ = 7 1 3 ݑ ݊ − 6 1 2 ]݊[ݑ Ans: ܺ ݖ = ݖ]݊[ݔି ஶ ୀିஶ = 7 1 3 ݖି − 6 1 2 ݖି ஶ ୀ ஶ ୀ Sum converges, ݖ > ଵ ଷ and ݖ > ଵ ଶ = 7 1 − 1 3 ݖିଵ − 6 1 − 1 2 ݖିଵ Prof: Sarun Soman, MIT, Manipal 39
- 40. transform ROC must not include any poles ROC , ݖ > ଵ ଶ Find z transform and ROC ݔ ݊ = 1 2 Ans: ݔ ݊ = 1 2 ݑ ݊ + 1 2 ି ݊−[ݑ − 1] ܺ ݖ = 1 1 − 1 2 ݖିଵ + 1 2 ି ݖି ିଵ ୀିஶ ݖ 2 ି ିଵ ୀିஶ Let ݇ = −݊ ݖ 2 ି ஶ ୀଵ 2 ݖ − 1 ஶ ୀ Sum converges ଶ ௭ < 1, ݖ < 2 1 1 − 2ݖିଵ − 1 Prof: Sarun Soman, MIT, Manipal 40
- 41. ݖ transform 2ݖିଵ 1 − 2ݖିଵ ܺ ݖ = 1 1 − 1 2 ݖିଵ + 2ݖିଵ 1 − 2ݖିଵ ROC 1 2 < ݖ < 2 Find the z transform of ݔ ݊ = ߜ[݊] Ans: ܺ ݖ = ߜ[݊]ݖି ஶ ୀିஶ = 1 ROC No zeros and poles, ROC is all z plane ݔ ݊ = ߜ ݊ − ݇ , ݇ > 0 Ans: ܺ ݖ = ߜ[݊ − ݇]ݖି ஶ ୀିஶ = (1)ݖି ROC all z-plane except ݖ = 0 Note: If ݔ ݊ of finite duration, then ROC is entire z-plane except possibly ݖ = 0 or ݖ = ∞ Prof: Sarun Soman, MIT, Manipal 41
- 42. z transform Prof: Sarun Soman, MIT, Manipal 42