Signals & systems Ch.3 Fourier Transform of Signals and LTI System

1. Signals & systemsCh.3 Fourier Transform of Signals and LTI System

2. Signals and systems in the Frequency domain Fourier transform Time [sec] Frequency [sec-1, Hz] KyungHee University

3. 3.1 Introduction • Orthogonal vector => orthonomal vector • What is meaning of magnitude of H? Any vector in the 2- dimensional space can be represented by weighted sum of 2 orthonomal vectors Fourier Transform(FT) Inverse FT KyungHee University

4. 3.1 Introduction cont’ • CDMA? Orthogonal? Any vector in the 4- dimensional space can be represented by weighted sum of 4 orthonomal vectors Orthonormal function? KyungHee University

5. 3.2 Complex Sinusoids and Frequency Response of LTI Systems cf) impulse response How about for complex z? (3.1) How about for complex s? (3.3) Magnitude to kill or not? Phase  delay KyungHee University

6. Fourier transform Time domain frequency domain discrete time Continuous time z-transform Laplace transform (periodic) - (discrete) (discrete) - (periodic) KyungHee University

7. 3.6 DTFT: Discrete-Time Fourier Transform (discrete) (periodic) (a-periodic) (continuous) (3.31) (3.32) KyungHee University

8. 3.6 DTFT Example 3.17 Example 3.17 DTFT of an Exponential Sequence Find the DTFT of the sequence Solution :  = 0.5  = 0.9 x[n] = nu[n]. magnitude  = 0.5  = 0.9 phase KyungHee University

9. 3.6 DTFT Example 3.18 Example 3.18 DTFT of a Rectangular Pulse Let Find the DTFT of Solution : (square)  (sinc) Figure 3.30 Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain. KyungHee University

10. 3.6 DTFT Example 3.18 KyungHee University

11. 3.6 DTFT Example 3.19 Example 3.19 Inverse DTFT of a Rectangular Spectrum Find the inverse DTFT of Solution : (sinc)  (square) Figure 3.31 (a) Rectangular pulse in the frequency domain. (b) Inverse DTFT in the time domain. KyungHee University

12. 3.6 DTFT Example 3.20-21 Example 3.20 DTFT of the Unit Impulse Find the DTFT of Solution : (impulse) - (DC) Example 3.21 Find the inverse DTFT of a Unit Impulse Spectrum. Solution : (impulse train) (impulse train) KyungHee University

13. 3.6 DTFT Example 3.23 Example 3.23 Multipath Channel : Frequency Response Solution : (a) a = 0.5ej2/3. (b) a = 0.9ej2/3. (a) a = 0.5ej2/3. (b) a = 0.9ej2/3. KyungHee University

14. 3.7 CTFT (continuous aperiodic)  (continuous aperiodic) CTFT (3.26) Inverse CTFT (3.35) Condition for existence of Fourier transform: KyungHee University

15. 3.7 CTFT Example 3.24 Example 3.24 FT of a Real Decaying Exponential Find the FT of Solution : Therefore, FT not exists. LPF or HPF? Cut-off from 3dB point? KyungHee University

16. 3.7 CTFT Example 3.25 Example 3.25 FT of a Rectangular Pulse Find the FT of x(t). Solution : (square)  (sinc) Example 3.25. (a) Rectangular pulse. (b) FT. KyungHee University

17. 3.7 CTFT Example 3.25 Example 3.26 Inverse FT of an Ideal Low Pass Filter!! Fine the inverse FT of the rectangular spectrum depicted in Fig.3.42(a) and given by Solution : (sinc) -- (square) KyungHee University

18. 3.7 CTFT Example 3.27-28 Example 3.27 FT of the Unit Impulse Solution : (impulse) - (DC)  Example 3.28 Inverse FT of an Impulse Spectrum Find the inverse FT of Solution : (DC)  (impulse) KyungHee University

19. 3.7 CTFT Example 3.29 Example 3.29 Digital Communication Signals Rectangular (wideband) Separation between KBS and SBS. Narrow band Figure 3.44 Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse. KyungHee University

20. 3.7 CTFT Example 3.29 Solution : Figure 3.45 BPSK (a) rectangular pulse shapes (b) raised-cosine pulse shapes. the same power constraints KyungHee University

21. 3.7 CTFT Example 3.29 rectangular pulse. One sinc Raised cosine pulse 3 sinc’s The narrower main lobe, the narrower bandwidth. But, the more error rate as shown in the time domain Figure 3.47 sum of three frequency-shifted sinc functions. KyungHee University

22. Fourier transform Time domain frequency domain Discrete time Continuous time KyungHee University

23. 3.9.1 Linearity Property KyungHee University

24. 3.9.1 Symmetry Properties • Real and Imaginary Signals (real x(t)=x*(t))  (conjugate symmetric) (3.37) (3.38) KyungHee University

25. 3.9.2 Symmetry Properties of FT • EVEN/ODD SIGNALS (even) (real) (odd)  (pure imaginary)   For even x(t), real KyungHee University

26. 3.10 Convolution Property (convolution)  (multiplication) But given change the order of integration  KyungHee University

27. 3.10 Convolution Property Example 3.31 Example 3.31 Convolution problem in the frequency domain Input to a system with impulse response Find the output Solution:  KyungHee University

28. 3.10 Convolution Property Example 3.32 Example 3.32 Find inverse FT’S by the convolution property Use the convolution property to find x(t), where  Ex 3.32 (p. 261). (a) Rectangular z(t). (b) KyungHee University

29. 3.10.2 Filtering Continuous time Discrete time(periodic with 2π LPF HPF BPF Figure 3.53 (p. 263) Frequency dependent gain (power spectrum) kill or not (magnitude) KyungHee University

30. 3.10 Convolution Property Example 3.34 Example 3.34 Identifying h(t) from x(t) and y(t) The output of an LTI system in response to an input is . Find frequency response and the impulse response of this system. Solution: But  But note  KyungHee University

31. 3.10 Convolution Property Example 3.35 EXAMPLE 3.35 Equalization(inverse) of multipath channel or Consider again the problem addressed in Example 2.13. In this problem, a distorted received signal y[n] is expressed in terms of a transmitted signal x[n] as   Then  KyungHee University

32. 3.11 Differentiation and Integration Properties EXAMPLE 3.37 The differentiation property implies that KyungHee University

33. 3.11 Differentiation and Integration Properties • 예제한 두개 KyungHee University

34. 3.11.2 DIFFERENTIATION IN FREQUENCY Differentiate w.r.t. ω, Then, Example 3.40 FT of a Gaussian pulse Use the differentiation-in-time and differentiation-in-frequency properties for the FT of the Gaussian pulse, defined by and depicted in Fig. 3.60. and   Then (But, c=?) Figure 3.60 (p. 275) Gaussian pulse g(t). KyungHee University

35. Laplacetransform and z transform KyungHee University

36. 3.11.3 Integration  Ex) Prove Note where a=0 We know  since linear  Fig. a step fn. as the sum of a constant and a signum fn. KyungHee University

37. Differentiation and Integration Properties Common Differentiation and Integration Properties. KyungHee University

38. 3.12.1 Time-Shift Property Fourier transform of time-shifted z(t) = x(t-t0) Note that x(t-t0) = x(t) * δ(-t0) and Table 3.7 Time-Shift Properties of Fourier Representations KyungHee University

39. 3.12 Time-and Frequency-Shift Properties Example)  Figure 3.62  KyungHee University

40. 3.12.2 Frequency-Shift Property Recall Table 3.8 Frequency-Shift Properties KyungHee University

41. 3.12.2 Frequency-Shift Property Example 3.42 FT by Using the Frequency-Shift Property Solution: We may express as the product of a complex sinusoid and a rectangular pulse   KyungHee University

42. 3.12 Shift Properties Ex. 3.43 Example 3.43 Using Multiple Properties to Find an FT Sol) Let and Then we may write By the convolution and differentiation properties The transform pair    KyungHee University

43. 3.12 Shift Properties Ex. 3.43 Example 3.43 Using Multiple Properties to Find an FT Sol) Let and Then we may write By the convolution and differentiation properties The transform pair   s  KyungHee University

44. 3.13 Inverse FT: Partial-Fraction Expansions • 3.13.1 Inverse FT by using N roots,   partial fraction  KyungHee University

45. 3.13 Inverse FT: Partial-Fraction Expansions • 3.13.1 Inverse FT by using Let then N roots,   partial fraction  KyungHee University

46. Inverse FT: Partial-Fraction Expansions KyungHee University

47. 3.13.2 Inverse DTFT • 3.13.2 where Then KyungHee University

48. Inverse FT: Partial-Fraction Expansions KyungHee University

49. 3.13.2 Inverse DTFT by z-transform • 3.13.2 where Then KyungHee University

50. 3.13 Inverse FT Example 3.45 Example 3.45 Inversion by Partial-Fraction Expansion Solution: Using the method of residues described in Appendix B, We obtain And Hence, 2011.5.4 KyungHee University