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# Chapter 5 The Discrete-Time Fourier Transform

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1. In this chapter, we will learn--- • The Discrete-Time Fourier Transform (DTFT) • The Fourier Transform for Periodic Signals • Properties of the DTFT

2. Notation: • CFS: ( the continuous Fourier series,FS) • DFS: ( the discrete Fourier series) • CTFT: ( the continuous time Fourier Transform,FT) • DTFT: ( the discrete time Fourier Transform)

3. 5.1 Representation of Aperiodic Signals: DTFT 5.1.1 Development of DTFT (from DFS to DTFT) (1) Fourier series (periodic signal)

4. x[n] ak Fourier series (periodic signal)

5. Fourier transform (aperiodic signal)

6. x[n] 0 0 (2) Fourier transform (aperiodic signal)

7. Since over When So DTFT

8. Where is the spacing of the samples in the frequency domain. The coefficients akare proportional to X(ejw) samples of

9. When

10. The discrete-time Fourier transform pair Synthesis equation Analysis equation • That means, x[n] is composed of ejwn at different frequencies w with “amplitude” X(ejw)dw/2p ; X(ejw)is referred to the spectrum of x[n].

11. Note: 1. Relation between of DFS and DTFT Discrete <--> periodic Periodic <--> discrete Discrete <--> periodic Aperiodic <--> continuous

12. 2. Comparing DTFT to FS Duality Continuous-time、Discrete-frequency Discrete-time、Continuous-frequency Eq.(5.9) can be regarded to FS in frequency-domain; Eq.(5.8) is just that x[n] is the FS coefficients of X(ejw).

13. 3. Relation between FT and DTFT Let and so

14. Example Poisson formula If then

15. Proof :

16. Note: That implies It is a Fourier Series in Frequency-Domain.

17. 5.1.2 Examples of DTFT Example 5.1

18. Example 5.2

19. We can conclude: real and even real and even

20. Example 5.3 Similar to a sinc, but periodic with a period 2π.

21. Note: If the periodization of x[n] which period N is made, We can get DFS coefficients: Just same as (3.104) in P218

22. 1 w 0 n 0 Example 5.4

23. Example 5.5 Ideal DT LPF A discrete sinc in time domain

24. or 5.1.3 Convergence of DTFT • Convergence conditions: –– Absolutely summable — Finite energy

25. Gibbs phenomenon in DTFT?

26. W=p/4 W=3p/8 W=3p/4 W=p/2 W=p W=7p/8

27. There are not any behaviors like the Gibbs phenomenon in evaluation the discrete-time Fourier transform synthesis equation. (see page 368 fig5.7)

28. 5.2 DTFT for periodic signals Recall CT result: What about DT: a) We expect an impulse (of area 2π) at ω=ωo b) But X(ejω) must be periodic with period 2π, In fact

29. w 1 w 0 0 n Ifw0=0

30. Now consider a periodic sequence See page 370 Fig5.9

31. Example 1. 2.

32. 3.

33. 5.3 Properties of DTFT 5.3.1 Periodicity 5.3.2 Linearity 5.3.3 Time Shifting and Frequency Shifting

34. Example

35. 5.3.4 Conjugation and Conjugate Symmetry If x[n] is real 5.3.5 Differencing and Accumulation

36. if n is a multiple of k if n is not a multiple of k. 5.3.6 Time Reversal 5.3.7 Time Expansion

37. 5.3.8 Differentiation in Frequency 5.3.7 Parseval’s Relation

38. 5.4 The Convolution Property P386 Ex5.14

39. 5.5 The Multiplication Property Periodic convolution Homework: P400-5.2* 5.10*