1. Signal and Systems Prof. H. Sameti Chapter 5: The Discrete Time Fourier Transform Examples of the DT Fourier Transform Properties of the DT Fourier Transform The Convolution Property and its Implication and Uses DTFT Properties and Examples Duality in FS & FT

2. The Discrete-Time Fourier Transform Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 2

3. DTFT Derivation (Continued) Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 3

4. DTFT Derivation (Home Stretch) Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 4 Any 2π Interval in ω

5. DT Fourier Transform Pair Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 5

6. Convergence Issues  Synthesis Equation: None, since integrating over a finite interval.  Analysis Equation: Need conditions analogous to CTFT, e.g. Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 6

7. Examples Parallel with the CT examples in Lecture #8 Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 7

8. More Examples Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 8

9. Still More Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 9

10. Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 10

11. DTFTs of Sums of Complex Exponentials Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 11

12. DTFT of Periodic Signals Book Chapter5: Section1 Computer Engineering Department, Signals and Systems 12 Linearity of DTFT

13. Example #1:DT sine function Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 13

14. Example #2: DT periodic impulse train Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 14 — Also periodic impulse train – in the frequency domain!

15. Properties of the DT Fourier Transform Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 15 — Analysis equation — Synthesis equation 1) Periodicity: — Different from CTFT 2) Linearity:

16. More Properties 3) Time Shifting: 4) Frequency Shifting: - Important implications in DT because of periodicity Example Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 16

17. Still More Properties 5) Time Reversal: 6) Conjugate Symmetry: Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 17 a nd

18. Yet Still More Properties Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 18 7) Time Expansion Recall CT property: Time scale in CT is infinitely fine But in DT:x[n/2] makes no sense x[2n] misses odd values of x[n] But we can “slow” a DT signal down by inserting zeros: k —an integer ≥ 1 x (k) [n] — insert (k - 1) zeros between successive values Insert two zeros in this example (k=3)

19. Time Expansion (continued) Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 19 — Stretched by a factor of k in time domain -compressed by a factor of k in frequency domain

20. Is There No End to These Properties? Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 20 8) Differentiation in Frequency multiply by j on both sides Multiplication by n Differentiation In frequency 9) Perseval’s Relation Total energy in time domain Total energy in frequency domain

21. The Convolution Property Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 21 h[n] Example #1:

22. Example#2: Ideal Lowpass Filter Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 22

23. Book Chapter5: Section1 Computer Engineering Department, Signal and Systems 23 Example #3:

24. Convolution Property Example Computer Engineering Department, Signal and Systems Book Chapter#: Section# 24  Ratio of polynomials in  A, B determined by partial fraction expansion (PFE)

25. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 25

26. DT LTI system Described by LCCDE’s  From time-shifting property: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 26 Rational function of use PFE to get h[n]

27.  Example: First-order recursive system  With the condition of initial restcausal Book Chapter#: Section# Computer Engineering Department, Signal and Systems 27

28. DTFT Multiplication Property   Derivation: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 28 Periodic Convolution

29. Calculating Periodic Convolutions  Suppose we integrate from –π to π:  where Book Chapter#: Section# Computer Engineering Department, Signal and Systems 29

30.  Example: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 30

31. Duality in Fourier Analysis  Fourier Transform is highly symmetric  CTFT: Both time and frequency are continuous and in general aperiodic  Suppose f (.) and g (.) are two functions related by:  Then:  Let т = t and r = ω:  Let т = -ω and r = t: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 31 Same except for these differences

32. Example of CTFT duality Book Chapter#: Section# Computer Engineering Department, Signal and Systems 32

33. DTFS  Discrete & periodic in time Periodic & discrete in frequency Duality in DTFS  Suppose f(.) and g(.) are two functions related by:  Then:  Let m = n and r = -k:  Let r = n and m = k: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 33

34. Duality between CTFS and DTFT  CTFS:  DTFT: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 34 Periodic in time Discrete in frequency Discrete in time Periodic in frequency

35. CTFS-DTFT Duality  Suppose f(.) is a CT signal and g[.] a DT sequence related by :  Then: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 35 periodic with period 2π