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Signals and Systems Discrete Time Fourier Series.

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Presentation on theme: "Signals and Systems Discrete Time Fourier Series."— Presentation transcript:

1 Signals and Systems Discrete Time Fourier Series

2 Discrete-Time Fourier Series


4 The conventional (continuous-time) FS represent a periodic signal using an infinite number of complex exponentials, whereas the DFS represent such a signal using a finite number of complex exponentials

5 Example 1  DFS of a periodic impulse train  Since the period of the signal is N  We can represent the signal with the DFS coefficients as

6 Example 2  DFS of an periodic rectangular pulse train  The DFS coefficients

7 Properties of DFS  Linearity  Shift of a Sequence  Duality

8 Symmetry Properties

9 Symmetry Properties Cont’d

10 Periodic Convolution  Take two periodic sequences  Let’s form the product  The periodic sequence with given DFS can be written as  Periodic convolution is commutative

11 Periodic Convolution Cont’d  Substitute periodic convolution into the DFS equation  Interchange summations  The inner sum is the DFS of shifted sequence  Substituting

12 Graphical Periodic Convolution

13 DTFT to DFT

14 Sampling the Fourier Transform  Consider an aperiodic sequence with a Fourier transform  Assume that a sequence is obtained by sampling the DTFT  Since the DTFT is periodic resulting sequence is also periodic  We can also write it in terms of the z-transform  The sampling points are shown in figure  could be the DFS of a sequence  Write the corresponding sequence

15 DFT Analysis and Synthesis

16 DFT

17 DFT is Periodic with period N

18 Example 1

19 Example 1 (cont.) N=5

20 Example 1 (cont.) N>M

21 Example 1 (cont.) N=10

22 DFT: Matrix Form

23 DFT from DFS

24 Properties of DFT  Linearity  Duality  Circular Shift of a Sequence

25 Symmetry Properties

26 DFT Properties

27 Example: Circular Shift



30 Duality

31 Circular Flip

32 Properties: Circular Convolution

33 Example: Circular Convolution


35 illustration of the circular convolution process Example (continued)

36 Illustration of circular convolution for N = 8:

37 Example:

38 Example (continued)

39 Proof of circular convolution property:

40 Multiplication:

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