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Signals and Systems Discrete Time Fourier Series

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Discrete-Time Fourier Series

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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

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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

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

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Properties of DFS Linearity Shift of a Sequence Duality

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Symmetry Properties

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Symmetry Properties Cont’d

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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

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

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Graphical Periodic Convolution

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DTFT to DFT

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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

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DFT Analysis and Synthesis

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DFT

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DFT is Periodic with period N

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Example 1

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Example 1 (cont.) N=5

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Example 1 (cont.) N>M

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Example 1 (cont.) N=10

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DFT: Matrix Form

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DFT from DFS

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Properties of DFT Linearity Duality Circular Shift of a Sequence

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Symmetry Properties

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DFT Properties

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Example: Circular Shift

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Duality

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Circular Flip

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Properties: Circular Convolution

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Example: Circular Convolution

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illustration of the circular convolution process Example (continued)

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Illustration of circular convolution for N = 8:

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Example:

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Example (continued)

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Proof of circular convolution property:

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Multiplication:

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