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# Multirate Filtering: Theory Exercise

Summary: You will work through an example problem that explores the effects of sample-rate compression and expansion on the spectrum of a signal.

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## Multirate Theory Exercise

Consider a sampled signal with the DTFT Xω X ω shown in Figure 1.

Assuming U=D=3 U D 3 , use the relations between the DTFT of a signal before and after sample-rate compression and expansion (Equation 1 and Equation 2) to sketch the DTFT response of the signal as it passes through the multirate system of Figure 2 (without any filtering). Include both the intermediate response Wω W ω and the final response Yω Y ω . It is important to be aware that the translation from digital frequency ω ω to analog frequency depends on the sampling rate. Therefore, the conversion is different for Xω X ω and Wω W ω .

Wω=1D k =0D1Xω+2πkD W ω 1 D k 0 D 1 X ω 2 k D
(1)
Yω=WUω Y ω W U ω
(2)

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