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Multirate Filtering: Theory Exercise

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade

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.

Figure 1: DTFT of the input signal.
Figure 1 (prelab_input.png)

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=0D−1Xω+2πkD W ω 1 D k 0 D 1 X ω 2 k D (1)
Yω=WUω Y ω W U ω (2)
Figure 2: Multirate System
Figure 2 (prelab_sys.png)

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