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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. E-mail the authors

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 X⁢ω 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 W⁢ω W ω and the final response Y⁢ω 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 X⁢ω X ω and W⁢ω W ω .

W⁢ω=1D⁢∑ k =0D−1X⁢ω+2⁢π⁢kD W ω 1 D k 0 D 1 X ω 2 k D
(1)
Y⁢ω=W⁢U⁢ω Y ω W U ω
(2)
Figure 2: Multirate System
Figure 2 (prelab_sys.png)

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