The wavelet transform [4] has recently become popular for various signal-processing operat...

The wavelet transform [4] has recently become popular for various signal-processing operations. The analysis step involves applying a series of low-pass filters to the input signal. After each application of the filter, the signal is downsampled to retain only every other sample point so that the number of wavelet transform coefficients that remains is equal to the number of points in the input function.

In the synthesis stage, the wavelet coefficients are repeatedly upsampled and filtered to reconstruct the input signal. This problem will demonstrate that the filtering and upsampling steps of the wavelet transform can be interchanged.

To upsample a signal x[n], we form

(a) A filter with impulse response h[n] is applied to a signal x[n] to form y[n]. The signal y[n] is then upsampled by a factor of M to form yM[n]. Find YM(z).

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(b) Next the input signal x[n] is first upsampled by M to form xM[n]. The filter h[n] is also upsampled by M to form hM[n] and is then applied to xM[n]. Find the z-transform of xM[n] *hM[n].