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Question: The following sentence is not clear to me (especially why down-shift occurs). Could someone explain it please?
On the other hand, downsampling the output of the highpass filter down-shifts the high-frequency band and expands it to the full range $|\omega| < \pi$.
This is what happens when a signal is downsampled: you get a scaling of the frequency axis and the addition of shifted spectra (aliasing). The same happens to the lowpass filtered signal. The spectrum of the downsampled signals is given by Eq. $(4.77)$ in Oppenheim and Schafer's Discrete-time Signal Processing (3rd ed), which for downsampling factor $M=2$ can be written as
$$X_d(e^{j\omega})=\frac12\Big[X\left(e^{j\omega/2}\right)+X\left(e^{j(\omega/2-\pi)}\right)\Big]\tag{1}$$
The second term on the right-hand side of $(1)$ fills up the "empty spaces" in the frequency ranges that are eliminated by the lowpass and highpass filters, respectively.
