I'm currently studying the Fast Wavelet Transform. As I currently understand, the Fast Wavelet Transform is implemented as a QMF filter bank where the frequency resolution decreases as the signal is low-pass filtered and sub-sampled. Pictured below:
However, I'm curious as to why this "works".
Let's say I have a true 750Hz signal that is sampled at 2kHz. The amplitude of this 750Hz signal is 2Vpp.
The Fast Wavelet Transform says that I can high-pass filter with a pass region of 500Hz - 1000Hz, decimate by a factor of 2, and it will output coefficients that correspond to the signal frequency. Because I know the signal is constant frequency 750Hz, I would hope for constant coefficients of magnitude 1 at Level 1.
But I'm confused. Because we're subsampling at 1KHz (half of the original sample rate), we're essentially aliasing the 750Hz signal into the 0Hz - 500Hz range. It seems like the coefficient value will vary depending on where we're sampling. However, because we know the true signal is a constant 750Hz signal, it would be desired that the wavelet coefficient is also constant.
How exactly do the coefficients of the Fast Wavelet Transform correspond to the true frequency content of a signal?
Edit: Are there any good resources for understanding the general properties of Wavelet coefficients? e.g. should a wavelet coefficient remain constant if the signal has constant frequency content?