The downsampling step with discrete wavelet transform

Question

For one stage discrete wavelet transform (DWT), if we have a signal with 1000 samples occupying the frequency range from zero to 500 Hz, the output of the low-pass filter is a signal with frequency range 0-250 Hz, and the output of the high-pass filter is a signal with frequency range 250-500 Hz. Downsampling is then performed. This is right according to the sampling theory; the sampling rate must be at least twice the highest frequency of the signal. Since the output highest frequency of the low-pass filter is reduced to the half, downsampling its output by a factor of two can be performed. What about the output of the high-pass? its highest frequency is still 500, how can we perform downsampling? the bandwidth of the high-pass is 250, but its highest frequency is 500? according to the sampling theory, fs must be at least 2fm, what is fm? the highest frequency of the signal, or its bandwidth?

Answer 1

TL' DR:

When you down-sample a real signal that is sampled at $f_s$ by a factor of two, the new sampling rate will be $f_s/2$. The frequency span from $0$ to $f_s/4$ will be intact as it was originally, but any spectrum that was originally from $f_s/4$ to $f_s/2$ will now alias to the same $0$ to $f_s/4$ spectrum, which is now the primary Nyquist band. The result of bandpass filtering and down-sample by two includes a direct frequency translation from the original $f_s/2$ to DC. (This is the reason we must have anti-alias filters in the analog prior to an A/D Converter when sampling). The digital filter in this case serves as the anti-alias filter prior to resampling (decimation). Just as in A/D conversion, we can use bandpass filters to select higher Nyquist zones, which will alias to the primary Nyquist band.

Details for the weary:

I like to explain this starting with sampling an analog signal since the effects are identical once we understand the fundamentals of sampling in general. Consider the process of sampling a 3 Hz sinusoid with a 20 Hz sampling clock. The 3 Hz real sinusoid has a frequency tone at +/-3 Hz (consistent with $cos(2\pi 3t) = 0.5e^{j2\pi 3 t}+0.5e^{-j2\pi 3 t}$: each of those exponentials is an impulse in the frequency domain). This is the top spectrum shown. Note specifically how a copy of this appears around every multiple of the sampling frequency if we extended our digital frequency axis to $\pm \infty$ (the noted periodicity in frequency that occurs in discrete time systems).

Because of this replication we only need to be concerned with the frequency from $\pm f_s/2$ since it repeats exactly everywhere else, and further with real signals such as in the OP's case, the spectrum is Hermitian symmetric (same magnitude, opposite phase), so we really only need to be concerned with the frequency from DC to $f_s/2$, just as the OP has mentioned. However when we start dealing with really understanding aliasing effects, and multi-rate signal processing (decimation and interpolation), I find it very intuitive to keep this equivalent view in mind: the unrolled digital spectrum.

Now helpful to understand the aliasing effects when down-sampling (and aliasing when sampling in general) is the plot below showing that we will get the exact same digital spectrum with a 23 Hz sinusoid (or 43, 63, 83, ... Hz sinusoid). Each of these unique bands that can alias to our unique digital span from $-f_s/2$ to $+f_s/2$ is referred to as a Nyquist zone. If the reason for that is confusing, I provide further links at the bottom of this post on other answers here at StackExchange that detail this further.

So from this we gain two main points: First, in order to not have severe distortion or elevated noise from every Nyquist zone that will equally create our unique digital spectrum in the $-f_s/2$ to $+f_s/2$ range, we must filter the spectrum of choice prior to sampling the signal. Second, this filter can be a low pass filter such as the case of the 3 Hz sinusoid, or it can be a bandpass filter such as the case of the 23 Hz sinusoid- both will produce an identical sampled signal and it is our knowledge of what filter was used that allows us to distinguish which signal was actually sampled. Bandpass filtering and sampling is also referred to as "undersampling".

Now with that all in mind, consider the OP's case of a down-sample by two, with a low pass filter from DC to $f_s/4$. Note the identical operations, in that the primary spectrum (the first Nyquist zone) in all cases is replicated around every copy of the sampling rate, and how after the down-sample by 2, $f_s/2$ becomes the new sampling rate. We select what we want to have as the primary spectrum with filtering, here below is the result after a low-pass filter:

And the same but with a high pass filter from $f_s/4$ to $f_s/2$:

So the result of bandpass filtering and down-sample by two includes a direct frequency translation from the original $f_s/2$ to DC.

Details for the very interested:

To really understand this process universally to signal processing applications, it is imperative to go beyond the "real-only" signals and follow this through for the case of complex signals; understanding this will avoid a lot of misunderstandings, particularly with aliasing effects. I explain this all in greater detail in the following posts which are recommended for further reading:

Aliasing after downsampling

Higher order harmonics during sampling

does T = 1/F always hold?