Hi Alexander,
- Yes, you are correct that the downsampling causes aliasing (or can cause aliasing). However, the analysis filters and synthesis filters in the wavelet transform are designed in such a way that any aliasing is canceled upon reconstruction.
- You are only using two filters because you are using the so-called pyramid algorithm. You can think of the equivalent filters in terms of successive convolutions, but you should keep in the mind the so called Noble identities from multirate signal processing. That allows you to see the effect of interchanging downsampling with filtering (or on the synthesis, the effect of interchanging upsampling with filtering). Read up on the Noble identities, for one example, if you lowpass filter with the scaling filter, G(f), and then downsample by two followed by filtering with the wavelet filter, H(f) that is equivalent to the second-level wavelet coefficients (prior to decimating those by two). Schematically, that is X(f) -> G(f) -> downsample by two -> H(f). The noble identities will tell you that is equivalent to X(f) -> G(f)H(2f) -> downsample by two so the equivalent filter for the second-level wavelet coefficients is G(f)H(2f). Now let's look at that filter in MATLAB.
N = 64;
[G,H] = wfilters('sym4');
Gdft = fft(G,N);
Hdft = fft(H,N);
H2 = Hdft(1+mod(2*(0:N-1),N));
H2 = Gdft.*H2;
Now let's plot Hdft (1st-level wavelet filter response) and H2 (second-level wavelet filter response)
f = 0:1/N:1/2;
plot(f,abs(Hdft(1:N/2+1)));
hold on;
plot(f,abs(H2(1:N/2+1)),'r');
legend('First Level Wavelet','Second Level Wavelet');
xlim([0 1/2]);
grid on;
xlabel('Cycles/Sample');
