This example shows how to create a quadrature mirror filter associated with the db10 wavelet.

Compute the scaling filter associated with the db10 wavelet.

sF = dbwavf('db10');

dbwavf normalizes the filter coefficients so that the norm is equal to $1/\sqrt{2}$. Normalize the coefficients so that the filter has norm equal to 1.

G = sqrt(2)*sF;

Obtain the wavelet filter coefficients by using qmf. Plot the filters.

H = qmf(G);
subplot(2,1,1)
stem(G)
title('Scaling (Lowpass) Filter G')
grid on
subplot(2,1,2)
stem(H)
title('Wavelet (Highpass) Filter H')
grid on

Set the DWT extension mode to Periodization. Generate a random signal of length 64. Perform a single-level wavelet decomposition of the signal using G and H.

origmode = dwtmode('status','nodisplay');
dwtmode('per','nodisplay')
n = 64;
rng 'default'
sig = randn(1,n);
[a,d] = dwt(sig,G,H);

The lengths of the approximation and detail coefficients are both 32. Confirm that the filters preserve energy.

[sum(sig.^2) sum(a.^2)+sum(d.^2)]

ans = 1×2

   92.6872   92.6872

Compute the frequency responses of G and H. Zeropad the filters when taking the Fourier transform.

n = 128;
F = 0:1/n:1-1/n;
Gdft = fft(G,n);
Hdft = fft(H,n);

Plot the magnitude of each frequency response.

figure
plot(F(1:n/2+1),abs(Gdft(1:n/2+1)),'r')
hold on
plot(F(1:n/2+1),abs(Hdft(1:n/2+1)),'b')
grid on
title('Frequency Responses')
xlabel('Normalized Frequency')
ylabel('Magnitude')
legend('Lowpass Filter','Highpass Filter','Location','east')

Confirm the sum of the squared magnitudes of the frequency responses of G and H at each frequency is equal to 2.

sumMagnitudes = abs(Gdft).^2+abs(Hdft).^2;
[min(sumMagnitudes) max(sumMagnitudes)]

ans = 1×2

    2.0000    2.0000

Confirm that the filters are orthonormal.

df = [G;H];
id = df*df'

id = 2×2

    1.0000    0.0000
    0.0000    1.0000

Restore the original extension mode.

dwtmode(origmode,'nodisplay')

This example shows the effect of setting the phase parameter of the qmf function.

Obtain the decomposition low-pass filter associated with a Daubechies wavelet.

lowfilt = wfilters('db4');

Use the qmf function to obtain the decomposition low-pass filter for a wavelet. Then, compare the signs of the values when the qmf phase parameter is set to 0 or 1. The reversed signs indicates a phase shift of $$\pi$$ radians, which is the same as multiplying the DFT by $$e^{i\pi }$$.

p0 = qmf(lowfilt,0)

p0 = 1×8

    0.2304   -0.7148    0.6309    0.0280   -0.1870   -0.0308    0.0329    0.0106

p1 = qmf(lowfilt,1)

p1 = 1×8

   -0.2304    0.7148   -0.6309   -0.0280    0.1870    0.0308   -0.0329   -0.0106

Compute the magnitudes and display the difference between them. Unlike the phase, the magnitude is not affected by the sign reversals.

abs(p0)-abs(p1)

ans = 1×8

     0     0     0     0     0     0     0     0