qmf

Scaling and wavelet filter

Syntax

Y = qmf(X)

Y = qmf(X,P)

Description

Y = qmf(X) changes the signs of the even-indexed elements of the reversed vector filter coefficients X.

Y = qmf(X,P) changes the signs of the even-indexed elements of the reversed vector filter coefficients X if P is 0. If P is 1, the signs of the odd-indexed elements are reversed. Changing P changes the phase of the Fourier transform of the resulting wavelet filter by π radians.

Examples

Create Quadrature Mirror Filter

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

Obtain 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

Save the current extension mode. Set the extension mode to Periodization. Generate a random signal of length 64. Perform a single-level wavelet decomposition of the signal using G and H. For purposes of reproducibility, set the random seed to the default value.

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")
hold off

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")

Controlling Phase of Quadrature Mirror Filter

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

Obtain the decomposition lowpass filter associated with a Daubechies wavelet.

lowfilt = wfilters("db3");

Use the qmf function to obtain the decomposition lowpass 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 = 1×6

    0.3327   -0.8069    0.4599    0.1350   -0.0854   -0.0352

p1 = qmf(lowfilt,1)

p1 = 1×6

   -0.3327    0.8069   -0.4599   -0.1350    0.0854    0.0352

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 = 1×6

     0     0     0     0     0     0

Input Arguments

X — Filter coefficients

vector

Filter coefficients, specified as a vector.

Data Types: single | double

P — Phase parameter

0 (default) | 1

Phase parameter, specified as follows.

• 0 — Change signs of even-indexed elements of the reversed vector X

• 1 — Change signs of odd-indexed elements of the reversed vector X

Data Types: single | double

More About

Quadrature Mirror Filters

Let x be a finite energy signal. Two filters F₀ and F₁ are quadrature mirror filters (QMF) if, for any x,

$${\Vert y_0\Vert }^2+{\Vert y_1\Vert }^2={\Vert x\Vert }^2$$

where y₀ is a decimated version of the signal x filtered with F₀, so y₀ is defined by x₀ = F₀(x) and y₀(n) = x₀(2n). Similarly, y₁ is defined by x₁ = F₁(x) and y₁(n) = x₁(2n). This property ensures a perfect reconstruction of the associated two-channel filter banks scheme (see [1] p. 103).

For example, if F₀ is a Daubechies scaling filter with norm equal to 1 and F₁ = qmf(F₀), then the transfer functions F₀(z) and F₁(z) of the filters F₀ and F₁ satisfy the condition:

$$|F_0(z){|}^2+|F_1(z){|}^2=2.$$

References

[1] Strang, Gilbert, and Truong Nguyen. Wavelets and Filter Banks. Rev. ed. Wellesley, Mass: Wellesley-Cambridge Press, 1997.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a