QMFDESIGN Function

Designs a finite impulse response (FIR) quadrature mirror filter (QMF).

Usage

result = QMFDESIGN(k)

Input Parameters

k—An array containing the coefficients of a known polynomial factor of the desired QMF.

Returned Value

result—A filter structure containing the coefficients of the QMF. If the known polynomial factor K(z) has order M , the QMF filter has an order that is £ 2M – 1.

Keywords

None.

Discussion

A quadrature mirror filter H(z) is defined by the equation:

|H(z)|² + |H(–z)|² = 1

Given a known factor K(z) of H(z), QMFDESIGN finds the factor U(z) so that H(z) = K(z)U(z) satisfies the QMF equation above.

The algorithm used to solve this problem first solves the equation:

|K(z)|²X(z) + |K(–z)|²X(–z) = z^M

for X(z) using the Euclid algorithm discussed in Demeure and Mullis (1989). Next, a spectral factorization of X(z) is performed to obtain
X(z) = U(z)U(z^–1) using the algorithm discussed in Demeure and Mullis (1990). Finally, the QMF H(z) is determined as H(z) = K(z)U(z). By choosing:

K(z) = (1 + z^–1)^M

QMFDESIGN generates the Daubechies, compactly supported, orthonormal wavelet with M vanishing moments.

Another common choice is to select K(z) so that it has all its zeros on the unit circle in the left half-plane to approximate the ideal half-band lowpass filter.

Note:

The polynomial K(z) must be chosen so that K(z) and K(–z) are co-prime, and X(z) is positive on the unit circle. If K(z) and K(–z) are not co-prime, a message saying, Input polynomials are not co-prime appears. If X(z) is not positive, a message saying, Newton-Raphson iterations did not converge appears.

Example

In this example, the Daubechies QMF is designed with 3-zeros and z = – 1. The results are shown in Magnitude Response of Daubechies Quadrature Mirror Filter.

; The coefficients of the polynomial K(z) = (1+z^-1)³.

k = [1, 3, 3, 1]

; Design the quadrature mirror filter.

h = QMFDESIGN(kz)

hf = FREQRESP_Z(h, Outfreq = f)

; Plot the filter magnitude response versus frequency.

PLOT, f, ABS(hf), Title = 'Daubechies QMF Magnitude', $
XTitle = 'Frequency'

Magnitude Response of Daubechies Quadrature Mirror Filter

For Additional Information

Akansu and Haddad, 1992. Daubechies, 1992. Demeure and Mullis, 1990.

Duell, 1994. Vaidyanathan, 1993.