Quadrature mirror filter

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Bank of QMFs

The quadrature mirror filters (QMF) are two filters with frequency characteristics symmetric about 1/4 of sampling frequency (i.e. \pi/2). They are used especially in process of orthogonal discrete wavelet transform design.

Simple variant[change | edit source]

In notation of Z-transform, we can create the quadrature mirror filter H_1(z) to (original) filter H_0(z) by substitution z with -z in the transfer function of H_0(z).

H_1(z) = H_0(-z)\,

By doing it, the transfer characteristic of H_1(z) is shifted to H_0(z) by \pi.

| H_1(e^{j\omega}) | = | H_0(e^{j(\pi-\omega)}) |\,

Impulse characteristic is therefore

h_1[n] = (-1)^n h_0[n]\, for 0 \leq n < N\,, where N is filter length.

According to the picture above, the signal split and passed into these filters can be downsampled by a factor of two. Nevertheless, original signal can be still reconstructed by using reconstruction filters G_0(z) and G_1(z). Reconstruction filters are given by time reversal analysis filters.

G_0(z) = H_0(z^{-1})\,
G_1(z) = H_1(z^{-1})\,

Orthogonal filter banks[change | edit source]

For orthogonal discrete wavelet transform H_1(z) is given by

H_1(z) = z^{-N} H_0(-z^{-1})\,, where N is filter length.

Impulse characteristic is

h_1[n] = (-1)^n h_0[N-1-n]\, for 0 \leq n < N\,.

Reconstruction filters are still given by same equations.

G_0(z) = H_0(z^{-1})\,
G_1(z) = H_1(z^{-1})\,