Quadrature mirror filter

Bank of QMFs

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

Simple variant

In notation of Z-transform, we can create the quadrature mirror filter [math]\displaystyle{ H_1(z) }[/math] to (original) filter [math]\displaystyle{ H_0(z) }[/math] by substitution [math]\displaystyle{ z }[/math] with [math]\displaystyle{ -z }[/math] in the transfer function of [math]\displaystyle{ H_0(z) }[/math].

[math]\displaystyle{ H_1(z) = H_0(-z)\, }[/math]

By doing it, the transfer characteristic of [math]\displaystyle{ H_1(z) }[/math] is shifted to [math]\displaystyle{ H_0(z) }[/math] by [math]\displaystyle{ \pi }[/math].

[math]\displaystyle{ | H_1(e^{j\omega}) | = | H_0(e^{j(\pi-\omega)}) |\, }[/math]

Impulse characteristic is therefore

[math]\displaystyle{ h_1[n] = (-1)^n h_0[n]\, }[/math] for [math]\displaystyle{ 0 \leq n \lt N\, }[/math], where [math]\displaystyle{ N }[/math] 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 [math]\displaystyle{ G_0(z) }[/math] and [math]\displaystyle{ G_1(z) }[/math]. Reconstruction filters are given by time reversal analysis filters.

[math]\displaystyle{ G_0(z) = H_0(z^{-1})\, }[/math]

[math]\displaystyle{ G_1(z) = H_1(z^{-1})\, }[/math]

Orthogonal filter banks

For orthogonal discrete wavelet transform [math]\displaystyle{ H_1(z) }[/math] is given by

[math]\displaystyle{ H_1(z) = z^{-N} H_0(-z^{-1})\, }[/math], where [math]\displaystyle{ N }[/math] is filter length.

Impulse characteristic is

[math]\displaystyle{ h_1[n] = (-1)^n h_0[N-1-n]\, }[/math] for [math]\displaystyle{ 0 \leq n \lt N\, }[/math].

Reconstruction filters are still given by same equations.

[math]\displaystyle{ G_0(z) = H_0(z^{-1})\, }[/math]

[math]\displaystyle{ G_1(z) = H_1(z^{-1})\, }[/math]