Conjugate Quadrature Filters (CQF)

A class of causal, FIR, two-channel, criticially sampled, exact perfect-reconstruction filter-banks is the set of so-called Conjugate Quadrature Filters (CQF). In the z-domain, the CQF relationships are

$\displaystyle H_1(z) = z^{-(L-1)}H_0(-z^{-1})$

In the time domain, the analysis and synthesis filters are given by

$\begin{eqnarray*} h_1(n) &=& -(-1)^n h_0(L-1-n) \\ [0.1in] f_0(n) &=& h_0(L-1-n) \\ [0.1in] f_1(n) &=& -(-1)^n h_0(n) = - h_1(L-1-n) \end{eqnarray*}$

That is, $f_0=\hbox{\sc Flip}(h_0)$ for the lowpass channel, and the highpass channel filters are a modulation of their lowpass counterparts by . Again, all four analysis and synthesis filters are determined by the lowpass analysis filter . It can be shown that this is an orthogonal filter bank. The analysis filters and are power complementary, i.e.,

$\displaystyle \left\vert H_0{e^{j\omega}}\right\vert^2 + \left\vert H_1{e^{j\omega}}\right\vert^2 = 1 \qquad\hbox{(Power Complementary)}$

$\displaystyle {\tilde H}_0(z) H_0(z) + {\tilde H}_1(z) H_1(z) = 1 \qquad\hbox{(Power Complementary)}$

where ${\tilde H}_0(z)\isdef \overline{H}_0(z^{-1})$ denotes the paraconjugate of (for real filters ). The paraconjugate is the analytic continuation of $\overline{H_0(e^{j\omega})}$ from the unit circle to the plane. Moreover, the analysis filters are power symmetric, e.g.,

$\displaystyle {\tilde H}_0(z) H_0(z) + {\tilde H}_0(-z) H_0(-z) = 1 \qquad\hbox{(Power Symmetric)}$

The power symmetric case was introduced by Smith and Barnwell in 1984 [249].

With the CQF constraints, Eq. $\,$ (11.1) reduces to

$\displaystyle \hat{X}(z) = \frac{1}{2}[H_0(z)H_0(z^{-1}) + H_0(-z)H_0(-z^{-1})]X(z) \protect$

(12.8)

Let , such that is a spectral factor of the half-band filter (i.e., $P(e^{j\omega})$ is a nonnegative power response which is lowpass, cutting off near $\omega=\pi/4$ ). Then, (11.8) reduces to

$\displaystyle \hat{X}(z) = \frac{1}{2}[P(z) + P(-z)]X(z) = -z^{-(L-1)}X(z)$

(12.9)

The problem of the PR filter design has thus been reduced to designing one half-band filter, . It can be shown that any half-band filter can be written in the form $p(2n) = \delta(n)$ . That is, all non-zero even-idexed values of are set to zero.

A simple design of an FIR half-band filter would be to window a sinc function:

$\displaystyle p(n) = \frac{\hbox{sin}[\pi n/2]}{\pi n/2}w(n)$

(12.10)

where is any suitable window, such as the Kaiser window.

Note that as a result of (11.8), the CQF filters are power complementary. That is, they satisfy:

$\displaystyle \left\vert H_0(e^{j \omega})\right\vert^2 + \left\vert H_1(e^{j \omega})\right\vert^2 = 2$

Also note that the filters and are not linear phase. It can be shown that there are no two-channel perfect reconstruction filter banks that have all three of the following characteristics (except for the Haar filters):

FIR
orthogonal
linear phase

In this design procedure, we have chosen to satisfy the first two and give up the third.

By relaxing ``orthogonality'' to ``biorthogonality'', it becomes possible to obtain FIR linear phase filters in a critically sampled, perfect reconstruction filter bank. (See §12.2.)

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Critically Sampled Perfect Reconstruction Filter Banks
          Conjugate Quadrature Filters (CQF)