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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Critically Sampled Perfect Reconstruction Filter Banks
          Two-Channel Critically Sampled Filter Banks

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Two-Channel Critically Sampled Filter Banks


\begin{psfrags} % latex2html id marker 30363\psfrag{x(n)}{\normalsize $x(n)$} ... ...caption{Two-channel critically sampled filter bank.} \end{figure} \end{psfrags}

Let's begin with a simple two-channel case, with lowpass analysis filter $ H_0(z)$, highpass analysis filter $ H_1(z)$, lowpass synthesis filter $ F_0(z)$, and highpass synthesis filter $ F_1(z)$. This system is diagrammed in Fig.11.16. The outputs of the two analysis filters are then

$\displaystyle X_k(z) = H_k(z)X(z), \quad k=0,1. $

After downsampling, the signals become

$\displaystyle V_k(z) = \frac{1}{2}\left[X_k(z^{1/2}) + X_k(-z^{1/2})\right], \; k=0,1. $

After upsampling, the signals become
$\displaystyle Y_k(z) = V_k(z^2)$ $\displaystyle =$ $\displaystyle \frac{1}{2}[X_k(z) + X_k(-z)]$  
  $\displaystyle =$ $\displaystyle \frac{1}{2}[H_k(z)X(z) + H_k(-z)X(-z)],\; k=0,1.$  

After substitutions and rearranging, the output $ \hat{x}$ is a filtered replica plus an aliasing term:
$\displaystyle \hat{X}(z)$ $\displaystyle =$ $\displaystyle \frac{1}{2}[H_0(z)F_0(z) + H_1(z)F_1(z)]X(z)$  
  $\displaystyle +$ $\displaystyle \frac{1}{2}[H_0(-z)F_0(z) + H_1(-z)F_1(z)]X(-z)$  
    $\displaystyle \hbox{(Filter Bank Reconstruction)} \protect$ (12.1)

We require the second term (the aliasing term) to be zero for perfect reconstruction. This is arranged if we set
$\displaystyle F_0(z)$ $\displaystyle =$ $\displaystyle \quad\! H_1(-z)$  
$\displaystyle F_1(z)$ $\displaystyle =$ $\displaystyle -H_0(-z)$  
    $\displaystyle \hbox{(Aliasing Cancellation Constraints)} \protect$ (12.2)

Thus,
  • The synthesis lowpass filter $ F_0(z)$ is the rotation by $ \pi$ of the analysis highpass filter $ H_1(z)$ on the unit circle. If $ H_1(z)$ is highpass, cutting off at $ \omega=\pi/2$, then $ F_0(z)$ will be lowpass, cutting off at $ \pi/2$.
  • The synthesis highpass filter $ F_1(z)$ is the negative of the $ \pi$-rotation of the analysis lowpass filter $ H_0(z)$.
Note that aliasing is completely canceled by this choice of synthesis filters $ F_0,F_1$, for any choice of analysis filters $ H_0,H_1$.

For perfect reconstruction, we additionally need

$\displaystyle c$ $\displaystyle =$ $\displaystyle H_0(z)F_0(z) + H_1(z)F_1(z)$  
    $\displaystyle \hbox{(Filtering Cancellation Constraint)} \protect$ (12.3)

where $ c=Ae^{-j\omega D}$ is any constant $ A>0$ times a linear-phase term corresponding to $ D$ samples of delay.

Choosing $ F_0$ and $ F_1$ to cancel aliasing,

$\displaystyle c$ $\displaystyle =$ $\displaystyle H_0(z)H_1(-z) - H_1(z)H_0(-z)$  
    $\displaystyle \hbox{(Filtering and Aliasing Cancellation)} \protect$ (12.4)

Perfect reconstruction thus also imposes a constraint on the analysis filters, which is of course true for any band-splitting filter bank.

Let $ {\tilde H}$ denote $ H(-z)$. Then both constraints can be expressed in matrix form as

$\displaystyle \left[\begin{array}{cc} H_0 & H_1 \\ [2pt] {\tilde H}_0 & {\tilde... ... F_1 \end{array}\right]=\left[\begin{array}{c} c \\ [2pt] 0 \end{array}\right] $

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Previous: Critically Sampled Perfect Reconstruction Filter Banks
Next: Amplitude-Complementary 2-Channel Filter Bank
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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