Perfect Reconstruction Filters

From above we know that the output $x'[n]$ of a 2-channel filter bank can be a perfect reconstruction of the input $x[n]$ if the following conditions are satisfied (equation (1) in previous section):

$$\begin{array}{l} G_0(z)H_0(-z)+G_1(z)H_1(-z)=0 \\ G_0(z)H_0(z)+G_1(z)H_1(z)=2 \end{array}$$

Note that the subsampling rate does not have to be above the Nyquest frequency specified by the sampling theorem. Even if the signal is undersampled and alising exists, the perfect reconstruction of the signalit is guaranteed due to the upsampling and synthesis filters determined by the above equations.

As there are four function variables $H_0$, $H_1$, $G_0$ and $G_1$ in the two equations, there exist many possible designs for the filter banks. Here are three particular ones:

• Quadrature mirror filters (QMFs)

  We let
  

  $$H_1(z)=H_0(-z),\;\;\;G_0(z)=H_0(z),\;\;\;G_1(z)=-H_0(-z)$$
  
  
  both of the two equations above can be written in terms of $H_0(z)$. The first equation above becomes:
  
  $$G_0(z)H_0(-z)+G_1(z)H_1(-z)=H_0(z)H_0(-z)-H_0(-z)H_0(z)=0$$
  
  
  and the second equation becomes:
  
  $$G_0(z)H_0(z)+G_1(z)H_1(z)=H_0(z)H_0(z)-H_0(-z)H_0(-z)\stackrel{\triangle}{=}2$$
  
  
  where $H_0(z)$ is so chosen that $H^2_0(z)-H^2_0(-z)=2$ to satisfy the requirement for perfect reconstruction.

• Conjugate quadrature filters (CQFs)

  We let
  

  $$G_0(z)=H_0(z^{-1}),\;\;\;G_1(z)=zH_0(-z),\;\;\;H_1(z)=z^{-1}H_0(-z^{-1})$$
  
  
  and both of the two equations above can be written in terms of $H_0$. The first equation above becomes:
  
  $$G_0(z)H_0(-z)+G_1(z)H_1(-z)=H_0(z^{-1})H_0(-z)-zH_0(-z)z^{-1}H_0(z^{-1})=0$$
  
  
  and the second equation becomes:
  

  $$G_0(z)H_0(z)+G_1(z)H_1(z)=H_0(z^{-1})H_0(z)+zH_0(-z)z^{-1}H_0(-z^{-1})$$

  $$= H_0(z^{-1})H_0(z)+H_0(-z)H_0(-z^{-1})\stackrel{\triangle}{=}2$$

  
  
  where $H_0(z)$ is so chosen that the second expression is 2 to satisfy the requirement for perfect reconstruction.

• Orthonormal (fast wavelet transform) filter

  We let
  

  $$H_0(z)=G_0(z^{-1}),\;\;\;H_1(z)=G_1(z^{-1}),\;\;\;G_1(z)=-z^{-2k+1}G_0(-z^{-1})$$
  
  
  and both of the two equations above can be written in terms of $G_0(z)$. The first equation above becomes:
  

  $$G_0(z)H_0(-z)+G_1(z)H_1(-z)=G_0(z)G_0(-z^{-1})+G_1(z)G_1(-z^{-1})$$

  $$= G_0(z)G_0(-z^{-1})-z^{-2k+1}G_0(-z^{-1})z^{2k-1}G_0(z)$$

  $$= G_0(z)G_0(-z^{-1})-G_0(-z^{-1})G_0(z)=0$$

  
  
  and the second equation becomes:
  

  $$G_0(z)H_0(z)+G_1(z)H_1(z)=G_0(z)G_0(z^{-1})+G_1(z)G_1(z^{-1})$$

  $$= G_0(z)G_0(z^{-1})+[-z^{-2k+1}G_0(-z^{-1})][-z^{2k-1}G_0(-z)]$$

  $$= G_0(z)G_0(z^{-1})+G_0(-z^{-1})G_0(-z)\stackrel{\triangle}{=}2$$

  
  
  where $G_0(z)$ is so chosen that the second expression is 2 to satisfy the requirement for perfect reconstruction. Note that $P(z)\stackrel{\triangle}{=}G_0(z)G_0(z^{-1})$ is the Z-transform of the autocorrelation $p[n]=\sum_m g_0[m]g_0[m+n]$, and the second equation becomes
  
  $$P(z)+P(-z)=2\;\;\;\;\mbox{i.e.}\;\;\;\frac{1}{2}[P(z)+P(-z)]=1$$
  
  
  Replacing $z$ by $z^{1/2}$, we get
  
  $$\frac{1}{2}[P(z^{1/2})+P(-z^{1/2})]=1$$
  
  
  Consider the down sampled version of the function $g'_0[n]=g_0[2n]$, and its autocorrelation $p'[n]=p[2n]$. As in Z domain we have:
  
  $$P'(z)=\frac{1}{2}[P(z^{1/2})+P(-z^{1/2})]=1$$
  
  
  in time domain we have
  
  $$p'[n]=\sum_m g'[m] g'[n+m]=\sum_m g[m] g[2n+m]=\delta[n]$$
  
  
  i.e., the down-sampled version of $g_0[n]$ is orthonormal.

Example: There are different ways to design the FIR filter orthonormal impulse response $g_0[n]$ for the two-channel filter bank.

The conditions for perfect construction filters listed above can be inverse Z-transformed to get:

$$H_i(z)=G_i(z^{-1}) \leftrightarrow h_i[n]=g_i[-n],\;\;\;\;(i=0,1)$$

$$G_1(z)=-z^{-2k+1}G_0(-z^{-1}) \leftrightarrow g_1[n]=(-1)^n g_0[2k-1-n]$$

i.e., $h_i$ is the time-reversed version of $g_i$ ($i=0,1$), and $g_1$ is both time reversed and modulated version of $g_0$. Once $g_0$ is determined, the rest can all be determined.