Critically Sampled Perfect Reconstruction Filter Banks

A Perfect Reconstruction (PR) filter bank is any filter bank whose reconstruction is the original signal, possibly delayed, and possibly scaled by a constant. In this context, critical sampling (also called ``maximal downsampling'') means that the downsampling factor is the same as the number of filter channels. For the STFT, this implies $ R=M=N$ (with $ M>N$ for Portnoff windows).

As derived in Chapter 7, the Short-Time Fourier Transform (STFT) is a PR filter bank whenever the Constant-OverLap-Add (COLA) condition is met by the analysis window $ w$ and the hop size $ R$. However, only the rectangular window case with no zero-padding is critically sampled (OLA hop size = FBS downsampling factor = $ N$). Advanced audio compression algorithms (``perceptual audio coding'') are based on critically sampled filter banks, for obvious reasons.

Important Point: We normally do not require critical sampling for audio analysis, digital audio effects, and music applications. We normally only need it when compression is a requirement.

Two-Channel Critically Sampled Filter Banks

% latex2html id marker 26853\psfrag{x(n)}{\normalsize $x(n)$} ...
...caption{Two-channel critically sampled filter bank.}

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.10.16. 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:
$\displaystyle \hat{X}(z)$ $\displaystyle =$ $\displaystyle \frac{1}{2}[H_0(z)F_0(z) + H_1(z)F_1(z)]X(z)$  
  $\displaystyle +$ $\displaystyle \frac{1}{2}[H_0(-z)F_0(z) + H_1(-z)F_1(z)]X(-z)$  
    $\displaystyle \hbox{(Filter Bank Reconstruction)}
\protect$ (11.1)

We require the second term (the aliasing term) to be zero for perfect reconstruction. This is arranged if we set
$\displaystyle F_0(z)$ $\displaystyle =$ $\displaystyle \quad\! H_1(-z)$  
$\displaystyle F_1(z)$ $\displaystyle =$ $\displaystyle -H_0(-z)$  
    $\displaystyle \hbox{(Aliasing Cancellation Constraints)}
\protect$ (11.2)

  • 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

$\displaystyle c$ $\displaystyle =$ $\displaystyle H_0(z)F_0(z) + H_1(z)F_1(z)$  
    $\displaystyle \hbox{(Filtering Cancellation Constraint)}
\protect$ (11.3)

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,

$\displaystyle c$ $\displaystyle =$ $\displaystyle H_0(z)H_1(-z) - H_1(z)H_0(-z)$  
    $\displaystyle \hbox{(Filtering and Aliasing Cancellation)}
\protect$ (11.4)

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]

Amplitude-Complementary 2-Channel Filter Bank

Perhaps the most natural choice of analysis filters for our two-channel, critically sampled filter bank, is an amplitude-complementary lowpass/highpass pair, i.e.,

$\displaystyle H_1(z) = 1-H_0(z)

where we impose the unity dc gain constraint $ H_0(1)=1$. Amplitude-complementary thus means constant overlap-add (COLA) on the unit circle in the $ z$ plane.

Plugging the COLA constraint into the Filtering and Aliasing Cancellation constraint (10.4) gives

A e^{-j\omega D} &=& H_0(z)[1-H_0(-z)] - [1-H_0(z)]H_0(-z) \\ ...
...n} \\ [5pt]
2h_0(n), & \hbox{$n$\ odd} \\
\end{array} \right.

Points to note:

  • Even-indexed terms of the impulse response are unconstrained, since they subtract out in the constraint.
  • For perfect reconstruction, exactly one odd-indexed term must be nonzero in the lowpass impulse response $ h_0(n)$. The simplest choice is $ h_0(1)\neq 0$.
Thus, the lowpass-filter impulse response can be anything of the form

$\displaystyle h_0 = [h_0(0), \bold{h_0(1)}, h_0(2), 0, h_0(4), 0, h_0(6), 0, \ldots] \protect$ (11.5)


$\displaystyle h_0 = [h_0(0), 0, h_0(2), \bold{h_0(3)}, h_0(4), 0, h_0(6), 0, \ldots]

etc. The corresponding highpass-filter impulse response is then

$\displaystyle h_1(n) = \delta(n) - h_0(n).

The first example (10.5) above goes with the highpass-filter

$\displaystyle h_1 = [1-h_0(0), -h_0(1), -h_0(2), 0, -h_0(4), 0, -h_0(6), 0, \ldots]

and similarly for the other example.

The above class of amplitude-complementary filters can be characterized in general as follows:

H_0(z) &=& E_0(z^2) + h_0(o) z^{-o}, \quad E_0(1)+h_0(o)=1, \,...
...{$o$\ odd}\\
H_1(z) &=& 1-H_0(z) = 1 - E_0(z^2) - h_0(o) z^{-o}

In summary, we see that an amplitude-complementary lowpass/highpass analysis filter pair yields perfect reconstruction (aliasing and filtering cancellation) when there is exactly one odd-indexed term in the impulse response of $ h_0(n)$.

Unfortunately, the channel filters are so constrained in form that it is impossible to make a high quality lowpass/highpass pair. This happens because $ E_0(z^2)$ repeats twice around the unit circle. Since we assume real coefficients, the frequency response, $ E_0(e^{j2\omega})$ is magnitude-symmetric about $ \omega=\pi/2$ as well as $ \pi$. This is not good since we only have one degree of freedom, $ h_0(o) z^{-o}$, with which we can break the $ \pi/2$ symmetry to reduce the high-frequency gain and/or boost the low-frequency gain. This class of filters cannot be expected to give high quality lowpass or highpass behavior.

To achieve higher quality lowpass and highpass channel filters, we will need to relax the amplitude-complementary constraint (and/or filtering cancellation and/or aliasing cancellation) and find another approach.

Haar Example

Before we leave this case (amplitude-complementary, two-channel, critically sampled, perfect reconstruction filter banks), let's see what happens when $ H_0(z)$ is the simplest possible lowpass filter having unity dc gain, i.e.,

$\displaystyle H_0(z) = \frac{1}{2} + \frac{1}{2}z^{-1}.

This case is obtained above by setting $ E_0(z^2)=1/2$, $ o=1$, and $ h_0(1)=1/2$.

The polyphase components of $ H_0(z)$ are clearly

$\displaystyle E_0(z^2)=E_1(z^2)=1/2.

Choosing $ H_1(z)=1-H_0(z)$ and choosing $ F_0(z)$ and $ F_1(z)$ for aliasing cancellation, the four filters become

H_0(z) &=& \frac{1}{2} + \frac{1}{2}z^{-1} = E_0(z^2)+z^{-1}E_...
...F_1(z) &=& -H_0(-z) = -\frac{1}{2} + \frac{1}{2}z^{-1} = -H_1(z)

Thus, both the analysis and reconstruction filter banks are scalings of the familiar Haar filters (``sum and difference'' filters $ (1\pm z^{-1})/\sqrt{2}$).

The frequency responses are

H_0(e^{j\omega}) &=& \quad\,F_0(e^{j\omega}) = \frac{1}{2} + \...
...ega}= j e^{-j\frac{\omega}{2}} \sin\left(\frac{\omega}{2}\right)

which are plotted in Fig.10.17.

Figure 10.17: Amplitude responses of the two channel filters in the Haar filter bank.

Polyphase Decomposition of Haar Example

% latex2html id marker 26987\psfrag{x(n)}{\normalsize $x(n)$} ...
...ion{Two-channel polyphase filter
bank and inverse.}

Let's look at the polyphase representation for this example. Starting with the filter bank and its reconstruction (see Fig.10.18), the polyphase decomposition of $ H_0(z)$ is

$\displaystyle H_0(z) = E_0(z^2) + z^{-1}E_1(z^2) = \frac{1}{2}+\frac{1}{2}z^{-1}

Thus, $ E_0(z^2)=E_1(z^2)=1/2$, and therefore

$\displaystyle H_1(z) = 1-H_0(z) = E_0(z^2)-z^{-1}E_1(z^2).

We may derive polyphase synthesis filters as follows:

\hat{X}(z) &=& \left[F_0(z)H_0(z) + F_1(z)H_1(z)\right] X(z)\\...
...)-H_1(z)\right] + z^{-1}\left[H_0(z) + H_1(z)\right]\right\}X(z)

The polyphase representation of the filter bank and its reconstruction can now be drawn as in Fig.10.19. Notice that the reconstruction filter bank is formally the transpose of the analysis filter bank [247]. A filter bank that is inverted by its own transpose is said to be an orthogonal filter bank, a subject to which we will return §10.3.8.

figure[htbp] \includegraphics{eps/poly2chan}

figure[htbp] \includegraphics{eps/poly2chanfast}

Commuting the downsamplers (using the noble identities from §10.2.5), we obtain Figure 10.20. Since $ E_0(z)=E_1(z)=1/2$, this is simply the OLA form of an STFT filter bank for $ N=2$, with $ N=M=R=2$, and rectangular window $ w=[1/2,1/2]$. That is, the DFT size, window length, and hop size are all 2, and both the DFT and its inverse are simply sum-and-difference operations.

Quadrature Mirror Filters (QMF)

The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:

$\displaystyle H_1(z) = H_0(-z)\quad \hbox{(QMF Symmetry Constraint)} \protect$ (11.6)

That is, the filter for channel 1 is constrained to be a $ \pi$-rotation of filter 0 along the unit circle. In the time domain, $ h_1(n) = (-1)^n h_0(n)$, i.e., all odd-index coefficients are negated. If $ H_0$ is a lowpass filter cutting off near $ \omega=\pi/2$ (as is typical), then $ H_1$ is a complementary highpass filter. The exact cut-off frequency can be adjusted along with the roll-off rate to provide a maximally constant frequency-response sum.

Two-channel QMFs have been around since at least 1976 [51], and appear to be the first critically sampled perfect reconstruction filter banks. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (10.6). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (10.6) [266].

Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by

F_0(z) &=& \quad\! H_1(-z) = \quad\! H_0(z)\\ [0.1in]
F_1(z) &=& -H_0(-z) = -H_1(z),

the perfect reconstruction requirement reduces to

$\displaystyle \hbox{constant}$ $\displaystyle =$ $\displaystyle H_0(z)F_0(z) + H_1(z)F_1(z) = H_0^2(z) - H_0^2(-z)$  
    $\displaystyle \hbox{(QMF Perfect Reconstruction Constraint)}
\protect$ (11.7)

Now, all four filters are determined by $ H_0(z)$.

It is easy to show using the polyphase representation of $ H_0(z)$ (see [266]) that the only causal FIR QMF analysis filters yielding exact perfect reconstruction are two-tap FIR filters of the form

H_0(z) &=& c_0 z^{-2n_0} + c_1 z^{-(2n_1+1)}\\
H_1(z) &=& c_0 z^{-2n_0} - c_1 z^{-(2n_1+1)}

where $ c_0$ and $ c_1$ are constants, and $ n_0$ and $ n_1$ are integers. Therefore, only weak channel filters are available in the QMF case ( $ H_1(z)=H_0(-z)$), as we saw in the amplitude-complementary case. On the other hand, very high quality IIR solutions are possible. See [266, pp. 201-204] for details. In practice, approximate ``pseudo QMF'' filters are more practical, which only give approximate perfect reconstruction. We'll return to this topic in §10.7.1.

The Haar filters, which we saw gave perfect reconstruction in the amplitude-complementary case, are also examples of a QMF filter bank:

H_0(z) &=& 1 + z^{-1}\\ [0.1in]
H_1(z) &=& 1 - z^{-1}

In this example, $ c_0=c_1=1$, and $ n_0=n_1=0$.

Linear Phase Quadrature Mirror Filter Banks

Linear phase filters delay all frequencies by equal amounts, and this is often a desirable property in audio and other applications. A filter phase response is linear in $ \omega$ whenever its impulse response $ h_0(n)$ is symmetric, i.e.,

$\displaystyle h_0(-n) = h_0(n)

in which case the frequency response can be expressed as

$\displaystyle H_0(e^{j\omega}) = e^{-j\omega N/2}\left\vert H_0(e^{j\omega})\right\vert

Substituting this into the QMF perfect reconstruction constraint (10.7) gives

$\displaystyle \hbox{constant} = e^{-j\omega N}\left[
\left\vert H_0(e^{j\omega})\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].

When $ N$ is even, the right hand side of the above equation is forced to zero at $ \omega=\pi/2$. Therefore, we will only consider odd $ N$, for which the perfect reconstruction constraint reduces to

$\displaystyle \hbox{constant} = e^{-j\omega N}\left[
\left\vert H_0(e^{j\omega})\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right]

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are power complementary. See [266] for further details.

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

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)

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 $ (-1)^n$. Again, all four analysis and synthesis filters are determined by the lowpass analysis filter $ H_0(z)$. It can be shown that this is an orthogonal filter bank. The analysis filters $ H_0(z)$ and $ H_1(z)$ 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 $ H_0(z)$ (for real filters $ H_0$). The paraconjugate is the analytic continuation of $ \overline{H_0(e^{j\omega})}$ from the unit circle to the $ z$ plane. Moreover, the analysis filters $ H_0(z)$ 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 [256].

With the CQF constraints, Eq.$ \,$(10.1) reduces to

$\displaystyle \hat{X}(z) = \frac{1}{2}[H_0(z)H_0(z^{-1}) + H_0(-z)H_0(-z^{-1})]X(z) \protect$ (11.8)

Let $ P(z) = H_0(z)H_0(-z)$, such that $ H_0(z)$ is a spectral factor of the half-band filter $ P(z)$ (i.e., $ P(e^{j\omega})$ is a nonnegative power response which is lowpass, cutting off near $ \omega=\pi/4$). Then, (10.8) reduces to

$\displaystyle \hat{X}(z) = \frac{1}{2}[P(z) + P(-z)]X(z) = -z^{-(L-1)}X(z)$ (11.9)

The problem of the PR filter design has thus been reduced to designing one half-band filter, $ P(z)$. 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 $ p(n)$ are set to zero.

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

$\displaystyle p(n) = \frac{\hbox{sin}[\pi n/2]}{\pi n/2}w(n)$ (11.10)

where $ w(n)$ is any suitable window, such as the Kaiser window.

Note that as a result of (10.8), 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 $ H_0$ and $ H_1$ 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):
  1. FIR
  2. orthogonal
  3. 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 §10.9.)

Orthogonal Two-Channel Filter Banks

Recall the reconstruction equation for the two-channel, critically sampled, perfect-reconstruction filter-bank:

\hat{X}(z) &=& \frac{1}{2}[H_0(z)F_0(z) + H_1(z)F_1(z)]X(z)
...\\ [0.1in]
&+& \frac{1}{2}[H_0(-z)F_0(z) + H_1(-z)F_1(z)]X(-z)

This can be written in matrix form as

$\displaystyle \hat{X}(z) = \frac{1}{2} \left[\begin{array}{c} F_0(z) \\ [2pt] F...
\left[\begin{array}{c} X(z) \\ [2pt] X(-z) \end{array}\right]

where the above $ 2 \times 2$ matrix, $ \bold{H}_m(z)$, is called the alias component matrix (or analysis modulation matrix). If

$\displaystyle {\tilde {\bold{H}}}_m(z)\bold{H}_m(z) = 2\bold{I}

where $ {\tilde {\bold{H}}}_m(z)\isdef \bold{H}_m^T(z^{-1})$ denotes the paraconjugate of $ \bold{H}_m(z)$, then the alias component (AC) matrix is lossless, and the (real) filter bank is orthogonal.

It turns out orthogonal filter banks give perfect reconstruction filter banks for any number of channels. Orthogonal filter banks are also called paraunitary filter banks, which we'll study in polyphase form in §10.5 below. The AC matrix is paraunitary if and only if the polyphase matrix (defined in the next section) is paraunitary [266].

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Perfect Reconstruction Filter Banks
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Polyphase Filtering