Paraunitary Filter Banks

Paraunitary filter banks form an interesting subset of perfect reconstruction (PR) filter banks. We saw above that we get a PR filter bank whenever the synthesis polyphase matrix $ \bold{R}(z)$ times the analysis polyphase matrix $ \bold{E}(z)$ is the identity matrix $ \bold{I}$ , i.e., when

$\displaystyle \bold{P}(z) \isdefs \bold{R}(z)\bold{E}(z) \eqsp \bold{I}.$ (12.70)

In particular, if $ \bold{R}(z)$ is the paraconjugate of $ \bold{E}(z)$ , we say the filter bank is paraunitary.

Paraconjugation is the generalization of the complex conjugate transpose operation from the unit circle to the entire $ z$ plane. A paraunitary filter bank is therefore a generalization of an orthogonal filter bank. Recall that an orthogonal filter bank is one in which $ \bold{E}(e^{j\omega})$ is an orthogonal (or unitary) matrix, to within a constant scale factor, and $ \bold{R}(e^{j\omega})$ is its transpose (or Hermitian transpose).

Lossless Filters

To motivate the idea of paraunitary filters, let's first review some properties of lossless filters, progressing from the simplest cases up to paraunitary filter banks:

A linear, time-invariant filter $ H(z)$ is said to be lossless (or allpass) if it preserves signal energy. That is, if the input signal is $ x(n)$ , and the output signal is $ y(n) = (h\ast x)(n)$ , then we have

$\displaystyle \sum_{n=-\infty}^{\infty} \left\vert y(n)\right\vert^2 \eqsp \sum_{n=-\infty}^{\infty} \left\vert x(n)\right\vert^2.$ (12.71)

In terms of the $ L2$ signal norm $ \left\Vert\,\,\cdot\,\,\right\Vert _2$4.10.1), this can be expressed more succinctly as

$\displaystyle \left\Vert\,y\,\right\Vert _2^2 \eqsp \left\Vert\,x\,\right\Vert _2^2.$ (12.72)

Notice that only stable filters can be lossless since, otherwise, $ \left\Vert\,y\,\right\Vert=\infty$ . We further assume all filters are causal for simplicity.

It is straightforward to show that losslessness implies

$\displaystyle \left\vert H(e^{j\omega})\right\vert \eqsp 1, \quad \forall \omega.$ (12.73)

That is, the frequency response must have magnitude 1 everywhere on the unit circle in the $ z$ plane. Another way to express this is to write

$\displaystyle \overline{H(e^{j\omega})} H(e^{j\omega}) \eqsp 1, \quad\forall\omega,$ (12.74)

and this form generalizes to $ {\tilde H}(z)H(z)$ over the entire the $ z$ plane.

The paraconjugate of a transfer function may be defined as the analytic continuation of the complex conjugate from the unit circle to the whole $ z$ plane:

$\displaystyle {\tilde H}(z) \isdefs \overline{H}(z^{-1})$ (12.75)

where $ \overline{H}(z)$ denotes complex conjugation of the coefficients only of $ H(z)$ and not the powers of $ z$ . For example, if $ H(z)=1+jz^{-1}$ , then $ \overline{H}(z) = 1-jz^{-1}$ . We can write, for example,

$\displaystyle \overline{H}(z) \isdefs \overline{H\left(\overline{z}\right)}$ (12.76)

in which the conjugation of $ z$ serves to cancel the outer conjugation.

We refrain from conjugating $ z$ in the definition of the paraconjugate because $ \overline{z}$ is not analytic in the complex-variables sense. Instead, we invert $ z$ , which is analytic, and which reduces to complex conjugation on the unit circle.

The paraconjugate may be used to characterize allpass filters as follows:

A causal, stable, filter $ H(z)$ is allpass if and only if

$\displaystyle {\tilde H}(z) H(z) \eqsp 1.$ (12.77)

Note that this is equivalent to the previous result on the unit circle since

$\displaystyle {\tilde H}(e^{j\omega}) H(e^{j\omega}) \eqsp \overline{H}(1/e^{j\omega})H(e^{j\omega}) \eqsp \overline{H(e^{j\omega})}H(e^{j\omega}).$ (12.78)

To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices.

A $ p\times q$ transfer function matrix $ \bold{H}(z)$ is said to be lossless if it is stable and its frequency-response matrix $ \bold{H}(e^{j\omega})$ is unitary. That is,

$\displaystyle \bold{H}^*(e^{j\omega})\bold{H}(e^{j\omega}) \eqsp \bold{I}_q$ (12.79)

for all $ \omega$ , where $ \bold{I}_q$ denotes the $ q\times q$ identity matrix, and $ \bold{H}^\ast(e^{j\omega})$ denotes the Hermitian transpose (complex-conjugate transpose) of $ \bold{H}(e^{j\omega})$ :

$\displaystyle \bold{H}^*(e^{j\omega}) \isdefs \overline{\bold{H}^T(e^{j\omega})}$ (12.80)

Note that $ \bold{H}^*(e^{j\omega})\bold{H}(e^{j\omega})$ is a $ q\times q$ matrix product of a $ q\times p$ times a $ p\times q$ matrix. If $ q>p$ , then the rank must be deficient. Therefore, we must have $ p\geq q$ . (There must be at least as many outputs as there are inputs, but it's ok to have extra outputs.)

A lossless $ p\times q$ transfer function matrix $ \bold{H}(z)$ is paraunitary, i.e.,

$\displaystyle {\tilde {\bold{H}}}(z) \bold{H}(z) \eqsp \bold{I}_q$ (12.81)

Thus, every paraunitary matrix transfer function is unitary on the unit circle for all $ \omega$ . Away from the unit circle, paraunitary $ \bold{H}(z)$ is the unique analytic continuation of unitary $ \bold{H}(e^{j\omega})$ .


Lossless Filter Examples

The simplest lossless filter is a unit-modulus gain

$\displaystyle H(z) \eqsp e^{j\phi}$ (12.82)

where $ \phi$ can be any phase value. In the real case $ \phi$ can only be 0 or $ \pi$ , hence $ H(z)=\pm 1$ .

A lossless FIR filter can only consist of a single nonzero tap:

$\displaystyle H(z) \eqsp e^{j\phi} z^{-K} \protect$ (12.83)

for some fixed integer $ K$ , where $ \phi$ is again some constant phase, constrained to be 0 or $ \pi$ in the real-filter case. We consider only causal filters here, so $ K\geq 0$ .

Every finite-order, single-input, single-output (SISO), lossless IIR filter (recursive allpass filter) can be written as

$\displaystyle H(z) \eqsp e^{j\phi} z^{-K} \frac{z^{-N}{\tilde A}(z)}{A(z)}$ (12.84)

where $ K\geq 0$ , $ A(z) = 1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_N
z^{-N}$ , and $ {\tilde A}(z)\isdef \overline{A}(z^{-1})$ . The polynomial $ {\tilde A}(z)$ can be obtained by reversing the order of the coefficients in $ A(z)$ , conjugating them, and multiplying by $ z^N$ . (The factor $ z^{-N}$ above serves to restore negative powers of $ z$ and hence causality.) Such filters are generally called allpass filters.

The normalized DFT matrix is an $ N\times N$ order zero paraunitary transformation. This is because the normalized DFT matrix, $ \bold{W}=[W_N^{nk}]/\sqrt{N}$ , $ n,k=0,\ldots,N-1$ , where $ W_N\isdeftext
e^{-j2\pi/N}$ , is a unitary matrix:

$\displaystyle \frac{\bold{W}^\ast}{\sqrt{N}} \frac{\bold{W}}{\sqrt{N}} \eqsp \bold{I}_N$ (12.85)


Properties of Paraunitary Filter Banks

Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let $ \bold{H}(z)$ denote the $ p\times q$ matrix transfer function of a paraunitary system. In the square case ($ p=q$ ), the matrix determinant, $ \det[\bold{H}(z)]$ , is an allpass filter. Therefore, if a square $ \bold{H}(z)$ contains FIR elements, its determinant is a simple delay: $ \det[\bold{H}(z)]=z^{-K}$ for some integer $ K$ .

An $ N$ -channel analysis filter bank can be viewed as an $ N\times 1$ MIMO filter:

$\displaystyle \bold{H}(z) \eqsp \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]$ (12.86)

A paraunitary filter bank must therefore satisfy

$\displaystyle {\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp 1.$ (12.87)

More generally, we allow paraunitary filter banks to scale and/or delay the input signal:

$\displaystyle {\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp c_K z^{-K}$ (12.88)

where $ K$ is some nonnegative integer and $ c_K\neq 0$ .


We can note the following properties of paraunitary filter banks:

  • The synthesis filter bank is simply the paraconjugate of the analysis filter bank:

    $\displaystyle \bold{F}(z) \eqsp {\tilde {\bold{H}}}(z)$ (12.89)

    That is, since the paraconjugate is the inverse of a paraunitary filter matrix, it is exactly what we need for perfect reconstruction.

  • The channel filters $ H_k(z)$ are power complementary:

    $\displaystyle \left\vert H_1(e^{j\omega})\right\vert^2 + \left\vert H_2(e^{j\omega})\right\vert^2 + \cdots + \left\vert H_N(e^{j\omega})\right\vert^2 \eqsp 1$ (12.90)

    This follows immediately from looking at the paraunitary property on the unit circle.

  • When $ \bold{H}(z)$ is FIR, the corresponding synthesis filter matrix $ {\tilde {\bold{H}}}(z)$ is also FIR.

  • When $ \bold{H}(z)$ is FIR, each synthesis filter, $ F_k(z) =
{\tilde {\bold{H}}}_k(z),\, k=1,\ldots,N$ , is simply the $ \hbox{\sc Flip}$ of its corresponding analysis filter $ H_k(z)=\bold{H}_k(z)$ :

    $\displaystyle f_k(n) \eqsp h_k(L-n)$ (12.91)

    where $ L$ is the filter length. (When the filter coefficients are complex, $ \hbox{\sc Flip}$ includes a complex conjugation as well.) This follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients. As we observed in (11.83) above (§11.5.2), only trivial FIR filters of the form $ H(z) = e^{j\phi} z^{-K}$ can be paraunitary in the single-input, single-output (SISO) case. In the MIMO case, on the other hand, paraunitary systems can be composed of FIR filters of any order.

  • FIR analysis and synthesis filters in paraunitary filter banks have the same amplitude response. This follows from the fact that $ \hbox{\sc Flip}(h) \;\leftrightarrow\;\overline{H}$ , i.e., flipping an FIR filter impulse response $ h(n)$ conjugates the frequency response, which does not affect its amplitude response $ \vert H(e^{j\omega})\vert$ .

  • The polyphase matrix $ \bold{E}(z)$ for any FIR paraunitary perfect reconstruction filter bank can be written as the product of a paraunitary and a unimodular matrix, where a unimodular polynomial matrix $ \bold{U}(z)$ is any square polynomial matrix having a constant nonzero determinant. For example,

    $\displaystyle \bold{U}(z) \eqsp
\left[\begin{array}{cc} 1+z^3 & z^2 \\ [2pt] z & 1 \end{array}\right] $

    is unimodular. See [287, p. 663] for further details.


Examples

Consider the Haar filter bank discussed in §11.3.3, for which

$\displaystyle \bold{H}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right].$ (12.92)

The paraconjugate of $ \bold{H}(z)$ is

$\displaystyle {\tilde {\bold{H}}}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1+z & 1 - z \end{array}\right]$ (12.93)

so that

$\displaystyle {\tilde {\bold{H}}}(z) \bold{H}(z) \eqsp \frac{1}{2} \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] \left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right] \eqsp 1.$ (12.94)

Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when $ {\tilde {\bold{H}}}(z)$ is $ N\times 1$ , power-complementary and paraunitary are the same property. For more about paraunitary filter banks, see Chapter 6 of [287].


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