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

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

In terms of the $\ensuremath{L_2}$ signal norm $\left\Vert\,\,\cdot\,\,\right\Vert _2$ (§4.10.1), this can be expressed more succinctly as

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

$$\left\vert H(\ejo )\right\vert \eqsp 1, \quad \forall \omega.$$ (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

$$\overline{H(\ejo )} H(\ejo ) \eqsp 1, \quad\forall\omega,$$ (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:

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

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

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

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

$${\tilde H}(\ejo ) H(\ejo ) \eqsp \overline{H}(1/\ejo )H(\ejo ) \eqsp \overline{H(\ejo )}H(\ejo ).$$ (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 $\bold{H}(\ejo )$ is unitary. That is,

$$\bold{H}^*(\ejo )\bold{H}(\ejo ) \eqsp \bold{I}_q$$ (12.79)

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

$$\bold{H}^*(\ejo ) \isdefs \overline{\bold{H}^T(\ejo )}$$ (12.80)

Note that $\bold{H}^*(\ejo )\bold{H}(\ejo )$ 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.,

$${\tilde {\bold{H}}}(z) \bold{H}(z) \eqsp \bold{I}_q$$ (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 $\bold{H}(\ejo )$ .