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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})$ .

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Lossless Filter Examples
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