Lossless Filter Examples

The simplest lossless filter is a unit-modulus gain

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

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

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

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