Lossless Filter Examples

1. The simplest lossless filter is a unit-modulus gain
   $$H(z) = e^{j\phi}$$
   where $\phi$ can be any phase value. In the real case $\phi$ can only be 0 or $\pi$ , hence $H(z)=\pm 1$ .

2. A lossless FIR filter can only consist of a single nonzero tap:
   $$H(z) = e^{j\phi} z^{-K}$$
   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 $K\geq 0$ .

3. Every finite-order, single-input, single-output (SISO), lossless IIR filter (recursive allpass filter) can be written as
   $$H(z) = e^{j\phi} z^{-K} \frac{z^{-N}{\tilde A}(z)}{A(z)}$$
   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.

4. 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 $W_N\isdef e^{-j2\pi/N}$ , is a unitary matrix:
   $$\frac{\bold{W}^\ast}{\sqrt{N}} \frac{\bold{W}}{\sqrt{N}} = \bold{I}_N$$