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The simplest lossless filter is a unit-modulus gain

$\displaystyle H(z) = e^{j\phi}$
where $\phi$ can be any phase value. In the real case $\phi$ can only be 0 or $\pi$ , hence $H(z)=\pm 1$ .
A lossless FIR filter can only consist of a single nonzero tap:

$\displaystyle H(z) = e^{j\phi} z^{-K}$
for some fixed integer , 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

$\displaystyle H(z) = e^{j\phi} z^{-K} \frac{z^{-N}{\tilde A}(z)}{A(z)}$
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 , conjugating them, and multiplying by . (The factor $z^{-N}$ above serves to restore negative powers of 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 $W_N\isdef e^{-j2\pi/N}$ , is a unitary matrix:

$\displaystyle \frac{\bold{W}^\ast}{\sqrt{N}} \frac{\bold{W}}{\sqrt{N}} = \bold{I}_N$

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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