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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Analog Filters
       Analog Allpass Filters
          Lossless Analog Filters

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Lossless Analog Filters

As discussed in §B.2, the an allpass filter can be defined as any filter that preserves signal energy for every input signal $ x(t)$. In the continuous-time case, this means

$\displaystyle \left\Vert\,x\,\right\Vert _2^2 \isdef \int_{-\infty}^\infty \le... ...\infty \left\vert y(t)\right\vert^2 dt \isdef \left\Vert\,y\,\right\Vert _2^2 $

where $ y(t)$ denotes the output signal, and $ \left\Vert\,y\,\right\Vert$ denotes the L2 norm of $ y$. Using the Rayleigh energy theorem (Parseval's theorem) for Fourier transforms [87], energy preservation can be expressed in the frequency domain by

$\displaystyle \left\Vert\,X\,\right\Vert _2 = \left\Vert\,Y\,\right\Vert _2 $

where $ X$ and $ Y$ denote the Fourier transforms of $ x$ and $ y$, respectively, and frequency-domain L2 norms are defined by

$\displaystyle \left\Vert\,X\,\right\Vert _2 \isdef \sqrt{\frac{1}{2\pi}\int_{-\infty}^\infty \left\vert X(j\omega)\right\vert^2 d\omega}. $

If $ h(t)$ denotes the impulse response of the allpass filter, then its transfer function $ H(s)$ is given by the Laplace transform of $ h$,

$\displaystyle H(s) = \int_0^{\infty} h(t)e^{-st}dt, $

and we have the requirement

$\displaystyle \left\Vert\,X\,\right\Vert _2 = \left\Vert\,Y\,\right\Vert _2 = \left\Vert\,H\cdot X\,\right\Vert _2. $

Since this equality must hold for every input signal $ x$, it must be true in particular for complex sinusoidal inputs of the form $ x(t) = \exp(j2\pi f_xt)$, in which case [87]

\begin{eqnarray*} X(f) &=& \delta(f-f_x)\\ Y(f) &=& H(j2\pi f_x)\delta(f-f_x), \end{eqnarray*}

where $ \delta(f)$ denotes the Dirac ``delta function'' or continuous impulse function (§E.4.3). Thus, the allpass condition becomes

$\displaystyle \left\Vert\,X\,\right\Vert _2 = \left\Vert\,Y\,\right\Vert _2 = \left\vert H(j2\pi f_x)\right\vert\cdot\left\Vert\,X\,\right\Vert _2 $

which implies

$\displaystyle \left\vert H(j\omega)\right\vert = 1, \quad \forall\, \omega\in(-\infty,\infty). \protect$ (E.15)

Suppose $ H$ is a rational analog filter, so that

$\displaystyle H(s) = \frac{B(s)}{A(s)} $

where $ B(s)$ and $ A(s)$ are polynomials in $ s$:

\begin{eqnarray*} B(s) &=& b_M s^M + b_{M-1}s^{M-1} + \cdots + b_1 s + b_0\\ A(s) &=& s^N + a_{N-1}s^{N-1} + \cdots + a_1 s + a_0 \end{eqnarray*}

(We have normalized $ B(s)$ so that $ A(s)$ is monic ($ a_N=1$) without loss of generality.) Equation (E.15) implies

$\displaystyle \left\vert A(j\omega)\right\vert = \left\vert B(j\omega)\right\vert, \quad \forall\, \omega\in(-\infty,\infty). \protect$

If $ M=N=0$, then the allpass condition reduces to $ \vert b_0\vert=\vert a_0\vert=1$, which implies

$\displaystyle b_0 = e^{j\phi} a_0 = e^{j\phi} $

where $ \phi\in[-\pi,\pi)$ is any real phase constant. In other words, $ b_0$ can be any unit-modulus complex number. If $ M = N = 1$, then the filter is allpass provided

$\displaystyle \left\vert b_1j\omega + b_0\right\vert = \left\vert j\omega + a_0\right\vert, \quad \forall\, \omega\in(-\infty,\infty). $

Since this must hold for all $ \omega$, there are only two solutions:
  1. $ b_0=a_0$ and $ b_1=1$, in which case $ H(s)=B(s)/A(s)=1$ for all $ s$.
  2. $ b_0=\overline{a_0}$ and $ b_1=1$, i.e.,

    $\displaystyle B(j\omega)=e^{j\phi}\overline{A(j\omega)}. $

Case (1) is trivially allpass, while case (2) is the one discussed above in the introduction to this section.

By analytic continuation, we have

$\displaystyle 1 = \left\vert H(j\omega)\right\vert = \left\vert H(j\omega)\right\vert^2 = \left. H(s)\overline{H(s)}\right\vert _{s=j\omega} $

If $ h(t)$ is real, then $ \overline{H(j\omega)} = H(-j\omega)$, and we can write

$\displaystyle 1 = \left. H(s)H(-s)\right\vert _{s=j\omega}. $

To have $ H(s)H(-s)=1$, every pole at $ s=p$ in $ H(s)$ must be canceled by a zero at $ s=p$ in $ H(-s)$, which is a zero at $ s=-p$ in $ H(s)$. Thus, we have derived the simplified ``allpass rule'' for real analog filters.

Previous: Analog Allpass Filters
Next: Matrix Filter Representations

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About the Author: 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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