Group delay of all-pass filters

We have already discussed (page ) all-pass filters, i.e., filters with constant unit spectra. They can be written as $P(Z) \overline{P}(1/Z) = 1$.In the frequency domain, P(Z) can be expressed as $e^{i \phi (\omega)}$, where $\phi$ is real and is called the ``phase shift." Clearly, $P \overline{P} = 1$ for all real $\phi$.It is an easy matter to make a filter with any desired phase shift--we merely Fourier transform $e^{i \phi (\omega)}$ into the time domain. If $\phi(\omega)$ is arbitrary, the resulting time function is likely to be two-sided. Since we are interested in physical processes that are causal, we may wonder what class of functions $\phi(\omega)$ corresponds to one-sided time functions. The answer is that the group delay $\tau_g = d\phi / d\omega$ of a causal all-pass filter must be positive.

Proof that $d\phi /d\omega \gt 0$ for a causal all-pass filter is found in FGDP; there is no need to reproduce the algebra here. The proof begins from equation () and uses the imaginary part of the logarithm to get phase. Differentiation with respect to $\omega$ yields a form that is recognizable as a spectrum and hence is always positive.

A single-pole, single-zero all-pass filter passes all frequency components with constant gain and a phase shift that can be adjusted by the placement of the pole. Taking Z₀ near the unit circle causes most of the phase shift to be concentrated near the frequency where the pole is located. Taking the pole farther away causes the delay to be spread over more frequencies. Complicated phase shifts or group delays can be built up by cascading single-pole filters.

The above reasoning for a single-pole, single-zero all-pass filter also applies to many roots, because the phase of each will add, and the sum of $\tau_g = d\phi / d\omega \gt 0$ will be greater than zero.

The Fourier dual to the positive group delay of a causal all-pass filter is that the instantaneous frequency of a certain class of analytic signals must be positive. This class of analytic signals is made up of all those with a constant envelope function, as might be approximated by field data after the process of automatic gain control.

EXERCISES:

1. Let x_t be some real signal. Let y_t =x_t+3 be another real signal. Sketch the phase as a function of frequency of the cross-spectrum X(1/Z)Y(Z) as would a computer that put all arctangents in the principal quadrants $-\pi /2 < \arctan < \pi /2$.Label the axis scales.
2. Sketch the amplitude, phase, and group delay of the all-pass filter $(1 - \overline{Z_0} Z)/(Z_0 - Z)$,where $Z_0 = (1 + \epsilon)e^{i\omega_0}$ and $\epsilon$ is small. Label important parameters on the curve.
3. Show that the coefficients of an all-pass, phase-shifting filter made by cascading $(1 - \overline{Z_0} Z)/$ (Z₀ - Z) with $(1 - Z_0 Z)/(\overline{Z_0} - Z)$ are real.
4. A continuous signal is the impulse response of a continuous-time, all-pass filter. Describe the function in both time and frequency domains. Interchange the words ``time" and ``frequency" in your description of the function. What is a physical example of such a function? What happens to the statement, the group delay of an all-pass filter is positive?
5. A graph of the group delay $\tau_g (\omega)$shows $\tau_g$ to be positive for all $\omega$.What is the area under $\tau_g$ in the range $0 < \omega < 2\pi$? (HINT: This is a trick question you can solve in your head.)