Need for all-pass filters; importance of normalization

Why an all-pass filter is needed

An all-pass filter is a filter whose magnitude is constant ($|T(j\omega)| = K$) at all frequencies, while its phase changes with frequency. Its zeros are mirror images of its poles about the $j\omega$ axis.

$$T_1(s) = \frac{s-a}{s+a}, \qquad T_2(s) = \frac{s^2 - \frac{\omega_0}{Q}s + \omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$$

For the first-order section, $|T_1(j\omega)| = 1$, the phase is $\theta = 180^\circ - 2\tan^{-1}(\omega/a)$ (falling from $180^\circ$ at DC to $0^\circ$ at high frequency), and the delay is $\tau = \dfrac{2a}{a^2+\omega^2}$. The sign-flipped form $\dfrac{a-s}{a+s}$ has $\theta = -2\tan^{-1}(\omega/a)$ (0° to −180°) and the same delay.

      jw
      |
  x   |   o      x = pole, o = zero
 -a   |   +a     (mirror images)
 -----+-------> sigma

It does not change the magnitude, but it changes the phase and delay, and that is exactly why it is used:

• Delay equalization: sharp filters (Chebyshev, elliptic) and channels (telephone lines, cables) have non-flat group delay. An all-pass section in cascade adds delay where it is lacking so the total delay becomes flat, removing pulse distortion and inter-symbol interference in data links, video and audio.
• Phase shifters / phase correction: e.g. 90° phase-splitting networks in single-sideband (SSB) modulators, phase compensation in control and measurement systems.
• Time delay circuits: a cascade of all-pass sections approximates a pure delay $e^{-s\tau}$ for analog delay lines.
• Building other filters: a notch or band-pass can be made by adding the input to the all-pass output; used in phasers and audio effects.
• Hilbert transformers and phase-locked systems in communication.

Practical example: a modem uses a 5th-order Chebyshev low-pass filter for its band limit. Its delay peaks near the cutoff, so pulses near the band edge arrive late and overlap the next symbol. A second-order all-pass section with $\omega_0$ near the cutoff is added; the magnitude is unchanged but the total delay becomes flat, and the eye diagram opens again.

Importance of normalization and denormalization

Normalization means designing the filter at a reference impedance of $1\ \Omega$ and a reference frequency of $1$ rad/s. Denormalization (scaling) converts this prototype into the real filter.

• Design tables (Butterworth, Chebyshev, Bessel poles and element values) are given only for normalized filters; one table serves every application.
• Element values near 1 are easy to compute and avoid numerical errors with very large or small numbers.
• Frequency transformations (LP to HP, BP, BS) are applied to the normalized low-pass prototype.
• Denormalization then gives practical values:

Example: normalized $C = 1$ F, scaled to $\omega_c = 1000$ rad/s and 10 kΩ gives $C' = 1/(1000\times10^4) = 0.1\ \mu$F.