Handbook for Sound Engineers

786 Chapter 23

23.1.8 Order

The order of a filter is determined by the number of
reactive elements in the circuit. These can either be
inductive or capacitive and generally only include those
added for purposes of the frequency response within the
audio band and not for stability or RF suppression. If all
of the elements act as either low pass or high pass, the
roll off in the stop band will approach 6 dB per octave
per order. A fourth-order low pass will have a roll-off of
24 dB per octave above the cutoff frequency, but a
fourth-order band pass will have 12 dB per octave on
either side of the center frequency.

23.1.9 Phase Angle

The phase angle at a particular frequency is a measure
of the relative time for a particular frequency to pass
through a system from input to output. Phase angle is a
relative measure and is usually expressed in degrees
where 360q represents one wavelength. In most
formulas, phase is used in terms of radians where S
represents one wavelength. The instantaneous phase of
a sinusoidal signal is given by

(23-3)

23.1.10 Phase Delay

The phase delay of a system at a given frequency is the
equivalent time offset that would induce the same phase
offset as measured on a sinusoid of the same frequency.

(23-4)

23.1.11 Group Delay

A filter can exhibit a group delay over a group of
frequencies covering a section of the audio spectrum if
those frequencies are all subject to the same time delay.
The group delay is given as the first derivative of the
phase with respect to frequency

. (23-5)

The threshold of perceptibility for group delay has

been shown to be between 1 to 3 ms over the 500 Hz to

4 kHz range of the audio spectrum.^1

23.1.12 Transient Response

The transient response of a filter is the time response to

an input stimulus. Impulse and step inputs are common

stimuli for this measurement. Narrow bandwidth filters,

when subjected to rapidly changing input, ring because

it takes a certain amount of time for the energy in the

network to change upon application or removal of the

signal. Ringing can most clearly be seen as a damped

tail on a signal after it has been removed, Fig. 23-2.

23.1.13 Minimum Phase

A minimum phase system is one for which the phase

shift at each frequency can be uniquely determined from

the magnitude response using the Hilbert transform. A

filter with more than one path from input to output, in

which the different branches have a different group

delay, will be a linear time invariant (LTI) system but

may be nonminimum phase.

23.2 Passive Filters

Passive filters do not have any amplification compo-

nents in the circuit and as such cannot put out more

energy than is put in. A passive filter can never have a

boost in the energy response, although with some reso-

nant circuits, instantaneous voltages may be higher than

the input voltage. In this case the output impedance will

rise, preventing any significant current from being

driven. To build a passive filter with boost, we must

construct a filter that cuts all other frequencies and then

use a separate amplifier to increase the overall gain.

23.2.1 First-Order L and C Networks

Inductor- and capacitor-based filter networks may be

analyzed in terms of their impedances by reducing the

circuit to its resistance and reactance components.

DZ= t

= 2 Suft

Wp D

Z

----=

D

2 Sf

=--------

Wg d

dZ

=------ -IZ

Figure 23-2. Ringing of a filter after the removal of a signal.

Ringing

Input cut at this time