In signal processing, a comb filter adds a delayed version of a signal to itself,
causing constructive and destructive interference. The frequency response of
a comb filter consists of a series of regularly spaced spikes, giving the
appearance of a comb. Applications
Comb filters are used in a variety of signal processing applications. These
include: Cascaded integrator–comb filters,
commonly used for anti-aliasing during interpolation and decimation operations
that change the sample rate of a discrete-time system.
2D and 3D comb filters implemented in hardware for PAL and NTSC television
decoders. The filters work to reduce artifacts such as dot crawl.
Audio effects, including echo, flanging, and digital waveguide synthesis. For
instance, if the delay is set to a few milliseconds, a comb filter can be used
to model the effect of acoustic standing waves in a cylindrical cavity or in a
vibrating string. In astronomy the astro-comb promises to
increase the precision of existing spectrographs by nearly a hundredfold.
In acoustics, comb filtering can arise in some unwanted ways. For instance,
when two loudspeakers are playing the same signal at different distances from
the listener, there is a comb filtering effect on the signal. In any enclosed
space, listeners hear a mixture of direct sound and reflected sound.
Because the reflected sound takes a longer path, it constitutes a delayed
version of the direct sound and a comb filter is created where the two combine
at the listener. Technical discussion
Comb filters exist in two different forms, feedforward and feedback; the
names refer to the direction in which signals are delayed before they are
added to the input. Comb filters may be implemented in
discrete time or continuous time; this article will focus on discrete-time
implementations; the properties of the continuous-time comb filter are very
similar. = Feedforward form=
The general structure of a feedforward comb filter is shown on the right. It
may be described by the following difference equation:
where is the delay length, and is a scaling factor applied to the delayed
signal. If we take the Z transform of both sides of the equation, we obtain:
We define the transfer function as: Frequency response
To obtain the frequency response of a discrete-time system expressed in the Z
domain, we make the substitution . Therefore, for our feedforward comb
filter, we get: Using Euler's formula, we find that the
frequency response is also given by Often of interest is the magnitude
response, which ignores phase. This is defined as:
In the case of the feedforward comb filter, this is:
Notice that the term is constant, whereas the term varies periodically.
Hence the magnitude response of the comb filter is periodic.
The graphs to the right show the magnitude response for various values of
, demonstrating this periodicity. Some important properties:
The response periodically drops to a local minimum, and periodically rises to
a local maximum. For positive values of , the first
minimum occurs at half the delay period and repeat at even multiples of the
delay frequency thereafter: . The levels of the maxima and minima are
always equidistant from 1. When , the minima have zero amplitude.
In this case, the minima are sometimes known as nulls.
The maxima for positive values of coincide with the minima for negative
values of , and vice versa. Impulse response
The feedforward comb filter is one of the simplest finite impulse response
filters. Its response is simply the initial impulse with a second impulse
after the delay. Pole–zero interpretation
Looking again at the Z-domain transfer function of the feedforward comb filter:
we see that the numerator is equal to zero whenever . This has solutions,
equally spaced around a circle in the complex plane; these are the zeros of
the transfer function. The denominator is zero at , giving poles at . This
leads to a pole–zero plot like the ones shown below.
= Feedback form= Similarly, the general structure of a
feedback comb filter is shown on the right. It may be described by the
following difference equation: If we rearrange this equation so that
all terms in are on the left-hand side, and then take the Z transform, we
obtain: The transfer function is therefore:
Frequency response If we make the substitution into the
Z-domain expression for the feedback comb filter, we get:
The magnitude response is as follows: Again, the response is periodic, as the
graphs to the right demonstrate. The feedback comb filter has some properties
in common with the feedforward form: The response periodically drops to a
local minimum and rises to a local maximum.
The maxima for positive values of coincide with the minima for negative
values of , and vice versa. For positive values of , the first
minimum occurs at 0 and repeats at even multiples of the delay frequency
thereafter: . However, there are also some important
differences because the magnitude response has a term in the denominator:
The levels of the maxima and minima are no longer equidistant from 1. The maxima
have an amplitude of . The filter is only stable if is
strictly less than 1. As can be seen from the graphs, as increases, the
amplitude of the maxima rises increasingly rapidly.
Impulse response The feedback comb filter is a simple
type of infinite impulse response filter. If stable, the response simply
consists of a repeating series of impulses decreasing in amplitude over
time. Pole–zero interpretation
Looking again at the Z-domain transfer function of the feedback comb filter:
This time, the numerator is zero at , giving zeros at . The denominator is
equal to zero whenever . This has solutions, equally spaced around a
circle in the complex plane; these are the poles of the transfer function. This
leads to a pole–zero plot like the ones shown below.
= Continuous-time comb filters= Comb filters may also be implemented in
continuous time. The feedforward form may be described by the following
equation: where is the delay. This has the
following transfer function: The feedforward form consists of an
infinite number of zeros spaced along the jω axis.
The feedback form has the equation: and the following transfer function:
The feedback form consists of an infinite number of poles spaced along
the jω axis. Continuous-time implementations share
all the properties of the respective discrete-time implementations.
See also Filter
Digital filter Fabry–Pérot interferometer
References