Comb filter

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 (CIC) 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 (and occasionally software) 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

File:Comb filter feedforward.svg

Feedforward comb filter structure

The general structure of a feedforward comb filter is shown on the right. It may be described by the following difference equation:

$\ y[n] = x[n] + \alpha x[n-K] \,$

where $K$ is the delay length (measured in samples), and $\alpha$ is a scaling factor applied to the delayed signal. If we take the Z transform of both sides of the equation, we obtain:

$\ Y(z) = (1 + \alpha z^{-K}) X(z) \,$

We define the transfer function as:

$\ H(z) = \frac{Y(z)}{X(z)} = 1 + \alpha z^{-K} = \frac{z^K + \alpha}{z^K} \,$

Frequency response

File:Comb filter response ff pos.svg

Feedforward magnitude response for various positive values of $\alpha$ and K=1

File:Comb filter response ff neg.svg

Feedforward magnitude response for various negative values of $\alpha$ and K=1

To obtain the frequency response of a discrete-time system expressed in the Z domain, we make the substitution $z = e^{j \Omega}$. Therefore, for our feedforward comb filter, we get:

$\ H(e^{j \Omega}) = 1 + \alpha e^{-j \Omega K} \,$

Using Euler's formula, we find that the frequency response is also given by

$\ H(e^{j \Omega}) = \left[1 + \alpha \cos(\Omega K)\right] - j \alpha \sin(\Omega K) \,$

Often of interest is the magnitude response, which ignores phase. This is defined as:

$\ | H(e^{j \Omega}) | = \sqrt{\Re\{H(e^{j \Omega})\}^2 + \Im\{H(e^{j \Omega})\}^2} \,$

In the case of the feedforward comb filter, this is:

$\ | H(e^{j \Omega}) | = \sqrt{(1 + \alpha^2) + 2 \alpha \cos(\Omega K)} \,$

Notice that the $(1 + \alpha^2)$ term is constant, whereas the $2 \alpha \cos(\Omega K)$ 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 $\alpha$, demonstrating this periodicity. Some important properties:

• The response periodically drops to a local minimum (sometimes known as a notch), and periodically rises to a local maximum (sometimes known as a peak).
• For positive values of $\alpha$, the first minimum occurs at half the delay period and repeat at even multiples of the delay frequency thereafter: $f = \frac{1}{2 K}, \frac{3}{2 K}, \frac{5}{2 K} ...$.
• The levels of the maxima and minima are always equidistant from 1.
• When $\alpha = \pm 1$, the minima have zero amplitude. In this case, the minima are sometimes known as nulls.
• The maxima for positive values of $\alpha$ coincide with the minima for negative values of $\alpha$, 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:

$\ H(z) = \frac{z^K + \alpha}{z^K} \,$

we see that the numerator is equal to zero whenever $z^K = -\alpha$. This has $K$ solutions, equally spaced around a circle in the complex plane; these are the zeros of the transfer function. The denominator is zero at $z^K = 0$, giving $K$ poles at $z = 0$. This leads to a pole–zero plot like the ones shown below.

File:Comb filter pz ff pos.svg Pole–zero plot of feedfoward comb filter with $K = 8$ and $\alpha = 0.5$

File:Comb filter pz ff neg.svg Pole–zero plot of feedfoward comb filter with $K = 8$ and $\alpha = -0.5$

Feedback form

File:Comb filter feedback.svg

Feedback comb filter structure

Similarly, the general structure of a feedback comb filter is shown on the right. It may be described by the following difference equation:

$\ y[n] = x[n] + \alpha y[n-K] \,$

If we rearrange this equation so that all terms in $y$ are on the left-hand side, and then take the Z transform, we obtain:

$\ (1 - \alpha z^{-K}) Y(z) = X(z) \,$

The transfer function is therefore:

$\ H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 - \alpha z^{-K}} = \frac{z^K}{z^K - \alpha} \,$

Frequency response

File:Comb filter response fb pos.svg

Feedback magnitude response for various positive values of $\alpha$ and K=2

File:Comb filter response fb neg.svg

Feedback magnitude response for various negative values of $\alpha$ and K=2

If we make the substitution $z = e^{j \Omega}$ into the Z-domain expression for the feedback comb filter, we get:

$\ H(e^{j \Omega}) = \frac{1}{1 - \alpha e^{-j \Omega K}} \,$

The magnitude response is as follows:

$\ | H(e^{j \Omega}) | = \frac{1}{\sqrt{(1 + \alpha^2) - 2 \alpha \cos(\Omega K)}} \,$

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 $\alpha$ coincide with the minima for negative values of $\alpha$, and vice versa.
• For positive values of $\alpha$, the first minimum occurs at 0 and repeats at even multiples of the delay frequency thereafter: $f = 0, \frac{1}{K}, \frac{2}{K} ...$.

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 $1 \over 1 - \alpha$.
• The filter is only stable if $|\alpha|$ is strictly less than 1. As can be seen from the graphs, as $|\alpha|$ 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:

$\ H(z) = \frac{z^K}{z^K - \alpha} \,$

This time, the numerator is zero at $z^K = 0$, giving $K$ zeros at $z = 0$. The denominator is equal to zero whenever $z^K = \alpha$. This has $K$ 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.

File:Comb filter pz fb pos.svg Pole–zero plot of feedback comb filter with $K = 8$ and $\alpha = 0.5$

File:Comb filter pz fb neg.svg Pole–zero plot of feedback comb filter with $K = 8$ and $\alpha = -0.5$

Continuous-time comb filters

Comb filters may also be implemented in continuous time. The feedforward form may be described by the following equation:

$\ y(t) = x(t) + \alpha x(t - \tau) \,$

where $\tau$ is the delay (measured in seconds). This has the following transfer function:

$\ H(s) = 1 + \alpha e^{-s \tau} \,$

The feedforward form consists of an infinite number of zeros spaced along the jω axis.

The feedback form has the equation:

$\ y(t) = x(t) + \alpha y(t - \tau) \,$

and the following transfer function:

$\ H(s) = \frac{1}{1 - \alpha e^{-s \tau}} \,$

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 (signal processing)
• Digital filter
• Fabry–Pérot interferometer