Bertrand's ballot theorem

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-aliasingTemplate:Dn 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

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

Feedforward magnitude response for various positive values of $\alpha$

Feedforward magnitude response for various negative values of $\alpha$

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

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

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

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

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: $f=\frac{1}{2K},\frac{3}{2K},\frac{5}{2K}\mathrm{...}$.
• 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.

Pole-zero plot of feedfoward comb filter with $K=8$ and $\alpha =0.5$

Pole-zero plot of feedfoward comb filter with $K=8$ and $\alpha =-0.5$

Feedback form

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

Feedback magnitude response for various positive values of $\alpha$

Feedback magnitude response for various negative values of $\alpha$

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: $f=0,\frac{1}{K},\frac{2}{K}\mathrm{...}$.

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 $\frac{1}{1-\alpha }$.
• 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.

Pole-zero plot of feedback comb filter with $K=8$ and $\alpha =0.5$

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:

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

and the feedback form by:

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

where $\tau$ is the delay (measured in seconds).

They have the following frequency responses, respectively:

$H(\omega )=1+\alpha e^{-j\omega \tau }$

$H(\omega )=\frac{1}{1-\alpha e^{-j\omega \tau }}$

Continuous-time implementations share all the properties of the respective discrete-time implementations.

See also

• Filter (signal processing)
• Digital filter
• Fabry-Pérot interferometer

References

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