Comb filter

Comb filters are employed in a variety of signal processing applications, including:

• 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) in PAL and NTSC analog television decoders, reduce artifacts such as dot crawl.
• Audio signal processing, including delay, flanging, physical modelling synthesis and digital waveguide synthesis. If the delay is set to a few milliseconds, a comb filter can 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 as an unwanted artifact. For instance, two loudspeakers playing the same signal at different distances from the listener, create a comb filtering effect on the audio.^[1] In any enclosed space, listeners hear a mixture of direct sound and reflected sound. The reflected sound takes a longer, delayed path compared to the direct sound, and a comb filter is created where the two mix at the listener.^[2] Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1 rule of thumb that neighboring mics should be separated at least three times the distance from its source to the mic.^{[citation needed]}

The general structure of a feedforward comb filter is described by the difference equation:

${\displaystyle y[n]=x[n]+\alpha x[n-K]}$ 

where ${\displaystyle K}$  is the delay length (measured in samples), and α is a scaling factor applied to the delayed signal. The z transform of both sides of the equation yields:

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

The transfer function is defined as:

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

The frequency response of a discrete-time system expressed in the z-domain is obtained by substitution ${\displaystyle z=e^{j\omega },}$  where ${\displaystyle j}$  is the imaginary unit and ${\displaystyle \omega }$  is angular frequency. Therefore, for the feedforward comb filter:

${\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}}$ 

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

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

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

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

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

${\displaystyle {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega K)}}\end{aligned}}}$ 

The ${\displaystyle (1+\alpha ^{2})}$  term is constant, whereas the ${\displaystyle 2\alpha \cos(\omega K)}$  term varies periodically. Hence the magnitude response of the comb filter is periodic.

The graphs show the periodic magnitude response for various values of ${\displaystyle \alpha .}$  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 or a tooth).
• For positive values of ${\displaystyle \alpha ,}$  the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter:

$${\displaystyle {\begin{aligned}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\end{aligned}}}$$ 

• The levels of the maxima and minima are always equidistant from 1.
• When ${\displaystyle \alpha =\pm 1,}$  the minima have zero amplitude. In this case, the minima are sometimes known as nulls.
• The maxima for positive values of ${\displaystyle \alpha }$  coincide with the minima for negative values of ${\displaystyle \alpha }$ , and vice versa.

The feedforward comb filter is one of the simplest finite impulse response filters.^[3] Its response is simply the initial impulse with a second impulse after the delay.

Looking again at the z-domain transfer function of the feedforward comb filter:

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

the numerator is equal to zero whenever z^K = −α. 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.

Similarly, the general structure of a feedback comb filter is described by the difference equation:

${\displaystyle y[n]=x[n]+\alpha y[n-K]}$ 

This equation can be rearranged so that all terms in ${\displaystyle y}$  are on the left-hand side, and then taking the z transform:

${\displaystyle \left(1-\alpha z^{-K}\right)Y(z)=X(z)}$ 

The transfer function is therefore:

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

By substituting ${\displaystyle z=e^{j\omega }}$  into the feedback comb filter's z-domain expression:

${\displaystyle H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,}$ 

the magnitude response becomes:

${\displaystyle \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.}$ 

Again, the response is periodic, as the graphs 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 ${\displaystyle \alpha }$  coincide with the minima for negative values of ${\displaystyle \alpha ,}$  and vice versa.
• For positive values of ${\displaystyle \alpha ,}$  the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:

${\displaystyle {\begin{aligned}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \end{aligned}}}$ 

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 ⁠1/1 − α⁠.
• The filter is only stable if |α| is strictly less than 1. As can be seen from the graphs, as |α| increases, the amplitude of the maxima rises increasingly rapidly.

The feedback comb filter is a simple type of infinite impulse response filter.^[4] If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.

Looking again at the z-domain transfer function of the feedback comb filter:

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

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 = α. 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.

Comb filters may also be implemented in continuous time which can be expressed in the Laplace domain as a function of the complex frequency domain parameter ${\displaystyle s=\sigma +j\omega }$  analogous to the z domain. Analog circuits use some form of analog delay line for the delay element. Continuous-time implementations share all the properties of the respective discrete-time implementations.

The feedforward form may be described by the equation:

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

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

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

The feedforward form consists of an infinite number of zeros spaced along the jω axis (which corresponds to the Fourier domain).

The feedback form has the equation:

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

and the following transfer function:

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

The feedback form consists of an infinite number of poles spaced along the jω axis.

• Dirac comb
• Fabry–Pérot interferometer

•   Media related to Comb filters at Wikimedia Commons