Comb filter

Frequency response of the amplitudes (amplitude response) of the comb filter

A comb filter (English comb filter ) is a filter , the signals from certain groups of frequency filters. In contrast to low- pass and high-pass , it is characterized by several filter frequencies with the same frequency spacing. The amplitude response (level versus frequency) of the frequency response of the comb filter has a comb-like appearance, which is where the comb filter gets its name.

A comb filter frequency response can always be clearly recognized at its zeros on a signal (amplitude response) displayed in the frequency range, but this does not necessarily say anything about the audibility of the mostly undesired effect if it is in the audible range.

Comb filters in sound engineering

If a signal is additively superimposed with a time-delayed copy of the same ("mixed" in the sense of " mixer "), a comb-filtered signal is created. With frequencies and their multiples, the period duration of which corresponds to the delay time, double the signal amplitude is obtained (constructive interference ), while the frequencies exactly in between cancel each other out (destructive interference). With a lower strength of the delayed signal, the comb filter effect is correspondingly lower - the differences between maxima and minima in the amplitude response are less pronounced.

In sound engineering practice, for example, delayed signals are often unintentionally superimposed - and thus also comb filter effects: When recording a microphone, reflections in the room with a delay of between 2 and 15 ms can lead to conspicuous, unpleasantly disruptive sound colorations. One should speak of an "audible" comb filter effect here, because only this is disturbing. These are mostly floor reflections or wall reflections. After superimposition of direct sound and reflection, as indicated above, regular areas of extinction and amplification arise in the frequency response. (Incidentally, the principle is also a cause of fading in the case of multipath propagation of radio waves - whereby the observed frequency band is narrow there .) The evenly alternating minima and maxima with a difference of a few dB are similar to the structure of a harmonic sound. Our ears have got used to the comb filters during 'natural hearing' and the brain-hearing system largely suppresses this effect. Electrically generated comb filters are noticeably more disruptive.

With added strong reflections, musical instruments often sound "potty" (high frequency components missing, dull sound). The color of the sound becomes particularly annoying when the sound source is moving, with the pitch character changing, such as “u-ü-i” or “i-ü-u”. This effect is also generated electronically with phasing or flanging , which is often used in light music for special effects . The color of the sound is particularly annoying with speech, for example when reading on a reflecting table.

The use of support microphones can also lead to these comb filter discolorations when recording music if a sound source is recorded with two microphones and one is placed closer than the other. Even when mixing effect sounds with the "dry" original signal in the mixer, these comb filter effects can arise due to a signal delay in the effects device.

To reduce the audible comb filter effect when recording sound with several support microphones mixed together, see the three-to-one rule .

Comb filter in optics

Longitudinal laser modes with a Gaussian gain profile in a resonator. Representation: amplitude as a function of frequency

Depending on the design, certain wavelengths and their multiples are particularly amplified in an optical resonator , because a standing wave is only produced between the mirrors for certain wavelengths . For the possible light wavelengths λ in a laser resonator, the following applies:

${\displaystyle 2L=N\cdot \lambda }$

Here, N is a natural number and L the cavity length (distance between the resonator mirrors). The consequence of this is that a laser always generates several wavelengths if no countermeasures are taken. This effect is a prerequisite for the function of a frequency comb .

Comb filters in television technology

Advanced PAL Comb Filter-II (APCF-II, Motorola MC141627FT)

In television technology, the comb filter produces a softer and clearer picture. It is about the separation of the color signal and the black / white image in composite signals, in which these two parts are mixed together; A comb filter can take advantage of the fact that black and white information repeats itself with the line frequency and thus reduces the disturbing dot crawl effect. Comb filters are mainly used in high-quality receivers, while most inexpensive televisions have simple low-pass filters .

Comb filters in signal processing

With this simple FIR filter , the signal is delayed by K time steps, weakened by the factor α and added again

Amount of the frequency response for different values ​​of α

Pole-zero distribution

Comb filters can be implemented both as continuous-time and discrete-time filters. In analog technology, the time offset is implemented using a delay line. In the following, only time-discrete comb filters of variable order without feedback, such as those used in the field of digital signal processing, are considered.

A digital comb filter without feedback, as shown in the structure in the adjacent figure, has the following function as an impulse response :

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

Here y [n] represents the output sequence, x [ n ] the input sequence of the measured values, which are sampled and stored at regular intervals . The delay time is described by the filter order K , α represents a scaling factor with values ​​between −1 and +1, which in the simplest case is set equal to +1 and defines the "ripple" of the absolute frequency response.

With the Z-transformation , the discrete impulse response can be transformed into the spectral range to Y ( z )

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

to get the transfer function H ( z ):

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

Using z = e ^jΩ , where Ω represents the frequency related to the sampling frequency, the frequency response of this comb filter results as:

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

The resulting frequency response

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

is shown in the second figure with different values ​​for α. It can be seen that at α = 1, the waviness is maximum.

The pole-zero distribution in the complex plane shows in the figure on the left for an 8th order comb filter ( K = 8) and α = 0.5 an eight-fold pole at the origin and 8 zeros near the unit circle . If α = 1, the zeros are on the unit circle. The uniform position of the zeros on the unit circle at α = 1 results in a characteristic, comb-like transfer function that gives this filter its name.

literature

• Karl-Dirk Kammeyer , Kristian Kroschel: Digital signal processing. Filtering and spectral analysis with MATLAB exercises . 6th, corrected and supplemented edition. Teubner, Wiesbaden 2006, ISBN 3-8351-0072-6 , p. 162-163 .

See also

• Cascaded Integrator Comb Filter

Web links

• Amplitudes + phase response of a comb filter The Fourier transformation (PDF file; 946 kB)
• Comb filter when removing a sound source from a wall (PDF file; 300 kB)
• Comb filter effect when recording sound (PDF file; 123 kB)
• Comb filter effect during microphone mixing (PDF file; 195 kB)
• Analysis and evaluation of comb filter effects in runtime stereophony with more than two microphones (PDF file; 2.99 MB)