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In signal processing a comb filter adds a delayed version of a signal processing to itself, causing Destructive interference#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 Filter (CIC) filters, commonly used for anti-aliasing during interpolation and decimation (signal processing) 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 effect , including Echo (phenomenon) 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 Acoustics standing waves in a cylindrical cavity or Karplus-Strong string synthesis However, comb filtering can sometimes arise in unwanted ways. For instance, comb filtering will occur where two loudspeakers are playing the same signal at different distances from the listener.lt;/ref> Comb filtering occurs in acoustics In any enclosed space, listeners hear a mixture of direct sound and reflected sound. Reflected sound, taking a longer path, constitutes a delayed version of the direct sound and a comb filter is created where the two combine at the listener.lt;/ref>

Technical discussion

Comb filters exist in two different forms, [[feed-forward]]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

Image:Comb filter feedforward.png The general structure of a feedforward comb filter is shown on the right. It may be described by the following difference equation :$\backslash \; yn]\; xn]\; +\; \backslash alpha\; xn-K]\; \backslash ,$ where $K$ is the delay length (measured in samples), and $\backslash alpha$ is a scaling factor applied to the delayed signal. If we take the Z transform of both sides of the equation, we obtain: :$\backslash \; Y(z)\; (1\; +\; \backslash alpha\; z^-K\})\; X(z)\; \backslash ,$ We define the Z transform#Transfer function as: :$\backslash \; H(z)\; \backslash fracY(z)\}X(z)\}\; 1\; +\; \backslash alpha\; z^-K\}\; \backslash fracz^K\; +\; \backslash alpha\}z^K\}\; \backslash ,$

Frequency response

Image:Comb filter response ff pos.png Image:Comb filter response ff neg.png To obtain the frequency response of a discrete-time system expressed in the Z domain, we make the substitution $z\; e^j\; \backslash omega\}$. Therefore, for our feedforward comb filter, we get: :$\backslash \; H(e^j\; \backslash omega\})\; 1\; +\; \backslash alpha\; e^-j\; \backslash omega\; K\}\; \backslash ,$ Using Euler's formula we find that the frequency response is also given by :$\backslash \; H(e^j\; \backslash omega\})\; \backslash left1\; +\; \backslash alpha\; \backslash cos(\backslash omega\; K)\backslash right]\; -\; j\; \backslash alpha\; \backslash sin(\backslash omega\; K)\; \backslash ,$ Often of interest is the magnituderesponse, which ignores phase. This is defined as: :$\backslash \; |\; H(e^j\; \backslash omega\})\; |\; \backslash sqrt\backslash Re\backslash H(e^j\; \backslash omega\})\backslash \}^2\; +\; \backslash Im\backslash H(e^j\; \backslash omega\})\backslash \}^2\}\; \backslash ,$ In the case of the feedforward comb filter, this is: :$\backslash \; |\; H(e^j\; \backslash omega\})\; |\; \backslash sqrt(1\; +\; \backslash alpha^2)\; +\; 2\; \backslash alpha\; \backslash cos(\backslash omega\; K)\}\; \backslash ,$ Notice that the $(1\; +\; \backslash alpha^2)$ term is constant, whereas the $2\; \backslash alpha\; \backslash cos(\backslash omega\; K)$ term varies periodic function Hence the magnitude response of the comb filter is periodic. The graphs to the right show the magnitude response for various values of $\backslash alpha$, 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. *The levels of the maxima and minima are always equidistant from 1. *When $\backslash alpha\; \backslash pm\; 1$, the minima have zero amplitude. In this case, the minima are sometimes known as nulls *The maxima for positive values of $\backslash alpha$ coincide with the minima for negative values of $\backslash alpha$, and vice versa.

Impulse response

The feedforward comb filter is one of the simplest finite impulse response filters.lt;/ref> 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: :$\backslash \; H(z)\; \backslash fracz^K\; +\; \backslash alpha\}z^K\}\; \backslash ,$ we see that the numerator is equal to zero whenever $z^K\; -\backslash alpha$. This has $K$ solutions, equally spaced around a circle in the complex plane these are the zero (complex analysis) of the transfer function. The denominator is zero at $z^K\; 0$, giving $K$ pole (complex analysis) at $z\; 0$. This leads to a pole-zero plot like the ones shown below. | | Image:Comb filter pz ff pos.svg | Image:Comb filter pz ff neg.svg |}

Feedback form

Image:Comb filter feedback.png Similarly, the general structure of a feedback comb filter is shown on the right. It may be described by the following difference equation :$\backslash \; yn]\; xn]\; +\; \backslash alpha\; yn-K]\; \backslash ,$ 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: :$\backslash \; (1\; -\; \backslash alpha\; z^-K\})\; Y(z)\; X(z)\; \backslash ,$ The transfer function is therefore: :$\backslash \; H(z)\; \backslash fracY(z)\}X(z)\}\; \backslash frac1\}1\; -\; \backslash alpha\; z^-K\}\}\; \backslash fracz^K\}z^K\; -\; \backslash alpha\}\; \backslash ,$

Frequency response

Image:Comb filter response fb pos.png Image:Comb filter response fb neg.png If we make the substitution $z\; e^j\; \backslash omega\}$ into the Z-domain expression for the feedback comb filter, we get: :$\backslash \; H(e^j\; \backslash omega\})\; \backslash frac1\}1\; -\; \backslash alpha\; e^-j\; \backslash omega\; K\}\}\; \backslash ,$ The magnitude response is as follows: :$\backslash \; |\; H(e^j\; \backslash omega\})\; |\; \backslash frac1\}\backslash sqrt(1\; +\; \backslash alpha^2)\; -\; 2\; \backslash alpha\; \backslash cos(\backslash omega\; K)\}\}\; \backslash ,$ 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 $\backslash alpha$ coincide with the minima for negative values of $\backslash alpha$, and vice versa. 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 filter is only BIBO stability if $|\backslash alpha|$ is strictly less than 1. As can be seen from the graphs, as $|\backslash alpha|$ increases, the amplitude of the maxima rises increasingly rapidly.

Impulse response

The feedback comb filter is a simple type of infinite impulse response filter.lt;/ref> 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: :$\backslash \; H(z)\; \backslash fracz^K\}z^K\; -\; \backslash alpha\}\; \backslash ,$ This time, the numerator is zero at $z^K\; 0$, giving $K$ zeros at $z\; 0$. The denominator is equal to zero whenever $z^K\; \backslash alpha$. 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. | | Image:Comb filter pz fb pos.svg | Image:Comb filter pz fb neg.svg |}

Continuous-time comb filters

Comb filters may also be implemented in continuous time The feedforward form may be described by the following equation: :$\backslash \; y(t)\; x(t)\; +\; \backslash alpha\; x(t\; -\; \backslash tau)\; \backslash ,$ and the feedback form by: :$\backslash \; y(t)\; x(t)\; +\; \backslash alpha\; y(t\; -\; \backslash tau)\; \backslash ,$ where $\backslash tau$ is the delay (measured in seconds). They have the following frequency responses, respectively: :$\backslash \; H(\backslash omega)\; 1\; +\; \backslash alpha\; e^-j\; \backslash omega\; \backslash tau\}\; \backslash ,$ :$\backslash \; H(\backslash omega)\; \backslash frac1\}1\; -\; \backslash alpha\; e^-j\; \backslash omega\; \backslash tau\}\}\; \backslash ,$ Continuous-time implementations share all the properties of the respective discrete-time implementations.

See also

*Filter (signal processing) *Digital filter

References

Category:Signal processing Category:Filter theory ca:Filtre pinta de:Kammfilter es:Filtro comb fr:Filtre en peigne it:Filtro comb nl:Kamfilter ja:コムフィルタ pt:Filtro Comb ru:Гребенчатый фильтр sv:Kamfilter zh:梳状滤波器