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Encyclopedia > Finite impulse response

A finite impulse response (FIR) filter is a type of a digital filter. It is 'finite' because its response to an impulse ultimately settles to zero. This is in contrast to infinite impulse response filters which have internal feedback and may continue to respond indefinitely. An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... IIR (infinite impulse response) is a property of signal processing systems. ...

Contents

Discussion

We start the discussion by stating the difference equation which defines how the input signal is related to the output signal In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

 y[n]=b_0 x[n] + b_1 x[n-1] + cdots + b_P x[n-P]

where P is the filter order, x[n] is the input signal, y[n] is the output signal and bi are the filter coefficients. The previous equation can also be expressed as

 y[n] = sum_{i=0}^{P} b_i x[n-i]

To find the impulse response we set The Impulse response from a simple audio system. ...

 x[n] = delta[n]

where δ[n] is the Kronecker delta impulse. The impulse response for an FIR filter follows as In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

begin{align} h[n] &= sum_{i=0}^{P}b_i delta[n-i]  &= b_n end{align}

The Z-transform of the impulse response yields the transfer function of the FIR filter In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

begin{align} H(z) &= Z{h[n]}  &= sum_{n=-infty}^{infty} h[n] z^{-n}  &= sum_{i=0}^{P}b_i,z^{-i} end{align}

The transfer function allows us to judge whether or not a system is BIBO stable. To be specific the BIBO stability criterion requires all poles of the transfer function to have an absolute value smaller than one. In other words all poles must be located within a unit circle in the z-plane. To find the poles of the transfer function we have to extend it with frac{z^{P}}{z^{P}} and arrive at In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ...

Hleft(zright)=frac{sum_{i=0}^{P}b_i z^{P-i}}{z^{P}}

The FIR transfer function contains P poles at z = 0. Since all poles are at the origin, all poles are located within the unit circle of the z-plane; therefore all FIR filters are stable.


Properties

A FIR filter has a number of useful properties which sometimes make it preferable to an infinite impulse response filter. FIR filters: IIR (infinite impulse response) is a property of signal processing systems. ...

  • Are inherently stable. This is due to the fact that all the poles are located at the origin and thus are located within the unit circle.
  • Require no feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation.
  • Can have linear phase

Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at . ...

Filter design

To design a filter means to select the coefficients such that the system has specific characteristics. The required characteristics are stated in filter specifications. Most of the time filter specifications refer to the frequency response of the filter. There are different methods to find the coefficients from the specifications:

  1. Window design method
  2. Frequency Sampling method
  3. Weighted least squares design
  4. Minimax design
  5. In practice the equiripple design is often used.

Software packages like MATLAB and GNU Octave provide convenient ways to apply these different methods. MATLAB is a numerical computing environment and programming language. ... For other uses of the word octave see Octave (disambiguation) Octave is a free computer program for performing numerical computations, which is mostly compatible with MATLAB. It is part of the GNU project. ...


Moving-average example

A moving-average filter is a very simple FIR filter. The filter coefficients are found via the following equation:

b_{i}=frac{1}{P+1} for i=0,1,dots,P

To provide a more specific example, we select the filter order:

 P=2

The impulse response of the resulting filter is:

h[n]=frac{1}{3}delta[n]+frac{1}{3}delta[n-1]+frac{1}{3}delta[n-2]

The following figure shows the block diagram of such a second-order moving-average filter.

Block Diagram of Simple FIR filter
Block Diagram of Simple FIR filter

To discuss stability and spectral topics we take the z-transform of the impulse response: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

H(z)=frac{1}{3}+frac{1}{3}z^{-1}+frac{1}{3}z^{-2}=frac{1}{3}frac{z^{2}+z+1}{z^{2}}

The following figure shows the pole-zero diagram of the filter. Two poles are located at the origin, and two zeros are located at z_{1}=-frac{1}{2}+jfrac{sqrt{3}}{2}, z_{2}=-frac{1}{2}-jfrac{sqrt{3}}{2} Pole-Zero Diagram Image File history File links Download high resolution version (1200x900, 18 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ...


The frequency response is:

H(e^{jomega})=frac{1}{3}+frac{1}{3}e^{-jomega}+frac{1}{3}e^{-j2omega}

The following figure shows the absolute value of the frequency response. Clearly, the moving-average filter leaves low frequencies unaffected and blocks high frequencies. This is a typical low-pass filter characteristic.

Amplitude response

The following figure shows the phase response. Phase response Image File history File links Download high resolution version (1200x900, 12 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (1200x900, 11 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ...


See also

An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... In electronics and signal processing, a filter is a device or process that modifies a signal. ... IIR (infinite impulse response) is a property of signal processing systems. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... Filter design is the process of working out a filter (in the sense in which the term is used in signal processing, statistics, and applied mathematics), often a linear shift-invariant filter, which satisfies a set of requirements, some of which are contradicting. ... Please wikify (format) this article or section as suggested in the Guide to layout and the Manual of Style. ...

External links


  Results from FactBites:
 
Finite Impulse Response filters (291 words)
A Finite Impulse Response (FIR) filter produces an output, y(n), that is the weighted sum of the current and past inputs, x(n).
The filter coefficents are a windowed sinc funtion, plotted in figure 5 and the amplitude response is plotted in figure 6.
FIR filters are computationally expensive to implement but need not introduce phase distortions - useful in processing high quality speech.
Finite Impulse Response Filter - Hydrogenaudio Knowledgebase (233 words)
Figure: Impulse response h(n) of a digital filter and frequency response of the digital filter (lowpass filter).
To implement the fir filter, an input signal x(n) is convolved with fir filter's impulse response h(n), resulting in a filtered output signal y(n).
Recursive filters (iir; infinite impulse response) are an extension of this, using previously calculated values from the output, besides points from the input.
  More results at FactBites »

 
 

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