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Encyclopedia > Filter design

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. The problem to be solved is to find a realization of the filter which met each of the requirements to a sufficient degree to make it useful. Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... To meet Wikipedias quality standards, this article may require cleanup. ...

The filter design process can be described as an optimization problem where each requirement contributes with a term to an error function which should be minimized. Certain parts of the design process can be automated, but normally it is necessary to use an experienced electrical engineer to get a good result.

Typical requirements which are considered in the design process are

• The filter should have a specific frequency function
• The filter should have a specific impulse response
• The filter should be causal
• The filter should be stable
• The filter should be localized
• The computational complexity of the filter should be low
• The filter should be implemented in a particular hardware or software.

In signal processing, a causal filter is one whose output depends only on past and present inputs. ...

### The frequency function

Typical examples of frequency function are

• A low-pass filter is used to block unwanted high-frequency signals.
• A high-pass filter passes high frequencies fairly well; it is helpful as a filter to block any unwanted low frequency components.
• A band-pass filter passes a limited range of frequencies.
• A band-stop filter passes frequencies above or below a certain range. This is the least common filter.
• A low-shelf filter passes all frequencies, but boosts or cuts frequencies below the cutoff frequency by specified amount.
• A high-shelf filter passes all frequencies, but boosts or cuts frequencies above the cutoff frequency by specified amount.
• A peak EQ filter makes a peak or a dip in the frequency response, commonly used in graphic equalizers.
• An all-pass filter passes through all frequencies unchanged, but changes the phase of the signal. This is a filter commonly used in phaser effects.

An important parameter is the required frequency response. In particular, the steepness and complexity of the response curve is a deciding factor for the filter order and feasibility. For information about computer bandwidth management, see Equalization (computing). ... A phaser is an audio signal processor used to filter a signal by attenuating a series of notches in the frequency spectrum. ... The factual accuracy of this article is disputed. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...

A first order filter will only have a single frequency-dependent component. This means that the slope of the frequency response is limited to 6 dB per octave. For many purposes, this is not sufficient. To achieve steeper slopes, higher order filters are required. Look up Slope in Wiktionary, the free dictionary. ... The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ...

In relation to the desired frequency function, there may also be an accompanying weighting function which describes, for each frequency, how important it is that the resulting frequency function approximates the desired one. The larger weight, the more important is a close approximation.

### The impulse response

There is a direct correspondence between the filter's frequency function and its impulse response, the former is the Fourier transform of the latter. This means that any requirement on the frequency function is a requirement on the impulse response, and vice versa.

However, in certain applications it may be the filter's impulse response which is explicit and the design process then aims at producing an as close approximation as possible to the requested impulse response given all other requirements.

### Causality

In order to be implementable, any time-dependent filter must be causal: the filter response only depends on the current and past inputs. A standard approach is to leave this requirement until the final step. If the resulting filter is not causal, it can be made causal by introducing an appropriate time-shift (or delay). If the filter is a part of a larger system (which it normally is) these types of delays have to be introduced with care since they affect the operation of the entire system. In signal processing, a causal filter is one whose output depends only on past and present inputs. ...

### Stability

A stable filter assures that every limited input signal produces a limited filter response. A filter which does not meet this requirement may in some situations prove useless or even harmful. Certain design approaches can guarantee stability, for example by using only feed-forward circuits such as an FIR filter. On the other hand, filter based on feedback circuits have other advantages and may therefore be preferred, even if this class of filters include unstable filters. In this case, the filters must be carefully designed in order to avoid instability. In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ...

### Locality

In certain applications we have to deal with signals which contain components which can be described as local phenomena, for example pulses or steps, which have certain time duration. A consequence of applying a filter to a signal is, in intuitive terms, that the duration of the local phenomena is extended by the width of the filter. This implies that it is sometimes important to keep the width of the filter's impulse response function as short as possible.

According to the uncertainty relation of the Fourier transform, the product of the width of the filter's impulse response function and the width of its frequency function must exceed a certain constant. This means that any requirement on the filter's locality also implies a bound on its frequency function's width. Consequently, it may not be possible to simultaneously meet requirements on the locality of the filter's impulse response function as well as on its frequency function. This is a typical example of contradicting requirements.

### Computational complexity

A general desire in any design is that the number of operations (additions and multiplications) needed to compute the filter response is as low as possible. In certain applications, this desire is a strict requirement, for example due to limited computational resources, limited power resources, or limited time. The last limitation is typical in real-time applications.

There are several ways in which a filter can have different computational complexity. For example, the order of a filter is more or less proportional to the number of operations. This means that by choosing a low order filter, the computation time can be reduced.

For discrete filters the computational complexity is more or less proportional to the number of filter coefficients. If the filter has many coefficients, for example in the case of multidimensional signals such as tomography data, it may be relevant to reduce the number of coefficients by removing those which are sufficiently close to zero.

### Other considerations

It must also be decided how the filter is going to be implemented:

An analog filter handles analog stimuli (e. ... An analog sampled filter an electronic filter that is a hybrid between an analog and a digital filter. ... An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... A mechanical filter is an electrical filter based on a mechanical element such as a ceramic resonator or crystal, as opposed to a tuned circuit using capacitors and inductors. ...

#### Analog filters

The design of linear analog filters is for the most part covered in the linear filter section. A linear filter applies a linear operator to a time-varying input signal. ...

#### Digital filters

Digital filters are implemented according to one of two basic principles, according to how they respond to an impulse: An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ...

• Infinite impulse response (IIR)
• Finite impulse response (FIR)

IIR (infinite impulse response) is a property of signal processing systems. ... A finite impulse response (FIR) filter is a type of a digital filter. ...

#### Sample rate

Unless the sample rate is fixed by some outside constraint, selecting a suitable sample rate is an important design decision. A high rate will require more in terms of computational resources, but less in terms of anti-aliasing filters. Interference and beating with other signals in the system may also be an issue. The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies. ...

#### Anti-aliasing

For any digital filter design, it is crucial to analyze and avoid aliasing effects. Often, this is done by adding analog anti-aliasing filters at the input and output, thus avoiding any frequency component above the Nyquist frequency. The complexity (i.e., steepness) of such filters depends on the required signal to noise ratio and the ratio between the sampling rate and the highest frequency of the signal. On statistics, signal processing, and related disciplines, aliasing is an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled. ... In digital signal processing, anti-aliasing is the technique of minimizing aliasing (jagged or blocky patterns) when representing a high-resolution signal at a lower resolution. ... The Nyquist frequency, named after Harry Nyquist or the Nyquistâ€“Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. ... The phrase signal-to-noise ratio, often abbreviated SNR or S/N, is an engineering term for the ratio between the magnitude of a signal (meaningful information) and the magnitude of background noise. ... The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ...

#### IIR

IIR filters are the digital counterpart to analog filters. They use feedback, and will normally require less computing resources than an FIR filter of similar performance. Due to the feedback, high order IIR filters may have problems with instability and arithmetic overflow, and require careful design to avoid such pitfalls. Additionally, they have an inherent frequency-dependent phase shift, which can be a problem in many situations. 2nd order IIR filters are often called 'biquads' and a common implementation of higher order filters is to cascade biquads. A useful reference for computing biquad coefficients is the RBJ Audio EQ Cookbook. IIR (infinite impulse response) is a property of signal processing systems. ... It has been suggested that this article or section be merged with Feedback loop. ... Originally, the word computing was synonymous with counting and calculating, and a science and technology that deals with the original sense of computing mathematical calculations. ... Instability in systems is generally characterized by some of the outputs or internal states growing without bounds. ... The term arithmetic overflow or simply overflow has the following meanings. ... Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...

#### FIR

FIR filters do not use feedback, and are inherently stable. FIR filter coefficients are normally symmetrical, and that makes them phase neutral by nature. It is also easier to avoid overflow. The main disadvantage is that they may require significantly more processing and memory resources than cleverly designed IIR variants. FIR filters are generally easier to design: The Remez exchange algorithm is one suitable method for designing quite good filters semi-automatically. A finite impulse response (FIR) filter is a type of a digital filter. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Typically, processing describes the act of taking something through an established and usually routine set of procedures to convert it from one form to another, as a manufacturing procedure (processing milk into cheese) or administrative procedure (processing paperwork to grant a mortgage loan). ... This article or section does not cite its references or sources. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ...

## Theoretical basis

Parts of the design problem relate to the fact that certain requirements are described in the frequency domain while others are expressed in the signal domain and that these may contradict. For example, it is not possible to obtain a filter which has both an arbitrary impulse response and arbitrary frequency function. Other effects which refer to relations between the signal and frequency domain are

• The uncertainty principle between the signal and frequency domains
• The variance extension theorem
• The asymptotic behaviour of one domain versus discontinuities in the other

### The uncertainty principle

As stated in the uncertainty principle, the product of the width of the frequency function and the width impulse response cannot be smaller than a specific constant. This implies that if a specific frequency function is requested, corresponding to a specific frequency width, the minimum width of the filter in the signal domain is set. Vice versa, if the maximum width of the response is given, the smallest possible width in the frequency domain is also set. This is a typical example of contradicting requirements where a useful compromise has to be found. In quantum physics, the Heisenberg uncertainty principle or the Heisenberg indeterminacy principle â€” the latter name given to it by Niels Bohr â€” states that when measuring conjugate quantities, which are pairs of observables of a single elementary particle, increasing the accuracy of the measurement of one quantity increases the uncertainty of...

### The variance extension theorem

Let $sigma^{2}_{s}$ be the variance of the input signal and let $sigma^{2}_{f}$ be the variance of the filter. The variance of the filter response, $sigma^{2}_{r}$, is then given by

$sigma^{2}_{r}$ = $sigma^{2}_{s}$ + $sigma^{2}_{f}$

This means that the σr > σf and implies that the localization of various features such as pulses or steps in the response is limited by the filter width in the signal domain. If a precise localization is requested, we need a filter of small width in the signal domain and, via the uncertainty principle, its width in the frequency domain cannot be arbitrary small.

### Discontinuities versus asymptotic behaviour

Let f(t) be a function and let F(ω) be its Fourier transform. There is a theorem which states that if a the first derivative of F which is discontinuous has order $n geq 0$, then f has an asymptotic decay like t n − 1.

A consequence of this theorem is that the frequency function of a filter should be as smooth as possible to allow its impulse response to have a fast decay, and thereby a short width.

## Methodology

One method to finding a discrete FIR filter is filter optimization described in Knutsson et al. In its basic form this approach requires that an ideal frequency function of the filter FI(ω) is specified together with a frequency weighting function W(ω) and set of coordinates xk in the signal domain where the filter coefficients are located.

An error function $varepsilon$ is defined as

$varepsilon = | W cdot (F_{I} - mathcal{F} { f }) |^{2}$

where f(x) is the discrete filter and $mathcal{F}$ is the discrete-time Fourier transform defined on the specified set of coordinates. The norm used here is, formally, the usual norm on L2 spaces. This means that $varepsilon$ measures the deviation between the requested frequency function of the filter, FI, and the actual frequency function of the realized filter, $mathcal{F} { f }$. However, the deviation is also subject to the weighting function W before the error function is computed. A discrete-time Fourier transform (or DTFT) is a Fourier transform of a function of an integer (discrete) time variable n with an unbounded domain. ...

Once the error function is established, the optimal filter is given by the coefficients f(x) which minimize $varepsilon$. This can be done by solving the corresponding least squares problem. In practice, the L2 norm has to be approximated by means of a suitable sum over discrete points in the frequency domain. In general, however, these points should be significantly more than the number of coefficients in the signal domain to obtain a useful approximation.

## References

• A. Antoniou (1993). Digital Filters: Analysis, Design, and Applications. McGraw-Hill, New York, NY.
• S.W.A. Bergen and A. Antoniou (2005). "Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function". EURASIP Journal on Applied Signal Processing 2005 (12): pp 1910-1922.
• A.G. Deczky (October 1972). "Synthesis of Recursive Digital Filters Using the Minimum p-Error Criterion". IEEE Trans. Audio Electroacoust AU-20: pp. 257-263.
• J.K. Kaiser (1974). "Nonrecursive Digital Filter Design Using the Io-sinh Window Function". Proc. 1974 IEEE Int. Symp. Circuit Theory, pp 20-23.
• H. Knutsson, M. Andersson and J. Wiklund (June 1999). "Advanced Filter Design". Proc. Scandinavian Symposium on Image Analysis, Kangerlussuaq, Greenland.
• S.K. Mitra (1998). Digital Signal Processing: A Computer-Based Approach. McGraw-Hill, New York, NY.
• A.V. Oppenheim and R.W. Schafer (1999). Discrete-Time Signal Processing. Prentice-Hall, Upper Saddle River, NJ.
• T.W. Parks and J.H. McClellan (March 1972). "Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase". IEEE Trans. Circuit Theory CT-19: pp. 189-194.
• L.R. Rabiner, J.H. McClellan, and T.W. Parks (April 1975). "FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation". Proc. IEEE 63: pp. 595-610.

An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ...

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