What Does Filter Order Mean

Signal processing blueprint process

Filter blueprint is the procedure of designing a bespeak processing filter that satisfies a set of requirements, some of which may exist conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient degree to make it useful.

The filter design process tin can be described every bit an optimization trouble where each requirement contributes to an mistake function that should be minimized. Sure parts of the design process tin be automated, but normally an experienced electrical engineer is needed to go a good upshot.

The design of digital filters is a deceptively circuitous topic.^[ane] Although filters are hands understood and calculated, the practical challenges of their design and implementation are significant and are the subject of avant-garde enquiry.

Typical design requirements [edit]

Typical requirements which are considered in the design process are:

• The filter should have a specific frequency response
• The filter should take a specific phase shift or group delay
• The filter should accept a specific impulse response
• The filter should be causal
• The filter should be stable
• The filter should be localized (pulse or step inputs should result in finite time outputs)
• The computational complexity of the filter should be low
• The filter should be implemented in particular hardware or software

The frequency function [edit]

An of import parameter is the required frequency response. In particular, the steepness and complexity of the response bend is a deciding factor for the filter order and feasibility.

A start-order recursive filter volition only take a single frequency-dependent component. This ways that the gradient of the frequency response is limited to half dozen dB per octave. For many purposes, this is non sufficient. To reach steeper slopes, higher-lodge filters are required.

In relation to the desired frequency part, at that place may besides 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.

Typical examples of frequency function are:

• A depression-pass filter is used to cut unwanted loftier-frequency signals.
• A high-pass filter passes high frequencies fairly well; it is helpful as a filter to cut any unwanted low-frequency components.
• A ring-pass filter passes a limited range of frequencies.
• A band-stop filter passes frequencies above and below a certain range. A very narrow band-cease filter is known equally a notch filter.
• A differentiator has an aamplitude response proportional to the frequency.
• A low-shelf filter passes all frequencies, but increases or reduces frequencies below the shelf frequency by specified amount.
• A loftier-shelf filter passes all frequencies, but increases or reduces frequencies higher up the shelf frequency by specified amount.
• A peak EQ filter makes a peak or a dip in the frequency response, unremarkably used in parametric equalizers.

Phase and group delay [edit]

• An all-pass filter passes through all frequencies unchanged, but changes the phase of the bespeak. Filters of this blazon can be used to equalize the grouping filibuster of recursive filters. This filter is also used in phaser effects.
• A Hilbert transformer is a specific all-pass filter that passes sinusoids with unchanged aamplitude merely shifts each sinusoid phase past ±90°.
• A fractional filibuster filter is an all-pass that has a specified and constant grouping or stage delay for all frequencies.

The impulse response [edit]

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

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

In some cases information technology may even be relevant to consider a frequency function and impulse response of the filter which are chosen independently from each other. For case, nosotros may desire both a specific frequency office of the filter and that the resulting filter have a small constructive width in the signal domain as possible. The latter status can exist realized by considering a very narrow function as the wanted impulse response of the filter even though this function has no relation to the desired frequency function. The goal of the design process is then to realize a filter which tries to run into both these contradicting design goals as much as possible.

Causality [edit]

In order to exist implementable, any time-dependent filter (operating in real time) must exist causal: the filter response only depends on the current and past inputs. A standard approach is to leave this requirement until the last step. If the resulting filter is not causal, it can exist fabricated causal past introducing an appropriate fourth dimension-shift (or delay). If the filter is a part of a larger system (which it commonly is) these types of delays have to exist introduced with intendance since they affect the operation of the entire system.

Filters that do not operate in real fourth dimension (e.one thousand. for epitome processing) can exist non-causal. This e.g. allows the blueprint of zero delay recursive filters, where the group delay of a causal filter is canceled by its Hermitian non-causal filter.

Stability [edit]

A stable filter assures that every limited input signal produces a limited filter response. A filter which does not come across this requirement may in some situations prove useless or even harmful. Certain design approaches can guarantee stability, for example by using just feed-frontwards circuits such as an FIR filter. On the other mitt, filters based on feedback circuits have other advantages and may therefore be preferred, even if this class of filters includes unstable filters. In this example, the filters must be advisedly designed in society to avert instability.

Locality [edit]

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

According to the uncertainty relation of the Fourier transform, the production of the width of the filter'south impulse response function and the width of its frequency office must exceed a certain abiding. This means that whatsoever requirement on the filter's locality likewise implies a jump on its frequency role's width. Consequently, it may non be possible to simultaneously see requirements on the locality of the filter'south impulse response function as well equally on its frequency function. This is a typical example of contradicting requirements.

Computational complexity [edit]

A general want in whatsoever 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 express computational resources, limited ability 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 case, the gild of a filter is more than or less proportional to the number of operations. This means that by choosing a low society filter, the computation time can be reduced.

For detached filters the computational complexity is more than or less proportional to the number of filter coefficients. If the filter has many coefficients, for example in the example 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 cypher. In multirate filters, the number of coefficients by taking advantage of its bandwidth limits, where the input signal is downsampled (e.g. to its critical frequency), and upsampled after filtering.

Another result related to computational complication is separability, that is, if and how a filter can be written as a convolution of two or more simpler filters. In detail, this consequence is of importance for multidimensional filters, eastward.g., second filter which are used in image processing. In this example, a meaning reduction in computational complexity tin can be obtained if the filter can be separated equally the convolution of one 1D filter in the horizontal management and one 1D filter in the vertical direction. A event of the filter blueprint procedure may, e.g., be to approximate some desired filter equally a separable filter or every bit a sum of separable filters.

Other considerations [edit]

It must besides exist decided how the filter is going to exist implemented:

• Analog filter
• Analog sampled filter
• Digital filter
• Mechanical filter

Analog filters [edit]

The design of linear analog filters is for the most office covered in the linear filter section.

Digital filters [edit]

Digital filters are classified into one of two basic forms, according to how they answer to a unit of measurement impulse:

• Finite impulse response, or FIR, filters express each output sample as a weighted sum of the last N input samples, where N is the social club of the filter. FIR filters are normally non-recursive, pregnant they practice not use feedback and equally such are inherently stable. A moving average filter or CIC filter are examples of FIR filters that are normally recursive (that use feedback). If the FIR coefficients are symmetrical (oftentimes the case), then such a filter is linear stage, so information technology delays signals of all frequencies as which is important in many applications. Information technology is as well straightforward to avoid overflow in an FIR filter. The main disadvantage is that they may crave significantly more processing and memory resources than cleverly designed IIR variants. FIR filters are mostly easier to pattern than IIR filters - the Parks-McClellan filter design algorithm (based on the Remez algorithm) is one suitable method for designing quite good filters semi-automatically. (Meet Methodology.)
• Infinite impulse response, or IIR, filters are the digital counterpart to analog filters. Such a filter contains internal state, and the output and the adjacent internal state are determined by a linear combination of the previous inputs and outputs (in other words, they apply feedback, which FIR filters normally practise not). In theory, the impulse response of such a filter never dies out completely, hence the proper name IIR, though in exercise, this is not true given the finite resolution of computer arithmetic. IIR filters usually require less calculating resources than an FIR filter of similar functioning. All the same, due to the feedback, high social club IIR filters may have problems with instability, arithmetic overflow, and limit cycles, and require conscientious design to avoid such pitfalls. Additionally, since the stage shift is inherently a non-linear office of frequency, the time delay through such a filter is frequency-dependent, which tin be a trouble in many situations. 2nd order IIR filters are oftentimes 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.

Sample rate [edit]

Unless the sample rate is fixed past some outside constraint, selecting a suitable sample rate is an important pattern decision. A high rate will require more in terms of computational resources, but less in terms of anti-aliasing filters. Interference and chirapsia with other signals in the arrangement may also be an issue.

Anti-aliasing [edit]

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 in a higher place 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 charge per unit and the highest frequency of the betoken.

Theoretical basis [edit]

Parts of the design problem chronicle to the fact that sure requirements are described in the frequency domain while others are expressed in the time domain and that these may disharmonize. For example, it is not possible to obtain a filter which has both an arbitrary impulse response and capricious frequency role. Other effects which refer to relations between the time and frequency domain are

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

The incertitude principle [edit]

As stated by the Gabor limit, an uncertainty principle, the product of the width of the frequency function and the width of the impulse response cannot be smaller than a specific abiding. This implies that if a specific frequency function is requested, respective 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, this determines the smallest possible width in the frequency. This is a typical case of contradictory requirements where the filter design process may try to find a useful compromise.

The variance extension theorem [edit]

Let ${\displaystyle \sigma _{s}^{two}}$ be the variance of the input point and allow ${\displaystyle \sigma _{f}^{ii}}$ exist the variance of the filter. The variance of the filter response, ${\displaystyle \sigma _{r}^{2}}$ , is so given by

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

This ways that ${\displaystyle \sigma _{r}>\sigma _{f}}$ and implies that the localization of diverse features such as pulses or steps in the filter response is express by the filter width in the point domain. If a precise localization is requested, we need a filter of pocket-size width in the signal domain and, via the doubt principle, its width in the frequency domain cannot be arbitrary small.

Discontinuities versus asymptotic behaviour [edit]

Let f(t) be a function and let ${\displaystyle F(\omega )}$ be its Fourier transform. There is a theorem which states that if the outset derivative of F which is discontinuous has order ${\displaystyle north\geq 0}$ , then f has an asymptotic disuse similar ${\displaystyle t^{-n-1}}$ .

A upshot of this theorem is that the frequency function of a filter should be as smooth as possible to allow its impulse response to accept a fast disuse, and thereby a brusque width.

Methodology [edit]

1 mutual method for designing FIR filters is the Parks-McClellan filter design algorithm, based on the Remez exchange algorithm. Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order North. The algorithm then finds the prepare of N coefficients that minimize the maximum deviation from the platonic. Intuitively, this finds the filter that is as close as you tin get to the desired response given that you can use only N coefficients. This method is particularly easy in practise and at least one text^[two] includes a programme that takes the desired filter and N and returns the optimum coefficients. One possible drawback to filters designed this way is that they contain many modest ripples in the passband(s), since such a filter minimizes the tiptop error.

Another method to finding a discrete FIR filter is filter optimization described in Knutsson et al., which minimizes the integral of the foursquare of the mistake, instead of its maximum value. In its bones form this approach requires that an ideal frequency role of the filter ${\displaystyle F_{I}(\omega )}$ is specified together with a frequency weighting function ${\displaystyle W(\omega )}$ and set up of coordinates ${\displaystyle x_{k}}$ in the signal domain where the filter coefficients are located.

An error office ${\displaystyle \varepsilon }$ is defined as

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

where ${\displaystyle f(x)}$ is the discrete filter and ${\displaystyle {\mathcal {F}}}$ is the discrete-time Fourier transform defined on the specified set of coordinates. The norm used hither is, formally, the usual norm on ${\displaystyle Fifty^{2}}$ spaces. This means that ${\displaystyle \varepsilon }$ measures the deviation betwixt the requested frequency function of the filter, ${\displaystyle F_{I}}$ , and the actual frequency part of the realized filter, ${\displaystyle {\mathcal {F}}\{f\}}$ . However, the deviation is also subject to the weighting role ${\displaystyle West}$ before the mistake function is computed.

Once the error function is established, the optimal filter is given by the coefficients ${\displaystyle f(x)}$ which minimize ${\displaystyle \varepsilon }$ . This can exist done past solving the corresponding least squares problem. In practice, the ${\displaystyle Fifty^{2}}$ norm has to be approximated by ways of a suitable sum over discrete points in the frequency domain. In general, yet, these points should be significantly more than the number of coefficients in the bespeak domain to obtain a useful approximation.

Simultaneous optimization in both domains [edit]

The previous method can be extended to include an additional error term related to a desired filter impulse response in the signal domain, with a corresponding weighting office. The ideal impulse response can exist chosen independently of the ideal frequency office and is in practice used to limit the effective width and to remove ringing effects of the resulting filter in the signal domain. This is done by choosing a narrow ideal filter impulse response role, e.g., an impulse, and a weighting role which grows fast with the distance from the origin, e.m., the distance squared. The optimal filter tin still be calculated by solving a simple to the lowest degree squares problem and the resulting filter is then a "compromise" which has a total optimal fit to the ideal functions in both domains. An of import parameter is the relative forcefulness of the two weighting functions which determines in which domain it is more than important to have a adept fit relative to the ideal function.

Meet as well [edit]

• Digital filter
• Prototype filter
• Finite impulse response#Filter design

References [edit]

1. ^ Valdez, Thousand.E. "Digital Filters". GRM Networks. Retrieved 13 July 2020.
2. ^ Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory and Awarding of Digital Signal Processing (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.) ISBN 0-13-914101-4

• A. Antoniou (1993). Digital Filters: Analysis, Blueprint, and Applications (2 ed.). McGraw-Hill, New York, NY. ISBN978-0-07-002117-4.
• A. Antoniou (2006). Digital Signal Processing: Signals, Systems, and Filters. McGraw-Loma, New York, NY. doi:10.1036/0071454241. ISBN978-0-07-145424-seven.
• S.Due west.A. Bergen; A. Antoniou (2005). "Design of Nonrecursive Digital Filters Using the Ultraspherical Window Role". EURASIP Journal on Applied Signal Processing. 2005 (12): 1910. doi:ten.1155/ASP.2005.1910.
• A.One thousand. Deczky (October 1972). "Synthesis of Recursive Digital Filters Using the Minimum p-Error Criterion". IEEE Trans. Audio Electroacoustics. AU-20 (4): 257–263. doi:x.1109/TAU.1972.1162392.
• J.K. Kaiser (1974). "Nonrecursive Digital Filter Design Using the I ₀-sinh Window Function". Proc. 1974 IEEE Int. Symp. Circuit Theory (ISCAS74). San Francisco, CA. pp. 20–23.
• H. Knutsson; M. Andersson; J. Wiklund (June 1999). "Advanced Filter Blueprint". Proc. Scandinavian Symposium on Epitome Analysis, Kangerlussuaq, Greenland.
• S.K. Mitra (1998). Digital Signal Processing: A Calculator-Based Approach. McGraw-Hill, New York, NY. ISBN978-0-07-286546-2.
• A.V. Oppenheim; R.Due west. Schafer; J.R. Buck (1999). Discrete-Fourth dimension Signal Processing. Prentice-Hall, Upper Saddle River, NJ. ISBN978-0-thirteen-754920-7.
• T.Westward. Parks; J.H. McClellan (March 1972). "Chebyshev Approximation for Nonrecursive Digital Filters with Linear Stage". IEEE Trans. Circuit Theory. CT-xix (2): 189–194. doi:x.1109/TCT.1972.1083419.
• L.R. Rabiner; J.H. McClellan; T.W. Parks (April 1975). "FIR Digital Filter Blueprint Techniques Using Weighted Chebyshev Approximation". Proc. IEEE. 63 (iv): 595–610. doi:10.1109/PROC.1975.9794.

External links [edit]

• An extensive list of filter design articles and software at Circuit Sage
• A list of digital filter pattern software at dspGuru
• Analog Filter Pattern Demystified
• Yehar's digital audio processing tutorial for the braindead! This paper explains only (between others topics) filters design theory and give some examples

What Does Filter Order Mean,

Source: https://en.wikipedia.org/wiki/Filter_design

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