In: Advanced Math
Report on this topic Design filtering
Content included...
Abstract
Introduction
Principles
Code design
Result and discussion
Conclusion
Reference
Abstract :Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which are contradictory. 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 can be described as an optimization problem where each requirement contributes to an error function which should be minimized. Certain parts of the design process can be automated, but normally an experienced electrical engineer is needed to get a good result.
Introduction:. Typical requirements which are considered in the design process are: The filter should have a specific frequency response The filter should have a specific phase shift or group delay The filter should have 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
An all-pass filter passes through all frequencies unchanged, but changes the phase of the signal. Filters of this type can be used to equalize the group delay 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 amplitude but shifts each sinusoid phase by ±90°. A fractional delay filter is an all-pass that has a specified and constant group or phase delay for all frequencies. The impulse response Edit There is a direct correspondence between the filter's frequency function and its impulse response: the former is the Fourier transform of the latter. That 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 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 it may even be relevant to consider a frequency function and impulse response of the filter which are chosen independently from each other. For example, we may want both a specific frequency function of the filter and that the resulting filter have a small effective width in the signal domain as possible. The latter condition can be 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 meet both these contradicting design goals as much as possible. Causality Edit In order to be implementable, any time-dependent filter (operating in real time) 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. Filters that do not operate in real time (e.g. for image processing) can be non-causal. This e.g. allows the design 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 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, filters based on feedback circuits have other advantages and may therefore be preferred, even if this class of filters includes unstable filters. In this case, the filters must be carefully designed in order to avoid instability. Locality Edit 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 Edit 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. 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 issue related to computational complexity is separability, that is, if and how a filter can be written as a convolution of two or more simpler filters. In particular, this issue is of importance for multidimensional filters, e.g., 2D filter which are used in image processing. In this case, a significant reduction in computational complexity can be obtained if the filter can be separated as the convolution of one 1D filter in the horizontal direction and one 1D filter in the vertical direction. A result of the filter design process may, e.g., be to approximate some desired filter as a separable filter or as a sum of separable filters.
Theoretical basis Edit 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 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 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, this determines the smallest possible width in the frequency. This is a typical example of contradictory requirements where the filter design process may try to find a useful compromise.
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 function. The ideal impulse response can be chosen independently of the ideal frequency function 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 function, e.g., an impulse, and a weighting function which grows fast with the distance from the origin, e.g., the distance squared. The optimal filter can still be calculated by solving a simple least squares problem and the resulting filter is then a "compromise" which has a total optimal fit to the ideal functions in both domains. An important parameter is the relative strength of the two weighting functions which determines in which domain it is more important to have a good fit relative to the ideal function.
References
Edit ^ Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory and Application of Digital Signal Processing (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.) ISBN 0-13-914101-4 A. Antoniou (1993). Digital Filters: Analysis, Design, and Applications (2 ed.). McGraw-Hill, New York, NY. ISBN 0-07-002117-1. A. Antoniou (2006). Digital Signal Processing: Signals, Systems, and Filters. McGraw-Hill, New York, NY. doi:10.1036/0071454241. ISBN 0-07-145424-1. S.W.A. Bergen; A. Antoniou (2005). "Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function". EURASIP Journal on Applied Signal Processing. 2005 (12): 1910. doi:10.1155/ASP.2005.1910. A.G. Deczky (October 1972). "Synthesis of Recursive Digital Filters Using the Minimum p-Error Criterion". IEEE Trans. Audio Electroacoustics. AU-20 (4): 257–263. doi:10.1109/TAU.1972.1162392. J.K. Kaiser (1974). "Nonrecursive Digital Filter Design Using the I0-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 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. ISBN 0-07-286546-6. A.V. Oppenheim; R.W. Schafer; J.R. Buck (1999). Discrete-Time Signal Processing. Prentice-Hall, Upper Saddle River, NJ. ISBN 0-13-754920-2. T.W. Parks; J.H. McClellan (March 1972). "Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase". IEEE Trans. Circuit Theory. CT-19 (2): 189–194. doi:10.1109/TCT.1972.1083419. L.R. Rabiner; J.H. McClellan; T.W. Parks (April 1975). "FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation". Proc. IEEE. 63 (4): 595–610. doi:10.1109/PROC.1975.9794.