Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication , in radio , television , audio recording , radar , control systems , music synthesis , image processing , and computer graphics.
There are many different bases of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Filters may be:. Linear continuous-time circuit is perhaps the most common meaning for filter in the signal processing world, and simply "filter" is often taken to be synonymous. These circuits are generally designed to remove certain frequencies and allow others to pass. Circuits that perform this function are generally linear in their response, or at least approximately so.
Any nonlinearity would potentially result in the output signal containing frequency components not present in the input signal. The modern design methodology for linear continuous-time filters is called network synthesis. Some important filter families designed in this way are:. The difference between these filter families is that they all use a different polynomial function to approximate to the ideal filter response. This results in each having a different transfer function. Another older, less-used methodology is the image parameter method.
Filters designed by this methodology are archaically called "wave filters". Some important filters designed by this method are:. One important application of filters is in telecommunication. Many telecommunication systems use frequency-division multiplexing , where the system designers divide a wide frequency band into many narrower frequency bands called "slots" or "channels", and each stream of information is allocated one of those channels. The people who design the filters at each transmitter and each receiver try to balance passing the desired signal through as accurately as possible, keeping interference to and from other cooperating transmitters and noise sources outside the system as low as possible, at reasonable cost.
Multilevel and multiphase digital modulation systems require filters that have flat phase delay—are linear phase in the passband—to preserve pulse integrity in the time domain,  giving less intersymbol interference than other kinds of filters. On the other hand, analog audio systems using analog transmission can tolerate much larger ripples in phase delay , and so designers of such systems often deliberately sacrifice linear phase to get filters that are better in other ways—better stop-band rejection, lower passband amplitude ripple, lower cost, etc.
Filters can be built in a number of different technologies. The same transfer function can be realised in several different ways, that is the mathematical properties of the filter are the same but the physical properties are quite different. Often the components in different technologies are directly analogous to each other and fulfill the same role in their respective filters. For instance, the resistors, inductors and capacitors of electronics correspond respectively to dampers, masses and springs in mechanics. Likewise, there are corresponding components in distributed-element filters.
Digital signal processing allows the inexpensive construction of a wide variety of filters. The signal is sampled and an analog-to-digital converter turns the signal into a stream of numbers.
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A computer program running on a CPU or a specialized DSP or less often running on a hardware implementation of the algorithm calculates an output number stream. This output can be converted to a signal by passing it through a digital-to-analog converter. There are problems with noise introduced by the conversions, but these can be controlled and limited for many useful filters.
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Due to the sampling involved, the input signal must be of limited frequency content or aliasing will occur. In the late s, engineers realized that small mechanical systems made of rigid materials such as quartz would acoustically resonate at radio frequencies, i. Some early resonators were made of steel , but quartz quickly became favored. The biggest advantage of quartz is that it is piezoelectric.
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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.
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. This e. 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.
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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. 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.
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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.
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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.