Wave filtering is the operation of filtering the frequency of a specific band in the signal, which is an important measure to suppress and prevent interference. Filtering can be divided into classical filtering and modern filtering.
Chinese name
wave filtering
Foreign name
Wave filtering
Substantive
Operation of filtering the frequency of a specific band in the signal
Filtering is tosignalThe operation of frequency filtering in a specific band is an important measure to suppress and prevent interference.It is based on the results of observing one random process to estimate another related random processprobabilityTheory and method.
origin
The term filtering originates fromCommunication Theory, which is extracted from the received signal containing interferenceuseful signal A technology of.The "received signal" is equivalent to the observed random process, and the "useful signal" is equivalent to the estimated random process.For example, useRadar trackingAircraft, the measured aircraft position data contains measurement errors and othersRandom interferenceHow to use these data to estimate the position, speed, acceleration, etc. of the aircraft at every moment as accurately as possible and predict the future position of the aircraft is a filtering and prediction problem.Such problems exist in a large number in electronic technology, aerospace science, control engineering and other scientific and technological departments.The earliest consideration in history isWiener filtering , later RE. Kalman and RS. Boussy proposed in the 1960sKalman filtering 。Now it is normalnonlinear filtering The research on the problem is quite active.
Basic concepts
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wave filtering
Filtering is an important concept in signal processing. There are two kinds of filtering: classical filtering and modern filtering.
Classical filtering
wave filtering
The concept of classical filtering is based onFourier analysisAnd transformation.according toAdvanced mathematicsIn theory, any signal satisfying certain conditions can be regarded as superposition of infinite sine waves.In other words, the engineering signal is a linear superposition of sine waves with different frequencies. The sine waves with different frequencies that make up the signal are called the frequency components of the signal orharmonicComposition.
wave filter
Only signal components within a certain frequency range are allowed to pass through normally, while blocking another partfrequency divisionThe circuit through which the rate component passes is called a classical filter orfilter circuit 。In fact, any oneElectronic systemAll have their own bandwidth (limit on the highest frequency of the signal), and the frequency characteristics reflect the basic characteristics of the electronic system.The filter is an engineering application circuit designed according to the influence of circuit parameters on circuit bandwidth.
Modern filtering
wave filtering
Pair with analog electronic circuitanalog signalThe basic principle of filtering is to use the frequency characteristics of the circuit to select the frequency components of the signal.When filtering according to frequency, the signal is regarded as an analog signal superimposed by sine waves with different frequencies, and signal filtering is realized by selecting different frequency components.
wave filtering
1. When the higher frequency components of the signal are allowed to pass through the filter, this filter is called a high pass filter.
2. When a lower frequency component of the signal is allowed to pass through a filter, this filter is called a low-pass filter.
3. Let the cutoff frequency of the low frequency band be fp1, and the cutoff frequency of the high frequency band be fp2:
1) A filter whose frequency is between fp1 and fp2 and can be attenuated by signals of other frequencies is called a band-pass filter.
2) Conversely, if the frequency is attenuated between fp1 and fp2, the filter that can pass outside is called band stop filter.
wave filtering
The behavior characteristic of ideal filter is usually described by amplitude frequency characteristic diagram, which is also called amplitude frequency characteristic of filter circuit.
Filtering problem and classification
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wave filtering
For filters, the frequency range with non-zero gain amplitude is called passband, or passband for short, and the frequency range with zero gain amplitude is called stopband.For example, for LP, from - w1 to w1, it is called LP passband, and other frequency parts are called stopband.The passband indicates that it can pass through the filter without attenuationsignal frequency The stopband represents the frequency component of the signal attenuated by the filter.The gain obtained by the signal in the passband is called passband gain, and the attenuation obtained by the signal in the stopband is called stopband attenuation.In engineering practice, dB is generally used as the amplitude gain unit of the filter.
wave filtering
It can be divided into continuous time filtering and discrete time filtering according to whether filtering is performed over a whole period of time or only at some sampling points.Time parameter set of the formerTIt can be the real half axis [0, ∞) orReal axis(-∞,∞);LatterTMay be taken asNonnegative integer set{0,1,2,...} or integer set {..., - 2, - 1, 0,1,2,...}.set upX={X,t∈T={Y,t∈T）Poor, namelyXIs the estimated process, which cannot be directly observed;YFor the observed process, it includesXSome information about.Express the time of arrival withtAll the observed data up to, if one of the data in can be foundfunction?(), make itMean square errorTo a minimum, it is calledXOptimal filtering of t;If the range of minimum value is limited tolinear function, calledXLinear optimal filtering of t.It can be proved that the optimal filter islinearThe optimal filtering exists only with probability 1.For the former, Min t isXTAboutσ() (GeneratedσDomain)Conditional expectation, recorded as for the latter, if the mean value E is further setXT 呏 EYIf t 呏 0, then pity t isXT in the tensionedHilbert spaceProjection on, recorded as if（X,Y）Yestwo-dimensionalNormal process, the optimal filtering is consistent with the linear optimal filtering.
wave filtering
For the convenience of application and description, the above definitions are sometimes classified in more detail.Let τ be a definitereal numberorintegerAnd consider the estimated process.According to τ=0, τ>0, τ<0, they are respectively called optimal filtering, (τ step) prediction or extrapolation, (τ step) smoothing or interpolation, and the corresponding errors andMean square errorAnd these problems are collectively called filtering problems.The main topic of the filtering problem is to study which types of stochastic processesXandY, can and how to use some analytic expression of observation results, ordifferential equation , orRecursive formulaTo express and further study their various properties.In addition, the one-dimensional stochastic process mentioned aboveX、Y, can be extended to multidimensional random processes.
Wiener filtering
wave filtering
In history, the first consideration is the wide smooth process (seeStationary process）OflinearThe general model of prediction and filtering isYt=Xt+Nt，Including（X，N）Is a two-dimensional wide stationary process or sequence, and its spectrumdistribution function It is known that the mean value is zero.Suppose from - ∞ to timetAll up toYThe values of have been observedXτ - step linear prediction and its applicationMean square error。If limited to considerationN=0, τ>0, it becomes under the condition of error free observationXOf itselflinearPrediction problem;IfN≠ 0, τ ≤ 0, then it will change from noiseNInterference reception signalYExtract useful signals fromXThe filtering problem of.From 1939 to 1941,Alas KolmogorovutilizeStationary seriesWald decomposition of (seeStationary process）The general theory and processing method of linear prediction are given, which is then extended to continuous time stationary processes.N. WienerIn 1942Stationary seriesThe explicit expression of linear optimal prediction and filtering, namely Wiener filtering formula, is derived by using spectral decomposition when the spectral density of the process exists and meets some regular conditions. It has been applied in air defense fire control, electronic engineering and other departments.The above model was extended to only a limited time in the 1950ssectionThe application scope of stationary processes and some special non-stationary processes that are observed within the scope of the observation also extends to more fields.So far, it still processes various dynamic data (such asmeteorological, hydrology, seismic exploration, etc.) and one of the powerful tools for predicting the future.
wave filtering
wave filtering
Wiener filtering The formula is derived from the spectral decomposition of stationary processes, which is difficult to be extended to more general non-stationary processes and multidimensional cases, so its application scope is limited.On the other hand, it is not easy to calculate the filter value and newObservationsIt is relatively simple to obtain new filter values, especially it can not meet the needs of rapid processing of large amounts of data on electronic computers.
Kalman filtering
wave filtering
With the development of high-speed electronic computers and measurementArtificial satellite orbitAnd navigation, R.E. Kalman and RS. Bousey put forward a new type oflinearThe model and method of filtering are generally called Kalman filtering.The basic assumption is that the estimated processXIs the finite order multidimensional under the influence of random noiselineardynamical system Output of the observedYT isXPart of t or itslinear functionThe superposition with the measurement noise is notrequirementStability, but it is required that the noise value at different times is irrelevant.In addition, observation only needs to start from a certain time, rather than an infinite observation interval.More importantly, adapted to the characteristics of electronic computers, the Kalman filter formula does not represent the estimated value as an obvious function of the observed value, but gives one of itsrecursive algorithm (i.e. real-time algorithm).Specifically, for discrete time filtering, as long as theXDimension, you cantThe filtered value table of the time becomes the filtered value of the previous time and the observed value of this timeYSome kind of tlinear combination。For continuous time filtering, we can giveYThe linear stochastic differential equation that t should satisfy.When it is necessary to continuously increase the observation results and output filter valuesalgorithmIt accelerates the speed of data processing and reduces the data storage.Kalman also demonstrated that if thelinear systemSatisfying some "controllability" and "observability" (which are two important concepts proposed by Kalman in modern control theory), then the optimal filter must be "asymptotically stable".Generally speaking, from the initialerror、Rounding errorAnd other inaccuraciesFiltering timeAnd gradually disappear or tend to be stable, so as not to form error accumulation.This is very important in practical application.
wave filtering
Kalman filtering There are also various forms of generalization, such as relaxing the restrictions on noise irrelevance and using linear systems to approximatenonlinear system , as well as the so-called "adaptive filtering", and has been increasingly widely used.
nonlinear filtering
wave filtering
As previously explained, the general nonlinear optimal filtering can be reduced to the problem of finding conditional expectations.In principle, conditional expectation can be used in the case of limited multiple observationsBayesian formulaTo calculate.But even in relatively simple occasions, the results obtained in this way are quite complicated, which is inconvenient for both practical application and theoretical research.AndKalman filtering Similarly, people also hope to give some kind of nonlinear filteringrecursive algorithm Or the stochastic differential equation it satisfies.But generally they do not exist, so it is necessary toXAndYApply appropriate restrictions.The research work of nonlinear filtering is quite active, which involves many modern achievements of stochastic process theory, such as general theory of stochastic processmartingale, stochastic differential equationPoint processEtc.One of the most important problems is to study under what conditions there is a martingale M, so that at any time, M andYBoth contain the same information;Such M is calledYThe innovation process of.For a class of so-called "conditionsNormal process”The strictly realizable recurrence formula of nonlinear optimal filtering has been given.In practical applications, various linear approximation methods are often used for nonlinear filtering problems.
