Impulsive Noise (779807), страница 2
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In a Poisson model, the probability of occurrence of k impulsivenoise in a time interval of T is given byP(k , T ) =(λT ) k −λTek!(12.16)where λ is a rate function with the following properties:Prob(one impulse in a small time interval û9 ) = λû9Prob(zero impulse in a small time interval û9 ) = 1− λû9(12.17)It is assumed that no more than one impulsive noise can occur in a timeinterval ∆t. In a Poisson–Gaussian model, the pdf of an impulsive noiseni(m) in a small time interval of ∆t is given byf NPG (ni (m) ) = (1 − û9 ) δ (ni (m) ) + û9 f N (ni (m) )I(12.18)where f N (ni (m) ) is the Gaussian pdf of Equation (12.14).12.2.3 A Binary-State Model of Impulsive NoiseAn impulsive noise process may be modelled by a binary-state model asshown in Figure 12.4. In this binary model, the state S0 corresponds to the“off” condition when impulsive noise is absent; in this state, the modelemits zero-valued samples.
The state S1 corresponds to the “on” condition;in this state the model emits short-duration pulses of random amplitude andduration. The probability of a transition from state Si to state Sj is denotedby aij. In its simplest form, as shown in Figure 12.5, the model ismemoryless, and the probability of a transition to state Si is independent ofthe current state of the model. In this case, the probability that at time t+1363Statistical Models for Impulsive Noisea = α01a = α11S0S1a =1 - α00a =1 - α10Figure 12.5 A binary-state model of an impulsive noise generator.a00S0a 10a 02a 01a12a20S2S1a 21a11a 22Figure 12.6 A 3-state model of impulsive noise and the decaying oscillationsthat often follow the impulses.the signal is in the state S0 is independent of the state at time t, and is givenbyP s (t + 1) = S 0 s (t ) = S 0 = P s (t + 1) = S 0 s (t ) = S1 = 1 − α (12.19)() ()where st denotes the state at time t. Likewise, the probability that at timet+1 the model is in state S1 is given by() ()P s ( t + 1) = S 1 s ( t ) = S 0 = P s ( t + 1) = S 1 s ( t ) = S 1 = α(12.20)In a more general form of the binary-state model, a Markovian statetransition can model the dependencies in the noise process.
The model thenbecomes a 2-state hidden Markov model considered in Chapter 5.In one of its simplest forms, the state S1 emits samples from a zero-meanGaussian random process. The impulsive noise model in state S1 can beconfigured to accommodate a variety of impulsive noise of different shapes,364Impulsive Noisedurations and pdfs.
A practical method for modelling a variety of impulsivenoise is to use a code book of M prototype impulsive noises, and theirassociated probabilities [(ni1, pi1), (ni2 , pi2), ..., (niM , piM)], where pjdenotes the probability of impulsive noise of the type nj. The impulsivenoise code book may be designed by classification of a large number of“training” impulsive noises into a relatively small number of clusters. Foreach cluster, the average impulsive noise is chosen as the representative ofthe cluster.
The number of impulses in the cluster of type j divided by thetotal number of impulses in all clusters gives pj, the probability of animpulse of type j.Figure 12.6 shows a three-state model of the impulsive noise and thedecaying oscillations that might follow the noise. In this model, the state S0models the absence of impulsive noise, the state S1 models the impulsivenoise and the state S2 models any oscillations that may follow a noise pulse.12.2.4 Signal to Impulsive Noise RatioFor impulsive noise the average signal to impulsive noise ratio, averagedover an entire noise sequence including the time instances when theimpulses are absent, depends on two parameters: (a) the average power ofeach impulsive noise, and (b) the rate of occurrence of impulsive noise.
LetPimpulse denote the average power of each impulse, and Psignal the signalpower. We may define a “local” time-varying signal to impulsive noiseratio asPsignal ( m )SINR ( m ) =(12.21)Pimpulse b ( m )The average signal to impulsive noise ratio, assuming that the parameterα is the fraction of signal samples contaminated by impulsive noise, can bedefined asPsignalSINR =(12.22)α PimpulseNote that from Equation (12.22), for a given signal power, there are manypair of values of α and PImpulse that can yield the same average SINR.365Median FiltersSliding Winow ofLength 3 SamplesImpulsive noise removedNoise-free samples distorted by the median filterFigure 12.7 Input and output of a median filter.
Note that in addition to suppressingthe impulsive outlier, the filter also distorts some genuine signal components.12.3 Median FiltersThe classical approach to removal of impulsive noise is the median filter.The median of a set of samples {x(m)} is a member of the set xmed(m) suchthat; half the population of the set are larger than xmed(m) and half aresmaller than xmed(m).
Hence the median of a set of samples is obtained bysorting the samples in the ascending or descending order, and then selectingthe mid-value. In median filtering, a window of predetermined length slidessequentially over the signal, and the mid-sample within the window isreplaced by the median of all the samples that are inside the window, asillustrated in Figure 12.7.The output xˆ (m) of a median filter with input y(m) and a medianwindow of length 2K+1 samples is given byxˆ (m) = y med (m)= median [y (m − K ),, y (m), , y (m + K )](12.23)The median of a set of numbers is a non-linear statistics of the set, withthe useful property that it is insensitive to the presence of a sample with anunusually large value, a so-called outlier, in the set.
In contrast, the mean,and in particular the variance, of a set of numbers are sensitive to the366Impulsive Noisepresence of impulsive-type noise. An important property of median filters,particularly useful in image processing, is that they preserves edges orstepwise discontinuities in the signal. Median filters can be used forremoving impulses in an image without smearing the edge information; thisis of significant importance in image processing. However, experimentswith median filters, for removal of impulsive noise from audio signals,demonstrate that median filters are unable to produce high-quality audiorestoration. The median filters cannot deal with “real” impulsive noise,which are often more than one or two samples long.
Furthermore, medianfilters introduce a great deal of processing distortion by modifying genuinesignal samples that are mistaken for impulsive noise. The performance ofmedian filters may be improved by employing an adaptive threshold, so thata sample is replaced by the median only if the difference between thesample and the median is above the threshold: y ( m)xˆ (m) = y med (m)if y (m) − y med (m) < k θ (m)otherwise(12.24)where θ(m) is an adaptive threshold that may be related to a robust estimateof the average of y (m) − y med (m) , and k is a tuning parameter.
Medianfilters are not optimal, because they do not make efficient use of priorknowledge of the physiology of signal generation, or a model of the signaland noise statistical distributions. In the following section we describe aautoregressive model-based impulsive removal system, capable ofproducing high-quality audio restoration.12.4 Impulsive Noise Removal Using Linear Prediction ModelsIn this section, we study a model-based impulsive noise removal system.Impulsive disturbances usually contaminate a relatively small fraction α ofthe total samples.
Since a large fraction, 1–α, of samples remain unaffectedby impulsive noise, it is advantageous to locate individual noise pulses, andcorrect only those samples that are distorted. This strategy avoids theunnecessary processing and compromise in the quality of the relativelylarge fraction of samples that are not disturbed by impulsive noise. Theimpulsive noise removal system shown in Figure 12.8 consists of twosubsystems: a detector and an interpolator. The detector locates the positionof each noise pulse, and the interpolator replaces the distorted samples367Impulsive Noise Removal Using LP ModelsSignal + impulsive noiseSignalInterpolatorLinearpredictionanalysisInverse filter1 : Impulse presentPredictor coefficientsMatched filter0 : Noiseless signalThresholddetectorNoisy excitationDetector subsystemRobust power estimatorFigure 12.8 Configuration of an impulsive noise removal system incorporating adetector and interpolator subsystems.using the samples on both sides of the impulsive noise.
The detector iscomposed of a linear prediction analysis system, a matched filter and athreshold detector. The output of the detector is a binary switch and controlsthe interpolator. A detector output of “0” signals the absence of impulsivenoise and the interpolator is bypassed. A detector output of “1” signals thepresence of impulsive noise, and the interpolator is activated to replace thesamples obliterated by noise.12.4.1 Impulsive Noise DetectionA simple method for detection of impulsive noise is to employ an amplitudethreshold, and classify those samples with an amplitude above the thresholdas noise. This method works fairly well for relatively large-amplitudeimpulses, but fails when the noise amplitude falls below the signal.Detection can be improved by utilising the characteristic differencesbetween the impulsive noise and the signal. An impulsive noise, or a shortduration pulse, introduces uncharacteristic discontinuity in a correlatedsignal.
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