Impulsive Noise (779807), страница 3

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The discontinuity becomes more detectable when the signal is368Impulsive Noisedifferentiated. The differentiation (or, for digital signals, the differencing)operation is equivalent to decorrelation or spectral whitening. In thissection, we describe a model-based decorrelation method for improvingimpulsive noise detectability. The correlation structure of the signal ismodelled by a linear predictor, and the process of decorrelation is achievedby inverse filtering. Linear prediction and inverse filtering are covered inChapter 8. Figure 12.9 shows a model for a noisy signal. The noise-freesignal x(m) is described by a linear prediction model asPx ( m )= ∑ a k x ( m − k )+ e( m )(12.25)k =1where a=[a1, a2, ...,aP]T is the coefficient vector of a linear predictor oforder P, and the excitation e(m) is either a noise-like signal or a mixture of arandom noise and a quasi-periodic train of pulses as illustrated in Figure12.9. The impulsive noise detector is based on the observation that linearpredictors are a good model of the correlated signals but not theuncorrelated binary-state impulsive-type noise.

Transforming the noisysignal y(m) to the excitation signal of the predictor has the followingeffects:(a) The scale of the signal amplitude is reduced to almost that of theoriginal excitation signal, whereas the scale of the noise amplituderemains unchanged or increases.(b) The signal is decorrelated, whereas the impulsive noise is smearedand transformed to a scaled version of the impulse response of theinverse filter.White noisePeriodic impulsetrainMixtureni (m)=n(m)b(m)ExcitationselectionSpeechLinear prediction x(m)filterNoisy speechy(m) = x(m)+ni (m)Coefficients“Hidden” modelcontrolFigure 12.9 Noisy speech model. The signal is modelled by a linear predictor.Impulsive noise is modelled as an amplitude-modulated binary-state process.369Impulsive Noise Removal Using LP ModelsBoth effects improve noise delectability.

Speech or music is composed ofrandom excitations spectrally shaped and amplified by the resonances ofvocal tract or the musical instruments. The excitation is more random thanthe speech, and often has a much smaller amplitude range. Theimprovement in noise pulse detectability obtained by inverse filtering canbe substantial and depends on the time-varying correlation structure of thesignal. Note that this method effectively reduces the impulsive noisedetection to the problem of separation of outliers from a random noiseexcitation signal using some optimal thresholding device.12.4.2 Analysis of Improvement in Noise DetectabilityIn the following, the improvement in noise detectability that results frominverse filtering is analysed.

Using Equation (12.25), we can rewrite a noisysignal model asy( m) = x (m) + ni ( m)=P∑ ak x (m − k )+ e( m) + ni (m)(12.26)k =1where y(m), x(m) and ni(m) are the noisy signal, the signal and the noiserespectively. Using an estimate aˆ of the predictor coefficient vector a, thenoisy signal y(m) can be inverse-filtered and transformed to the noisyexcitation signal v(m) asPv(m) = y(m) − ∑ aˆk y(m − k)k =1(12.27)P= x(m ) + ni (m) − ∑ (ak − a˜k )[x(m − k) + ni (m − k)]k =1where a˜k is the error in the estimate of the predictor coefficient. UsingEquation (12.25) Equation (12.27) can be rewritten in the following form:v(m) = e(m) + ni (m) +PP∑ a˜k x(m − k) − ∑ aˆ k ni (m − k)k =1k =1(12.28)370Impulsive NoiseFrom Equation (12.28) there are essentially three terms that contribute tothe noise in the excitation sequence:(a) the impulsive disturbance ni(m) which is usually the dominant term;(b) the effect of the past P noise samples, smeared to the present time bythe action of the inverse filtering, ∑ aˆk ni (m − k) ;(c) the increase in the variance of the excitation signal, caused by theerror in the parameter vector estimate, and expressed by the term∑ a˜k x(m − k).The improvement resulting from the inverse filter can be formulated asfollows.

The impulsive noise to signal ratio for the noisy signal is given byimpulsive noise power E[ni2 (m)]=signal powerE[ x 2 (m)](12.29)where E[·] is the expectation operator. Note that in impulsive noisedetection, the signal of interest is the impulsive noise to be detected fromthe accompanying signal. Assuming that the dominant noise term in thenoisy excitation signal v(m) is the impulse ni(m), the impulsive noise toexcitation signal ratio is given byimpulsive noise power E[ni2 (m)]=excitation powerE[e 2 (m)](12.30)The overall gain in impulsive noise to signal ratio is obtained, by dividingEquations (12.29) and (12.30), asE[ x 2 (m)]= gainE[e 2 (m)](12.31)This simple analysis demonstrates that the improvement in impulsive noisedetectability depends on the power amplification characteristics, due toresonances, of the linear predictor model.

For speech signals, the scale ofthe amplitude of the noiseless speech excitation is on the order of 10–1 to10–4 of that of the speech itself; therefore substantial improvement in371Impulsive Noise Removal Using LP Modelsimpulsive noise detectability can be expected through inverse filtering ofthe noisy speech signals.Figure 12.10 illustrates the effect of inverse filtering in improving thedetectability of impulsive noise.

The inverse filtering has the effect that thesignal x(m) is transformed to an uncorrelated excitation signal e(m),whereas the impulsive noise is smeared to a scaled version of the inversefilter impulse response [1, -a1, ...,-aP], as indicated by the term∑ aˆk ni (m − k) in Equation (12.28). Assuming that the excitation is a whitenoise Gaussian signal, a filter matched to the inverse filter coefficients mayenhance the delectability of the smeared impulsive noise from the excitationsignal.t(a)t(b)(c)tFigure 12.10 Illustration of the effects of inverse filtering on detectability of Impulsivenoise: (a) Impulsive noise contaminated speech with 5% impulse contamination at anaverage SINR of 10dB, (b) Speech excitation of impulse-contaminated speech, and(c) Speech excitation of impulse-free speech.372Impulsive Noise12.4.3 Two-Sided Predictor for Impulsive Noise DetectionIn the previous section, it was shown that impulsive noise detectability canbe improved by decorrelating the speech signal.

The process ofdecorrelation can be taken further by the use of a two-sided linearprediction model. The two-sided linear prediction of a sample x(m) is basedon the P past samples and the P future samples, and is defined by theequationPPk =1k =1x ( m )= ∑ a k x ( m − k )+ ∑ a k + P x ( m + k )+ e( m )(12.32)where ak are the two-sided predictor coefficients and e(m) is the excitationsignal.

All the analysis used for the case of one-sided linear predictor can beextended to the two-sided model. However, the variance of the excitationinput of a two-sided model is less than that of the one-sided predictorbecause in Equation (12.32) the correlations of each sample with the future,as well as the past, samples are modeled. Although Equation (12.32) is anon-causal filter, its inverse, required in the detection subsystem, is causal.The use of a two-sided predictor can result in further improvement in noisedetectability.12.4.4 Interpolation of Discarded SamplesSamples irrevocably distorted by an impulsive noise are discarded and thegap thus left is interpolated.

For interpolation imperfections to remaininaudible a high-fidelity interpolator is required. A number of interpolatorsfor replacement of a sequence of missing samples are introduced in Chapter10. The least square autoregressive (LSAR) interpolation algorithm ofSection 10.3.2 produces high-quality results for a relatively small numberof missing samples left by an impulsive noise. The LSAR interpolationmethod is a two-stage process.

In the first stage, the available samples onboth sides of the noise pulse are used to estimate the parameters of a linearprediction model of the signal. In the second stage, the estimated modelparameters, and the samples on both sides of the gap are used to interpolatethe missing samples. The use of this interpolator in replacement of audiosignals distorted by impulsive noise has produced high-quality results.373Robust Parameter Estimation12.5 Robust Parameter EstimationIn Figure 12.8, the threshold used for detection of impulsive noise from theexcitation signal is derived from a nonlinear robust estimate of theexcitation power. In this section, we consider robust estimation of aparameter, such as the signal power, in the presence of impulsive noise.A robust estimator is one that is not over-sensitive to deviations of theinput signal from the assumed distribution.

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