Impulsive Noise (779807), страница 4
Текст из файла (страница 4)
In a robust estimator, an inputsample with unusually large amplitude has only a limited effect on theestimation results. Most signal processing algorithms developed foradaptive filtering, speech recognition, speech coding, etc. are based on theassumption that the signal and the noise are Gaussian-distributed, andemploy a mean square distance measure as the optimality criterion. Themean square error criterion is sensitive to non-Gaussian events such asimpulsive noise.
A large impulsive noise in a signal can substantiallyCost functionInfluence function∂ E[e 2 (m)]∂θE [e 2(m)]Mean squared errorθE [ABS (e 2 (m))]θ∂ E[ABS (e2 (m ))]∂θMean absolute valueθθE [Φ(e(m))]∂ E[Φ(e(m))]∂θMean squarederrorAbsolute valueAbsolute valueθθ∂ E[Ψ (e( m))]∂θE [Ψ (e(m))]Bi-weigth functionθθFigure 12.11 Illustration of a number of cost of error functions and thecorresponding influence functions.374Impulsive Noiseovershadow the influence of noise-free samples.Figure 12.11 illustrates the variations of several cost of error functionswith a parameter θ.
Figure 12.11(a) shows a least square error cost functionand its influence function. The influence function is the derivative of thecost function, and, as the name implies, it has a direct influence on theestimation results. It can be seen from the influence function of Figure12.11(a) that an unbounded sample has an unbounded influence on theestimation results.A method for introducing robustness is to use a non-linear function andlimit the influence of any one sample on the overall estimation results. Theabsolute value of error is a robust cost function, as shown by the influencefunction in Figure 12.11(b).
One disadvantage of this function is that it isnot continuous at the origin. A further drawback is that it does not allow forthe fact that, in practice, a large proportion of the samples are notcontaminated with impulsive noise, and may well be modelled withGaussian densities.Many processes may be regarded as Gaussian for the sample valuesthat cluster about the mean. For such processes, it is desirable to have aninfluence function that limits the influence of outliers and at the same timeis linear and optimal for the large number of relatively small-amplitudesamples that may be regarded as Gaussian-distributed. One such function isHuber's function, defined as e 2 (m)ψ [e(m)] = k e(m)if e(m) ≤ k(12.33)otherwiseHuber's function, shown in Figure 12.11(c), is a hybrid of the least meansquare and the absolute value of error functions.
Tukeys bi-weight function,which is a redescending robust objective function, is defined as {1−[1 − e 2 (m)]3 } 6ψ [e(m)] = 1 6if e(m) ≤ 1otherwise(12.34)As shown in Figure 12.11(d), the influence function is linear for smallsignal values but introduces attenuation as the signal value exceeds somethreshold. The threshold may be obtained from a robust median estimate ofthe signal power.375Restoration of Archived Gramophone Records601000800406004002020000-200-20-400-600-40-800-60-1000050100150200250300350400450050050100150200(a)250300350400450500350400450500(b)10008080060600404002020000-200-20-400-40-600-60-800-80-10000-10050100150200250(d)300350400450500050100150200250300(c)Figure 12.12 (a) A noisy audio signal from a 78 rpm record, (b) Noisy excitationsignal, (c) Matched filter output, (d) Restored signal.12.6 Restoration of Archived Gramophone RecordsThis Section describes the application of the impulsive noise removalsystem of Figure 12.8 to the restoration of archived audio records.
As thebandwidth of archived recordings is limited to 7–8 kHz, a low-pass, antialiasing filter with a cutoff frequency of 8 kHz is used to remove the out ofband noise. Playedback signals were sampled at a rate of 20 kHz, anddigitised to 16 bits. Figure 12.12(a) shows a 25 ms segment of noisy musicand song from an old 78 rpm gramophone record. The impulsiveinterferences are due to faults in the record stamping process, granularitiesof the record material or physical damage. This signal is modelled by apredictor of order 20. The excitation signal obtained from the inverse filterand the matched filter output are shown in Figures 12.12(b) and (c)376Impulsive Noiserespectively. Close examination of these figures show that some of theambiguities between the noise pulses and the genuine signal excitationpulses are resolved after matched filtering.The amplitude threshold for detection of impulsive noise from theexcitation signal is adapted on a block basis, and is set to k σ e2 , where σ e2 isa robust estimate of the excitation power.
The robust estimate is obtained bypassing the noisy excitation signal through a soft nonlinearity that rejectsoutliers. The scalar k is a tuning parameter; the choice of k reflects a tradeoff between the hit rate and the false-alarm rate of the detector. As kdecreases, smaller noise pulses are detected but the false detection rate alsoincreases. When an impulse is detected, a few samples are discarded andreplaced by the LSAR interpolation algorithm described in Chapter 10.Figure 12.12(d) shows the signal with the impulses removed.
The impulsivenoise removal system of Figure 12.8 was successfully applied to restorationof numerous examples of archived gramophone records. The system is alsoeffective in suppressing impulsive noise in examples of noisy telephoneconversations.12.7 SummaryThe classic linear time-invariant theory on which many signal processingmethods are based is not suitable for dealing with the non-stationaryimpulsive noise problem. In this chapter, we considered impulsive noise asa random on/off process and studied several stochastic models for impulsivenoise, including the Bernoulli–Gaussian model, the Poisson–Gaussian andthe hidden Markov model (HMM). The HMM provides a particularlyinteresting framework, because the theory of HMM studied in Chapter 5 iswell developed, and also because the state sequence of an HMM of noisecan be used to provide an estimate of the presence or the absence of thenoise.
By definition, an impulsive noise is a short and sharp eventuncharacteristic of the signal that it contaminates. In general, differencingoperation enhance the detectibility of impulsive noise. Based on thisobservation, in Section 12.4, we considered an algorithm based on a linearprediction model of the signal for detection of impulsive noise.In the next Chapter we expand the materials we considered in this chapterfor the modelling, detection, and removal of transient noise pulses.Bibliography377BibliographyDEMPSTER A.P., LAIRD N.M and RUBIN D.B. (1971) Maximum likelihoodfrom Incomplete Data via the EM Algorithm. Journal of the RoyalStatistical Society, Ser. 39, pp. 1–38.GODSIL S.
(1993) Restoration of Degraded Audio Signals, CambridgeUniversity Press.GALLAGHER N.C. and WISE G.L. (1981) A Theoretical Analysis of theProperties of Median Filters. IEEE Trans. Acoustics, Speech and SignalProcessing, ASSP-29, pp. 1136–1141JAYNAT N.S. (1976) Average and Median Based Smoothing for ImprovingDigital Speech Quality in the Presence of Transmission Errors.
IEEETrans. Commun. pp. 1043–1045, Sept.KELMA V.C and LAUB A.J. (1980) The Singular Value Decomposition : ItsComputation and Some Applications. IEEE Trans. Automatic Control,AC-25, pp. 164–176.KUNDA A., MITRA S. and VAIDYANATHAN P. (1984) Applications of TwoDimensional Generalised Mean Filtering for Removal of ImpulsiveNoise from Images. IEEE Trans. Acoustics, Speech and SignalProcessing, ASSP, 32, 3, pp. 600–609, June.MILNER B.P. (1995) Speech Recognition in Adverse Environments. PhDThesis, University of East Anglia, UK.NIEMINEN, HEINONEN P.
and NEUVO Y. (1987) Suppression and Detection ofImpulsive Type Interference using Adaptive Median Hybrid Filters.IEEE. Proc. Int. Conf. Acoustics, Speech and Signal Processing,ICASSP-87, pp. 117–120.TUKEY J.W. (1971) Exploratory Data Analysis. Addison Wesley, Reading,MA.RABINER L.R., SAMBUR M.R. and SCHMIDT C.E. (1984) Applications of aNonlinear Smoothing Algorithm to Speech Processing. IEEE Trans.ASSP-32, 3, June.VASEGHI S.V. and RAYNER P.J.W. (1990) Detection and Suppression ofImpulsive Noise in Speech Communication Systems. IEE Proc-ICommunications Speech and Vision, pp. 38–46, February.VASEGHI S.V.
and MILNER B.P. (1995) Speech Recognition in ImpulsiveNoise, Inst. of Acoustics, Speech and Signal Processing. IEEE Proc. Int.Conf. Acoustics, Speech and Signal Processing, ICASSP-95, pp. 437–440..
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
