Transient Noise Pulses (Vaseghi - Advanced Digital Signal Processing and Noise Reduction)

Advanced Digital Signal Processing and Noise Reduction, Second Edition.Saeed V. VaseghiCopyright © 2000 John Wiley & Sons LtdISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)13TRANSIENT NOISE PULSES13.1 Transient Noise Waveforms13.2 Transient Noise Pulse Models13.3 Detection of Noise Pulses13.4 Removal of Noise Pulse Distortions13.5 SummaryTransient noise pulses differ from the short-duration impulsive noisestudied in the previous chapter, in that they have a longer durationand a relatively higher proportion of low-frequency energy content,and usually occur less frequently than impulsive noise. The sources oftransient noise pulses are varied, and may be electromagnetic, acoustic ordue to physical defects in the recording medium.

Examples of transientnoise pulses include switching noise in telephony, noise pulses due toadverse radio transmission environments, noise pulses due to on/offswitching of nearby electric devices, scratches and defects on damagedrecords, click sounds from a computer keyboard, etc.

The noise pulseremoval methods considered in this chapter are based on the observationthat transient noise pulses can be regarded as the response of thecommunication channel, or the playback system, to an impulse. In thischapter, we study the characteristics of transient noise pulses and considera template-based method, a linear predictive model and a hidden Markovmodel for the modelling and removal of transient noise pulses. The subjectof this chapter closely follows that of Chapter 12 on impulsive noise.Transient Noise Waveforms37913.1 Transient Noise WaveformsTransient noise pulses often consist of a relatively short sharp initial pulsefollowed by decaying low-frequency oscillations as shown in Figure 13.1.The initial pulse is usually due to some external or internal impulsiveinterference, whereas the oscillations are often due to the resonance of thecommunication channel excited by the initial pulse, and may be consideredas the response of the channel to the initial pulse.

In a telecommunicationsystem, a noise pulse originates at some point in time and space, and thenpropagates through the channel to the receiver. The noise pulse is shapedby the channel characteristics, and may be considered as the channel pulseresponse. Thus we expect to be able to characterize the transient noisepulses with a similar degree of consistency to that of characterizing thechannels through which the pulses propagate.As an illustration of the distribution of a transient noise pulse in timeand frequency, consider the scratch pulses from a damaged gramophonerecord shown in Figures 13.1 and 13.2.

Scratch noise pulses are acousticmanifestations of the response of the stylus and the associated electromechanical playback system to a sharp physical discontinuity on therecording medium. Since scratches are essentially the impulse response ofthe playback mechanism, it is expected that for a given system, variousscratch pulses exhibit a similar characteristics. As shown in Figure 13.1, atypical scratch waveform often exhibits two distinct regions:(a) the initial high-amplitude pulse response of the playback system tothe physical discontinuity on the record medium; this is followed by(b) decaying oscillations that cause additive distortion.The initial pulse is relatively short and has a duration on the order of 1–5ms, whereas the oscillatory tail has a longer duration and may last up to 50ms. Note in Figure 13.1 that the frequency of the decaying oscillationsdecreases with time.

This behaviour may be attributed to the nonlinearmodes of response of the electro-mechanical playback system excited by thephysical scratch discontinuity. Observations of many scratch waveformsfrom damaged gramophone records reveal that they have a well-definedprofile, and can be characterised by a relatively small number of typicaltemplates.Transient Noise Pulses380n(m)mFigure 13.1 The profile of a transient noise pulse from a scratched gramophonerecord.(a)(b)Figure 13.2 An example of (a) the time-domain waveform and (b) the spectrogramof transient noise scratch pulses in a damaged gramophone record.Transient Noise Pulse Models381A similar argument can be used to describe the transient noise pulses inother systems as the response of the system to an impulsive noise.

Figure13.2(a) (b) show the time-domain waveform and the spectrogram of asection of music and song with scratch-type noise. Note that as the scratchdefect on the record was radial, the scratch pulses occure periodically with aperiod of 78 pulses per scratch per minute. As can be seen, there were in facttwo scratches on the record.The observation that transient noise pulses exhibit certain distinct,definable and consistent characteristics can be used for the modellingdetection and removal of transient noise pulses.13.2 Transient Noise Pulse ModelsTo a first approximation, a transient noise pulse n(m) can be modelled asthe impulse response of a linear time-invariant filter model of the channelasn(m) = ∑ hk Aδ (m − k ) = Ahm(13.1)kwhere A is the amplitude of the driving impulse and hk is the channelimpulse response.

A burst of overlapping, or closely spaced, noise pulsescan be modelled as the response of a channel to a sequence of impulses as()n(m) = ∑ hk ∑ A j δ (m − T j ) − k = ∑ A j hm−T jkj(13.2)jwhere it is assumed that the jth transient pulse is due to an impulse ofamplitude Aj at time Tj. In practice, a noise model should be able to dealwith the statistical variations of a variety of noise and channel types.

In thissection, we consider three methods for modelling the temporal, spectraland durational characteristics of a transient noise pulse process:(a) a template-based model;(b) a linear-predictive model;(c) a hidden Markov model.382Transient Noise Pulses13.2.1 Noise Pulse TemplatesA widely used method for modelling the space of a random process is tomodel the process as a collection of signal clusters, and to design a codebook of templates containing the “centroids” of the clusters. The centroidsrepresent various typical forms of the process. To obtain the centroids, thesignal space is partitioned into a number of regions or clusters, and the“centre” of the space within each cluster is taken as a centroid of the signalprocess.Similarly, a code book of transient noise pulses can be designed bycollecting a large number of training examples of the noise, and then usinga clustering technique to group, or partition, the noise database into anumber of clusters of noise pulses.

The centre of each cluster is taken as acentroid of the noise space. Clustering techniques can be used to obtain anumber of prototype templates for the characterisation of a set of transientnoise pulses. The clustering of a noise process is based on a set of noisefeatures that best characterise the noise. Features derived from themagnitude spectrum are commonly used for the characterisation of manyrandom processes. For transient noise pulses, the most important featuresare the pulse shape, the temporal–spectral characteristics of the pulse, thepulse duration and the pulse energy profile. Figure 13.3 shows a number oftypical noise pulses. The design of a code book of signal templates isdescribed in Chapter 4.n(m)n(m)mmn(m)n(m)mFigure 13.3 A number of prototype transient pulses.m383Transient Noise Pulse Models13.2.2 Autoregressive Model of Transient Noise PulsesModel-based methods have the advantage over template-based methodsthat overlapped noise pulses can be modelled as the response of the modelto a number of closely spaced impulsive inputs.

In this section, we consideran autoregressive (AR) model of transient noise pulses. The AR model fora single noise pulse n(m) can be described asPn(m) = ∑ c k n(m − k )+ Aδ (m)(13.3)k =1where ck are the AR model coefficients, and the excitation is an impulsefunction δ(m) of amplitude A. A number of closely spaced and overlappingtransient noise pulses can be modelled as the response of the AR model toa sequence of impulses:PMk =1jn ( m) = ∑ c k n ( m − k ) + ∑ A j δ ( m − T j )(13.4)where it is assumed that Tj is the start of the jth pulse in a burst of Mexcitation pulses.An improved AR model for transient noise, proposed by Godsill, isdriven by a two-state excitation: in the state S0, the excitation is a zeromean Gaussian process of small variance σ 02 , and in the state S1, theexcitation is a zero-mean Gaussian process of relatively larger varianceσ12 >> σ 02 .

In the state S1 a short-duration, and relatively large-amplitude,excitation generates a linear model of the transient noise pulse. In the stateS0 the model generates a low-amplitude excitation that partially models theinaccuracies of approximating a transient noise pulse by a linear predictivemodel. The binary-state excitation signal can be expressed as[]en (m) = σ 1b(m) + σ 0 b (m) u (m)(13.5)where u(m) is an uncorrelated zero-mean unit-variance Gaussian process,and b(m) indicates the state of the excitation signal: b(m)=1 indicates thatthe excitation has a variance of σ12 , and b(m)=0 (or its binary complement384Transient Noise Pulsesb (m)= 1 ) indicates the excitation has a smaller variance of σ 02 .

The timevarying variance of en(m) can be expressed asσ e2n (m) = σ 12 b(m) + σ 02 b (m)(13.6)Assuming that the excitation pattern b(m) is given, and that the excitationamplitude is Gaussian, the pdf of an N-sample long noise pulse n is givenby1 1 T T −1f N ( n) =exp−ΛnCCnee(13.7)nn1/ 2N /22Λ(2π )en e nwhere C is a matrix of coefficients of the AR model of the noise (asdescribed in Section 8.4), and Λ enen is the diagonal covariance matrix ofthe input to the noise model.