Transient Noise Pulses (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 3

The adaptivescaling coefficient w is estimated as follows. The correlation of the noisysignal y(m) with the delayed noise pulse template n (m − D) givesN −1N −1m=0m =0∑ y (m)n (m − D) = ∑ [ x(m)+wn (m − D)]n (m − D)N −1N −1m =0m =0(13.22)= ∑ x ( m ) n ( m − D ) + w∑ n ( m − D ) n ( m − D )where N is the pulse template length. Since the signal x(m) and the noisen(m) are uncorrelated, the term Σ x(m) n (m − D) on the right hand side ofEquation (13.22) is small, and we have392Transient Noise Pulsesw≈∑ x ( m) n ( m − D )m∑ n 2 (m − D)(13.23)mNote when a false detection of a noise pulse occurs, the cross-correlationterm and hence the adaptation coefficient w could be small.

This will keepthe signal distortion resulting from false detections to a minimum.Samples that are irrevocably distorted by the initial scratch pulse arediscarded and replaced by one of the signal interpolators introduced inChapter 10. When there is no noise pulse, the coefficient w is zero, theinterpolator is bypassed and the input signal is passed through unmodified.Figure 13.7(b) shows the result of processing the noisy signal of Figure13.7(a). The linear oscillatory noise is completely removed by the adaptivesubtraction method. For this signal 80 samples irrevocably distorted by theinitial scratch pulse were discarded and interpolated.13.4.2 AR-based Restoration of Signals Distorted by NoisePulsesA model-based approach to noise detection/removal provides a morecompact method for characterisation of transient noise pulses, and has theadvantage that closely spaced pulses can be modelled as the response of themodel to a number of closely spaced input impulses.

The signal x(m) ismodelled as the output of an AR model of order P1 asP1x ( m ) = ∑ a k x ( m − k ) + e( m )(13.24)k =1Assuming that e(m) is a zero-mean uncorrelated Gaussian process withvariance σ e2 , the pdf of a vector x of N successive signal samples of anautoregressive process with parameter vector a is given byf X ( x) = − 1 x T A T Ax exp 2σ 2(2πσ e2 ) N / 2e1(13.25)393Removal of Noise Pulse Distortionswhere the elements of the matrix A are composed of the coefficients ak ofthe linear predictor model as described in Section 8.4. In Equation (13.25),it is assumed that the P1 initial samples are known.

The AR model for asingle noise pulse waveform n(m) can be written asP2n(m) = ∑ c k n(m − k ) + Aδ (m)(13.26)k =1where ck are the model coefficients, P2 is the model order, and theexcitation is a assumed to be an impulse of amplitude A. A number ofclosely spaced and overlapping noise pulses can be modelled asP2Mk =1jn( m ) = ∑ a k n( m − k ) + ∑ A j δ ( m − T j )(13.27)where it is assumed that Tk is the start of the kth excitation pulse in a burstof M pulses. A linear predictor model proposed by Godsill is driven by abinary-state excitation. The excitation waveform has two states: in state“0”, the excitation is a zero-mean Gaussian process of variance σ 02 , and instate “1”, the excitation is a zero-mean Gaussian process of varianceσ12 >> σ 02 . In state “1”, the model generates a short-duration largeamplitude excitation that largely models the transient pulse. In state “0”,the model generates a low excitation that partially models the inaccuraciesof approximating a nonlinear system by an AR model.

The compositeexcitation signal can be written as[]en (m) = b(m)σ 1 +b (m)σ 0 u (m)(13.28)where u(m) is an uncorrelated zero-mean Gaussian process of unit variance,b(m) is a binary sequence that indicates the state of the excitation, andb (m) is the binary complement of b(m). When b(m)=1 the excitation2variance is σ12 and when b(m)=0, the excitation variance is σ 0 . Thebinary-state variance of en(m) can be expressed asσ e2n (m) = b(m)σ 12 + b (m)σ 02(13.29)394Transient Noise PulsesAssuming that the excitation pattern b=[b(m)] is given, the pdf of an Nsample noise pulse x isf N ( n | b )=1(2π ) N / 2 Λenen 1exp  − n T C T Λe−n1en C n 1/ 2 2(13.30)where the elements of the matrix C are composed of the coefficients ck ofthe linear predictor model as described in Section 8.4.

The posterior pdf ofthe signal x given the noisy observation y, fX|Y(x|y),can be expressed, usingBayes’ rule, as1f Y X (y| x) f X (x)f Y (y) |1f N (y − x) f X (x)=f Y (y)f X | Y (x| y) =(13.31)For a given observation fY(y) is a constant. Substitution of Equations(13.30) and (13.25) in Equation (13.31) yieldsf X |Y ( x y ) =11f Y ( y ) (2πσ ) N Λeen en1/ 2 11× exp  − ( y − x )T C T Λe−1e C ( y − x ) −x T A T Ax  2n n2σ e2(13.32)The MAP solution obtained by maximisation of the log posterior functionwith respect to the undistorted signal x is given by(xˆ MAP = A T A σ e2 + C T Λe−n1en C)−1C T Λe−n1en C y(13.33)Summary39513.5 SummaryIn this chapter, we considered the modelling, detection and removal oftransient noise pulses.

Transient noise pulses are non-stationary eventssimilar to impulsive noise, but usually occur less frequently and have alonger duration than impulsive noise. An important observation in themodelling of transient noise is that the noise can be regarded as the impulseresponse of a communication channel, and hence may be modelled by oneof a number of statistical methods used in the of modelling communicationchannels. In Section 13.2, we considered several transient noise pulsemodels including a template-based method, an AR model-based methodand a hidden Markov model. In Sections 13.2 and 13.3, these models wereapplied to the detection and removal of noise pulses.BibliographyGODSILL S.J.

(1993), The Restoration of Degraded Audio Signals. PhDThesis, Cambridge University.VASEGHI S.V. (1987), Algorithm for Restoration of Archived GramophoneRecordings. Ph.D. Thesis, Cambridge University..