Transient Noise Pulses (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 2

The diagonal elements of Λ enen are given byEquation (13.6).13.2.3 Hidden Markov Model of a Noise Pulse ProcessA hidden Markov model (HMM), described in Chapter 5, is a finite statestatistical model for non-stationary random processes such as speech ortransient noise pulses. In general, we may identify three distinct states for atransient noise pulse process:(a) the periods during which there are no noise pulses;(b) the initial, and often short and sharp, pulse of a transient noise;(c) the decaying oscillatory tail of a transient pulse.Figure 13.4 illustrates a three-state HMM of transient noise pulses.

Thestate S0 models the periods when the noise pulses are absent. In this state,the noise process may be zero-valued. This state can also be used to modela different noise process such as a white noise process. The state S1 modelsthe relatively sharp pulse that forms the initial part of many transient noisepulses. The state S2 models the decaying oscillatory part of a noise pulsethat usually follows the initial pulse of a transient noise.

A code book ofwaveforms in states S1 and S2 can model a variety of different noise pulses.Note that in the HMM model of Figure 13.4, the self-loop transition385Detection of Noise Pulsesa00S0a10a 02a 01 a12a 20S1S2a 21a11a 22Figure 13.4 A three-state model of a transient noise pulse process.provides a mechanism for the modelling of the variations in the duration ofeach noise pulse segment. The skip-state transitions provide a mechanismfor the modelling of those noise pulses that do not exhibit either the initialnon-linear pulse or the decaying oscillatory part.A hidden Markov model of noise can be employed for both thedetection and the removal of transient noise pulses.

As described in Section13.3.3, the maximum-likelihood state-sequence of the noise HMMprovides an estimate of the state of the noise at each time instant. Theestimates of the states of the signal and the noise can be used for theimplementation of an optimal state-dependent signal restoration algorithm.13.3 Detection of Noise PulsesFor the detection of a pulse process n(m) observed in an additive signalx(m), the signal and the pulse can be modelled asy(m)= b(m)n(m)+ x(m)(13.8)where b(m) is a binary “indicator” process that signals the presence orabsence of a noise pulse.

Using the model of Equation (13.8), the detectionof a noise pulse process can be considered as the estimation of theunderlying binary-state noise-indicator process b(m). In this section, we386Transient Noise Pulsesconsider three different methods for detection of transient noise pulses,using the noise template model within a matched filter, the linear predictivemodel of noise, and the hidden Markov model described in Section 13.2.13.3.1 Matched Filter for Noise Pulse DetectionThe inner product of two signal vectors provides a measure of thesimilarity of the signals.

Since filtering is basically an inner productoperation, it follows that the output of a filter should provide a measure ofsimilarity of the filter input and the filter impulse response. The classicalmethod for detection of a signal is to use a filter whose impulse response ismatched to the shape of the signal to be detected.

The derivation of amatched filter for the detection of a pulse n(m) is based on maximisation ofthe amplitude of the filter output when the input contains the pulse n(m).The matched filter for the detection of a pulse n(m) observed in a“background” signal x(m) is defined asH ( f )= KN*( f )PXX ( f )(13.9)where PXX(f) is the power spectrum of x(m) and N*(f) is the complexconjugate of the spectrum of the noise pulse. When the “background”signal process x(m) is a zero mean uncorrelated signal with variance σ 2x ,the matched filter for detection of the transient noise pulse n(m) becomesH ( f )=Kσ x2N*( f )(13.10)The impulse response of the matched filter corresponding to Equation(13.10) is given byh ( m) = C n( − m)(13.11)where the scaling factor C is given by C = K σ x2 . Let z(m) denote theoutput of the matched filter.

In response to an input noise pulse, the filteroutput is given by the convolution relationz ( m) = C n( − m ) ∗ n ( m)(13.12)387Detection of Noise Pulseswhere the asterisk * denotes convolution. In the frequency domainEquation (13.12) becomesZ ( f ) = N ( f )H ( f ) = C N ( f )(13.13)2The matched filter output z(m) is passed through a non-linearity and adecision is made on the presence or the absence of a noise pulse as1bˆ(m)=0if z (m) ≥ threshold(13.14)otherwiseIn Equation (13.14), when the matched filter output exceeds a threshold,the detector flags the presence of the signal at the input. Figure 13.5 showsa noise pulse detector composed of a bank of M different matched filters.The detector signals the presence or the absence of a noise pulse.

If a pulseis present then additional information provide the type of the pulse, themaximum cross-correlation of the input and the noise pulse template, and atime delay that can be used to align the input noise and the noise template.This information can be used for subtraction of the noise pulse from thenoisy signal as described in Section 13.4.1.Noise pulse+ signal...Pulse type MMaximum correlation detectorPulse type 1Pulsepresent/absentPulsetypePulsecorrelationPulsedelayFigure 13.5 A bank of matched filters for detection of transient noise pulses.388Transient Noise Pulses13.3.2 Noise Detection Based on Inverse FilteringThe initial part of a transient noise pulse is often a relatively short andsharp impulsive-type event, which can be used as a distinctive feature forthe detection of the noise pulses.

The detectibility of a sharp noise pulsen(m), observed in a correlated “background” signal y(m), can often beimproved by using a differencing operation, which has the effect ofenhancing the relative amplitude of the impulsive-type noise. Thedifferencing operation can be accomplished by an inverse linear predictormodel of the background signal y(m). An alternative interpretation is thatthe inverse filtering is equivalent to a spectral whitening operation: itaffects the energy of the signal spectrum whereas the theoretically flatspectrum of the impulsive noise is largely unaffected. The use of an inverselinear predictor for the detection of an impulsive-type event was consideredin detail in Section 12.4. Note that the inverse filtering operation reducesthe detection problem to that of detecting a pulse in additive white noise.13.3.3 Noise Detection Based on HMMIn the three-state hidden Markov model of a transient noise pulse process,described in Section 13.2.3, the states S0, S1 and S2 correspond to thenoise-absent state, the initial noise pulse state, and the decaying oscillatorynoise state respectively.

As described in Chapter 5, an HMM, denoted byM, is defined by a set of Markovian state transition probabilities andGaussian state observation pdfs. The statistical parameters of the HMM ofa noise pulse process can be obtained from a sufficiently large number oftraining examples of the process.Given an observation vector y=[y(0), y(1), ..., y(N–1)], the maximumlikelihood state sequence s=[s(0), s(1), ..., s(N–1)], of the HMM M isobtained ass ML = arg max f Y | S ( y | s, M )s(13.15)where, for a hidden Markov model, the likelihood of an observationsequence fY|S(y|s,λ) can be expressed asRemoval of Noise Pulse Distortions389f Y |S (y (0), y (1), , y ( N − 1) s (0),s (1), ,s ( N − 1) )= π s (0) f s ( 0) ( y (0) ).a s ( 0),s (1) f s (1) ( y (1) ) a s ( N −2),s ( N −1) f s ( N −1) ( y ( N − 1) )(13. 16)where π s(i) is the initial state probability, as(i ),s( j) is the probability of atransition from state s(i) to state s(j), and f s(i) ( y(i)) is the state observationpdf for the state s(i).

The maximum-likelihood state sequence sML, derivedusing the Viterbi algorithm, is an estimate of the underlying states of thenoise pulse process, and can be used as a detector of the presence orabsence of a noise pulse.13.4 Removal of Noise Pulse DistortionsIn this section, we consider two methods for the removal of transient noisepulses: (a) an adaptive noise subtraction method and (b) an autoregressive(AR) model-based restoration method. The noise removal methods assumethat a detector signals the presence or the absence of a noise pulse, andprovides additional information on the timing and the underlying the statesof the noise pulse13.4.1 Adaptive Subtraction of Noise PulsesThe transient noise removal system shown in Figure 13.6 is composed of amatched filter for detection of noise pulses, a linear adaptive noisesubtractor for cancellation of the linear transitory part of a noise pulse, andan interpolator for the replacement of samples irrevocably distorted by theinitial part of each pulse.

Let x(m), n(m) and y(m) denote the signal, thenoise pulse and the noisy signal respectively; the noisy signal model isy(m)= x(m) + b(m) n(m)(13.17)where the binary indicator sequence b(m) indicates the presence or theabsence of a noise pulse. Assume that each noise pulse n(m) can bemodelled as the amplitude-scaled and time-shifted version of the noisepulse template n (m) so that390Transient Noise PulsesSignal estimateSignal + noise pulsey(m) = x(m) + n(m)Interpolator1 : Noise pulse present0 : Noise pulse absentMatched filterdetectorw =Noise pulsetemplate^x(m)rsyryyDelayFigure 13.6 Transient noise pulse removal system.n ( m)≈ w n ( m − D )(13.18)where w is an amplitude scalar and the integer D denotes the relative delay(time shift) between the noise pulse template and the detected noise. FromEquations (13.17) and (13.18) the noisy signal can be modelled:y (m)≈ x(m)+ wn (m − D)(13.19)From Equation (13.19) an estimate of the signal x(m) can be obtained bysubtracting an estimate of the noise pulse from that of the noisy signal:xˆ (m)= y (m)−wn (m − D)(13.20)where the time delay D required for time-alignment of the noisy signaly(m) and the noise template n (m) is obtained from the cross-correlationfunction CCF asD=arg max [CCF ( y (m), n (m − k ) )](13.21)kWhen a noise pulse is detected, the time lag corresponding to themaximum of the cross-correlation function is used to delay and time-alignthe noise pulse template with the noise pulse.

The template energy isadaptively matched to that of the noise pulse by an adaptive scalingcoefficient w. The scaled and time-aligned noise template is subtracted391Removal of Noise Pulse Distortions(a)(b)Figure 13.7 (a) A signal from an old gramophone record with a scratch noisepulse. (b) The restored signal.from the noisy signal to remove linear additive distortions.