Impulsive Noise (779807)
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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)12IMPULSIVE NOISE12.1Impulsive Noise12.2Statistical Models for Impulsive Noise12.3Median Filters12.4Impulsive Noise Removal Using Linear Prediction Models12.5Robust Parameter Estimation12.6Restoration of Archived Gramophone Records12.7SummaryImpulsive noise consists of relatively short duration “on/off” noisepulses, caused by a variety of sources, such as switching noise, adversechannel environments in a communication system, dropouts or surfacedegradation of audio recordings, clicks from computer keyboards, etc. Animpulsive noise filter can be used for enhancing the quality andintelligibility of noisy signals, and for achieving robustness in patternrecognition and adaptive control systems.
This chapter begins with a studyof the frequency/time characteristics of impulsive noise, and then proceedsto consider several methods for statistical modelling of an impulsive noiseprocess. The classical method for removal of impulsive noise is the medianfilter. However, the median filter often results in some signal degradation.For optimal performance, an impulsive noise removal system should utilise(a) the distinct features of the noise and the signal in the time and/orfrequency domains, (b) the statistics of the signal and the noise processes,and (c) a model of the physiology of the signal and noise generation.
Wedescribe a model-based system that detects each impulsive noise, and thenproceeds to replace the samples obliterated by an impulse. We also considersome methods for introducing robustness to impulsive noise in parameterestimation.356Impulsive Noise12.1 Impulsive NoiseIn this section, first the mathematical concepts of an analog and a digitalimpulse are introduced, and then the various forms of real impulsive noisein communication systems are considered.The mathematical concept of an analog impulse is illustrated in Figure12.1. Consider the unit-area pulse p(t) shown in Figure 12.1(a).
As the pulsewidth ∆ tends to zero, the pulse tends to an impulse. The impulse functionshown in Figure 12.1(b) is defined as a pulse with an infinitesimal timewidth as1 / û, t ≤ û / 2δ (t ) = limit p(t ) = (12.1)û→00,tû/2>The integral of the impulse function is given by∞1∫ δ (t ) dt = û× û =1(12.2)−∞The Fourier transform of the impulse function is obtained as∞û( f ) = ∫ δ (t )e − j 2πft dt = e0 =1(12.3)−∞where f is the frequency variable. The impulse function is used as a testfunction to obtain the impulse response of a system.
This is because asδ(t)p(t)1/∆∆(a)As ∆∆(f)0tt(b)f(c)Figure 12.1 (a) A unit-area pulse, (b) The pulse becomes an impulse as(c) The spectrum of the impulse function.û→0,357Impulsive Noiseshown in Figure 12.1(c), an impulse is a spectrally rich signal containing allfrequencies in equal amounts.A digital impulse δ (m) , shown Figure 12.2(a), is defined as a signalwith an “on” duration of one sample, and is expressed as: 1,δ ( m) = 0,m=0m≠0(12.4)where the variable m designates the discrete-time index. Using the Fouriertransform relation, the frequency spectrum of a digital impulse is given by∆( f ) =∞∑ δ (m)e− j 2πfm =1.0,− ∞< f <∞(12.5)m = −∞In communication systems, real impulsive-type noise has a duration that isnormally more than one sample long. For example, in the context of audiosignals, short-duration, sharp pulses, of up to 3 milliseconds (60 samples ata 20 kHz sampling rate) may be considered as impulsive-type noise.
Figures12.1(b) and 12.1(c) illustrate two examples of short-duration pulses andtheir respective spectra.ni1(m) =δ (m)Ni1 (f)⇔(a)mfni2(m)Ni2 (f)⇔(b)mfni3(m)Ni3 (f)⇔(c)mfFigure 12.2 Time and frequency sketches of (a) an ideal impulse, and (b) and (c)short-duration pulses.358Impulsive Noiseni1(m)ni2(m)mm(a)ni3(m)(b)m(c)Figure 12.3 Illustration of variations of the impulse response of a non-linearsystem with increasing amplitude of the impulse.In a communication system, an impulsive noise originates at somepoint in time and space, and then propagates through the channel to thereceiver.
The received noise is shaped by the channel, and can beconsidered as the channel impulse response. In general, the characteristicsof a communication channel may be linear or non-linear, stationary or timevarying. Furthermore, many communication systems, in response to alarge-amplitude impulse, exhibit a nonlinear characteristic.Figure 12.3 illustrates some examples of impulsive noise, typical ofthose observed on an old gramophone recording. In this case, thecommunication channel is the playback system, and may be assumed timeinvariant.
The figure also shows some variations of the channelcharacteristics with the amplitude of impulsive noise. These variations maybe attributed to the non-linear characteristics of the playback mechanism.An important consideration in the development of a noiseprocessing system is the choice of an appropriate domain (time or thefrequency) for signal representation. The choice should depend on thespecific objective of the system.
In signal restoration, the objective is toseparate the noise from the signal, and the representation domain must bethe one that emphasises the distinguishing features of the signal and thenoise. Impulsive noise is normally more distinct and detectable in the timedomain than in the frequency domain, and it is appropriate to use timedomain signal processing for noise detection and removal.
In signalclassification and parameter estimation, the objective may be to compensatefor the average effects of the noise over a number of samples, and in somecases, it may be more appropriate to process the impulsive noise in thefrequency domain where the effect of noise is a change in the mean of thepower spectrum of the signal.359Impulsive Noise12.1.1 Autocorrelation and Power Spectrum of Impulsive NoiseImpulsive noise is a non-stationary, binary-state sequence of impulses withrandom amplitudes and random positions of occurrence. The non-stationarynature of impulsive noise can be seen by considering the power spectrum ofa noise process with a few impulses per second: when the noise is absentthe process has zero power, and when an impulse is present the noise poweris the power of the impulse. Therefore the power spectrum and hence theautocorrelation of an impulsive noise is a binary state, time-varying process.An impulsive noise sequence can be modelled as an amplitude-modulatedbinary-state sequence, and expressed asni (m) = n(m) b(m)(12.6)where b(m) is a binary-state random sequence of ones and zeros, and n(m)is a random noise process.
Assuming that impulsive noise is an uncorrelatedrandom process, the autocorrelation of impulsive noise may be defined as abinary-state process:rnn (k, m) = E[ni (m)ni (m + k )] = σ n2 δ (k )b(m)(12.7)where δ(k) is the Kronecker delta function. Since it is assumed that thenoise is an uncorrelated process, the autocorrelation is zero for k ≠ 0 ,therefore Equation (12.7) may be written asrnn (0, m) = σ n2 b(m)(12.8)Note that for a zero-mean noise process, rnn(0,m) is the time-varyingbinary-state noise power.
The power spectrum of an impulsive noisesequence is obtained, by taking the Fourier transform of the autocorrelationfunction Equation (12.8), asPN I N I ( f, m) =σ n2 b(m)(12.9)In Equation (12.8) and (12.9) the autocorrelation and power spectrum areexpressed as binary state functions that depend on the “on/off” state ofimpulsive noise at time m.360Impulsive Noise12.2 Statistical Models for Impulsive NoiseIn this section, we study a number of statistical models for thecharacterisation of an impulsive noise process. An impulsive noisesequence ni(m) consists of short duration pulses of a random amplitude,duration, and time of occurrence, and may be modelled as the output of afilter excited by an amplitude-modulated random binary sequence asP −1ni (m) = ∑ hk n(m − k )b(m − k )(12.10)k =0Figure 12.4 illustrates the impulsive noise model of Equation (12.10).
InEquation (12.10) b(m) is a binary-valued random sequence model of thetime of occurrence of impulsive noise, n(m) is a continuous-valued randomprocess model of impulse amplitude, and h(m) is the impulse response of afilter that models the duration and shape of each impulse. Two importantstatistical processes for modelling impulsive noise as an amplitudemodulated binary sequence are the Bernoulli-Gaussian process and thePoisson–Gaussian process, which are discussed next.12.2.1 Bernoulli–Gaussian Model of Impulsive NoiseIn a Bernoulli-Gaussian model of an impulsive noise process, the randomtime of occurrence of the impulses is modelled by a binary Bernoulliprocess b(m) and the amplitude of the impulses is modelled by a GaussianBinary sequence b(m)Amplitude modulatedbinary sequencen(m) b(m)h(m)Impulsive noisesequence nI(m)Amplitude modulatingsequence n(m)Impulse shapingfilterFigure 12.4 Illustration of an impulsive noise model as the output of a filterexcited by an amplitude-modulated binary sequence.361Statistical Models for Impulsive Noiseprocess n(m).
A Bernoulli process b(m) is a binary-valued process that takesa value of “1” with a probability of α and a value of “0” with a probabilityof 1–α. Τhe probability mass function of a Bernoulli process is given byαPB (b(m) )= 1 − αfor b(m)=1for b(m)=0.(12.11)A Bernoulli process has a meanµ b = E [(b(m) )] =α(12.12)and a variance()σ b2 = E b(m) − µ b 2 =α (1 − α )(12.13)A zero-mean Gaussian pdf model of the random amplitudes of impulsivenoise is given byf N (n(m) )= n 2 ( m) exp −2 2π σ n 2σ n 1(12.14)where σ n2 is the variance of the noise amplitude.
In a Bernoulli–Gaussianmodel the probability density function of an impulsive noise ni(m) is givenbyf NBG (ni (m) )= (1 − α )δ (ni (m) )+α f N (ni (m) )(12.15)where δ (ni (m) ) is the Kronecker delta function. Note that the functionf NBG (ni (m) ) is a mixture of a discrete probability mass function δ (ni (m) )and a continuous probability density function f N (ni (m) ) .An alternative model for impulsive noise is a binary-state Gaussianprocess (Section 2.5.4), with a low-variance state modelling the absence ofimpulses and a relatively high-variance state modelling the amplitude ofimpulsive noise.362Impulsive Noise12.2.2 Poisson–Gaussian Model of Impulsive NoiseIn a Poisson–Gaussian model the probability of occurrence of an impulsivenoise event is modelled by a Poisson process, and the distribution of therandom amplitude of impulsive noise is modelled by a Gaussian process.The Poisson process, described in Chapter 2, is a random event-countingprocess.
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