Spectral Subtraction (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 2

The filter acts as a SNR-dependentattenuator. The attenuation at each frequency increases with the decreasingSNR, and conversely decreases with the increasing SNR.The least mean square error linear filter for noise removal is the Wienerfilter covered in chapter 6. Implementation of a Wiener filter requires thepower spectra (or equivalently the correlation functions) of the signal andthe noise process, as discussed in Chapter 6. Spectral subtraction is used as asubstitute for the Wiener filter when the signal power spectrum is notavailable.

In this section, we discuss the close relation between the Wienerfilter and spectral subtraction. For restoration of a signal observed inuncorrelated additive noise, the equation describing the frequency responseof the Wiener filter was derived in Chapter 6 asW( f )=E [| Y ( f ) | 2 ] − E [| N ( f ) | 2 ]E [| Y ( f ) | 2 ](11.18)340Spectral SubtractionA comparison of W(f) and H(f), from Equations (11.18) and (11.17), showsthat the Wiener filter is based on the ensemble-average spectra of the signaland the noise, whereas the spectral subtraction filter uses the instantaneousspectra of the noisy signal and the time-averaged spectra of the noise. Inspectral subtraction, we only have access to a single realisation of theprocess. However, assuming that the signal and noise are wide-sensestationary ergodic processes, we may replace the instantaneous noisy signalspectrum | Y ( f ) | 2 in the spectral subtraction equation (11.18) with the timeaveraged spectrum | Y ( f ) | 2 , to obtainH( f )=| Y ( f ) |2 −| N ( f ) |2| Y ( f ) |2(11.19)For an ergodic process, as the length of the time over which the signals areaveraged increases, the time-averaged spectrum approaches the ensembleaveraged spectrum, and in the limit, the spectral subtraction filter ofEquation (11.19) approaches the Wiener filter equation (11.18).

In practice,many signals, such as speech and music, are non-stationary, and only alimited degree of beneficial time-averaging of the spectral parameters can beexpected.11.2 Processing DistortionsThe main problem in spectral subtraction is the non-linear processingdistortions caused by the random variations of the noise spectrum. FromEquation (11.12) and the constraint that the magnitude spectrum must havea non-negative value, we may identify three sources of distortions of theinstantaneous estimate of the magnitude or power spectrum as:(a) the variations of the instantaneous noise power spectrum about themean;(b) the signal and noise cross-product terms;(c) the non-linear mapping of the spectral estimates that fall below athreshold.The same sources of distortions appear in both the magnitude and the powerspectrum subtraction methods.

Of the three sources of distortions listed341Processing Distortions|y(f)|Distortion in the form of asharp trough in signal spectra.Distortions in the form ofIsolated “musical” noise.fFigure 11.3 Illustration of distortions that may result from spectral subtraction.above, the dominant distortion is often due to the non-linear mapping of thenegative, or small-valued, spectral estimates. This distortion produces ametallic sounding noise, known as “musical tone noise” due to their narrowband spectrum and the tin-like sound. The success of spectral subtractiondepends on the ability of the algorithm to reduce the noise variations and toremove the processing distortions.

In its worst, and not uncommon, case theresidual noise can have the following two forms:(a) a sharp trough or peak in the signal spectra;(b) isolated narrow bands of frequencies.In the vicinity of a high amplitude signal frequency, the noise-inducedtrough or peak is often masked, and made inaudible, by the high signalenergy. The main cause of audible degradations is the isolated frequencycomponents also known as musical tones or musical noise illustrated inFigure 11.3.

The musical noise is characterised as short-lived narrow bandsof frequencies surrounded by relatively low-level frequency components. Inaudio signal restoration, the distortion caused by spectral subtraction canresult in a significant deterioration of the signal quality. This is particularlytrue at low signal-to-noise ratios. The effects of a bad implementation ofsubtraction algorithm can result in a signal that is of a lower perceivedquality, and lower information content, than the original noisy signal.342Spectral SubtractionNoisy signal spaceNoise-free signal space(a)fhfhfhfl(b)flAfter subtraction ofthe noise mean(c)flNoise inducedchange in the meanFigure 11.4 Illustration of the distorting effect of spectral subtraction on the space ofthe magnitude spectrum of a signal.11.2.1 Effect of Spectral Subtraction on Signal DistributionFigure 11.4 is an illustration of the distorting effect of spectral subtractionon the distribution of the magnitude spectrum of a signal.

In this figure, wehave considered the simple case where the spectrum of a signal is dividedinto two parts; a low-frequency band fl and a high-frequency band fh. Eachpoint in Figure 11.4 is a plot of the high-frequency spectrum versus the lowfrequency spectrum, in a two-dimensional signal space. Figure 11.4(a)shows an assumed distribution of the spectral samples of a signal in the twodimensional magnitude–frequency space. The effect of the random noise,shown in Figure 11.4(b), is an increase in the mean and the variance of thespectrum, by an amount that depends on the mean and the variance of themagnitude spectrum of the noise. The increase in the variance constitutes anirrevocable distortion. The increase in the mean of the magnitude spectrumcan be removed through spectral subtraction. Figure 11.4(c) illustrates thedistorting effect of spectral subtraction on the distribution of the signalspectrum.

As shown, owing to the noise-induced increase in the variance ofthe signal spectrum, after subtraction of the average noise spectrum, aproportion of the signal population, particularly those with a low SNR,become negative and have to be mapped to non-negative values. As shownthis process distorts the distribution of the low-SNR part of the signalspectrum.Processing Distortions34311.2.2 Reducing the Noise VarianceThe distortions that result from spectral subtraction are due to the variationsof the noise spectrum. In Section 9.2 we considered the methods of reducingthe variance of the estimate of a power spectrum.

For a white noise processwith variance σ n2 , it can be shown that the variance of the DFT spectrum ofthe noise N(f) is given by2Var [ | N ( f ) | 2 ] ≈ PNN( f )=σ n4(11.20)and the variance of the running average of K independent spectralcomponents is 1 K −1 1 21(11.21)Var  ∑ | N i ( f ) | 2  ≈ PNN( f )≈ σ n4K K i =0 KFrom Equation (11.21), the noise variations can be reduced by timeaveraging of the noisy signal frequency components.

The fundamentallimitation is that the averaging process, in addition to reducing the noisevariance, also has the undesirable effect of smearing and blurring the timevariations of the signal spectrum. Therefore an averaging process shouldreflect a compromise between the conflicting requirements of reducing thenoise variance and of retaining the time resolution of the non-stationaryspectral events. This is important because time resolution plays an importantpart in both the quality and the intelligibility of audio signals.In spectral subtraction, the noisy signal y(m) is segmented into blocksof N samples. Each signal block is then transformed via a DFT into a blockof N spectral samples Y(f). Successive blocks of spectral samples form atwo-dimensional frequency–time matrix denoted by Y(f,t) where the variablet is the segment index and denotes the time dimension. The signal Y(f,t) canbe considered as a band-pass channel f that contains a time-varying signalX(f,t) plus a random noise component N(f,t).

One method for reducing thenoise variations is to low-pass filter the magnitude spectrum at eachfrequency. A simple recursive first-order digital low-pass filter is given by| Y LP ( f , t ) | = ρ | Y LP ( f , t − 1) | + (1− ρ )| Y ( f , t ) |(11.22)where the subscript LP denotes the output of the low-pass filter, and thesmoothing coefficient ρ controls the bandwidth and the time constant of thelow-pass filter.344Spectral SubtractionSpectral magnitudeThreshold levelTimeWindow lengthSliding window: Deleted: SurviveFigure 11.5 Illustration of a method for identification and filtering of “musical noise”.11.2.3 Filtering Out the Processing DistortionsAudio signals, such as speech and music, are composed of sequences ofnon-stationary acoustic events.

The acoustic events are “born”, have avarying lifetime, disappear, and then reappear with a different intensity andspectral composition. The time–varying nature of audio signals plays animportant role in conveying information, sensation and quality. The musicaltone noise, introduced as an undesirable by-product of spectral subtraction,is also time-varying. However, there are significant differences between thecharacteristics of most audio signals and so-called musical noise.

Thecharacteristic differences may be used to identify and remove some of themore annoying distortions. Identification of musical noise may be achievedby examining the variations of the signal in the time and frequency domains.The main characteristics of musical noise are that it tends to be relativelyshort-lived random isolated bursts of narrow band signals, with relativelysmall amplitudes.Using a DFT block size of 128 samples, at a sampling rate of 20 kHz,experiments indicate that the great majority of musical noise tends to last nomore than three frames, whereas genuine signal frequencies have aconsiderably longer duration. This observation was used as the basis of aneffective “musical noise” suppression system. Figure 11.5 demonstrates amethod for the identification of musical noise.

Each DFT channel isexamined to identify short-lived frequency events. If a frequency componenthas a duration shorter than a pre-selected time window, and an amplitudesmaller than a threshold, and is not masked by signal components in theadjacent frequency bins, then it is classified as distortion and deleted.Non-Linear Spectral Subtraction34511.3 Non-Linear Spectral SubtractionThe use of spectral subtraction in its basic form of Equation (11.5) maycause deterioration in the quality and the information content of a signal.For example, in audio signal restoration, the musical noise can causedegradation in the perceived quality of the signal, and in speech recognitionthe basic spectral subtraction can result in deterioration of the recognitionaccuracy. In the literature, there are a number of variants of spectralsubtraction that aim to provide consistent performance improvement acrossa range of SNRs.