Spectral Subtraction (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)Noisy signal spaceNoise-free signal space11fhfhfhflAfter subtraction ofthe noise meanflSPECTRAL SUBTRACTION11.1 Spectral Subtraction11.2 Processing Distortions11.3 Non-Linear Spectral Subtraction11.4 Implementation of Spectral Subtraction11.5 SummarySpectral subtraction is a method for restoration of the power spectrumor the magnitude spectrum of a signal observed in additive noise,through subtraction of an estimate of the average noise spectrum fromthe noisy signal spectrum.

The noise spectrum is usually estimated, andupdated, from the periods when the signal is absent and only the noise ispresent. The assumption is that the noise is a stationary or a slowly varyingprocess, and that the noise spectrum does not change significantly inbetween the update periods.

For restoration of time-domain signals, anestimate of the instantaneous magnitude spectrum is combined with thephase of the noisy signal, and then transformed via an inverse discreteFourier transform to the time domain. In terms of computationalcomplexity, spectral subtraction is relatively inexpensive. However, owingto random variations of noise, spectral subtraction can result in negativeestimates of the short-time magnitude or power spectrum.

The magnitudeand power spectrum are non-negative variables, and any negative estimatesof these variables should be mapped into non-negative values. This nonlinear rectification process distorts the distribution of the restored signal.The processing distortion becomes more noticeable as the signal-to-noiseratio decreases. In this chapter, we study spectral subtraction, and thedifferent methods of reducing and removing the processing distortions.fl334Spectral Subtraction11.1 Spectral SubtractionIn applications where, in addition to the noisy signal, the noise is accessibleon a separate channel, it may be possible to retrieve the signal by subtractingan estimate of the noise from the noisy signal.

For example, the adaptivenoise canceller of Section 1.3.1 takes as the inputs the noise and the noisysignal, and outputs an estimate of the clean signal. However, in manyapplications, such as at the receiver of a noisy communication channel, theonly signal that is available is the noisy signal. In these situations, it is notpossible to cancel out the random noise, but it may be possible to reduce theaverage effects of the noise on the signal spectrum. The effect of additivenoise on the magnitude spectrum of a signal is to increase the mean and thevariance of the spectrum as illustrated in Figure 11.1. The increase in thevariance of the signal spectrum results from the random fluctuations of thenoise, and cannot be cancelled out.

The increase in the mean of the signalspectrum can be removed by subtraction of an estimate of the mean of thenoise spectrum from the noisy signal spectrum. The noisy signal model inthe time domain is given byy(m)= x(m) + n(m)56 x106 x104422500-2-2-4-4-602004006008001000-6012002020151510105500(11.1)20040060080010001200050100150200250050100150200250Figure 11.1 Illustrations of the effect of noise on a signal in the time and thefrequency domains.335Spectral Subtractionwhere y(m), x(m) and n(m) are the signal, the additive noise and the noisysignal respectively, and m is the discrete time index. In the frequencydomain, the noisy signal model of Equation (11.1) is expressed asY ( f )= X ( f )+ N ( f )(11.2)where Y(f), X(f) and N(f) are the Fourier transforms of the noisy signal y(m),the original signal x(m) and the noise n(m) respectively, and f is thefrequency variable.

In spectral subtraction, the incoming signal x(m) isbuffered and divided into segments of N samples length. Each segment iswindowed, using a Hanning or a Hamming window, and then transformedvia discrete Fourier transform (DFT) to N spectral samples. The windowsalleviate the effects of the discontinuities at the endpoints of each segment.The windowed signal is given byyw ( m) = w (m) y( m)= w(m)[ x (m)+ n(m)]= x w ( m)+ nw ( m)(11.3)The windowing operation can be expressed in the frequency domain asYw ( f )=W ( f ) * Y ( f )= X w ( f )+ N w ( f )(11.4)where the operator * denotes convolution. Throughout this chapter, it isassumed that the signals are windowed, and hence for simplicity we dropthe use of the subscript w for windowed signals.Figure 11.2 illustrates a block diagram configuration of the spectralsubtraction method.

A more detailed implementation is described in Section11.4. The equation describing spectral subtraction may be expressed asbXˆ ( f ) = Y ( f ) b −α N ( f ) b(11.5)where | Xˆ ( f ) |b is an estimate of the original signal spectrum | X ( f ) |b and| N ( f ) |b is the time-averaged noise spectra. It is assumed that the noise is awide-sense stationary random process. For magnitude spectral subtraction,the exponent b=1, and for power spectral subtraction, b=2. The parameter α336Spectral SubtractionXˆ ( f )PostsubtractionprocessingY(f)y(m)DFTxˆ (m)IDFTNoise estimateFigure 11.2 A block diagram illustration of spectral subtraction.in Equation (11.5) controls the amount of noise subtracted from the noisysignal.

For full noise subtraction, α=1 and for over-subtraction α>1. Thetime-averaged noise spectrum is obtained from the periods when the signalis absent and only the noise is present as| N ( f ) |b =1KK −1∑ | N i ( f ) |b(11.6)i =0In Equation (11.6), |Ni(f)| is the spectrum of the ith noise frame, and it isassumed that there are K frames in a noise-only period, where K is avariable. Alternatively, the averaged noise spectrum can be obtained as theoutput of a first order digital low-pass filter as| N i ( f ) | b = ρ | N i −1 ( f ) | b + (1− ρ ) | N i ( f ) | b(11.7)where the low-pass filter coefficient ρ is typically set between 0.85 and0.99. For restoration of a time-domain signal, the magnitude spectrumestimate | Xˆ ( f ) | is combined with the phase of the noisy signal, and thentransformed into the time domain via the inverse discrete Fourier transformasxˆ ( m) =N −1∑| Xˆ ( k ) | e jθY ( k ) e− j2πkmN(11.8)k =0where θ Y (k) is the phase of the noisy signal frequency Y(k).

The signalrestoration equation (11.8) is based on the assumption that the audible noiseis mainly due to the distortion of the magnitude spectrum, and that the phasedistortion is largely inaudible. Evaluations of the perceptual effects ofsimulated phase distortions validate this assumption.337Spectral SubtractionOwing to the variations of the noise spectrum, spectral subtraction mayresult in negative estimates of the power or the magnitude spectrum. Thisoutcome is more probable as the signal-to-noise ratio (SNR) decreases. Toavoid negative magnitude estimates the spectral subtraction output is postprocessed using a mapping function T[·] of the form | Xˆ ( f ) |T [ | Xˆ ( f ) | ] =  fn[| Y ( f ) |]if | Xˆ ( f ) | > β | Y ( f ) |otherwise(11.9)For example, we may chose a rule such that if the estimate| Xˆ ( f ) | > 0.01 | Y ( f ) | (in magnitude spectrum 0.01 is equivalent to –40 dB)then | Xˆ ( f )| should be set to some function of the noisy signal fn[Y(f)].

In itssimplest form, fn[Y(f)]=noise floor, where the noise floor is a positiveconstant. An alternative choice is fn[|Y(f)|]=β |Y(f)|. In this case,| Xˆ ( f ) | if | Xˆ ( f ) | > β | Y ( f ) |T [ | Xˆ ( f ) | ] = otherwise β | Y ( f ) |(11.10)Spectral subtraction may be implemented in the power or the magnitudespectral domains. The two methods are similar, although theoretically theyresult in somewhat different expected performance.11.1.1 Power Spectrum SubtractionThe power spectrum subtraction, or squared-magnitudesubtraction, is defined by the following equation:| Xˆ ( f ) | 2 = | Y ( f ) | 2 − | N ( f ) | 2spectrum(11.11)where it is assumed that α, the subtraction factor in Equation (11.5), isunity.

We denote the power spectrum by E [| X ( f ) | 2 ] , the time-averaged2power spectrum by X ( f ) and the instantaneous power spectrum by2X ( f ) . By expanding the instantaneous power spectrum of the noisy338Spectral Subtraction2signal Y ( f ) , and grouping the appropriate terms, Equation (11.11) may berewritten as| Xˆ ( f ) | 2 =| X ( f )| 2 +  | N ( f ) | 2 − | N ( f ) | 2  + X * ( f ) N ( f ) + X ( f ) N * ( f ) Noise variationsCross products(11.12)Taking the expectations of both sides of Equation (11.12), and assumingthat the signal and the noise are uncorrelated ergodic processes, we haveE [| Xˆ ( f ) | 2 ] = E [| X ( f ) | 2 ](11.13)From Equation (11.13), the average of the estimate of the instantaneouspower spectrum converges to the power spectrum of the noise-free signal.However, it must be noted that for non-stationary signals, such as speech,the objective is to recover the instantaneous or the short-time spectrum, andonly a relatively small amount of averaging can be applied.

Too muchaveraging will smear and obscure the temporal evolution of the spectralevents. Note that in deriving Equation (11.13), we have not considered nonlinear rectification of the negative estimates of the squared magnitudespectrum.11.1.2 Magnitude Spectrum SubtractionThe magnitude spectrum subtraction is defined as| Xˆ ( f ) | = | Y ( f ) | − | N ( f ) |whereN( f )(11.14)is the time-averaged magnitude spectrum of the noise.Taking the expectation of Equation (11.14), we haveE [ | Xˆ ( f ) |] = E [ | Y ( f ) |] − E [ | N ( f ) | ]= E [ | X ( f ) + N ( f ) |] − E [ | N ( f ) | ]≈ E [ | X ( f ) |](11.15)339Spectral SubtractionFor signal restoration the magnitude estimate is combined with the phase ofthe noisy signal and then transformed into the time domain using Equation(11.8).11.1.3 Spectral Subtraction Filter: Relation to Wiener FiltersThe spectral subtraction equation can be expressed as the product of thenoisy signal spectrum and the frequency response of a spectral subtractionfilter as| Xˆ ( f ) | 2 =| Y ( f ) | 2 − | N ( f ) | 2(11.16)=H ( f ) | Y ( f ) |2where H(f), the frequency response of the spectral subtraction filter, isdefined asH ( f ) =1−| N ( f ) |2| Y ( f ) |22=| Y ( f ) | −| N ( f ) |(11.17)2| Y ( f ) |2The spectral subtraction filter H(f) is a zero-phase filter, with its magnituderesponse in the range 0 ≥ H ( f ) ≥ 1 .