Литература / Advanced Digital Signal Processing and Noise Reduction (Saeed V. Vaseghi) / 11 - Spectral subtraction

Advanced Digital Signal Processing and Noise Reduction, Second Edition.

Saeed V. Vaseghi

Copyright © 2000 John Wiley & Sons Ltd

ISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)

11

SPECTRAL SUBTRACTION

11.1Spectral Subtraction

11.2Processing Distortions

11.3Non-Linear Spectral Subtraction

11.4Implementation of Spectral Subtraction

11.5Summary

Spectral subtraction is a method for restoration of the power spectrum or the magnitude spectrum of a signal observed in additive noise, through subtraction of an estimate of the average noise spectrum from the noisy signal spectrum. The noise spectrum is usually estimated, and

updated, from the periods when the signal is absent and only the noise is present. The assumption is that the noise is a stationary or a slowly varying process, and that the noise spectrum does not change significantly inbetween the update periods. For restoration of time-domain signals, an estimate of the instantaneous magnitude spectrum is combined with the phase of the noisy signal, and then transformed via an inverse discrete Fourier transform to the time domain. In terms of computational complexity, spectral subtraction is relatively inexpensive. However, owing to random variations of noise, spectral subtraction can result in negative estimates of the short-time magnitude or power spectrum. The magnitude and power spectrum are non-negative variables, and any negative estimates of 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-noise ratio decreases. In this chapter, we study spectral subtraction, and the different methods of reducing and removing the processing distortions.

where y(m), x(m) and n(m) are the signal, the additive noise and the noisy signal respectively, and m is the discrete time index. In the frequency domain, the noisy signal model of Equation (11.1) is expressed as

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 the frequency variable. In spectral subtraction, the incoming signal x(m) is buffered and divided into segments of N samples length. Each segment is windowed, using a Hanning or a Hamming window, and then transformed via discrete Fourier transform (DFT) to N spectral samples. The windows alleviate the effects of the discontinuities at the endpoints of each segment. The windowed signal is given by

The windowing operation can be expressed in the frequency domain as

Yw ( f )=W ( f ) *Y ( f ) (11.4)

= X w ( f )+Nw ( f )

where the operator * denotes convolution. Throughout this chapter, it is assumed that the signals are windowed, and hence for simplicity we drop the use of the subscript w for windowed signals.

Figure 11.2 illustrates a block diagram configuration of the spectral subtraction method. A more detailed implementation is described in Section 11.4. The equation describing spectral subtraction may be expressed as

| N ( f ) |b is the time-averaged noise spectra. It is assumed that the noise is a wide-sense stationary random process. For magnitude spectral subtraction, the exponent b=1, and for power spectral subtraction, b=2. The parameter α

Noise estimate

Figure 11.2 A block diagram illustration of spectral subtraction.

in Equation (11.5) controls the amount of noise subtracted from the noisy signal. For full noise subtraction, α=1 and for over-subtraction α>1. The time-averaged noise spectrum is obtained from the periods when the signal is absent and only the noise is present as

In Equation (11.6), |Ni(f)| is the spectrum of the ith noise frame, and it is assumed that there are K frames in a noise-only period, where K is a variable. Alternatively, the averaged noise spectrum can be obtained as the output of a first order digital low-pass filter as

where the low-pass filter coefficient ρ is typically set between 0.85 and 0.99. For restoration of a time-domain signal, the magnitude spectrum

estimate | ˆ ( ) | is combined with the phase of the noisy signal, and then

X f

transformed into the time domain via the inverse discrete Fourier transform as

where θY (k) is the phase of the noisy signal frequency Y(k). The signal restoration equation (11.8) is based on the assumption that the audible noise is mainly due to the distortion of the magnitude spectrum, and that the phase distortion is largely inaudible. Evaluations of the perceptual effects of simulated phase distortions validate this assumption.

338 Spectral Subtraction

signal Y ( f ) 2 , and grouping the appropriate terms, Equation (11.11) may be rewritten as

Taking the expectations of both sides of Equation (11.12), and assuming that the signal and the noise are uncorrelated ergodic processes, we have

From Equation (11.13), the average of the estimate of the instantaneous power 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, and only a relatively small amount of averaging can be applied. Too much averaging will smear and obscure the temporal evolution of the spectral events. Note that in deriving Equation (11.13), we have not considered nonlinear rectification of the negative estimates of the squared magnitude spectrum.

11.1.2 Magnitude Spectrum Subtraction

The magnitude spectrum subtraction is defined as

For signal restoration the magnitude estimate is combined with the phase of the noisy signal and then transformed into the time domain using Equation (11.8).

11.1.3 Spectral Subtraction Filter: Relation to Wiener Filters

The spectral subtraction equation can be expressed as the product of the noisy signal spectrum and the frequency response of a spectral subtraction filter as

=H ( f ) | Y ( f ) |2

where H(f), the frequency response of the spectral subtraction filter, is defined as

H ( f ) = 1 − | N ( f ) |2 | Y ( f ) |2

(11.17)

=| Y ( f ) |2 −| N ( f ) |2 | Y ( f ) |2

The spectral subtraction filter H(f) is a zero-phase filter, with its magnitude response in the range 0≥ H ( f ) ≥1. The filter acts as a SNR-dependent

attenuator. The attenuation at each frequency increases with the decreasing SNR, and conversely decreases with the increasing SNR.

The least mean square error linear filter for noise removal is the Wiener filter covered in chapter 6. Implementation of a Wiener filter requires the power spectra (or equivalently the correlation functions) of the signal and the noise process, as discussed in Chapter 6. Spectral subtraction is used as a substitute for the Wiener filter when the signal power spectrum is not available. In this section, we discuss the close relation between the Wiener filter and spectral subtraction. For restoration of a signal observed in uncorrelated additive noise, the equation describing the frequency response of the Wiener filter was derived in Chapter 6 as

E[| Y ( f ) |2 ]

A comparison of W(f) and H(f), from Equations (11.18) and (11.17), shows that the Wiener filter is based on the ensemble-average spectra of the signal and the noise, whereas the spectral subtraction filter uses the instantaneous spectra of the noisy signal and the time-averaged spectra of the noise. In spectral subtraction, we only have access to a single realisation of the process. However, assuming that the signal and noise are wide-sense stationary ergodic processes, we may replace the instantaneous noisy signal spectrum | Y ( f ) |2 in the spectral subtraction equation (11.18) with the time-

averaged spectrum | Y ( f ) |2 , to obtain

| Y ( f ) |2

For an ergodic process, as the length of the time over which the signals are averaged increases, the time-averaged spectrum approaches the ensembleaveraged spectrum, and in the limit, the spectral subtraction filter of Equation (11.19) approaches the Wiener filter equation (11.18). In practice, many signals, such as speech and music, are non-stationary, and only a limited degree of beneficial time-averaging of the spectral parameters can be expected.

11.2 Processing Distortions

The main problem in spectral subtraction is the non-linear processing distortions caused by the random variations of the noise spectrum. From Equation (11.12) and the constraint that the magnitude spectrum must have a non-negative value, we may identify three sources of distortions of the instantaneous estimate of the magnitude or power spectrum as:

(a)the variations of the instantaneous noise power spectrum about the mean;

(b)the signal and noise cross-product terms;

(c)the non-linear mapping of the spectral estimates that fall below a threshold.

The same sources of distortions appear in both the magnitude and the power spectrum subtraction methods. Of the three sources of distortions listed

Noise induced change in the mean

Figure 11.4 Illustration of the distorting effect of spectral subtraction on the space of the magnitude spectrum of a signal.

11.2.1 Effect of Spectral Subtraction on Signal Distribution

Figure 11.4 is an illustration of the distorting effect of spectral subtraction on the distribution of the magnitude spectrum of a signal. In this figure, we have considered the simple case where the spectrum of a signal is divided into two parts; a low-frequency band fl and a high-frequency band fh. Each point 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 the spectrum, by an amount that depends on the mean and the variance of the magnitude spectrum of the noise. The increase in the variance constitutes an irrevocable distortion. The increase in the mean of the magnitude spectrum can be removed through spectral subtraction. Figure 11.4(c) illustrates the distorting effect of spectral subtraction on the distribution of the signal spectrum. As shown, owing to the noise-induced increase in the variance of the signal spectrum, after subtraction of the average noise spectrum, a proportion of the signal population, particularly those with a low SNR, become negative and have to be mapped to non-negative values. As shown this process distorts the distribution of the low-SNR part of the signal spectrum.