1. Introduction
In this paper, we report our work on suppression of acoustic noise. This problem has received considerable attention, since it is relevant to many important applications like speech recognition and compression, restoration of analog audio recordings, etc. Our work targets the problem of speech enhancement in particular, and all our experiments were carried on speech signals ([Lim83] contains a good overview on the subject).
Speech enhancement is an extremely difficult problem if we don't make any assumptions about the nature of the noise signal we aim to remove. In this project, we did restrict ourselves to additive, stationary noise, i.e. white noise (broadband noise, like tape hiss), colored noise, and different kinds of narrowband noises. We did investigate two algorithms that were reported to work well under these restrictive assumptions about the noise signal: spectral subtraction and Wiener filtering. Our experiments did confirm the performance of these two techniques, but we did notice differences in the applicability of these two methods. We did also notice that they are not equally good at removing noises of different natures. Given the kind of application that is considered and the characteristics of the noise that has to be reduced, the choice of either algorithm can have a great impact on performance.
This report is divided in four sections. Section two presents the theoretical background of spectral subtraction and discusses experimental results; section three does the same with Wiener filters. Finally, we summarize in the last section the advantages and disadvantages of both methods, and discuss some improvements.
2. Speech Enhancement by Spectral Subtraction
2.1. Theoretical Background
Spectral subtraction is a widely used algorithm in acoustic noise reduction, mainly because of its simplicity of implementation. It was introduced in the late '70s by Boll [Bol79], then generalized and improved by Berouti [BSM79]. Assuming an additive model of noise, and given the linearity property of the Fourier transform, we get:
where ,, and are the Fourier transform of the noisy signal, clean signal, and noise (respectively). Boll describes an algorithm where the shorttime noise spectrum is first estimated with spectra measured within a noiseonly segment (the problem of segmenting the signal into speech and nonspeech regions isn't really considered in [Bol79]), resulting in the expected amplitude of . The estimate of the clean signal spectrum is obtained as follows:
The second step is to compute the spectral subtraction estimator using the above amplitude and the phase of the noisy signal . It is widely accepted that the shorttime phase is of relative unimportance to estimate [LO79,Bol79]. The estimator is then computed:
Berouti [BSM79] generalizes the spectral subtraction technique by not only considering subtraction of amplitude spectra, but also power spectra, or more generally any power of the shorttime amplitude spectrum. Given ,,, the power spectra of the estimated clean signal, the noisy signal, and the noise (respectively), Berouti introduced two parameters in the spectral subtractor estimator, which is expressed as follows:
The parameter allows overestimating the power spectrum of noise, and raises the power of the power spectrum before subtraction.The practical use of these parameters are explained in the following section.
2.3. Experiments and Results
The first step in the application of the spectral subtraction method is to compute the shorttime Fourier transform of the noisy signal using the fast Fourier transform (FFT) and windowing the input signal with a Hanning window. For this, we did use the same parameters as in [Bol79] and set the length of the window and the FFT to 256, with a shift in steps of 128 points. This section exemplifies the use of spectral subtraction through the tentative denoising of a speech signal (wav) deteriorated by some narrowband noise (signaltonoise ratio of 10 dB). We assume for the moment that we have available a good estimate of the noise. To this end, we build an unreasonable scenario where the real noise is readily available for our computation of the estimate of the spectral characteristics of the noise. Thus, we can separately assess the value of spectral subtraction algorithms and noise estimate algorithms (the latters could depend on voice activity detectors).
By first applying the spectral subtraction method as presented in [Bol79], we get an amelioration of the signaltonoise ratio (SNR) and reach 14.58 dB (Fig. 2).Two problems immediately appears: a clear narrowband of noise still remains in the spectrum, even if our estimate of noise is correct, and listening to the enhanced signal, we can notice an undesirable new noise appearing. As explained by Berouti [BSM79], peaks and valleys exist in the noise spectrum, and once the estimate is subtracted, peaks remain as randomly occurring peaks, while valleys are set to zero. The peaks are "perceived as time varying tones which we refer to as musical noise." Several methods have been proposed to remove musical noise, and two of them are investigated in this section.
Fig. 2: a speech signal degradated by a narrowband noise [wav],
and the result of spectral subtraction [wav]).
To face the first problem, we can take advantage of the generalized model proposed by Berouti and subtract more than the estimated noise shorttime spectrum. We investigated oversubtraction of both amplitude and power, but it seems that only oversubtraction of power works well (Berouti simply concludes that power subtraction performs in general better than amplitude subtraction). We can see in Fig. 3(a) that oversubtraction of the power spectrum allows for a much stronger reduction of the narrowband noise present in the original signal, however the musical noise becomes clearly noticeable once power subtraction is used (we never noticed such a highlevel of residual noise with amplitude subtraction). With this oversubtraction method, the SNR has improved and reached 16.96 dB, even if as human listener we wouldn't consider this signal to be 'cleaner' (we believe the musical noise in [wav] (Fig.3 (a)) is disturbing for most listeners.)
Fig 3: In (a), noise is oversubtracted (=2.5)
to ameliorate the SNR, however random amplitude peaks now corrupt the signal
[wav]. In (b), the noise is further
oversubtracted (=6)
until its almost complete disappearance [wav]:
the signal becomes clean, but the intelligibility slightly suffers from
this operation. In (c), both oversubtraction (=2.5)
and residual noise reduction methods are applied to both reduce the original
narrowband noise and musical noise [wav].
Several approaches are known to attenuate or remove musical noise. Berouti's motivation for oversubtracting the noise estimate was to reduce the musical noise, and we can indeed set a quite high subtraction factor (=6) and almost completely eliminate residual noise. Berouti (who suggests to be in the range 36) concedes that the intelligibility of speech may suffer if the subtraction parameter is excessively high. It is already apparent both in the spectrogram (Fig. 3(b)) and in the speech signal that for a parameter set to 6, some spectral components of speech are seriously affected by the oversubtraction.
Another approach to remove musical noise was proposed by Boll [Bol79] (though Boll didn't call it musical noise, but residual noise). His residual noise reduction reduces "framebyframe randomness" by taking, for every analysis frame below the noise estimate, the minimal value among adjacent analysis frames. This tends to smooth out most random amplitude peaks (this is noticeable when we compare Fig 3(a) and Fig. 3(c)), but it seems that it introduces some slurring effect, and speech intelligibility is also decreased.
The choice between these different parameters really depends on the kind of application that is targeted. If the restored signal is intended to be listened by humans, high level of noise reduction is desirable, and the musical noise should be kept as little as possible. According to experiments described in [BSM79], s slight loss of intelligibility may be to tolerated in this case. In other applications (e.g. automatic speech recognition), loss of intelligibility should be avoided.
2.4. Noise Estimation
In this section, we made the unrealistic assumption that we have a good estimate of the noise spectrum. We will now consider the problem of estimating that spectrum from a noisy signal, without any prior information that some portions of the signal contain only noise. Two common approaches can be taken: we can either design a voice activity detection (VAD) which could help us finding nonspeech activity in the signal, or we can estimate the noise spectral information without explicitly identifying sections containing only noise [Hir93,Mar94]. As pointed out by Hirsch [Hir93], speaker activity detection "is a very difficult and ultimately unsolved problem for realistic situations with a varying noise level". Even in the case of stationary noise (as in our work), reliable VAD is extremely difficult, and one can't simply assume that a low energy level is a good indicator of absence of speech, since nonvoiced consonants have little energy.
Since reliable VAD is extremely difficult to make [Mar94], we did avoid this problem by estimating the noise spectrum on the full signal, without explicitely detecting nonspeech intervals. We did follow Hirsch's noise estimation technique [Hir93]. Following the assumption of stationary noise, he observed that the amplitude values appearing the most frequently in a given frequency band of a spectrum are good indicators of the noise amplitude in that band. Taking into account this observation, we did design a simple algorithm that discretizes the amplitude range (dBscaled) into 1024 possible values. Then, for every band of frequency, we detect the most frequently occurring amplitude level and consider it as the estimate of the noise expected amplitude. We report the result of denoising a speech signal containing noise in two narrowbands:
Fig 3: (a) represents the noisy signal [wav].
(b) shows the estimate the expected noise amplitude spectrum. Finally, (c)
represents the result of spectral (over)subtraction with residual noise
reduction [wav].
In Fig. 3(b), we can see that Hirsch's algorithm is able to make a coarse estimate of the noise spectral characteristics, and then this noise can quite accurately be removed from the original signal: the listenability is quite good, except that some musical noise still remains.
3. Speech Enhancement with Wiener Filters
When the original signal is not available, it is possible to estimate the
original signal through spectral subtraction or some other method.
In addition, the original signal can be estimated as P_{YY}P_{NN},
which is done in one of the programs. This is not a good estimate for the
Power Spectrum of the original signal, so an iterative method is used in
which the Wiener filter is first represented as P_{YY}P_{NN}/P_{YY},
and then the noisy signal Y is filtered to give a new estimate of the original
signal, and this is substituted into the formula. This process of obtaining
a new estimate of the original signal and using that in the filter is repeated
until the estimates of the original signal converge. This iterative method
was tried, but only resulted in a small reduction in the amount of noise
in the very low frequencies.
At first the power spectrums were obtained by first calculating the correlations
of the signals and taking the Fourier Transform of that. Next a different
approach was tried, the formula
4. Conclusion
4.1. Spectral Subtraction and Wiener Filtering
Overall, spectral subtraction seems to let us achieve better acoustic noise reduction, and does provide more parameters we can tune to achieve a better performance. To some extend, this explains why specral subtraction remained so popular. On the other hand, in certain settings, subtraction of the noise spectrum does create the annoying effect called 'musical noise'. Wiener filters are much less subject to this kind of signal corruption, but they offer much less flexibility, and in almost all cases where the amount of noise we did add was substential, we were never really able to fully remove the background noise (e.g. hiss). This is not the case with spectral subtraction: if a resonably limited loss of speech intelligibility is acceptable, the spectral subtraction technique seems to allow us to remove a fair amount of noise, including cancelling almost completely musical noise (this is also true for other kind of noise we did experiment with: colored noise and pink noise, but this isn't reported in the present document due to a lack of space).
To overcome the problem of musical noise, our initial idea was to combine Wiener filtering and spectral subtraction, as suggested by Vaseghi [Vas97]. Our assessment of the nature of the signal produced by the Wiener filter was that is does produce a sound that seems much more natural (at least compared to spectral subtractor's musical noise), with little degradation of intelligibility. The idea behind this combination of Wiener filters and spectral subtraction is that Wiener filters need an estimate of the clean signal in order to minimize the least mean square error, and for this purpose the noisy signal is used, since no better signal is readily available (iterative Wiener filtering can give better performance). As reported in section 3, we were successful in using the output of spectral subtraction to have a better estimate of the clean signal for the purpose of training a Wiener filter.
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