I Andrianakis

P R White

Speech spectral amplitude estimators using optimally shaped Gamma and Chi priors

Keywords: Speech Enhancement, MMSE, MAP, Gamma, Chi

In this paper, four STFT based speech enhancement algorithms are proposed. The algorithms enhance speech by estimating its short time spectral amplitude and are combinations of two estimators (MMSE and MAP) with two speech spectral amplitude priors (Gamma and Chi). The proposed priors have a shape parameter a, whose effect on the quality of speech is a focal point of our investigation. Rather than using a priori estimated values of a, we seek those values that maximise the quality of the enhanced speech, in an a posteriori fashion. The performance of the algorithms is first evaluated as a function of the shape parameter a and optimal values are then sought by means of a formal subjective listening test. Finally, the parallel examination of four speech enhancement algorithms offers an insight into the relative importance of the employed priors and estimators, as the proposed algorithms are only different with respect to these two elements.

Introduction

The continuous evolution of computers and digital systems has led to the widespread use of voice capturing and processing devices (e.g. mobile phones, hearing aids). The portability of such devices enables them to be deployed in environments where background noise conditions can be adverse. Background noise poses a serious problem for both voice-based communication and automated services. Speech quality and intelligibility can be seriously hindered and automatic speech recognition systems are far less robust to noise than humans. Speech enhancement algorithms can ameliorate to some extend the aforementioned problems.

The speech enhancement algorithms we examine here work in the frequency domain. The transformation is performed using the Short Time Fourier Transform (STFT), due to the nonstationary nature of speech. The main reasons for working in the frequency domain are the importance of the short time spectrum in the perception of speech [START_REF] Lim | Enhancement and bandwidth compression of noisy speech[END_REF] and the low computational cost of the transformation when the FFT algorithm is used. The task of the frequency domain speech enhancement algorithms is to produce an optimal, in some sense, estimate of the clean speech STFT, once the STFT of the noisy speech is observed.

One of the earliest algorithms that falls in the above category is the Wiener filter [START_REF] Lim | Enhancement and bandwidth compression of noisy speech[END_REF]. The same algorithm was also presented in its Bayesian context in [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF], where it was explicitly stated that the Wiener filter is the Minimum Mean Square Error (MMSE) estimator of the real (Re) and imaginary (Im) parts of the speech STFT coefficients, when these are assumed to follow a Gaussian distribution. Observing that the Re and Im parts of the clean speech STFT are better modelled by super-Gaussian densities, [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF] developed MMSE estimators of the Re and Im parts using the Laplacian and the two sided Gamma density.

Capitalising on the importance of the short time speech spectral amplitude relative to short time phase in speech perception, [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] developed a spectral amplitude MMSE estimator. The speech spectral amplitude coefficients were modelled using a Rayleigh density, which is an instance of the Chi density that we examine here (eq. ( 3)). A computationally more efficient, albeit approximate MAP estimator, which used the same modelling assumptions as in [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF], was proposed in [START_REF] Wolfe | Efficient alternatives to the Ephraim Malah suppression rule for audio signal enhancement[END_REF]. Speech spectral amplitude MAP estimators that employed instances of the Gamma (eq. ( 2)) and Chi (eq. ( 3)) densities were also proposed in [START_REF] Lotter | Noise reduction by joint maximum a posteriori spectral amplitude and phase estimation with super-gaussian speech modelling[END_REF] and in [START_REF] Dat | Generalized Gamma modeling of speech and its online estimation for speech enhancement[END_REF].

In this work we consider four speech enhancement algorithms which are a combination of the MMSE and MAP estimators with the Gamma and Chi priors 1 . Analytically, the algorithms we consider are: the MS1G (MMSE estimator, Gamma priors), the MS1C (MMSE estimator, Chi priors), the MP1G (MAP estimator, Gamma priors) and the the MP1C (MAP estimator, Chi priors). The algorithms that employ the MAP estimator have also been considered in [START_REF] Dat | Generalized Gamma modeling of speech and its online estimation for speech enhancement[END_REF][START_REF] Lotter | Noise reduction by joint maximum a posteriori spectral amplitude and phase estimation with super-gaussian speech modelling[END_REF]). The difference with our work is that while a single value of the priors' shape parameter was considered in the above studies, here we utilise a range of different shapes for the priors and seek the one that maximises the quality of speech. The MS1C and the MS1G algorithms are presented for the first time in this paper. Despite the fact that the MS1C yields the [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] algorithm as a special case, we view them as distinct algorithms because the former models the speech spectral amplitude with a Chi prior, while the latter with the Rayleigh. The instances of the algorithms considered here that have appeared previously in the literature are summarised in table 1. [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] MS1C a = 2 [START_REF] Wolfe | Efficient alternatives to the Ephraim Malah suppression rule for audio signal enhancement[END_REF] MP1C a = 2 [START_REF] Lotter | Noise reduction by joint maximum a posteriori spectral amplitude and phase estimation with super-gaussian speech modelling[END_REF]) MP1G a = 2 [START_REF] Dat | Generalized Gamma modeling of speech and its online estimation for speech enhancement[END_REF] MP1G a = 1.5, MP1C a = 1 Table 1 Instances of the proposed algorithms found in the literature Apart from introducing novel speech enhancement algorithms, the present work also investigates the effect of the shape parameter a on the quality of the enhanced speech and attempts to find a set of optimal values for the different algorithms. A method for the a priori estimation of a is first evaluated, which is compatible with the speech estimation model and is based on selecting speech data from narrow a priori SNR intervals, as proposed in [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF]. We find that the results of this method are not consistent and depend on the a priori SNR interval from which the speech data are selected. In view of the shortcomings of this estimation method, the strategy we follow is to seek the optimal values of a in an a posteriori fashion, based on the results of the speech enhancement algorithms. We first evaluate the proposed algorithms as a function of the shape parameter a and demonstrate its effect on the quality of speech. Secondly, we perform a formal subjective listening test, in which, the subjects are asked to select the values of a that maximise the speech quality. The parallel evaluation of four algorithms that are different only with respect to the estimator and the prior density used, offers a valuable insight on the relative importance of these two elements of a Bayesian estimation scheme.

The structure of the paper is as follows: In section 2 we explain the specifics of the estimation model and assess the assumption of independence between the amplitude and phase of the speech STFT samples. In section 3 we derive the proposed algorithms and discuss the selection of their parameters. In section 4 the performance of the algorithms is evaluated as a function of the priors' shape parameter a and the results of the subjective listening test are presented. Finally, section 5 draws the conclusions that have stemmed from this work.

Problem formulation

Let us denote by s(i) and n(i) the sampled speech and noise signals, which are assumed to be independent and zero mean. The noisy speech signal x(i) is modelled as the sum of s(i) and n(i). The transformation of x(i) to the STFT domain is achieved by windowing the first K samples with a tapered window h(i) and applying a K point DFT to the windowed data. The window is then shifted by J samples and the procedure is repeated for the remainder of the signal. The STFT transformation can be written as

X(k, l) = K-1 m=0 x(Jl + m) h(m) e -j2π mk K (1)
where k and l are the frequency bin and time frame indices, and j ≡ √ -1. According to the linear property of the Fourier transform, the relationship between the STFT's of x(i), s(i) and n

(i) is X(k, l) = S(k, l) + N(k, l).
The task of the speech enhancement algorithms is to produce an estimate of S(k, l) when only X(k, l) is observed. There are generally two approaches in tackling this estimation problem: the first is the separate the estimation of the Re and Im parts of S(k, l) and the second is the estimation of the amplitude and phase. In practice, the algorithms that follow the second approach estimate the amplitude only, which is then combined with the phase of the noisy speech to yield the enhanced speech signal. In this way the data that needs to be estimated is essentially halved. A reason for using the noisy speech phase ψ as an estimate for the phase of the clean speech φ is that the noisy speech phase minimises a least squares type error criterion E [1-cos(φ -ψ)], as it was shown in [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF]. In the same paper, it was also shown that the noisy phase exponential e jψ is the MMSE estimator of the clean speech phase exponential e jφ under the constraint | e jφ | = 1. Finally, [START_REF] Wang | The unimportance of phase in speech enhancement[END_REF] demonstrated that improving the estimates of phase did not result in substantial improvements in the quality of speech.

In this work we present algorithms that estimate the amplitude of the clean speech STFT, which is then combined with the phase of noisy speech to produce the enhanced speech signal. A number of algorithms that estimate the Re and Im parts of the speech STFT were investigated in [START_REF] Andrianakis | Bayesian algorithms for speech enhancement[END_REF] but they were generally shown to be inferior to the amplitude estimation algorithms. In the following, the amplitudes of the noisy and clean speech will be denoted as R ≡ |X| and A ≡ |S| respectively.

Speech Priors

The speech priors we propose in this work for modelling the speech STFT amplitude are the Gamma and the Chi. The motivation behind their employment is that they constitute generalisations of priors that have been used in the literature as it was summarised in table 1. The functional form of the Gamma density is given by [START_REF] Johnson | Continuous Univariate Distributions[END_REF]

p G (A) = 1 θ a Γ(a) A a-1 exp - A θ (2)
and the Chi density is given by

p C (A) = 2 θ a Γ(a/2) A a-1 exp - A 2 θ 2 (3)
The latter density is also found in [START_REF] Johnson | Continuous Univariate Distributions[END_REF] under the name Chi density with a degrees of freedom and scale parameter θ/ √ 2. Γ(.) denotes the Gamma function.

In the above densities, the parameter a controls their shape, while θ controls their scale. For a < 1 both densities have a singularity at zero, while for a > 1 the singularity is replaced by a zero. A well known instance of the Gamma density is the exponential, which is obtained for a = 1. The Chi density yields the half-Gaussian for a = 1 and the Rayleigh for a = 2 . Hence, the Gamma and Chi densities constitute generalisations of the densities mentioned above.

In this work, we model the clean speech phase with a uniform distribution, i.e. p(φ) = 1/2π. The validity of this model is confirmed by a simple inspection of a phase histogram of clean speech. Additionally, we assume that the speech spectral amplitude is independent from its phase, that is, p(A, φ) = p(A)p(φ). The validity of this assumption is assessed in the next section.

Assessment of the independence of the amplitude and phase for speech data

In the development of estimators for the STFT Re and Im parts of speech signals, it is assumed that the two components are mutually independent (e.g. [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF]. Similarly, the independence of the amplitude and the phase is assumed in the development of amplitude STFT estimators (e.g. [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF][START_REF] Lotter | Noise reduction by joint maximum a posteriori spectral amplitude and phase estimation with super-gaussian speech modelling[END_REF]. These assumptions are invoked because they significantly simplify the derivation of the estimators. However, these assumptions cannot hold simultaneously for data that follow distributions other than the complex Gaussian, while there is no evidence that this is the distribution of the speech STFT data. Therefore, the speech STFT coefficients must exhibit dependencies either between their Re and Im parts, or between their amplitude and phase. In [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF] it was reported that the amplitude and phase were statistically less dependent than the Re and Im parts. However, no results were given in support of this statement. In this section we evaluate these independence assumptions for clean speech STFT data by measuring the symmetric uncertainty coefficient [START_REF] Press | Numer-ical Recipes in C: the art of scientific computing[END_REF].

The symmetric uncertainty coefficient between two random variables (r.v.'s)

x and y is given by

U(x, y) = 2 H(x) + H(y) -H(x, y) H(x) + H(y) (4)
where H(.) is the entropy of a r.v. The symmetric uncertainty coefficient is a measure of independence between two r.v.'s, which is 0 for independent and 1 for fully dependent r.v.'s. The numerator of eq. ( 4) is the mutual information between x and y, which is denoted by I(x, y) [START_REF] Kraskov | Estimating mutual information[END_REF]. For the calculation of the mutual information we used the algorithm proposed in [START_REF] Kraskov | Estimating mutual information[END_REF]. H(x) and H(y) in the denominator are calculated using the same algorithm and exploiting the property I(x, x) = H(x) [START_REF] Press | Numer-ical Recipes in C: the art of scientific computing[END_REF].

The clean speech STFT data used in the evaluation was calculated from a clean speech database that consisted of 48 TIMIT sentences uttered by 3 male and 3 female speakers. The total duration of the speech data was 2 minutes and the sampling frequency 8 KHz. The transformation to the STFT domain was performed with Hamming windows of 256 samples and 75% overlap. For comparison we also calculated the symmetric uncertainty coefficient between the Re and Im and between the amplitude and phase of three test signals. The first of the test signals was drawn from a complex Gaussian distribution with independent Re and Im parts. The second was drawn from complex Laplacian distribution with independent Re and Im parts, and finally, the third test signal had exponentially distributed (1 sided Laplacian) amplitude, which was independent from its uniformly distributed phase. The analytic models of the above signals predict that the amplitude and phase of the first signal are independent, while the amplitude and phase of the second and the Re and Im parts of the third have some dependencies. The symmetric uncertainty coefficient results for the above data are shown in table 2. Table 2 shows that the symmetric uncertainty coefficient results 2 agree with the model predictions for the test data. It also shows that the amplitude and phase for the speech data are independent, while some dependencies seem to be present between the Re and Im parts. It is also worth mentioning that similar results were obtained when different overlap sizes were used in the calculation of the STFT, although these results are not shown here. The independence between the amplitude and the phase of the speech data might have been anticipated by considering that small shifts in time of the STFT analysis windows would affect the speech phase but not its amplitude. Conversely, a multiplication of the speech time waveform with an arbitrary constant, would have affected its spectral amplitude but not its phase. Both of these examples indicate some form of independence between the speech spectral amplitude and phase. Based on the results of this section we proceed in developing our estimators, assuming that the speech spectral amplitude is independent from its phase.

STFT estimation algorithms

Derivation of the MMSE and MAP estimators

In this section we derive the estimators of the clean speech STFT amplitude. The estimated amplitude is then combined with the phase of the noisy speech and the inversion of the resulting STFT gives the enhanced speech. We start by presenting the MMSE estimators.

2 The algorithm did not produce exactly 0 for the zeros shown in table 2. It instead produced either negative or very small values (< 1 × 10 -5 ) that varied between realisations for the test data. It is stated in [START_REF] Kraskov | Estimating mutual information[END_REF]) that these cases indicate independent r.v.'s, hence the zeros in the tables. For the used speech data the actual U(A, φ) was -6 × 10 -6 .

MMSE Estimators

The MMSE estimator is known to be equal to the mean of the posterior density of the r.v. we are estimating conditioned on the noisy r.v. (VanTrees, 1968). The estimator of the speech spectral amplitude conditioned on the noisy observations is given by [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF])

 = ∞ 0 2π 0 A p(R, ψ|A, φ) p(A, φ) dφdA ∞ 0 2π 0 p(R, ψ|A, φ) p(A, φ) dφdA (5)
where R is the amplitude of the noisy speech and ψ is its phase. Based on the results of section 2.3, we assume that the amplitude and phase of speech are independent, therefore we can factorise their joint density p(A, φ) = p(A)p(φ). We also assume that the phase is uniformly distributed, i.e. p(φ) = 1/2π.

The expression for the likelihood p(R, ψ|A, φ) can be obtained as follows: the joint density of the noisy speech STFT coefficients conditioned on the clean can be shown to be

p(X Re , X Im |S Re , S Im ) = p N Re (X Re -S Re )p N Im (X Im -S Im ) (6)
where p N Re , p N Im are the distributions of the Re and Im parts of noise, which are assumed to be zero mean Gaussians with variance σ 2 N /2. Applying the transformation X Re = R cos(ψ), X Im = R sin(ψ) [START_REF] Papoulis | Probability, random variables and stochastic processes[END_REF] and expressing S in its polar form we obtain

p(R, ψ|A, φ) = R πσ 2 N exp - R 2 + A 2 -2RA cos(ψ -φ) σ 2 N (7)
The integral with respect to φ, which is common in both the numerator and the denominator of eq. ( 5) can be written using Bayes' rule as

2π 0 p(R, ψ|A, φ)p(φ) dφ = p(R, ψ|A) (8)
The above integral can be expressed in terms of the modified Bessel function of the first kind I 0 (.), using the relation (Gradshteyn & Ryzhik, 1965, eq. 8.431.5)

I 0 (x) = 1 2π 2π 0 exp {x cos(φ)} dφ (9) to yield p(R, ψ|A) = R πσ 2 N exp - R 2 + A 2 σ 2 N I 0 ( 2RA σ 2 N ) ( 10
)
Substitution of eq. ( 8) in eq. ( 5) yields

 = ∞ 0 A p(R, ψ|A) p(A) dA ∞ 0 p(R, ψ|A) p(A) dA (11)
The above equation can yield the expressions for both the MMSE estimators we are examining, if in the place of p(A) we substitute the appropriate prior. Substituting the expression for the Gamma prior p G (A) from eq. ( 2) in eq. ( 11) we get

ÂMS1G = I G (a + 1) I G (a) (12)
where

I G (ν) = ∞ 0 A ν-1 exp - A 2 σ 2 N - A θ I 0 ( 2RA σ 2 N ) dA (13)
The above integral has no analytic solution for ν ∈ (0, ∞), which is the range of interest for our problem. To solve this problem we resorted to numerical integration. The integrand in I G is sufficiently smooth and allowed the convergence of the Adaptive Lobatto Quadrature [START_REF] Gander | Adaptive quadrature -revisited[END_REF] in a few iterations.

The MMSE estimator with the Chi priors can be obtained by substituting p C (A) from eq. ( 3) into eq. ( 11). The resulting expression is

ÂMS1C = I C (a + 1; 1 σ 2 N + 1 θ 2 ; 2R σ 2 N ) I C (a; 1 σ 2 N + 1 θ 2 ; 2R σ 2 N ) (14)
where (Gradshteyn & Ryzhik, 1965, eq. 6.631.1)

I C (ν; β; z) = ∞ 0 A ν-1 exp -βA 2 I 0 (zA)dA = Γ(ν/2) 2β ν/2 1 F 1 ( ν 2 ; 1; z 2 4β ) ( 15
)
The function 1 F 1 is the Confluent Hypergeometric Function (Gradshteyn & Ryzhik, 1965, eq. 9.210.1). Substituting the integral form of eq. ( 15) with its analytic in eq. ( 14) and doing the necessary simplifications, the estimator becomes:

ÂMS1C = σ 2 N ζ Γ( a+1 2 ) Γ( a 2 ) 1 F 1 ( a+1 2 ; 1; R 2 σ 2 N ζ) 1 F 1 ( a 2 ; 1; R 2 σ 2 N ζ) where ζ = θ 2 θ 2 + σ 2 N (16)
The calculation of 1 F 1 was performed by the routine provided in [START_REF] Barrowes | Matlab routines for computation of special functions[END_REF]. In order to alleviate the numerical problems that occur in the evaluation of the Confluent Hypergeometric Function for large values of its input arguments (typically for R 2 σ 2 N ζ > 700) the asymptotic expansions given in (Abramowitz & Stegun, 1965, eq. 13.5.1) were used, producing numerically stable results for all input ranges. A very similar estimator was derived in [START_REF] Loizou | Speech enhancement based on perceptually motivated Bayesian estimators of the magnitude spectrum[END_REF] from a perceptually motivated point of view. Additionally, for a = 2 the above estimator is equivalent to the well known Ephraim and Malah MMSE STSA algorithm [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF].

Gain curves for the MMSE algorithms are shown in figure 1. The 1 st and 3 rd rows show the gain as a function of the a priori SNR ξ, which is defined in section 3.2, and the instantaneous SNR γ -1, where γ is the a posteriori SNR, defined as γ = R 2 /σ 2 N as per [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF]. The 2 nd and 4 th rows show 2 dimensional gain curves for ξ = -5 and 5 dB for different values of the shape parameter a, along with the respective curves of the [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] of the gain curves is that for large values of a and low a priori SNR values the amount of suppression increases with increasing values of the a posteriori SNR. This 'counter-intuitive' behaviour was shown in [START_REF] Cappé | Elimination of the musical noise phenomenon with the Ephraim and Malah noise suppressor[END_REF] to aid the suppression of the musical character of the residual noise.

MAP Estimators

The MAP estimator is known to be the maximum of the posterior density of the unknown r.v. given the noisy measurements (VanTrees, 1968). In the spectral amplitude estimation problem this can be formulated as [START_REF] Wolfe | Efficient alternatives to the Ephraim Malah suppression rule for audio signal enhancement[END_REF])

 = arg max A (p(A|R, ψ)) (17)
Consecutive application of Bayes' theorem, yields for the posterior density

p(A|R, ψ) = p(R, ψ, A) p(R, ψ) = 2π 0 p(R, ψ|A, φ)p(A, φ) dφ p(R, ψ) (18)
As the denominator of the above equation is not a function of A, the MAP estimate can be found by maximising the numerator of eq. ( 18) only. Additionally, assuming that A and φ are independent, we can also factorise p(A, φ) as p(A)p(φ). Finally, calculating the phase integral using eq. ( 8), the MAP estimator becomes

 = arg max A (ln (p(R, ψ|A)p(A))) (19)
Following [START_REF] Mcaulay | Speech enhancement using a soft-decision noise suppression filter[END_REF] and [START_REF] Wolfe | Efficient alternatives to the Ephraim Malah suppression rule for audio signal enhancement[END_REF], we approximate the Bessel function that appears in p(R, ψ|A) with the following formula, which allows the derivation of the MAP estimators in a closed form

I 0 (x) ≈ 1 √ 2πx e x (20)
The above approximation has an error less than 5% for x > 3. Considering that in eq. ( 10) the argument of the Bessel function is 2RA σ 2

N

, we may conclude that the samples that contain speech will yield a high value for the above ratio, and therefore will not be affected significantly by the approximation.

The MAP estimator with the Gamma prior can be obtained by substituting p G (A) from eq. ( 2) into eq. ( 19). The expression that needs to be maximised becomes

ÂMP1G = arg max A ln(A (a-1.5) ) - (R -A) 2 σ 2 N - A θ (21)
Solving the above maximisation problem, the expression for the MAP estimator with the Gamma speech priors turns out

ÂMP1G = ζ + ζ 2 + (a -1.5)σ 2 N /2 1/2 where ζ = R 2 - σ 2 N 4θ (22)
The MAP estimator with the Chi priors can be obtained with a similar procedure. Substituting the prior (eq. ( 3)) and the likelihood (eq. ( 10)) into eq. ( 19), the expression that has to be maximised is

ÂMP1C = arg max A ln(A (a-1.5) ) - (R -A) 2 σ 2 N - A 2 θ 2 (23)
Maximisation of the above expression yields the estimator

ÂMP1C = ζ R 2 + ζ R 2 2 + (a -1.5)σ 2 N /2ζ 1/2 where ζ = θ 2 θ 2 + σ 2 N (24)
For a < 1.5 p(R, ψ|A)p(A) has a singularity at zero, which means that the MAP estimate (maximum of the posterior density) is zero. As this does not result in a useful algorithm, the strategy we follow in this case is to use the value of the local maximum, when this exists and suppress the noisy samples by a fixed amount (i.e. 50 dB) when there is no local maximum [START_REF] Andrianakis | MMSE speech spectral amplitude estimators with Chi and Gamma speech priors[END_REF]. Figure 2 shows three instances of the posterior density p(A|R, ψ) of the MP1G algorithm. In the first instance a is > 1.5, in which case there is always a global maximum. In the two next instances a is less than 1.5, so there is a singularity at zero but only in one case there is a local maximum.

Gain curves for the MAP algorithms are shown in figure 3. Again the 1 st and 3 rd curves show the gain as a function of ξ and γ -1, while the 2 nd and 4 th rows show the gain for ξ = -5 and 5 dB. The hard thresholding of the MAP algorithms described in the previous paragraph is visible in the gain curves for small values of a. For large values of a and small a priori SNR values, the 'counter-intuitive' behaviour of increasing the suppression for increasing the a posteriori SNR values, which was discussed in section 3.1.1, is also evident for the MAP algorithms. Finally, figures 1, 3 show that both the MMSE and MAP algorithms apply 0 dB of suppression for high values of ξ and γ. 3. Gain curves for the MAP algorithms. The 1 st and 3 rd rows show the suppression curves for the MP1G and the MP1C algorithms as a function of the a priori SNR ξ and instantaneous SNR γ -1. The 2 nd and 4 th rows show slices of the 3D plots for ξ = -5 and 5 dB. The dotted, dash-dotted and dashed lines correspond to a = 0.1, 1.5 and 3 respectively. The continuous line corresponds to the [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF]) (MS1C with a = 2) algorithm, which is plotted for comparison. All quantities are in dB.

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Parameter estimation

The speech priors we employ in this work have two parameters; the shape parameter a and the scale parameter θ. In this section we discuss their estimation. A natural set of values for both parameters would be those that provide the optimal fit of the priors to the clean speech data. The fitting can be performed by minimising the Kullback-Leibler (KL) divergence [START_REF] Kullback | Information Theory and Statistics[END_REF] between the priors and histograms created from a speech data. The definition of the KL divergence for the discrete case is

KL = N bin i=1 (p d (i) -p s (i)) ln p d (i) p s (i) ( 25
)
where p d (i) is the histogram of the speech data and p s (i) is the speech prior density. N bin is the number of histogram bins.

The drawback of using a fixed value for the scale parameter θ for the entire duration of an utterance is that the enhanced speech suffers from musical noise artefacts, as we found in [START_REF] Andrianakis | Bayesian algorithms for speech enhancement[END_REF]. A possible explanation is that using a fixed value for θ is equivalent to using a fixed value for the a priori SNR ξ, since θ 2 is proportional to ξ as we will shortly demonstrate. On the other hand, the adaptive estimation of the a priori SNR with a method such as the Decision Directed [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF], is known to aid the suppression of the musical noise artefacts [START_REF] Cappé | Elimination of the musical noise phenomenon with the Ephraim and Malah noise suppressor[END_REF]. We therefore propose to use the values of the a priori SNR ξ for the calculation of θ, in order to benefit from the noise suppression capabilities of methods for the estimation of the a priori SNR, such as the decision directed or the recursive estimation method [START_REF] Cohen | Relaxed statistical model for speech enhancement and a priori SNR estimation[END_REF]. The relationship between ξ and the scale parameter θ for each prior is described in the following.

The definition of the a priori SNR for an STFT sample is ξ

≡ E[|S| 2 ]/E[|N| 2 ]. Noting that E[A 2 ] ≡ E[|S| 2 ] and σ 2 N ≡ E[|N| 2 ],
the second moment of the speech spectral amplitude can be written as E[A 2 ] = σ 2 N ξ. Subsequently, θ can be related to the a priori SNR via its relation with the second moment of A for each prior. For the Gamma priors this expression is E[A 2 ] = θ 2 a(a + 1) and for the Chi priors the corresponding expression is E[A 2 ] = θ 2 a/2. The relations of θ to the a priori SNR ξ for each of the examined priors are summarised in table 3.

Gamma

Chi 3 Relation of θ to the a priori SNR ξ for the Gamma and Chi priors

θ 2 = σ 2 N ξ a(a + 1) θ 2 = 2σ 2 N ξ a Table
We now turn our attention to the estimation of a. The use of the a priori SNR for the estimation of θ means that fitting the priors to long term speech data for obtaining an estimate of a is no longer appropriate. The reason is that fitting the speech priors to long time data assumes that the values of a and θ remain constant for the duration of the signal, which is clearly not the case when the a priori SNR estimates and subsequently the values of θ change with time.

To overcome this problem [START_REF] Martin | Speech enhancement based on minimum mean-square error estimation and supergaussian priors[END_REF] and [START_REF] Lotter | Noise reduction by joint maximum a posteriori spectral amplitude and phase estimation with super-gaussian speech modelling[END_REF] proposed fitting the priors to data with a priori SNR estimates that belonged in narrow intervals. Here, we reproduce the results of this method, with the addition of considering data with a priori SNR estimates that belonged in several different intervals, as opposed to a single interval that was considered in the above studies. To obtain these data we corrupted speech with white Gaussian noise at very high global SNR (50 dB), applied the [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] algorithm (MS1C algorithm with a = 2) and extracted the a priori SNR estimates for each STFT sample using the decision directed method. The speech data used for this purpose was 256 sentences from the TIMIT database from 16 male and 16 female speakers. After removing the silent frames with a Voice Activity Detector (VAD) the total length of the data was 12.5 minutes. The sampling frequency was 8 KHz and the STFT was obtained with Hamming windows of 256 samples and 75% overlap.

Table 4 shows the results obtained by fitting the priors to data from narrow a priori SNR intervals. We present results obtained from data that had a priori SNR in three different intervals: 19-20, 49-50 and 79-80 dB. The first interval mainly consisted of weak speech components, such as consonants, while the last interval consisted of high amplitude data, typically found in the harmonics of the pitch frequency of vowels. 4 Parameter values that minimize the KL divergence when fitting the Chi and Gamma densities to amplitude data from a narrow variance interval.

The conclusion that can be drawn from the above table is that the shape of the priors that optimally fits the data depends on the a priori SNR interval. This does not allow the extraction of a single value of a that optimally fits the data. Additionally, a mapping between the value of the a priori SNR interval and the values of a cannot be obtained, as such a mapping depends on the global SNR, which is not known in general. Finally, the KL distance shows that the Gamma priors match closer the data with a priori SNRs in the intervals 19-20 and 49-50 dB, while the data with higher a priori SNR, are better modelled with the Chi priors.

The above observations indicate that finding a priori a value for the shape parameter that optimally fits the data, in a fashion that is compatible with the adaptive estimation of the scale parameter θ, as estimated from the a priori SNR, is not a straightforward task. In [START_REF] Andrianakis | Bayesian algorithms for speech enhancement[END_REF] we also implemented an adaptive method for the estimation of a, which was based on moment matching, similar to the one proposed in [START_REF] Dat | Generalized Gamma modeling of speech and its online estimation for speech enhancement[END_REF], and compatible with the adaptive estimation of θ. The success of this method was however moderate, as it resulted in a considerable amount of musical noise artefacts. In view of the difficulties encountered in trying to determine a priori optimal values for the shape parameter, the strategy we propose is to use a range of different values of a with the algorithms and evaluate their performance as a function of the shape parameter a. In this way, the optimal values of the shape parameter can be decided a posteriori, based on the results of the speech enhancement algorithms. Note also that an a posteriori evaluation of the statistical model was proposed in [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF], in order to sidestep the problems arising from the inaccessibility of the true statistical model of speech.

Results

General

In this section we present the results from the evaluation of the proposed algorithms. A focal point in our evaluation is the the effect of the shape parameter a in the performance of the algorithms. In order to investigate its effect, the algorithms' performance is first evaluated as a function of the value of the shape parameter a. In the second part of this section, optimal values for a are sought by means of a formal listening test.

For the evaluation of the proposed algorithms we used a subset of the TIMIT speech database that comprised of three male and three female speakers, each uttering 8 sentences. The speech segments were downsampled to 8 KHz and their total duration was 2 minutes and 10 seconds. The speech data was corrupted by computer generated white Gaussian noise and recorded car noise. The algorithms were evaluated with the average segmental SNR (SegSNR) [START_REF] Deller | Discrete-Time Processing of Speech Signals[END_REF] and the Perceptual Evaluation of Speech Quality (PESQ) [START_REF] Rix | Perceptual evaluation of speech quality (PESQ) -a new method for speech quality assessment of telephone networks and codecs[END_REF]) measure, which is the ITU recommendation P.862.

The transformation to the STFT domain was performed with Hamming windows of 256 samples and 75% overlap. The a priori SNR was estimated with the DD method [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF]. Additionally, a lower limit for the a priori SNR was set to -25 dB for perceptual reasons [START_REF] Cappé | Elimination of the musical noise phenomenon with the Ephraim and Malah noise suppressor[END_REF]. The estimates of the noise variance were obtained directly from the noise data, in order to isolate the effect of a noise estimation algorithm.

Performance evaluation as a function of a

Figures 4-5 present the objective measures results for all the considered algorithms for different input SegSNR levels and noise types. Because the signals that are enhanced using the same estimator are acoustically similar and to facilitate the presentation of the results, we will separately discuss the performance of the algorithms according to the estimator used and provide a comparison at the end.

In analysing the performance of the MAP estimators we can identify two discrete ranges of a, which depend on the existence (first range) or non existence (second range) of the singularity in the posterior distribution. The MAP algorithms with values from the first range preserve adequately the weaker speech components, especially for a ∼ 0.1. However, the residual noise has a strong musical character, which can be attributed to the sharp cut off characteristics of the respective MAP gain curves (figures 3(a), 3(f)). Although the MAP algorithms with values of a from the second range are less successful in recovering the weaker speech components, the residual noise has a more uniform character. Furthermore, its level, which increases with the value of a, can be adjusted so that the majority of the spurious spectral peaks are masked. On the basis of the uniform character of the residual noise, we prefer values of a from the second range for the MAP algorithms.

In order to provide a comparison between the examined MAP algorithms, we use values of a that result in equal levels of residual noise. To obtain these values we concatenate a speech utterance with a segment of white Gaussian noise of very low level (≈100 dB lower than the level of speech), which represents a segment of silence. The resulting signal is enhanced with the two MAP algorithms, adjusting a so that the output SegSNRs in the silence segment are equal. Informal listening tests suggest that the enhanced utterances obtained using the above values of a produce acoustically very similar results. The Chi prior generally preserves a few more weak speech spectral components, at the expense of a larger number of spurious spectral peaks. The Gamma prior on the other hand, suppresses some of the weaker spectral speech components, but at the same time, the fewer spurious spectral peaks reduce the amount of the perceived speech distortion. Additionally, the Gamma prior has a slightly faster response at the onset of speech after a segment of silence. Nevertheless, speech of very similar quality can be achieved with both priors and appropriate tuning of a, and this result is a consequence of the flexibility provided by the priors' shape parameter.

The MMSE algorithms provide an adequate preservation of the speech spectral components for small values of a (a ∼ 0.2). The resulting speech however, suffers from musical noise. As the value of a increases, the level of the residual noise increases as well. The advantage, however, is that the residual noise eventually loses its musical character and becomes uniform. The similar shape of the suppression curves of the MMSE algorithms with those of the MAP for low a priori SNR conditions and large values of a gives an indication about the uniform character of the residual noise. The MMSE algorithms have the additional advantage that the preservation of the speech spectral components is not affected by the value of a.

The differences in the quality of the speech enhanced with the two MMSE algorithms, when the values of the shape parameter a result in equal levels of residual noise, are again relatively subtle. The use of the Gamma prior results in a slightly better restoration of some weaker speech spectral components, especially at the onset of speech. The Chi prior on the other hand, results in smoother spectral peaks in the noise dominated regions of the spectrogram, and hence, the residual noise of the enhanced sentence is more uniform.

A comparison between the MAP and the MMSE algorithms reveals that the former result clearly in lower levels of residual noise. However, the preservation of the speech spectral components is better when the MMSE estimator is used and the resulting speech sounds less bandlimited and more natural. We should also note at this point that the computational complexity of the MMSE algorithms is generally higher compared to that of their MAP counterparts, because the former involve the calculation of special functions or numerical integration techniques. Nevertheless, an implementation of the MMSE algorithms by means of a look up table, could reduce significantly their computational load.

Subjective estimation of an optimal value of a

In this section we present the results from a formal subjective listening test that we carried out, in order to identify a set of values for the shape parameter that result in the highest quality of speech. A set of 20 subjects were asked to determine the value of a that provided the best enhanced speech quality, using 6 sentences that were corrupted with white Gaussian noise at 0 and 10 dB input SegSNR. The subjects were presented over headphones with the clean and the noisy speech, and could then adjust the value of the shape parameter and listen to the corresponding enhanced speech. No visual cues were given for the value of a, so the subjects had to base their decision solely on the audio samples. Additionally, the value of a was randomised for each new sentence so that the preferred values for one sentence could not affect the decision made for the others. Finally, each subject was presented with a sequence of audio samples in which the order of the sentences and the input SegSNR levels was random. The analysis in section 4.2 revealed that the two MAP algorithms produced speech of very similar quality when the shape parameter of the priors was tuned so that the level of the residual noise was equal between the two MAP algorithms. For this reason, only one MAP algorithm was evaluated, which was the MP1G. The latter was selected because it resulted in slightly more uniform noise than the MP1C for the same level of residual noise. For the same reasons, between the two examined MMSE algorithms only the MS1C was evaluated in the subjective test.

Figure 6 presents the results of the subjective test for the two different algorithms and the two different noise levels. The histograms show the occurrences of the different values of a for all the subjects and the presented sentences. It is noticeable that for the low input SegSNR the selections are concentrated around some particular values. For the MAP algorithm the majority of the values are around 2-4, while for the MMSE the most popular range is between 1 and 1.5. For the higher input SegSNR on the other hand, the selected values are considerably more spread.

To verify the validity of the above observation we performed a chi square significance test [START_REF] Moore | Introduction to the practice of statistics[END_REF]. The above test was used in order to check the hypothesis that the subjective test data for each algorithm and input SegSNR level came from a uniform distribution (null hypothesis). Rejection of the null hypothesis for the low SegSNR data and failure of rejection for the high SegSNR data would indeed confirm the larger spread of the high SegSNR data set. given by the formula

X 2 = N X i=1 (O i -Ōi ) 2 Ōi ( 26
)
where O i is the number of occurrences in the i th histogram bin and Ōi is the number of expected occurrences in the i th histogram bin according to the assumed distribution (uniform). The number of histogram bins was N X = 20, which satisfies the requirement for a minimum of 5 expected occurrences in each of the histogram bins, given that the total number of observations for each case was 120. The p-values shown in table 5 indicate that the null hypothesis can be safely rejected for the low input SegSNR condition for both algorithms, while for the high input SegSNR level there is not sufficient evidence for its rejection.

The differences in the shapes of the distributions for the two input SegSNR conditions can be attributed mainly to two reasons: The first is related to the fact that for the low input SegSNR condition the extreme values of a were not favoured, because either the musical noise was too intense (small a) or the residual noise was excessive (large a). For the high input SegSNR however, the effect of selecting a value of a closer to the extremes of the range was not as adverse, which generally made harder to pinpoint an optimal value for a and contributed to the flatter shape of the respective histograms.

The second reason was related to the spectral content of some particular sentences. Specifically, it was observed that for two out of six sentences the subjects consistently chose higher values of a for the high input SegSNR compared to their selections for the low input SegSNR condition. An example is shown in figure 7 where the histograms of the selected values for two sentences and two input SegSNR's are shown. The results correspond to the MP1G algorithm, while the first sentence is 'The cow wandered from the farmland and became lost', denoted as 'TC' and the second is 'Be careful not to plough over the flower beds' denoted as 'BC'. Note that for both input SegSNR levels the values selected for the first sentence are relatively similar (figures 7(a), 7(c)), while for the second sentence (figures 7(b), 7(d)) the values chosen for the high input SegSNR condition were significantly higher than those selected for A retrospective evaluation of the above sentences, in terms of inspecting the respective spectrograms and performing informal listening tests, revealed that the above observations may stem from differences in the distribution of the spectral energy of each sentence on the time frequency plane. For example, processing the 'BC' sentence with the MP1G algorithm and a = 4 at 10 dB input SegSNR resulted in a number of spurious spectral peaks, which were the result of the distribution of the weaker speech spectral components in the clean sentence. The spurious spectral peaks of the 'TC' sentence, when processed with the same algorithm and the same value of a, were considerably less. The spectrograms of the two sentences are shown in figure 8, where the spurious spectral peaks of the 'BC' that are perceived as musical noise are highlighted. The existence of more randomly placed spectral peaks in the 'BC' sentence, may have led the subjects to increase the level of the residual noise, by means of increasing the value of a, for masking purposes. For the low input SegSNR the majority of the weak speech spectral components were immersed in noise and their recovery was not possible, while any remaining spurious spectral peaks were masked from noise for much smaller values of a due to the higher overall background noise level. This resulted in more consistent choices of a for the different sentences in the low SegSNR condition and consequently, the values of a are more concentrated in the respective aggregate histograms of figure 6.

Based on the results of the subjective test we propose as optimal, the mean values of the parameter a extracted from the low SegSNR condition3 . These values were 2.6 for the MP1G and 1.4 for the MS1C algorithm. In table 6 we show the results in the objective measures for the two examined algorithms with the aforementioned values of a. Table 6 also shows the results for the remaining two MAP and MMSE algorithms for values of a that result in the same residual noise levels as the MP1G and MS1C respectively. Furthermore, the objective scores of the [START_REF] Ephraim | Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator[END_REF] and [START_REF] Ephraim | Speech enhancement using a minimum meansquare error log-spectral amplitude estimator[END_REF] algorithms, denoted as EM and EM-LSA respectively, are shown for comparison. Table 7 shows the respective results for the car noise.

The objective scores reveal that the MMSE algorithms generally score higher than their MAP counterparts. An exception is found for the SegSNR measure at 0 dB input SegSNR level. This should be attributed to the fact that the MAP algorithms achieve a lower residual noise level. The more adequate preservation of speech by the MMSE algorithms on the other hand, yields consistently higher results in the PESQ measure. Regarding the priors we note that the Gamma priors achieve higher SegSNR scores, mainly due to the better preservation of speech especially at its onset. Although the PESQ scores of the MMSE algorithms are almost identical for the different priors, the MP1C algorithm achieves slightly higher PESQ scores than the MP1G. We believe that this is the result of the MP1C algorithm being more successful in the preservation of a few extra speech spectral components, which nevertheless contribute more to the musical character of the residual noise rather than to the enhancement of the speech quality, as indicated by informal listening tests.

Comparing the examined algorithms with the EM and EM-LSA the following conclusions can be drawn. The EM algorithm is essentially the MS1C with a = 2. Using a = 1.4, as suggested by the subjective listening tests, lower levels of uniform residual noise are obtained compared to the traditional EM algorithm. There is also a consistent improvement in the objective scores when a is set to 1.4. The EM-LSA algorithm results in lower residual noise levels than the MS1C with a = 1.4, although the latter is slightly more successful in recovering the weaker speech spectral components. The greatest noise reduction however, is achieved with the MAP algorithms.

A final remark we wish to make is that the values of a selected via the subjective experiment do not correspond to those values that maximise the objective measures, particularly for the MMSE algorithms. We believe that this is due to the relatively poor performance of the objective measures in penalising the musical character of the residual noise. Note for example that the MMSE algorithms achieve the highest scores in the objective measures for a < 0.8. The listening experiment however, showed that the subjects selected considerably higher values (1.4 on average for the MP1C and 2.4 for the MP1G).

In this paper we proposed four algorithms that employ two families of generalised priors, the Gamma and the Chi. The proposed two priors were combined with two estimators, the MMSE and the MAP, in order to yield the four algorithms. The Gamma and the Chi priors have a parameter a, which determines their shape and influences heavily the performance of the estimators. The examination of the influence of the shape parameter a on the performance of the proposed algorithms was a central contribution of this paper.

Although a statistical method for the estimation of a exists, we found that it resulted in inconsistent estimates, which were dependent upon the selected speech data. For this reason, we preferred an a posteriori estimation of the optimal values of a, which was based on the performance of the speech enhancement algorithms.

The performance of the proposed algorithms was evaluated as a function of the value of a. The main conclusion was that a controls a trade off between the level of the residual noise and its musical character. The results of formal subjective listening tests conducted in order to identify a set of values for a that maximised the quality of the enhanced speech were reported. It was found that these values were generally different from those that maximised the scores in objective speech quality measures (SegSNR and PESQ). This discrepancy mainly stems from the poor ability of these objective measures to penalise the musical noise artefacts.

Another outcome of this research was that speech of very similar quality could be achieved with algorithms that used the two different priors but the same estimator. This is attributed to the flexibility that is provided by the shape parameter a in the priors, which allowed the listener to closely match the performance of algorithms that used the same estimator, but different priors.

The selection of the estimator on the other hand, had a significant impact on the quality of the enhanced speech. The MAP estimator resulted in lower residual noise levels than the MMSE. The latter estimator however, was more successful in the restoration of the speech spectral components and achieved higher scores in the objective speech quality measures, especially in the PESQ, even though these improvements came at the expense of a higher computational load. Both estimators however, were capable of producing enhanced speech free of musical noise artefacts.

Fig. 1 .

 1 Fig. 1. Gain curves for the MMSE algorithms. The 1 st and 3 rd rows show the suppression curves for the MS1G and the MS1C algorithms as a function of the a priori SNR ξ and instantaneous SNR γ -1. The 2 nd and 4 th rows show slices of the 3D plots for ξ = -5 and 5 dB. The dotted, dash-dotted and dashed lines correspond to a = 0.1, 1.5 and 3 respectively. The continuous line corresponds to the (Ephraim & Malah, 1984) (MS1C with a = 2) algorithm, which is plotted for comparison. All quantities are in dB.

Fig. 2 .

 2 Fig. 2. Three instances of the posterior density of MP1G algorithm. The arrows in figures (a) and (b) indicate the MAP estimate. In figures (b) and (c) the 'peaks' at zero are singular.

Fig. 5 .

 5 Fig. 5. SegSNR and PESQ scores for Car noise.

Fig. 6 .

 6 Fig.6. Histograms of values of a selected in the subjective experiment.

Fig. 7 .

 7 Fig. 7. Histograms of values of a selected in the subjective experiment for two different sentences and input SegSNR levels.the low SegSNR.

Fig. 8 .

 8 Fig. 8. Spectrograms of two different sentences enhanced with the MP1G algorithm and a = 4. The ellipses highlight the spurious spectral peaks that are perceived as musical noise. Noisy MP1C MP1G MS1C MS1G EM EM-LSA a = 1.6 a = 2.6 a = 1.4 a = 2.4 --

  Symmetric uncertainty coefficient results for the Re and Im and the amplitude and phase of test and speech STFT data.

		Speech Gaussian Laplacian Exp. Amp. &
			Re & Im Re & Im	Unif. Phase
	U(S Re , S Im )	0.03	0	0	0.01
	U(A, φ)	0	0	0.001	0
	Table 2

Table 5

 5 shows the results of the chi square test. The chi square statistic X 2 is Chi square significance test results for the two algorithms and input SegSNR levels.

		Input SegSNR		0 dB	10 dB
		Algorithm	X 2	p-value	X 2 p-value
		MP1G	103.0 1.5 × 10 -13 16.3	0.63
	Table 5	MS1C	68.0	2 × 10 -7	23.7	0.17

  Results in the objective measures obtained with values of a selected from the subjective test. The corrupting noise was white Gaussian.

				6 a = 1.4 a = 2.4	-	-
		0	7.03	7.19	6.71	6.96	6.27	6.82
	SegSNR	10	12.70	12.96	13.03	13.38	12.71	12.99
		20	20.09	20.40	20.69	21.09	20.49	20.62
		2.11	2.79	2.74	2.81	2.82	2.76	2.81
	PESQ	2.80	3.27	3.24	3.33	3.34	3.28	3.34
		3.46	3.79	3.78	3.82	3.82	3.75	3.82
	Table 6

  Results in the objective measures obtained with values of a selected from the subjective test. The speech was corrupted with car noise.

		Noisy MP1C MP1G MS1C	MS1G	EM EM-LSA
			a = 1.6 a = 2.6 a = 1.4 a = 2.4	-	-
		0	10.52	10.62	9.72	9.91	9.25	9.79
	SegSNR	10	16.63	16.79	16.61	16.86	16.31	16.54
		20	23.88	24.02	24.13	24.39	23.94	24.09
		2.89	3.37	3.36	3.37	3.38	3.31	3.38
	PESQ	3.49	3.81	3.80	3.82	3.82	3.77	3.81
		4.07	4.18	4.18	4.20	4.20	4.17	4.21
	Table 7

Part of this work has also been presented in(Andrianakis & White,

2006)

For the MP1G algorithm we actually used the mean value of the data that appear in the main 'hump' between 1 and

4.5.

The authors would like to thank Prof. Saeed Vaseghi, Esfandiar Zavarehei, Qin Yan, Dr. Ben Milner, Jonathan Darch and EPSRC Grant No. GR/S30238/01. The subjects that participated in the subjective listening test are also gratefully acknowledged.