INTERSPEECH 2013

Musical Noise Analysis for Bayesian Minimum Mean-Square Error Speech
Amplitude Estimators Based on Higher-Order Statistics
Hiroshi Saruwatari1 , Suzumi Kanehara1 , Ryoichi Miyazaki1 , Kiyohiro Shikano1 , Kazunobu Kondo2
1

2

Nara Institute of Science and Technology, Nara, 630-0192 Japan
YAMAHA Corporate Research & Development Center, Shizuoka, 438-0192 Japan
Abstract

generation in DD-approach-based WF, the MMSE STSA estimator and the minimum mean-square error log-spectral amplitude (MMSE LSA) estimator [17] by combining Breithaupt’s
approximation and our higher-order-statistics analysis. We also
compare the result of theoretical analysis and that of objective
experimental evaluation to indicate the validity of the proposed
closed-form analysis.

In this study, we perform a theoretical analysis of the amountof musical noise generated in Bayesian minimum mean-squareerror speech amplitude estimators. In our previous study, amusical noise assessment based on kurtosis has been successfully applied to spectral subtraction. However, it is difficult toapply this approach to the methods with a decision-directed apriori SNR estimator because it corresponds to a nonlinear recursive process for noise power spectral sequences. Therefore,
in this paper, we analyze musical noise generation by combining Breithaupt-Martin’s approximation and our higher-orderstatistics analysis. We also compare the result of theoreticalanalysis and that of objective experimental evaluation to indicate the validity of the proposed closed-form analysis.
Index Terms: noise reduction, musical noise, higher-order
statistics, kurtosis, decision-directed a priori SNR estimator

2. Related works
2.1. Mathematical metric of musical noise generation via
higher-order statistics [5, 6]
We speculate that the amount of musical noise is highly correlated with the number of isolated power spectral components
and their level of isolation. In this paper, we call these isolated components tonal components. Since such tonal components have relatively high power, they are strongly related to theweight of the tail of their probability density function (p.d.f.).
Therefore, quantifying the tail of the p.d.f. makes it possible tomeasure the number of tonal components. Thus, we adopt kurtosis, one of the most commonly used higher-order statistics,
to evaluate the percentage of tonal components among the total components. A large kurtosis value indicates a signal with aheavy tail, meaning that the signal has many tonal components.
Kurtosis is defined as

1. Introduction
Over the past decade, the number of applications of speechcommunication systems, such as TV conference systems andmobile phones, has increased. These systems, however, always suffer from a problem of deterioration of speech quality
under adverse noise conditions. Therefore, in speech signal
processing, noise reduction is a problem requiring urgent attention. Spectral subtraction (SS) [1], Wiener filter (WF) [2],and the minimum mean-square error short-time spectral amplitude (MMSE STSA) estimator [3] are the commonly used noise
reduction methods that have high noise reduction performance.
However, in these methods, artificial distortion, so-called musical noise, arises owing to nonlinear signal processing, leading
to a serious deterioration of sound quality [4].
Recently, an objective metric to measure how much musicalnoise is generated through nonlinear signal processing based onhigher-order statistics has been developed by some of the authors [5, 6, 7] and others [8, 9]. Using this metric, we havesuccessfully analyzed the amount of musical noise generated
via various types of SS-based methods [10, 11, 12, 13, 14, 15].
However, it is still difficult to theoretically analyze the noise reduction methods with a decision-directed a priori SNR estimator (hereafter, this is referred to as the DD approach), e.g., the
MMSE STSA estimator, because it corresponds to a nonlinear
recursive (infinite) process for noise power spectral sequences.
Several studies on the systematic analysis of the DD approach
have been provided [4, 16], but they did not use the explisit metric of musical noise generation like higher-order statistics so far.
In this paper, we focus on an approximated model [16] proposed by Breithaupt et al. We propose to analyze musical noise

kurt = µ4 /µ22 ,where kurt is the kurtosis and µm is the mth-order moment as
Z ∞
µm =

z m p(z)dz,(2)0

where p(z) is the p.d.f. of a signal z in the power spectral domain.
In this study, we apply such a kurtosis-based analysis to anoise-only time-frequency period of subject signals for the assessment of musical noise. Thus, this analysis should be conducted during, e.g., periods of silence during speech. This isbecause we aim to quantify the tonal components arising in thenoise-only part, which is the main cause of musical noise perception, and not in the target-speech-dominant part.
Although kurtosis can be used to measure the number of
tonal components, note that the kurtosis itself is not sufficient
to measure the amount of musical noise. This is obvious since
the kurtosis of some unprocessed noise signals, such as an interfering speech signal, is also high, but we do not recognize
speech as musical noise. Hence, we turn our attention to thechange in kurtosis between before and after signal processing
to identify only the musical-noise components. Thus, we adopt

This work was partly supported by JST CREST, Japan.

Copyright © 2013 ISCA

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25- 29 August 2013, Lyon, France

the kurtosis ratio as a measure to assess musical noise [5]. Thismeasure is defined as
kurtosis ratio = kurtproc /kurtorg ,2.4. MOSIE estimator [16]In this paper, we introduce the MOSIE estimator proposed byBreithaupt et al., which generalizes the MMSE STSA estimatorand MMSE LSA estimator [16]. The processed signal Y (f, τ )via the MOSIE estimator is written asqp
Y (f, τ ) = ξ(f, τ )/(ρ + ξ(f, τ )) PN̂ (f )
Γ(ρ + β2 ) Φ(1 − ρ − β2 , 1; −ν(f, τ )) 1/β j arg(X(f,τ ))e,Γ(ρ)Φ(1 − ρ, 1; −ν(f, τ ))
(10)

(3)

where kurtproc is the kurtosis of the processed signal and kurtorg
is the kurtosis of the observed signal. This measure increases as
the amount of generated musical noise increases. In Ref. [5], it
was reported that the kurtosis ratio is strongly correlated withthe human perception of musical noise.
2.2. Analysis of amount of noise reduction [12]where ν(f, τ ) = ξ(f, τ )γ(f, τ )(ρ + ξ(f, τ ))−1 , PN̂ (f ) is a
power spectral density estimated from the speech absence period of the observed signal, Φ(a, b; k) = 1 F1 (a, b; k) is theconfluent hypergeometric function, and Γ(·) is the gamma function. Also, β is an amplitude compression parameter with theerror function e(S(f, τ ), Y (f, τ )) = S β (f, τ )−Y β (f, τ ), andρ is a shape parameter of a chi-distribution that is used for modeling the p.d.f. of the speech amplitude. Equation (10) corresponds to the MMSE STSA estimator with ρ = 1, β = 1, andto the MMSE LSA estimator with ρ = 1, β → 0.

We analyze the amount of noise reduction via processing. Hereafter, we define the noise reduction rate (NRR) [18, 19] as ameasure of the noise reduction performance, which is definedas the output SNR in dB minus the input SNR in dB. The NRR
is
NRR = 10log10 (E[s2out ]/E[n2out ])/(E[s2in ]/E[n2in ]),(4)

where sin and sout are the input and output speech signals,
and nin and nout are the input and output noise signals, respectively. If we assume that the amount of noise reduction
is much larger than that of speech distortion in processing, i.e.,E[s2out ] ' E[s2in ], then
NRR ' 10log10 E[n2in ]/E[n2out ] = 10log10 µ1 /µ01 ,3. Theoretical analysis of higher-order
statistics in WF and MOSIE estimator
3.1. Motivation and strategy(5)

In our previous report [20, 21], we tried to calculate the kurtosis ratio in the MMSE STSA estimator based on the multidimensional numerical integration. It required, however, toohuge computation like Tera–Giga order multiply-accumulation,
resulting in several-week or -day calculations for each parameter setting. Therefore, in this paper we propose a newcomputational-cost-efficient closed-form analysis.
First, we model the noise signal power spectra x using thefollowing gamma distribution as

where µ1 is the 1st-order moment of observed signal power
spectra, and µ01 is the 1st-order moment of processed signal
power spectra.
2.3. DD-approach-based WF [2]We apply short-time Fourier analysis to the observed signal,
which is a mixture of target speech and noise, to obtain the timefrequency signal X(f, τ ) = S(f, τ ) + N (f, τ ), where X(f, τ )is the observed signal, f denotes the frequency subband, and τ
is the frame index. S(f, τ ) and N (f, τ ) denote the input speechand noise signals. The signal processing procedures of WF areformulated asp(x) = xη−1 exp(−x/θ)(θη Γ(η))−1 ,(11)

ξ(f, τ ) = E[|S(f, τ )|2 ]/E[|N (f, τ )|2 ],(7)

where η is the shape parameter corresponding to the type ofnoise, and θ is the scale parameter of the gamma distribution. If
the input signal is Gaussian noise, the p.d.f. of its power spectra
obeys the chi-square distribution with two degrees of freedom,which corresponds to the gamma distribution with η = 1. Also,
if the input signal is super-Gaussian noise, the p.d.f. of its power
spectra obeys the gamma distribution with η < 1. Hereafter,
to analyze the NRR and kurtosis ratio in WF and the MOSIE
estimator with the DD approach, we formulate the mth-ordermoment using several useful approximations.

γ(f, τ ) = |X(f, τ )| /E[|N (f, τ )| ].

(8)

3.2. Theoretical analysis of DD-approach-based WF

Y (f, τ ) = ξ(f, τ )/(ξ(f, τ ) + 1) · X(f, τ ),(6)

where Y (f, τ ) is the enhanced target speech signal. Also,ξ(f, τ ) and γ(f, τ ) are a priori and a posteriori SNRs, which
are defined as2

2

WF without the DD approach has been analized using higherorder statistics in our previous study [12], but no analysis for
DD-approach-based WF has been provided so far; therefore we
give it in this section. Since the DD approach (9) requires aninfinite number of samples from the past, we cannot estimate
how the p.d.f. of noise will change via WF. In this paper, we
introduce an approximation defined below as the estimation ofa priori SNR [16]In (7) and (8), we can commonly estimate E[|N (f, τ )|2 ]by averaging the noise power spectra in the speech absent time
period, or by using other estimation methods [3, 16]. However, since we cannot estimate E[|S(f, τ )|2 ] in advance, a priori
SNR ξ(f, τ ) is approximately calculated by the following DD
approach;ξ̂(f, τ ) =αγ(f, τ −1)G2 (f, τ−1)+(1−α)F [γ(f, τ )−1],(9)ξ̂(f, τ ) ≈ (1 − α)ξ ml (f, τ ),(12)

ξ (f, τ ) = max{0, γ(f, τ ) − 1},

(13)

ml

where α is a forgetting factor and F [·] is a flooring function.

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where ξ ml (f, τ ) corresponds to a maximum-likelihood estimateof the a priori SNR. The estimation of the processed signal via
WF is rewritten as follows, where we use (12) instead of (9);Y (f, τ ) ≈(1 − α)ξ ml (f, τ )X(f, τ ).
(1 − α)ξ ml (f, τ ) + 1

and c = 0.5772... is Euler’s constant.
Next, we introduce an approximation defined below as an
estimate of a priori SNR [16] such as the case of WF,ˆ τ ) ≈ hξ (f, τ ) ∗ ξ ml (f, τ ),
ξ(f,

(14)

where ∗ is an operator of convolution, and the variables in (23)are defined as

The p.d.f. of the observed signal, p(x), is transformed intop(y) by signal processing. We can calculate p(y) by considering a change of variables of the p.d.f. Suppose that a change ofvariables, y = g(x), is applied to convert an integral in termsof the variable x to an integral in terms of the variable y. The
converted p.d.f. p(y) can be written as p(y) = p(g −1 (y))|J|,
where |J| = |dg −1 /dy| is the Jacobian of the transformation.
We apply this to (11) to obtain the p.d.f. after processing, p(y).
Since x is the power spectral domain signal and its mean value
E[|N (f, τ )|2 ] is given by ηθ in the gamma distribution, thevariable y used for processing is expressed as2 (1−α)( xηθ −x)
if x ≥ ηθx −1)+1 ,y = (1−α)( ηθ
(15)

0,
if x < ηθ.

hξ (f, τ ) =(1 − α)exp(−λξ τ ),1
λξ =ln.
αR(β)

The mth-order moment of p(y) is given byZ ∞) 0h(y)η−1 exp(−h(y)θh (y)dy.
µm =ym
Γ(η)θηθη− (hξ (f, τ ) ∗ ξ ml (f, τ ))2 R(β)PN̂ (f ).

(16)

Km (|Y (f, τ )|2 ) =((1 − α)R(β)ηθ)m Km (ξ ml (f, τ ))

(17)

2 2

(1 − α)( tθ−1)+1θηwhere Km (·) is the mth-order cumulant of ·. Using the sum ofa geometric series, we can rewrite (27) asKm (|Y (f, τ )|2 ) =

(19)

2
X

Then, we apply (18) and (19) to (17) to obtain(−1)i−1 (1−α)im (R(β)ηθ)m Zi (m)Km ((ξ ml (f, τ ))i ),

i=1

(28)

m
Z 
t2
−t
(1 − α)m θm ∞ηµm =tη−1 exp(−t)dt.
t
Γ(η)
(1
−
α)
+
α
θη
η
(20)

where Zi (m) = exp(−imλξ )/(1 − exp(−mλξ ))i . Here,
we introduce an analogy that the sequence ξ ml (f, τ ) =max{0, γ(f, τ ) − 1} is generated from the process of normalized SS with the oversubtraction parameter of 1 and the flooring
parameter of 0. Thanks to this analogy, we can obtain the mthorder moment of ξ ml (f, τ ) in a closed form [12] as

Here, the integration term in (20) can be easily calculated using
a simple one-dimensional numerical integration method.
3.3. Theoretical analysis of MOSIE estimatorµm[SS] =Hereafter, we assume ρ = 1. First, in order to simplify theexpression (10), we use the first and second terms of the Taylor
series expansion ξ(f, τ )/(1 + ξ(f, τ )) ≈ ξ(f, τ ) − ξ 2 (f, τ ),which is valid for low SNR conditions with ξ(f, τ )  1. In
addition, we introduce some approximations [16], then (10) canbe rewritten asqpp|Y (f, τ )| ≈ ξ(f, τ ) − ξ 2 (f, τ ) PN̂ (f ) R(β), (21)where(
Γ(1+β/2)2/β ,R(β) =exp(−c),if β 6= 0
if β = 0(27)

τ1 =1 τ2 =1

m.

exp(−mλξ τ )− ((1 − α)2 R(β)ηθ)m Km ((ξ ml (f, τ ))2 )∞ X∞
X
exp(−mλξ τ1 )exp(−mλξ τ2 ),(18)(1 − α)( θθηt − θt)

∞
X
τ =1

We apply (18) to (15), then y m is expressed as(26)

Since the convolution calculation exists in (26), it is difficult to directly calculate the mth-order moment of the processedsignal via the MOSIE estimator. To solve this problem, first,
we calculate the cumulant of the processed signal, and then we
calculate the moment of the processed signal using the transformation formula of cumulants and moments. From (26), we canobtain the mth-order cumulant of the processed signal as

Let t = h(y)/θ, then dy = θ/h (y)dt and the range of theintegral does not change. Furthermore, h(y) is expressed as

ym =

(25)

|Y (f, τ )|2 ≈(hξ (f, τ ) ∗ ξ ml (f, τ ))R(β)PN̂ (f )0h(y) = tθ = x.

(24)

Equation (21) can be rewritten as follows by using (23) insteadof (9),

Since x > 0 and y > 0, the Jacobian is dx/dy = h0 (y) = |J|,
where h(·) is the inverse function of g(·). Consequently,p(y) = h(y)η−1 exp(−h(y)/θ)(Γ(η)θη )−1 h0 (y).

(23)

m
X
l=0

(−η)l

Γ(m + 1)Γ(η + m − l, η).
Γ(η)Γ(l + 1)Γ(m − l + 1)

(29)

Using the transformation formula from moments to cumulants,
Km ((ξ ml (f, τ ))i ) on the right-hand side of (28) can be expressed as
K1 ((ξ ml (f, τ ))i ) = µi[SS]



ml
i
2


K2 ((ξ (f, τ )) ) = µ2i[SS] −µi[SS]K3 ((ξ ml (f, τ ))i ) = µ3i[SS] −3µ2i[SS] µi[SS] +2µ3i[SS] ,
K4 ((ξ ml (f, τ ))i ) = µ4i[SS] −4µ3i[SS] µi[SS] −3µ22i[SS]+12µ2i[SS] µ2i[SS] −6µ4i[SS]
(30)

(22)

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Figure 1: Theoretical behavior and experimental results for WF:(a) noise reduction rate, and (b) log kurtosis ratio.
Figure 2: Theoretical behavior and experimental results for
MMSE STSA estimator and MMSE LSA estimator: (a) noise
reduction rate, and (b) log kurtosis ratio.

resulting in the closed-form calculation of Km (|Y (f, τ )|2 ) in(28). Using the transformation formula from cumulants to moments, we transform the cumulant (28) into the corresponding
moment. Finally, each order moment of the processed signal isgiven by

4.2. Resultsµ1 = K1 (|Y (f, τ )|2 )



2
2
2


µ2 = K2 (|Y (f, τ )| )+K1 (|Y (f, τ )| )2
µ4 = K4 (|Y (f, τ )| )+4K3 (|Y (f, τ )|2 )K1 (|Y (f, τ )|2 ) .
+3K22 (|Y (f, τ )|2 )+6K2 (|Y (f, τ )|2 )K12 (|Y (f, τ )|2 )+K14 (|Y (f, τ )|2 )(31)

4. Experiment4.1. Experimental conditions
We calculated the NRR and the kurtosis ratio using (20) and(31), respectively. The shape parameter η of the noise p.d.f.
is set to 1.0, and the forgetting factor α is varied from 0.8 to
0.99. Since the kurtosis of the processed signal changes exponentially, we depict the logarithm of the kurtosis ratio, which isreferred to as the log kurtosis ratio. If the log kurtosis ratio islarge, it denotes that much musical noise generated. If the log
kurtosis ratio equals zero, it means that there is no musical noise
generated.
In addition, we conducted a real noise reduction experimentin order to confirm the validity of our proposed theoretical analysis. The NRR and the log kurtosis ratio are calculated from
actual noise reduction results obtained from the observed signals and processed signals. In the evaluation experiment, thenoisy observed signals were generated by adding noise signals
to target speech signals with an SNR of 0 dB. The target speech
signals were the utterances of four speakers (four sentences).
The length of each signal was 15 s, and each signal was sampled at 16 kHz. The FFT size is 1024, and the frame shift lengthis 256. In these experiments, we calculated the noise prototype,i.e., the average of |N̂ (f, τ )|2 , in the first 10 s frames, where
the speech signal is absent.

Figures 1 and 2 show the theoretical behaviors calculated using (20) and (31), and the objective evaluations for each of thenoise reduction methods. Regarding the detailed behavior of
the NRR, all the estimators show the same tendency in that the
NRR becomes higher as the larger forgetting factor α is used(see Figs. 1(a) and 2(a)).
Regarding the log kurtosis ratio, WF shows a monotonic
increase of the kurtosis ratio as α increases (see Fig. 1(b)), indicating that the NRR and musical noise generation share a tradeoff relationship. Consequently, there is no justification of usinglarge α in WF. In contrast, we can see that the log kurtosis ratio
drops when large α is used in the MMSE STSA estimator and
MMSE LSA estimator (see Fig. 2(b)), whereas WF does notshow such a kurtosis-ratio drop. From the results, we can speculate that the kurtosis-ratio drop is the key factor of less musicalnoise property in the MOSIE estimator when we set a large α inthe DD approach, unlike WF. In addition, our approximated theoretical analysis can successfully explain the tendency in good
agreement with the experimantal results.

5. Conclusion
In this study, we performed a theoretical analysis of the amountof musical noise generated via WF and the MOISE estimatorwith the DD approach on the basis of higher-order statistics. In
particular, we derived higher-order statistics in the closed form
using the approximated model proposed by Breithaupt et al.
From this result, we can well explain the tendency of the behaviors in the NRR and kurtosis ratio for each method. Also,
we can speculate that the kurtosis-ratio drop is the key factor
of less musical noise property in the MOISE estimator when
we set a large forgetting factor in the DD approach, unlike WF(even when WF is based on the DD approach).

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