Abstract
We propose an efficient solution to the problem of sparse linear prediction analysis of the speech signal. Our method is based on minimization of a weighted l_{2}norm of the prediction error. The weighting function is constructed such that less emphasis is given to the error around the points where we expect the largest prediction errors to occur (the glottal closure instants) and hence the resulting cost function approaches the ideal l_{0}norm cost function for sparse residual recovery. We show that the efficient minimization of this objective function (by solving normal equations of linear least squares problem) provides enhanced sparsity level of residuals compared to the l_{1}norm minimization approach which uses the computationally demanding convex optimization methods. Indeed, the computational complexity of the proposed method is roughly the same as the classic minimum variance linear prediction analysis approach. Moreover, to show a potential application of such sparse representation, we use the resulting linear prediction coefficients inside a multipulse synthesizer and show that the corresponding multipulse estimate of the excitation source results in slightly better synthesis quality when compared to the classical technique which uses the traditional nonsparse minimum variance synthesizer.
1 Introduction
Linear prediction (LP) analysis is a ubiquitous analysis technique in current speech technology. The basis of LP analysis is the sourcefilter production model of speech. For voiced sounds in particular, the filter is assumed to be an allpole linear filter and the source is considered to be a semiperiodic impulse train which is zero most of the times, i.e., the source is a sparse time series. LP analysis results in the estimation of the allpole filter parameters representing the spectral shape of the vocal tract. The accuracy of this estimation can be evaluated by observing the extent in which the residuals (the prediction error) of the corresponding prediction filter resemble the hypothesized source of excitation [1] (a perfect impulse train in case of voiced speech). However, it is shown in [1] that even when the vocal tract filter follows an actual allpole model, this criterion of goodness is not fulfilled by the classical minimum variance predictor. Despite the theoretic physical significance, such sparse representation forms the basis for many applications in speech technology. For instance, a class of efficient parametric speech coders are based on the search for a sparse excitation sequence feeding the LP synthesizer [2].
It is argued in [3] that the reason behind the failure of the classical method in providing such sparse representation is that it relies on the minimization of l_{2}norm of prediction error. It is known that the l_{2}norm criterion is highly sensitive to the outliers [4], i.e., the points having considerably larger norms of error. Hence, l_{2}norm error minimization favors solutions with many small nonzero entries rather than the sparse solutions having the fewest possible nonzero entries [4]. Hence, l_{2}norm is not an appropriate objective function for the problems where sparseness constraints are incorporated. Indeed, the ideal solution for sparse residual recovery is to directly minimize the cardinality of this vector, i.e., the l_{0}norm of prediction error which yields a combinatorial optimization problem. Instead, to alleviate the exaggerative effect of l_{2}norm criterion at points with large norms of error, it is usual to consider the minimization of l_{1}norm as it puts less emphasis on outliers. l_{1}norm can be regarded as a convex relaxation of the l_{0}norm and its minimization problem can be recasted into a linear program and solved by convex programming techniques [5].
The l_{1}norm minimization of residuals is already proven to be beneficial for speech processing [68]. In [6], the stability issue of l_{1}norm linear programming is addressed and a method is introduced for both having an intrinsically stable solution as well as keeping the computational cost down. The approach is based the Burg method for autoregressive parameters estimation using the least absolute forwardbackward error.
In [7], the authors have compared the Burg method with their l_{1}norm minimization method using the modern interior points method and shown that the sparseness is not preserved with the Burg method. Later, they have proposed a reweighted l_{1}norm minimization approach in [8], to enhance the sparsity of the residuals and to overcome the mismatch between l_{0}norm minimization and l_{1}norm minimization while keeping the problem solvable with convex programming tools. Initially the l_{1}norm minimization problem is solved using the interior points method and then the resulted residuals are used iteratively, to reweight the l_{1}norm objective function such that less weight is given to the points having larger residual norms. The optimization problem is thus iteratively approaching the solution for the ideal l_{0}norm objective function. We also mention that, an interesting review is made in [9,10], on several solvers for the general problem of mixed l_{p}l_{0}norm minimization in the context of piecewise constant function approximation, which indeed their adaptation to the problem of sparse linear prediction analysis can be beneficial (particularly the stepwise jump penalization algorithm, which is shown to be highly efficient and reliable in detection of sparse events).
In this article, we propose a new and efficient solution to sparse LP analysis which is based on weighting of the l_{2}norm objective function so as to maintain the computational tractability of the final optimization problem and to avoid the computational burden of convex programming. The weighting function plays the most important role in our solution in maintaining the sparsity of the resulting residuals. We first extract from the speech signal itself, the points having the potential of attaining largest norms of residuals (the glottal closure instants) and then we construct the weighting function such that the prediction error is relaxed on these points. Consequently, the weighted l_{2}norm objective function can be minimized by the solution of normal equations of liner least squares problem. We show that our closedform solution provides better sparseness properties compared to the l_{1}norm minimization using the interior points method. Also, to show the usefulness of such sparse representation, we use the resulting prediction coefficients inside a multipulse excitation (MPE) coder and we show that the corresponding multipulse excitation source provides slightly better synthesis quality compared to the estimated excitation of the classical minimum variance synthesizer.
The article is organized as follows. In Section 2, we provide the general formulation of the LP analysis problem. In Section 3, we briefly review previous studies on sparse LP analysis and the numerical motivations behind them. We present our efficient solution in Section 4. In Section 5, the experimental results are presented and finally in Section 6, we draw our conclusion and perspectives.
2 Problem formulation
The consideration of the vocal tract filter in the sourcefilter production model as an allpole filter results in the wellknown autoregressive model for the speech signal x(n):
where a_{k}are the prediction coefficients, K is the order of prediction filter and r(n) is the prediction error or the residual. In the ideal case, when the {a_{k}} coefficients are perfectly estimated and the production mechanism verifies the allpole assumption, the residual should resemble the hypothesized excitation source. In case of voiced speech, it should be a perfect semiperiodic impulse train which is zero most of the times, i.e., it is a sparse time series. The linear prediction analysis problem of a frame of length N can be written in the general matrix form as the l_{p}norm minimization of the residual vector r:
where a is a vector representing the set {a_{k}} and
and N_{1 }= 1 and N_{2 }= N + K (For n < 1 and n > N, we put x (n) = 0). The l_{p}norm is defined as
3 Approaching the l_{0}norm
The ideal solution to the LP analysis problem of Equation 2 so as to retrieve the sparse excitation source of voiced sounds, is to directly minimize the number of nonzero elements of the residual vector, i.e., its cardinality or the socalled l_{0}norm [11]. As this problem is an NP hard optimization problem [8], its relaxed but more tractable versions (p = 1,2) are the most widely used.
Setting p = 2 results in the classical minimum variance LP analysis problem. Although the latter suggests the highest computational efficiency, it is known that this solution can not provide the desired level of sparsity, even when the vocal tract filter is truly an allpole filter [1]. It is known that l_{2}norm has an exaggerative effect on the points having larger values of prediction error (the socalled outliers). Consequently, the minimizer puts much effort on forcing down the value of these outliers, with the cost of more nonzero elements. Hence, the resulting residuals are not as sparse as desired.
It is known that this exaggerative effect on the outliers is reduced with the use of l_{1}norm and hence, its minimization could be a meliorative strategy w.r.t the minimum variance solution, in that the error on the outliers are less penalized [11]. The solution to the l_{1}norm minimization is not as easy as the classical minimum variance LP analysis problem but it can be solved by recasting the minimization problem into a linear program [12] and then using convex optimization tools [5]. However, it is argued in [6] that linear programming l_{1}norm minimization, suffers from stability and computational issues and instead, an efficient algorithm is introduced, based on a lattice filter structure in which the reflection coefficients are obtained using a Burg method with l_{1} criterion and the robustness of the method is shown to be interesting for voiced sound analysis. However, it is shown in [7] that the l_{1}norm Burg algorithms behaves somewhere in between the l_{2}norm and the l_{1}norm minimization. Instead, the authors have shown that enhanced sparsity level can be achieved using modern interior points method [5] of solving the linear program. They have shown interesting results of such analysis and have argued that the added computational burden is negligible considering the consequent simplifications (granted by such a sparse representation) in applications such as open and closed loop pitch analysis and algebraic excitation search.
An iteratively reweighted l_{1}norm minimization approach is consequently proposed by the same authors in [8] to enhance the sparsity of residuals, while keeping the problem solvable by convex techniques. The algorithm starts by plain l_{1}norm minimization and then, iteratively, the resulting residuals are used to reweight the l_{1}norm cost function such that the points having larger residuals (outliers) are less penalized and the points having smaller residuals are penalized heavier. Hence, the optimizer encourages small values to become smaller while augmenting the amplitude of outliers [13].
The enhanced sparsity properties of the reweighted l_{1}norm solution compared to the l_{1}norm minimization, and also the better performance of the l_{1}norm criterion compared to l_{2}norm criterion, can be explained numerically with the help of the graphical representation in Figure 1. There, the numerical effect of different residual values on l_{p}norm cost functions is graphically depicted. It can be seen that the penalty on outliers is increasing with p. Indeed, as p→0 the penalty of the corresponding cost function on nonzero values approaches l_{0}norm cost function (where any nonzero value is equally penalized and there is no penalization of larger values). This will force the minimization to include as many zeros as possible as their weight is zero. In case of the reweighted l_{1}norm solution [8], any residual is weighted by its inverse at each iteration and hence, the equal penalization of any nonzero value (as in l_{0}norm criterion) is achieved. In other words, if a point has a very large (resp. very small) residual, it will be less (resp. much more) penalized in the next iteration and so, the sparsity is enhanced iteratively.
4 The weighted l_{2}norm solution
We aim at developing an alternative and efficient optimization strategy which approximates the desired sparsity of the residuals. Our approach is based on the minimization of a weighted version of the l_{2}norm criterion. The weighting function plays the key role in maintaining the sparsity of the residuals. Other than pure numerical motivations on deemphasizing the exaggerative effect of l_{2}norm on outliers (as discussed in Section 3), the design of this function is motivated by the physical production process of the speech signal. We extract from the speech signal itself, the points which are physically susceptible of attaining larger values of residuals and we construct the weighting function such that the error at those outliers is less penalized.
The outliers of LP residuals have an important physical interpretation as their timepattern follows the pitch period of the speech signal. In other words, they follow the physical excitation source of the vocal tract system, which is a sparse sequence of glottal pulses separated by the pitch period. Indeed, the impulselike nature of this excitation source is reflected as effective discontinuities in the residual signal [14]: when no significant excitation is presented at the input of the vocal tract system, its output is resonating freely according to the hypothesized allpole model, and hence, it is maximally predictable by the parameters of the filter. On the other hand, the predictability is minimized when the significant excitations takes place and hence, the output signal would be under the influence of both the excitation source and the vocal tract filter. Consequently, LP residual contains clear peaks (outliers) around the instants of significant excitations of the vocal tract system. Hence, if we have apriori knowledge about these instants we can use this knowledge to impose constraints on the LP analysis problem, so as to relax the prediction error at those points. By doing so, we ensure that if any enhancement is achieved in the sparsity level of residuals, it also corresponds to the physical sparse source of excitation.
The instants of significant excitations of vocal tract are called the Glottal closure instants (GCI) [14]. The detection of GCIs has gained significant attention recently as it founds many interesting applications in pitchsynchronous speech analysis. Many methods are developed for GCI detection in adverse environment (a comparative study is provided in [15]) and the physical significance of the detected GCIs is validated by comparing them to the electro Glotto graph signal. In this article, we use the recent robust SEDREAMS algorithm [15]^{a}. The weighting function is then constructed such that less emphasize is given to the GCI points, and hence the exaggerative effect on the outliers of the residuals is canceled. We can now proceed to formalize the proposed solution.
4.1 Optimization algorithm
We opt for l_{2}norm cost function to preserve computational efficiency and then we cope with its exaggerative effect on outliers, by careful downweighting of the cost function at those points. Formally, we define following optimization problem for the recovery of sparse residuals:
where w(·) is the weighting function. Once w(·) is properly defined, the solution to Equation 3 is straightforward. Indeed, setting the derivative of the cost function to zero results in a set of normal equations which can be solved as in the classical l_{2}norm approach:
while in our case, R=(W⊙X)X^{T}, p=w⊙(X^{T}x), ⊙ denotes the elementwise product of the two matrices and:
It is interesting to mention that our experiments show that as long as the smoothness of the w(·) is maintained the stability of the solution is preserved. Indeed, the special form of the input vector Xin Equation 2, is the one used in autocorrelation formulation of LP analysis using l_{2}norm minimization. It is proven that autocorrelation formulation always results in a minimumphase estimate of the allpole filter, even if the real vocal tract filter is not minimum phase [1]. As our formulation is similar to the autocorrelation formulation, we can fairly expect the same behavior (though we don’t have a theoretical proof). This is indeed beneficial, as having a nonminimum phase spectral estimate results in saturations during synthesis applications. Our experiments show that such saturation indeed never happens. This is an interesting advantage of our method compared to l_{1}norm minimization methods which do not guaranty a minimum phase solution, unless if additional constraints are imposed to the problem [7].
4.2 The weighting function
The weighting function is expected to provide the lowest weights at the GCI points and to give equal weights (of one) to the remaining points. To put a smoothly decaying downweighting around GCI points and to have a controllable region of tolerance around them, a natural choice is to use a Gaussian function. We thus define the final weighting function as:
where T_{k }k =1 … N_{gci} denotes the detected GCI points and g(·) is a Gaussian function (
Figure 2. The weighting function A frame of a voiced sound along with the detected GCI locations and the constructed weighting function (with σ = 50 and κ = 1 ).
5 Experimental results
We first show the ability of our approach in retrieving sparse residuals for stationary voiced signals and also we show that it can provide a better estimation of the allpole vocaltract filter parameters. We then show how such sparse modeling can enhance the performance of a multipulse excitation estimation. All the results presented in this section are obtained using the following set of parameters for w(·): κ = 0 .9 and σ = 50. The choice of the parameters was obtained using a small development set (of few voiced frames) taken from the TIMIT database [18].
5.1 Sparsity of residuals for voiced sounds
We compare the performance of our weightedl_{2}norm solution with that of the classic l_{2}norm minimization and also the l_{1}norm minimization via convex programming. For minimization of the l_{1}norm, we use the publicly available l_{1}magic toolbox [12] which uses the primaldual interior points optimization [5]. Figure 3 shows the residuals obtained for all these different optimization strategies. It is clear that the weightedl_{2} and also l_{1}norm criteria achieve higher level of sparsity compared to the classic l_{2}norm criterion. Moreover, a closer look reveals that our weightedl_{2}norm solution shows better sparsity properties compared to the l_{1}norm minimization: in the former, each positive peak of residuals is followed by a single negative peak (of almost the same amplitude) while for the latter, any positive peak is surrounded by two negative peaks of smaller (but yet significant) values.
Figure 3. Comparison of the sparsity of residuals. The residuals of the LP analysis obtained from different optimization strategies. The prediction order is K = 13 and the frame length is N = 160.
This comparison can be further formalized by using a quantitative measure of sparsity. There exists plenty of such measures on which a review is provided in [19]. Among them, we use the kurtosis, as it satisfies three of the most important properties that are intuitively expected from a measure of sparsity: scale invariance, rising tide and Robin Hood [19]. Kurtosis is a measure of peakedness of a distribution and higher values of kurtosis implies higher level of sparsity. Table 1 shows the kurtosis of the residuals obtained from the three optimization strategies, averaged over 20 randomly selected sentences of both male and female speakers taken from TIMIT database. From the table, it is clear the our method achieves the highest level of sparsity as it obtains the highest values of the kurtosis.
Table 1. Comparison of the sparsity levels
5.2 Estimation of the allpole vocaltract filter
We also investigate the ability of our method in estimation of the allpole filter parameters. To do so, we generate a synthetic speech signal by exciting a known allpole system with a periodic sequence of impulses (at known locations). We then estimate these parameters from the synthetic signal by LP analysis using our method and the classical l_{2}norm method. Figure 4 shows the frequency response of the resulting estimates along with the frequencydomain representation of the synthetic excitation source. It can be seen that for the l_{2}norm minimizer, there is a clear shift in the peaks of the estimated filter towards the harmonics of the excitation source. Specifically, the first spectral peak is shifted toward the forth harmonic of the excitation source. Indeed, the effort of l_{2}norm minimizer in reducing great errors (the outliers due to the excitation source), has caused the estimated filter to be influenced by the excitation source. However, our weightedl_{2}norm minimization makes a very well estimation of the original allpole filter and there is no shift in the spectral peaks. This shows that our method effectively decouples the contributions of the excitation source and the allpole filter (as the source contribution is deemphasized by the weighting function).
Figure 4. Estimation of the allpole filter. (top) synthetic speech signal, (bottom) frequency response of the filters obtained with l_{2}norm and weightedl_{2}norm minimization (prediction order K = 13). Note that only the first half of the frequency axis is shown so as to enhance the presentation.
5.3 Multipulse excitation estimation
The sparseness of the excitation source is a fundamental assumption in the framework of linear predictive coding (LPC) where the speech is synthesized by feeding the estimated allpole filter by an estimate of the excitation source. The coding gain is achieved by considering a sparse representation for the excitation source. In the popular multipulse excitation (MPE) method [20,21], the synthesis filter is estimated through the classic l_{2}norm minimization and then a sparse multipulse excitation sequence is extracted through an iterative AnalysisbySynthesis procedure. However, as discussed in previous sections this synthesizer is not intrinsically a sparse one. Hence, it would be logical to expect that the employment of an intrinsically sparse synthesis filter, such as the one developed in this article, would enhance the quality of the synthesized speech using the corresponding multipulse estimate. Consequently, we compare the performance of the classical MPE synthesizer which uses minimum variance LPC synthesizer with the one whose synthesizer is obtained trough our weighted l_{2}norm minimization procedure. We emphasize that we follow exactly the same procedure for estimation of multipulse coders for both synthesizers, as in the classical MPE implementation in [21] (iterative minimization of perceptually weighted error of reconstruction).
We have tried to follow the same experimental protocol as in [22]. That is, we evaluate our method using about 1 h of clean speech signal randomly chosen from the TIMIT database (resampled to 8 kHz) uttered by speakers of different genders, accents and ages which provides enough diversity in the characteristics of the analyzed signals. Thirteen prediction coefficients are computed for frames of 20 ms (N=160) and the search for the multipulse sequence (10 pulses per frame) is performed as explained in [21]. We evaluate the quality of reconstructed speech in terms of SNR and the PESQ measure [23] which provides a score of perceptual quality in the range of 1 (the worst quality) to 5 (the best quality). The results are shown in Table 2, which shows that our method achieves slightly higher coding quality than the classical MPE synthesizer.
Table 2. Multipulse excitation coding
Finally, we emphasize that the superior performance of our weightedl_{2}norm solution in retrieving sparse residuals in Section 5.2, plus the slight improvement of the coding quality in Section 5.3, was achieved with roughly the same computational complexity as the classical l_{2}norm minimization (if we neglect the computational burden of the GCI detector). This is a great advantage compared to the computationally demanding l_{1}norm minimization via convex programming (as in [7] or in [8] where multiple reweighted l_{1}norm problems are solved) which also suffers from instability issues. Moreover, another important feature of our solution is that, during the coding experiment we observed that by using the Gaussian shape for the weighting function, the solution is always stable and it does not meet the instability issues as l_{1}norm minimization.
6 Conclusion
We introduced a simple and efficient solution to the problem of sparse residual recovery of the speech signal. Our approach is based on minimization of weighted l_{2}norm of the residuals. The l_{2}norm was used to preserve the simplicity and the efficiency of the solution, while the weighting function was designed to circumvent l_{2}norm’s exaggerative effect on larger residuals (the outliers). This is done by deemphasizing the error on the GCIs where, by considering the physical production mechanism of the speech, we expect the outliers to occur. We showed that our methodology provides better sparsity properties compared to the complex and computationally demanding l_{1}norm minimization via linear programming. Moreover, the method is interestingly immune to instability problems as opposed to l_{1}norm minimization. Also, we showed that such intrinsically sparse representation can result in slightly better synthesis quality by sparse multipulse excitation of the synthesis filter in MPEcoding framework. The performance, the efficiency and the stability of the proposed solution show a promising potential in speech processing as its application can be further investigated in a variety of applications in the general framework of speech synthesis. This would be the subject of our future communications.
Endnote
^{a}We opted for the SEDREAMS so as to benefit from its proven reliability, in order to focus on proofofconcept.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main contribution of this article is the introduction of a simple, efficient and stable solution for retrieving sparse residuals of the linear prediction analysis of the speech signal. Our solution is based on minimization of a weighted version of the l_{2}norm prediction error. The l_{2}norm was used to provide a simple and efficient solution, while the weighting function was designed to circumvent l_{2}norm’s exaggerative effect on the outliers of linear prediction error. The weighting function is thus designed such that less weight is given to the instants of significant excitation of vocal tract system (the glottal closure instants), as we expect the prediction error to be maximized at these points. Our method shows superior sparsity properties compared to the classical l_{2}norm and also the modern l_{1}norm minimization techniques. Our method has roughly the same computational complexity as the classical minimum variance solution (contrary to the computationally demanding l_{1}norm minimization using convex programming) and it does not suffer from the instability issues as l_{1}norm minimization. Moreover, we showed that our intrinsically sparse representation can slightly enhance the synthesis quality by sparse multipulse excitation of the synthesis filter in MPEcoding framework. Both authors read and approved the final manuscript.
Acknowledgements
Vahid Khanagha is funded by the INRIA CORDIS doctoral program.
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International Telecommunication Union: ITUT Recommendation p. 862: Perceptual evaluation of speech quality (PESQ) an objective method for endtoend speech quality assessment of narrowband telephone networks and speech codecs