quasi-Design of Switched Quantizers and Speech Coding Based on Quasi-Logarithmic Compandor

1 Abstract —This article investigates switched quantizers for speech signal depicted with Gaussian probability density function (PDF). Gaussian PDF is better for smaller frame lengths that are represented here. Companding technique results in constant Signal to Quantization Noise Ratio (SQNR). In this paper two approaches are present: quasi-logarithmic (QL) and piecewise uniform (PU) compandor. Simpler compandor directly affects the complexity of hardware realization and expense of given solution, but, on the other hand, also brings to weaker performances. Therefore, a smart choice has to be made. Usage of switched technique leads to better performances. This way, the quality of quantization is improved by dividing the dynamic range of variances into multiple subranges. For each subrange a separate quantizer is designed, with its support region’s amplitude. The optimal amplitude is numerically determined, whereby a single criterion is obtaining the maximal SQNR. Bit rates of these quantizers don’t depend on signal variance, as the fixed length codes are used. The performances of proposed quantizers are demonstrated on real speech signals from the reliable database. Comparison of obtained results with other recent solutions is done in order to show the efficiency of this model.

1 Abstract-This article investigates switched quantizers for speech signal depicted with Gaussian probability density function (PDF).Gaussian PDF is better for smaller frame lengths that are represented here.Companding technique results in constant Signal to Quantization Noise Ratio (SQNR).In this paper two approaches are present: quasi-logarithmic (QL) and piecewise uniform (PU) compandor.Simpler compandor directly affects the complexity of hardware realization and expense of given solution, but, on the other hand, also brings to weaker performances.Therefore, a smart choice has to be made.Usage of switched technique leads to better performances.This way, the quality of quantization is improved by dividing the dynamic range of variances into multiple subranges.For each subrange a separate quantizer is designed, with its support region's amplitude.The optimal amplitude is numerically determined, whereby a single criterion is obtaining the maximal SQNR.Bit rates of these quantizers don't depend on signal variance, as the fixed length codes are used.The performances of proposed quantizers are demonstrated on real speech signals from the reliable database.Comparison of obtained results with other recent solutions is done in order to show the efficiency of this model.Index Terms-Signal processing algorithms; Signal quantization; Speech processing; Signal to noise ratio.

I. INTRODUCTION
The design of quantizers for speech signal transmission mostly assumes that input signal can be well described with Gaussian or Laplacian probability density functions (PDF).Both approaches are represented in the recent researches [1]- [4].A quantizer designed according to the Gaussian PDF leads to better performances, i.e. higher Signal to Quantization Noise Ratio (SQNR), than a quantizer with Laplacian PDF.Another approach is a frame size: the quantizer with Gaussian PDF is used for smaller frames [5].Since this is the case in our paper, we will present a model with Gaussian PDF.Every quantizer is described with its codebook with a certain number of code words.They divide into fixed length codes (FLC), analysed in [1], and variable length codes (VLC), examined in [2], [3].In this research, we put a stress on FLC, whereby the code length is log2N for the codebook with N levels.Paper [2] proposes a model of QL quantizer with VLC; and [3] investigates piecewise uniform quantizer with N = 128 levels.Switched quantizers are elaborated in [2], [3], as the division of variance range into subranges, whereby for each subrange separate quantizer is designed, brings to the achievement of better performances.QL compandor is characterized with its nonlinear compression function, and PU compandor with linear approximation, i.e. linear splines, between samples.
The switched piecewise uniform quantizer's reduced complexity of designing and hardware realization gave us motivation to work on this paper.Here are also proposed both efficient approximations of numerical Q function, and methods of computing performances of piecewise uniform quantizer [6].In order to prove the efficiency of proposed solutions, we performed their software simulations on real speech signals from [7].

II. SWITCHED SCALAR QUASI-LOGARITHMIC QUANTIZER
In order to suppress mismatch of the weak signal to the uniform quantizer, a non-uniform quantization has been developed.One of the realization methods was proposed by [5], by introducing companding technique, which works as follows: first, input signal is being compressed with nonlinear compressor whose compression function is c(x), afterwards obtained compressed signal is quantized with uniform quantizer Q(x), and at the end, expanding, with an inverse compression function c -1 (x), is applied.The block diagram of this method is illustrated on Fig. 1.In order to achieve constant Signal to Quantization Noise Ratio (SQNR) in the wide range of input variances, quasi-logarithmic (QL) µ-law of compression can be applied where μ is a non-dimensional compression factor, and xmax is a support region's amplitude.The support region of scalar compandor is [xmax, xmax].Due to its symmetry, we are able to make some analyses just on the positive part.Determining the optimal support region's amplitude is very important for the compandor's design and serious examinations need to be done where x and 2 x are defined as: For large xmax and assuming Although QL quantizer provides almost constant SQNR in the wide dynamic range of variances, even better results can be achieved with switched quantization technique.The quality of quantization is improved by dividing the dynamic range of variances into multiple subranges.Each of these subranges has its specially adapted compression function.The most used adaptation parameter is an average power of the signal, i.e. variance of the speech signal.If processing method of input samples "frame by frame" is used for quantization, a goal is to achieve the highest quality in the wide supposed range of frame variances.
Switched technique starts with buffering j-th frame that has M samples xj, i, whereby i = 1, ..., M, as shown in Fig. 2.After buffering, a variance of j-th frame is being determined and then log-uniform quantized.On our disposal are k quantizers designed for variances 2  ˆp  , p = 1, …, k, that have  With the appliance of switched technique and after the statistical analyses the s-th quantizer is chosen among k available ones, whereby a designation where the current frame belongs is unambiguous.The design of each of k available quantizers is performed separately, and it implies the determination of optimal amplitude as max opt ˆ, with the aim of reaching minimal distortion, i.e. maximal SQNR, in the point ˆp

 
. For this criterion the optimal parameters of quantizer copt and µopt are computed with numerical Nelder-Mead method, as well as maximal SQNR.These values are shown in Table I    where quantizer's relative range factor cp is also different for every quantizer, contrary the parameter in (9) where it is the same for all of k quantizers.This support region is divided into 2l = 8 unequal segments whose thresholds are defined as where 0, ..., il  .Inside the segment quantization is uniform with t = N/2l cells.The step size of quantization (cell width) is where 0, ..., 1. il  The parameter μ is known in advance.The determination of optimal quantizer's relative range factor cp follows the aforesaid idea linked with (9), i.e. the optimal value should lead to the maximal SQNR.We will use different integer values from [cmin, cmax] and observe where average value of SQNR reaches maximum.Average SQNR is calculated in the large number of uniformly distributed points n among the wide range of variances B, as following   where Δi(σ) represents cell width, as defined in (13).Pi is а cumulative probability of i-th segment Here used function Q(x) is defined as numerical function and we compute it as [7]  where 1 x  .On the other side, overload distortion is calculated as [8]:  The bit rate follows the same formula (10).IV.

V. CONCLUSIONS
As expected, larger number of levels N, as well as larger number of quantizers k, imply higher SQNR.QL quantizer has better performances and higher complexity of hardware realization.For the software realization, recommendation is use QL quantizer, but for hardware realization piecewise uniform quantizer is recommended.
. The uniform quantizer is realized with N levels.The support region is divided into 2l = 8 segments, which are further divided into unequal cells.In total there are N  2 cells, whose lower and upper borders are xi-1 and xi, respectively, whereby i = 2, …, N1.Edge borders are x1 = xmax and xN-1 = xmax, and for full examination is set: x0 → ∞, xN → ∞.Each of these cells has its representation level yi.All samples bigger than xmax are represented with yN, and all samples smaller than xmax are represented with y1 = yN.The final element of this companding technique, expander, uses inverse compression function c -1 (x).A mathematically complex inverse function would imply more expensive compandor's hardware realization, so it's important to apply suitable compression function c(x).Speech signal can be depicted with Gaussian probability density function a measure of quantizer's quality, is defined as expected mean squared error between original and quantized signal.Granular distortion Dg is calculated within the support region [xmax, xmax].The overload distortion Dov measures performances out of that region.Total distortion is sum of these distortions: D = Dg + Dov, that are equal to: been log-uniformly distributed in the dynamic range of variances

Fig. 2 .
Fig. 2. The switched technique of the adaptive scalar quantization.

Figure 3 gives
Figure 3 gives SQNR of switched QL quantizer for k = 8 and k = 16 quantizers and N = 64 and N = 128 levels.

Fig. 3 .
Fig. 3. SQNR of switched QL quantizer.III.SWITCHED PIECEWISE UNIFORM QUANTIZERPiecewise uniform (PU) quantizer follows above mentioned theoretical settings of QL quantizer.The main difference is in the approximation between samples: QL uses non-linear splines and PU uses linear splines.In switched technique, as above defined, support region's amplitude is different for every quantizer p = 1, …, k compare our results with other proposed solutions, we also calculate SQNRmax (maximal value) as formulae for SQNR and distortion still apply.However, granular and overload distortion and calculated in a different way.Granular distortion for piecewise linear quantization and high bit rate is[3] Figure 4 shows the average bit rates for k = 8 and k = 16 quantizers and N = 64 and N = 128 levels.

Fig. 4 .
Fig. 4. Bit rate of switched QL quantizer.IV.EXPERIMENTAL RESULTS An analysis of proposed quantizers is based on real speech signals from the ITU-T database [7].Analysed speech signals are from different speakers talking in English.Signals are labelled as SP01-SP16, whereas first 8 speakers (SP01-SP08) are male and last 8 speakers (SP09-SP16) are female.Signal samples are coded with 16 bits, sampled at frequency equal to 8 kHz and filtered within the range 300 Hz-3400 Hz.Switched scalar quasi-logarithmic quantizers (SWLOG) and switched scalar piecewise linear quantizers (SWPPUN) with 2l = 8 linear segments are analysed.Optimal values of parameters c and µ, as shown in Table I, are used for the quantizer designing.Simulations are performed for N = 64 and N = 128 quantization levels, for switched quantization with k = 8 and k = 16 quantizers, and different frame lengths M. In Table III are shown values for N = 64, k = 16, l = 4. Average values for SQNR, computed in simulations, are shown in TableIV.
. An adaptation technique requires a transmission of additional information about the quantizer selected for the processing of current frame sized M, out of k available ones, which all have N levels.Bit rate [bits/sample] is equal to

TABLE I .
OPTIMAL VALUES FOR DESIGNATED NUMBER OF REPRESENTATION LEVELS N.
The obtained values for SQNRav and SQNRmax of switched piecewise uniform quantizer with N = 128 levels and different values of parameters k and μ are shown in TableII. .

TABLE II .
VALUES FOR SQNRav [dB] AND SQNRmax [dB] OF SWITCHED PIECEWISE UNIFORM QUANTIZER WITH N = 128LEVELS AND l = 4 SEGMENTS.