Lloyd-Max Quantization of Correlated Processes: How to Obtain Gains by Receiver-Sided Time-Variant Codebooks

Transcription

1 Lloyd-Max Quantization of Correlated Processes: How to Obtain Gains by Receiver-Sided Time-Variant Codebooks Sai Han and Tim Fingscheidt Institute for Communications Technology, Technische Universität Braunschweig Schleinitzstr. 22, Braunschweig, Germany Abstract Scalar non-uniform Lloyd-Max quantization (LMQ) is optimized for the minimum mean squared error satisfying the centroid condition, stating that the reconstruction level is the centroid of the area of the signal probability density function (PDF) in the respective quantization interval. For a given signal PDF, reconstruction levels are therefore time-invariant. Without predictive means at the encoder, scalar LMQ does not exploit source correlation at all. With the purpose of improving LMQ performance for correlated processes, we propose a novel scalar decoding approach employing a receiver-sided predictor inside a feedback loop. Based on the standard LMQ decision levels and utilizing the prediction error PDF shifted by the predictor output, a receiver-sided time-variant quantization codebook can be generated according to the centroid condition. Moreover, we give an analytical derivation of the variance of the prediction error. Simulations over an additive white Gaussian noise (AWGN) channel in error-free and error-prone transmission conditions show that the proposed approach outperforms the standard LMQ by about 1.0 db in terms of signal-to-noise ratio (SNR), especially for low rate quantization and highly correlated source processes. I. INTRODUCTION The Lloyd-Max quantizer (LMQ) is a well known scalar non-uniform quantizer optimized for the minimum mean squared error (MMSE) [1, 2]. Quantization codebook entries (i.e., reconstruction levels) turn out to be the centroid, or center of mass, of the signal probability density function (PDF) in the related quantization intervals, often called the centroid condition [3]. A direct consequence is that once the signal PDF is fixed, reconstruction levels are also time-invariant. Three categories of quantization approaches are identified in [4]: (1) Scalar quantization (SQ) [1, 2], or vector quantization (VQ) [5]; (2) quantizers with fixed-rate coding having the same codeword length [6 8], or quantizers with variable-rate coding having variable codeword length [9, 10]; (3) quantizers being memoryless having a fixed codebook [1, 2, 11], or those having memory utilizing the source correlation with either a time-invariant or a time-variant codebook [12 14]. It is well known that considerable redundancy is observed in speech anmages [4], with redundancy referring to statistical source correlation. Efficient quantization of correlated processes asks for VQ [5] or SQ with memory utilizing the statistical properties of the input signals (e.g., predictive quantization [15, 16] and transform coding [17]). Our study focuses on fixed-rate scalar quantization using adaptive codebooks, where the transmitter is assumed to be given and memoryless, but the receiver may now be freely designed, e.g., by including memory. Classical predictive quantization, typically employing the same predictor both at the transmitter and the receiver side, is widely usen differential pulse code modulation (DPCM) [13, 14] and adaptive differential pulse code modulation (AD- PCM) [18 20]. In the transmitter, the prediction error, which is the difference between the original signal ants predicted signal, is quantized. The transmitter-sided predictive estimation can either depend on the past original/unquantized signal (open-loop predictive quantization), or on the past reconstructed/quantized signal (closed-loop predictive quantization) with the quantizer being inside the prediction feedback loop. If the source process is an N-th order autoregressive process (AR(N)) with zero-mean Gaussian innovation, the optimal transmitter-sided predictor coefficients in open-loop predictive quantization are the same as the AR(N) source process coefficients. As a result, the variance of the transmitter-sided prediction error is identical to the variance of the innovation. However, the optimal predictor coefficients and the optimal prediction error variance in a closed-loop scheme are not supposed to be the same as the values in the open-loop scheme [3, 21]. As the prediction error is the input to the quantizer, also the scaling of the quantization codebook is further influenced and different either. As opposed to the negligible high rate quantization in the prediction loop, the above effects are even more prominent for low rate quantization [3, 21]. In order to improve performance of a given non-predictive scalar LMQ for correlated source processes, we propose a new decoding approach taking advantage of the source correlation from a Gaussian AR(N) process model. Different to predictive quantization, we assume a standard LMQ without any predictor in the encoder and employ a predictor in a feedback loop only in the decoder. In this approach, new reconstruction levels of the quantization codebook in the decoder can be generated on the basis of the prediction error PDF shifted by the predicted sample, resulting in a time-variant receiver-sided codebook. For similar reasons as outlined above, due to the feedback loop, the receiver-sided predictor coefficients and prediction error variance differ from the respective values of the Gaussian AR(N) source signal. In our previous work, a full numerical search for the optimal values of the predictor coefficient(s) and prediction error variance has been performed [22, 23]. In this paper, we present an analytical solution of the receiver-sided

2 AR(1) Signal Model Transmitter Receiver pũ(ũ) ũ + n ẽ n û + n ρ T LMQ i n Equivalent Channel î n û + n pê(ê n = û + n) dîn+1 dîn i = î n (6) T â û n 1 û n +1 pũ( û + n) = pê(ê n = û + n) ũ Fig. 1: Block diagram of the transmission system with the newly proposed receiver. prediction error variance for an AR(1) source process. The receiver-sided predictor coefficient can either be taken from the AR(1) source process or it can be found by a numerical search. The proposed decoder can advantageously employ either hard decisions or soft decisions [22, 24]. We focus on hard-decision decoding in this paper. A particular superiority of this approach is its system-compatible employment in the receiver making the best out of a given standard LMQ in the transmitter by utilizing the source correlation at the receiver. The paper is structured as follows: Section II presents the novel approach including encoder and decoder. Section III provides the formulae to calculate the variance of the receiversided prediction error. Simulation results are discussen Section IV. Conclusions are drawn in Section V. II. A. Overview TOWARDS A TIME-VARIANT LMQ CODEBOOK IN THE RECEIVER The block diagram of the transmission system with our newly proposed receiver is depicten Figure 1. A first order autoregressive process (AR(1)) is used as the source signal model, with the i.i.d. Gaussian innovation ẽ = (ẽ 0,ẽ 1,...,ẽ n,...) having zero mean, variance σẽ 2, and time index n N 0. Therefore, the correlated source samples ũ = (ũ 0,ũ 1,...,,...) fulfill = ẽ n + ρ 1, with the correlation coefficient ρ of the AR(1) process and ũ 0 = ẽ 0. The (unquantized) sample can be quantized to u n by an M bit LMQ codebook and further represented by a corresponding quantization index i n I = {0,1,...,2 M 1}. In this paper, the channel model is described as an equivalent channel, which comprises binary phase-shift keying (BPSK) modulation over an additive white Gaussian noise (AWGN) channel without any channel coding. After bit mapping, the corresponding bits are transmitted over the equivalent channel and further transformed to a received quantization index î n I. In the receiver, as outlinen Section II-C, the received sample û n equals a new time-variant LMQ reconstruction level computed from the receivendex î n and the decision levels of the standard LMQ. B. Encoder As can be seen from Figure 1, the standard LMQ is employen the encoder. Therefore, according to the centroid +1 û + n (given) Fig. 2: PDF of the unquantized signal and shifted PDF of the receiver-sided prediction error ê n. condition [3], the LMQ reconstruction level u (i) = ũ pũ(ũ)dũ pũ(ũ)dũ is the centroid of the region of the unquantized sample PDF pũ(ũ) in the i-th quantization interval [,+1 ] [12], with the decision levels and +1. The centrois marked as a black dot in the upper plot of Figure 2. As can be seen in Figure 1, the correlated Gaussian AR(1) signal model comprises a transmitter-sided prediction, with predicted sample ũ + n = a 1 having transmitter-sided predictor coefficient a = ρ and transmitter-sided prediction error ũ + n = ẽ n (equalling the innovation). If we are interesten the PDF of = ẽ n +ũ + n given a known predicted sample ũ + n at time index n, the signal PDF pũ() conditioned on a known deterministic value ũ + n becomes the transmittersided prediction error PDF pẽ() shifted by ũ + n. Therefore, we can write (1) pũ( ũ + n) = pẽ(ẽ n = ũ + n) = f( ), (2) with the shifted PDF pẽ(ẽ n = ũ + n) of the transmitter-sided prediction error being a function of for any given ũ + n. C. Decoder The newly proposed receiver for an AR(1) signal is depicted in Figure 1. For correlated processes, after the previous sample û n 1 has been estimated by the receiver, the prediction of the current sample can be carried out by a first order predictor û + n = â û n 1, (3) with the receiver-sided predictor coefficient â and û + n=0 = 0. We define a receiver-sided prediction error ê n = û + n. Replacing ũ + n in (2) by û + n, and consequently renaming ẽ n by ê n, and thereby adapting (2) to the receiver, we obtain pũ( û + n) = pê(ê n = û + n) = f( ), (4) with pê(ê n = û + n) denoting the receiver-sided prediction error PDF pê() shifted by the receiver-sided predictor output û + n, sketchen the lower graph of Figure 2.

3 Moreover, for a fixed received quantization index i at time n, the reconstruction levelu (i) given a known predicted sample û + n from (3) turns out to be (compare to (1)) u (i) n = pũ( û + n)d. (5) pũ( û + n)d In consequence, applying (4) to (5), the new time-variant codebook reconstruction levels can be calculated by u (i) n = pê(ê n = û + n)d. (6) pê(ê n = û + n)d The new centroid resulting in a new reconstruction level can be illustrated by comparing the two plots in Figure 2: First, the PDF shapes in the fixed quantization interval [,+1 ] are different. Moreover, the unquantized sample PDF (upper plot) has a larger variance than the (shifted) prediction error PDF. The shifted PDF of the receiver-sided prediction error in (6) can be obtained by pê(ê n = û + 1 n) = exp ( ( û + n) 2 ) 2πˆσê 2ˆσ ê 2, (7) with the mean û + n and the receiver-sided prediction error variance ˆσ ê 2, which will be deriven Section III. Besides, equation (6) is easily solved numerically with the help of the error function. In addition, as shown in Figure 1, the unquantized samples ũ are quantized by the standard LMQ. In other words, the correct quantization interval [,+1 ] (i.e., where the original sample falls in between) can be known by the receiver in error-free transmission conditions. As a result, the very same quantization interval should be used for decoding. Summing up the new proposed receiver process in Figure 1, the received quantization index î n I is used as i = î n in (6). Taking the decision levels dîn and dîn+1 from the standard LMQ, the received sample û n = u (i) n can be obtained by sequentially applying (3), (7), and (6). III. RECEIVER-SIDED PREDICTION ERROR Due to the prediction feedback loop in the receiver, the receiver-sided prediction error variance ˆσ ê 2 in (4) and (7) differs from the transmitter-sided prediction error variance σẽ 2 of the AR(1) signal model in (2). In this section, we present the derivation of ˆσ ê 2 in error-free transmission conditions for an AR(1) source process, finally resulting in (13). As mentionen [3], the predictor coefficients in closedloop predictive quantization are often chosen as the values in open-loop predictive quantization, because it is difficult to find the optimal coefficients given the quantized past. Additionally it is assumed that the reconstruction is reasonably good. Similarly, the variance of the receiver-sided prediction error is also difficult to obtain due to the feedback loop. In the following we adopt a simplified prediction scheme without feedback loop to model the receiver-sided prediction error in error-free transmission conditions, as it is shown in Figure 3. It represents an approximation of our newly proposed decoder Transmitter Model u n e LMQ n Receiver Prediction Path û + n T â û n 1 Fig. 3: A simplified transmission and receiver-sided prediction model without feedback loop (for error-free transmission). scheme from Figure 1, because the estimated sample û n is simply regarded as the standard LMQ output (û n = u n ), although our scheme will provide even better sample estimates as will be shown in Section IV. In consequence, the receiversided prediction error can be modeled as ê n = û + n = â û n 1 = â u n 1. (8) Considering the Lloyd-Max quantization error e LMQ n =u n, we further have û n ê n = â ( 1 +e LMQ n 1 ). (9) In order to calculate the variance of the receiver-sided prediction error, first we need the expectation value E{ê n } = E{ } â E{ 1 } â E{e LMQ = 0, (10) with the property of the LMQ that the mean of the total quantization error E{e LMQ is zero [3]. Consequently, the variance of the receiver-sided prediction error can be calculated by ˆσ ê 2 = E{(ê n E{ê n }) 2 } = E{ê 2 n}. (11) Applying (9) to (11), we obtain ˆσ ê 2 =E{( â ( 1 +e LMQ n 1 ))2 } =E{ũ 2 n +â 2 ( 1 +e LMQ n 1 )2 2â ( 1 +e LMQ n 1 )} =E{ũ 2 n +â 2 ũ 2 n 1 +â 2 (e LMQ n 1 )2 +2â 2 1 e LMQ n 1 2â 1 2â e LMQ =E{ũ 2 n}+â 2 E{ũ 2 n 1}+â 2 E{(e LMQ n 1 )2 } +2â 2 E{ 1 e LMQ 2âE{ 1 } 2âE{ e LMQ, (12) finally resulting in (see Appendix) ˆσ 2 ê = (1+â2 2âρ) σ 2 ẽ 1 ρ 2 (â2 2âρ) σ 2 ẽ SNR(M) (1 ρ 2 ) = ((1+â2 2âρ) SNR(M) (â 2 2âρ)) σẽ 2 SNR(M) (1 ρ 2, ) (13) with SNR(M) = σ2 ũ being the achievable signal-to-noise σ 2 e ratio of LMQ for Gaussian LMQ processes [12, Table 4.4 on page 135]. (These standard values can also be taken from Table I.) Note that SNR(M) is a linear entity, while the values from [12] are given in the log domain as db. Assuming the reconstruction is reasonably good, the receiver-sided predictor coefficient can be set to the correlation

4 M standard SNR SNR â = ρ ˆσ â=ρ proposed SNR â=â opt ˆσ opt TABLE I: SNR (in db) results of the standard LMQ and the proposed LMQ decoder for M bit quantized Gaussian AR(1) samples in error-free transmission, with predictor coefficient â = ρ = 0.9 and optimal predictor coefficient â opt by a full numerical search. ρ standard SNR SNR â = ρ ˆσ â=ρ proposed SNR â=â opt ˆσ opt TABLE II: SNR (in db) results of the standard LMQ and the proposed LMQ decoder for M = 2 bit quantized Gaussian AR(1) samples having different correlation coefficients ρ in error-free transmission, with predictor coefficient â = ρ and optimal predictor coefficient â opt by a full numerical search. coefficient of the Gaussian AR(1) process â = ρ. Then (13) simplifies to 10 M= 2 bit IV. A. Simulation Setup ˆσ 2 ê = 1+ ρ 2 σ 2 ẽ SNR(M) (1 ρ 2 ). (14) EVALUATION AND DISCUSSION A number of10 6 source samples is taken from the Gaussian AR(1) process with zero mean and unit variance (σ 2 ẽ = 1) innovation. The source samples are then quantized according to M {1,2,3,4,5} bit Lloyd-Max quantization codebooks, separately. The bit mapping of the quantization index is processed on the basis of the natural binary code (NBC) [12]. Assuming an AWGN channel with BPSK modulation and varying a given E b /N 0 ratio between 0 db to 10 db, the bits are transmitted over different noisy channel realizations, with E b denoting the signal energy per bit and N 0 denoting the noise power spectral density. Finally, we use the global signalto-noise ratio (SNR) (in db) with respect to the 10 6 source samples to evaluate the performance. In order to identify the optimal receiver-sided predictor coefficient â opt given the reconstructed past for error-free transmission conditions, a full numerical search over the number range ρ 0.1 â ρ in steps of is performed beforehand. For each â, the receiver-sided prediction error variance ˆσ ê 2 is obtained according to (13). The optimal value â opt is determined by the maximum SNR search result and made known to the receiver. B. Discussion The simulation results of the standard LMQ and the proposed LMQ for M bit quantized Gaussian AR(1) samples in error-free transmission conditions are providen Table I, with the correlation coefficient ρ = 0.9. For the proposed approach, â = ρ denotes that the receiver-sided predictor coefficient â is chosen to be the same as the correlation coefficient ρ and the corresponding prediction error variance ˆσ â=ρ 2 is taken from (14); on the other hand, â = â opt represents a full numerical search for the optimal â opt, while the corresponding prediction error variance ˆσ opt 2 is obtained from (13). Clear SNR gains can be observed by the proposed approach with ˆσ â=ρ 2. The best improvement is obtained for low rate quantization. SNR (db) M= 1 bit proposed with â=â opt proposed with â=ρ standard LMQ E b /N 0 (db) Fig. 4: SNR results for M=1 bit and 2 bit quantized Gaussian AR(1) samples with correlation coefficient ρ = 0.9. Further SNR gains can be achieved by the proposed approach with a full numerical search for â opt, however, the increase is practically insignificant. We also performed simulations for M = 2 bit quantized Gaussian AR(1) samples in error-free transmission conditions with various correlation coefficients ρ. As shown in Table II, better performance of the proposed approach is achievable for higher correlations. On the other hand, the same SNR is resulting from the standard and the proposed approach for uncorrelated samples (ρ = 0), due to the standard PDF pê(ê n = 0) pũ( ) acquiren the proposed approach, with â = ρ = 0, ˆσ 2 â=ρ = 1 or â = â opt = 0, ˆσ 2 opt = 1. Furthermore, our approach can also advantageously be applien error-prone transmission conditions. We performed two experiments for M = 1 bit and 2 bit quantized Gaussian AR(1) samples with correlation coefficient ρ = 0.9, respectively. In each experiment, the proposed approach can be either with a full numerical search for the optimal â opt in errorfree transmission (curve with asterisks) or with â = ρ = 0.9 (curve with circles). As can be seen in Figure 4, the SNR can be improved by up to 1.1 db using our proposed approach compared to the standard LMQ decoder. Comparing the two

5 curves with â = â opt and â = ρ, it is found again that choosing the predictor coefficient to be the source correlation coefficient achieves similar performance as performing a full numerical search forâ opt. Therefore we can state that adopting the predictor coefficient from the source process correlation coefficient (â = ρ) provides already most of the gains. Additionally, comparing the simulation results in Table 2 from [23] and the results in Table II, virtually the same performance is achieved for M = 2 bit by the two-dimensional full search as in [23], which implies that the approximation in Section III is valid. For M = 1 bit, however, the approximation leads to an SNR difference of about 0.2 db for ρ = 0.9. V. CONCLUSIONS In this paper, we propose a new approach utilizing the source correlation to improve scalar Lloyd-Max quantization (LMQ) performance. The new approach assumes a given nonpredictive standard LMQ encoder, while an additional receiversided predictor operating on previously reconstructed samples is integratento a feedback loop. The probability density function (PDF) of the source samples given the predictor output turns out to be the prediction error PDF shifted by the predicted sample. A time-variant reconstruction level can be obtained referring to the decision levels of the standard LMQ ants centroid condition. Moreover, we give an analytical derivation of the variance of the receiver-sided prediction error for Gaussian AR(1) processes. Performing a full numerical search for the optimal receiver-sided predictor coefficient or alternatively adopting the correlation coefficient from the AR(1) source processes fortunately leads to similar performance. Significant SNR improvements of about 1.0 db are observed in simulations over an AWGN channel, especially for highly correlated source processes with low rate quantization, both in error-free and error-prone transmission conditions. APPENDIX In the following, we will show how to calculate each term in the last line of (12). The LMQ property that the quantizer output is uncorrelated with the quantization error E{u n 1 e LMQ = 0 [3] results in E{ 1 e LMQ = E{(u n 1 e LMQ n 1 ) elmq = σ 2 e LMQ, (15) with σ 2 e being the variance of the Lloyd-Max quantization error. From LMQ the definition of the autocorrelation coefficient ρ we know that E{ 1 } = ϕũũ (1) = ρ ϕũũ (0) = ρ σ 2 ũ, (16) with σũ 2 being the variance of the unquantized samples ũ. Using (15), the last term in (12) can be obtained by E{ e LMQ = E{(ρ 1 +ẽ n ) e LMQ = ρ E{ 1 e LMQ +E{ẽ n e LMQ = ρ σ 2 e LMQ, (17) with E{ẽ n e LMQ = E{ẽ n} E{e LMQ = 0 because the innovation ẽ n is independent from the previous Lloyd-Max quantization error e LMQ n 1. Combining (15), (16), and (17), equation (12) becomes ˆσ 2 ê =σ 2 ũ +â 2 σ 2 ũ +â 2 σ 2 e LMQ 2â 2 σ 2 e LMQ 2âρσ2 ũ +2âρσ 2 e (18) LMQ =(1+â 2 2âρ) σũ 2 (â 2 2âρ) σ 2 e LMQ. Due to the known property of a Gaussian AR(1) process σ 2 ẽ = 1 ρ 2, we obtain σ 2 σũ 2 ũ = σ2 ẽ 1 ρ. Also, we have the signalto-noise ratio as a function of rate M [bits/sample] given as 2 SNR(M) = σ2 ũ, therefore σ 2 σ σ 2 e LMQ e = 2 LMQ ẽ SNR(M) (1 ρ 2 ). Note that SNR(M) varies for different quantizer bit rates M. Replacing σũ 2 and σ2 e in (18) we obtain (13). LMQ REFERENCES [1] S. Lloyd, Least Squares Quantization in PCM, IEEE Transactions on Information Theory, vol. 28, no. 2, pp , Mar [2] J. Max, Quantizing for Minimum Distortion, IRE Transactions on Information Theory, vol. 6, no. 1, pp. 7 12, Mar [3] A. Gersho and R. Gray, Vector Quantization and Signal Compression. Boston, Dordrecht, London: Kluwer Academic Publishers, [4] R. Gray and D. Neuhoff, Quantization, IEEE Transactions on Information Theory, vol. 44, no. 6, pp , Oct [5] Y. Linde, A. Buzo, and R. Gray, An Algorithm for Vector Quantizer Design, IEEE Transactions on Communications, vol. 28, no. 1, pp , Jan [6] B. Oliver, J. Pierce, and C. Shannon, The Philosophy of PCM, Proceedings IRE, vol. 36, no. 11, pp , Nov [7] D. Hui and D. Neuhoff, Asymptotic Analysis of Optimal Fixed- Rate Uniform Scalar Quantization, IEEE Transactions on Information Theory, vol. 47, no. 3, pp , Mar [8] S. Han, F. Pflug, and T. Fingscheidt, Improved AMR Wideband Error Concealment for Mobile Communications, in Proc. of EUSIPCO, Marrakech, Morocco, Sep. 2013, pp [9] T. Lookabaugh, E. Riskin, P. Chou, and R. Gray, Variable Rate Vector Quantization for Speech, Image, and Video Compression, IEEE Transactions on Communications, vol. 41, no. 1, pp , Jan [10] S. Han and T. Fingscheidt, Variable-Length Versus Fixed-Length Coding: On Tradeoffs for Soft-Decision Decoding, in Proc. of ICASSP, Florence, Italy, May. 2014, pp [11] W. Kleijn and R. Hagen, On Memoryless Quantization in Speech Coding, IEEE Signal Processing Letters, vol. 3, no. 8, pp , Aug [12] N. Jayant and P. Noll, Digital Coding of Waveforms. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., [13] N. Jayant, Digital Coding of Speech Waveforms: PCM, DPCM, and DM Quantizers, Proc. IEEE, vol. 62, no. 5, pp , May [14] R. A. McDonald, Signal-to-Noise and Idle Channel Performance of Differential Pulse Code Modulation Systems Particular Applications to Voice Signals, Bell System Technical Journal, vol. 45, no. 7, pp , Sep [15] P. Elias, Predictive Coding I, IRE Transactions on Information Theory, vol. 1, no. 1, pp , Mar [16] D. Arnstein, Quantization Error in Predictive Coders, IEEE Transactions on Communications, vol. 23, no. 4, pp , Apr [17] J. Huang and P. Schultheiss, Block Quantization of Correlated Gaussian Random Variables, IEEE Transactions on Communications Systems, vol. 11, no. 3, pp , Sep [18] P. Cummiskey, N. S. Jayant, and J. L. Flanagan, Adaptive Quantization in Differential PCM Coding of Speech, Bell System Technical Journal, vol. 52, no. 7, pp , Sep [19] ITU-T Recommendation G.726, 40, 32, 24, 16 kbit/s Adaptive Differential Pulse Code Modulation (ADPCM), ITU-T, Aug [20] ITU-T Recommendation G.722, 7 khz Audio-Coding Within 64 kbit/s, ITU, Nov

6 [21] P.-C. Chang and R. Gray, Gradient Algorithms for Designing Predictive Vector Quantizers, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 34, no. 4, pp , Aug [22] S. Han and T. Fingscheidt, Improving Scalar Quantization for Correlated Processes Using Adaptive Codebooks Only At the Receiver, in Proc. of EUSIPCO, Lisbon, Portugal, Sep. 2014, pp [23], Scalar Quantization With Optimized Receiver-Sided Adaptive Codebook Reconstruction Levels Controlled by a Predictor, in Proc. of 11th ITG Conference on Speech Communication, Erlangen, Germany, Sep. 2014, pp [24] T. Fingscheidt and P. Vary, Softbit Speech Decoding: A New Approach to Error Concealment, IEEE Transactions on Speech and Audio Processing, vol. 9, no. 3, pp , Mar