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Three-level delta modulation for Laplacian source coding.


A signal is considered to be nonstationary if its frequency or spectral contents are changing with respect to time. Nonstationary signals are in general unpredictable and very hard to model or forecast. In order to receive consistent, reliable results, the nonstationary signals can be transformed into quasi-stationary ones, which are characterized by a nearly constant mean and variance over time. Speech is an example of nonstationary signal, but its frequency content varies slowly with time. Thus, the signal can be analyzed over a short-duration interval (frame or window) where it is considered to be quasi-stationary. It was shown that the distribution of speech signal is best modeled by Laplacian distribution for frame lengths shorter than 200 ms [1].

Delta modulation ([DELTA]M) is an attractive technique that has been employed in various applications in signal processing. It is a simplified version of differential pulse code modulation (DPCM) and the class of predictive coding [26]. The standard AM architecture involves one-bit quantizer along with the fixed first-order predictor [5]. A good choice of parameters known as step size and sampling frequency can lead to an efficient implementation of [DELTA]M system.

A vast amount of investigations are conducted over the years to enhance the performance of the standard (fixed size) [DELTA]M. In the adaptive delta modulation (A[DELTA]M) the step size is adapted to signal statistics (e.g. variance) for each signal frame [7]. If better quality is required, the same adaptation can be employed using modified Pulse Code Modulation (PCM) configuration with higher number of quantization levels [8]. The analysis of nonadaptive and adaptive two-bit [DELTA]M systems is presented in [9], [10], emphasizing the performance improvement over the standard one at the cost of higher complexity. Another special type of delta modulation called sigma delta modulation (S[DELTA]M) has also been reported in the literature [11], where the short-term performance examination of S[DELTA]M has been conducted, showing that its performance is highly correlated with the variation of input signal. The solution presented in [12] offers gain in performance (i.e. higher resolution and linearity, increased dynamic range) by applying high-level (three-bit) quantization.

Three-level quantization has been used in different areas of signal processing [5], [13-15]. Its application to coding for low-energy sensors in sensor network systems is discussed in [13]. An effective implementation of three-level quantization in ECG signal compression has been demonstrated in [14]. Furthermore, its incorporation in SAM structure, for the purpose of arithmetic processing has been shown in [15]. To the best knowledge of the authors, an implementation of three-level quantization in delta modulation system has not yet been reported for coding of Laplacian sources. We propose a three-level scalar quantizer designed for variable length code-words (VLC) employing Huffman coding, since it is highly efficient when operating with low number of quantization levels [3]. The representative levels are determined as centroids of their respective cells, as in the case of the Lloyd-Max quantization [2-4]. The following criteria should be fulfilled: the criterion of minimal distortion and the simultaneous criterion of minimal distortion and minimal bit rate. The appropriate performance discussion of different types of quantization, when Huffman coding is incorporated can be found in [16].

Furthermore, we proposed two configurations of delta modulation, based on two variants of three-level quantization, applied in the forward adaptive coding scheme [7], [17]. In contrast to the backward adaptation technique, forward adaptation provides gain in Signal to Quantization Noise Ratio (SQNR) in the wide dynamic range [18], as well as less sensitivity to transmission error [2]; however, it requires transfer of side information to the receiving part. The system is operating on frame-by-frame basis, where the quantizer is adapted to the short-term estimate of the frame variance.

Memoryless Laplacian source is often used for modeling of speech, image, video and bio-medical signals. In this paper, we use speech signal to test the performance of proposed configurations in a real environment.

In the first configuration, the fixed first-order predictor is considered. We propose the choice of optimal fixedpredictor coefficient to provide the highest performance measured by SNR. In the second configuration the switched first-order predictor based on correlation is used instead of fixed predictor. One effective way to use the correlation between samples within a frame is studied in [19], where the analysis is done for high speech signal quality by employing DPCM scheme with the switched first-order predictor. In our solution, the switched predictor has two fixed values of predictor coefficients on disposal, one for weakly correlated and one for highly correlated frames. The main idea presented here implies determination of the correlation coefficient for a particular frame of the input signal, which is used for selection of the appropriate predictor coefficient.

The performances of the proposed schemes are evaluated using SNR and Perceptual Evaluation of Speech Quality (PESQ) [20], as well as the bit rate. We use the adaptive two-level (conventional scheme) and four-level delta modulation schemes as baselines. Furthermore, the attained PESQ scores are compared to the wideband rate distortion bound given in [21], which is the lower bound of performance of several standardized ADPCM speech codecs for wideband speech, including G.722 standard [22] and its extensions.

The remaining of this paper is organized as follows: in Section II we present a three-level scalar quantizer design. In Section III the proposed configurations based on delta modulation are introduced. In Section IV the experimental results obtained using a real speech signal are presented and discussed. Finally, concluding remarks are given in Section V.


A scalar quantizer with N=3 levels is specified by the set of real numbers [t.sub.1], [t.sub.2], called decision thresholds, satisfying -[infinity] = [t.sub.0] < [t.sub.1] < [t.sub.2] < [t.sub.N=3] = +[infinity] and set of numbers [y.sub.1], [y.sub.2], [y.sub.3], called representation levels, satisfying [y.sub.i] [member of] [[alpha].sub.i] = [[t.sub.i-1], [t.sub.i]), for i = 1, 2, 3. With [[alpha].sub.i] is denoted the quantization cell, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The quantizer is defined as many-to-one mapping Q : R [right arrow] {[y.sub.1], [y.sub.2], [y.sub.3]} defined by Q (x) = [y.sub.i] where x [member of] [[alpha].sub.i]. Additionally, for the assumed nonlinear input source, cell [[alpha].sub.2] forms the granular region and is called granular cell, while [[alpha].sub.1] and [[alpha].sub.3] constitute an overload region and are called overload cells.

In this paper, we deal with the symmetrical three-level quantizer involving zero level [y.sub.2]. Due to the symmetry, threshold [t.sub.1] and level [y.sub.1] in the negative part of the quantizer characteristic are symmetrical to their counterparts in the positive part, i.e. the equations -[t.sub.1] = [t.sub.2] and -[y.sub.1] = [y.sub.3] will hold. Hence, only the positive threshold and level need to be determined.

Assuming the Laplacian source with zero mean and unit variance, the probability density function (pdf) becomes [1-4].

p(x) = 1/[square root of 2] exp (-[square root of 2][absolute value of x]) (1)

Given pdf, the representative level [y.sub.3] can be determined as the centroid of the respective cell [[alpha].sub.3] = ([t.sub.2], [infinity]) [2-4].


The mean squared distortion D is usually used as a measure of irreversible error introduced during the quantization process. It can be expressed as the sum of distortions in the granular and overload region [2-4].


We use SQNR as a measure of performance. Assuming the unit variance case it is determined as [2-4].

SQNR = 10[log.sub.10] (1/D) (4)

Let us denote by [p.sub.1], [p.sub.2] and [p.sub.3] the probabilities of occurrence of representative levels [y.sub.1], [y.sub.2] and [y.sub.3], respectively.



The bit rate R can be calculated as [3].

R = [N = 3.summation over (i = 1)][p.sub.i][l.sub.i] (7)

where [l.sub.i] is the length of the Huffman codeword corresponding to the output yi.

Equations (2)-(7) show that performance of the proposed quantizer is strongly dependent on decision threshold [t.sub.2]. Therefore, its corresponding value will be numerically determined in accordance with the criterion of minimal distortion (criterion 1) and the simultaneous criterion of minimal distortion and minimal bit rate (criterion 2).

Fig. 1 illustrates the bit rate R versus the distortion D for the proposed quantization model. Point A on the curve indicates the position of a three-level quantizer designed according to the criterion 1 ([t.sub.2] = 0.707), whereas the point B denotes the position when it is designed with the criterion 2 ([t.sub.2] = 0.887). We observe that the bit rate approaches to one as distortion tends to variance of the input signal (case t2 [much greater than] 1) [23].

Fig. 2 shows SQNR obtained for the Lloyd-Max quantization with N = 2 and N = 4 levels having bit rates R = 1 bit/sample and R = 2 bit/sample, respectively [2], [3]. The marked points above the curve indicate the achieved performance of three-level quantizer designed according to the aforementioned criteria. Note that the proposed model of quantizer satisfying the criterion 1 provides 1.1 dB higher SQNR when compared to the expected SQNR value (specified by a point on a curve) for the same bit rate. In addition, when the criterion 2 is satisfied, the gain of 1.31 dB can be perceived.

The detailed description of delta modulation configuration with fixed and switched prediction is given in the following section, where the proposed quantizer satisfying both criteria is implemented.


A. Delta Modulation with a Fixed Predictor and an Adaptive Three-Level Quantizer

A simple delta modulation scheme with a fixed first-order predictor is depicted in Fig. 3. In this scenario, the quantizer described in section II is applied to the forward adaptive coding scheme that consists of a buffer, a variance estimator, a log-uniform quantizer having L levels and an adaptive three-level quantizer. The adaptation to the short-term estimate of the variance is performed for each frame of the signal difference. Consequently, it is necessary to modify the codebook frame-wise. The following procedure was conducted. The buffer was used for storage of M samples of the signal e [n], where M is a frame size and e[n] = x [n] - [??][n] is a prediction error. The predicted signal x[n] = a x x [n - 1] is provided at the output of the fixed predictor. The first sample in each frame x[1] is predicted using the last sample from previous frame x[0] (i.e. the overlap between frames is one sample), except the first frame where x[0] = 0.

The average variance a] for each frame is calculated in the variance estimator.

[[sigma].sup.2.sub.e] = [1/M][M.summation over (i = 1)][e.sup.2.sub.i] (8)

and used for adaptation to the short-term signal statistics. Information about the variance has to be available at the decoder side for parameters scaling, hence it is quantized using the log-uniform quantizer ([Q.sub.LU]). It was shown in [18] that such quantizer is preferred compared to the uniform one due to its improved performance in a wide dynamic range. It is designed to quantize logarithmic variance 10[log.sub.10] ([[sigma].sup.2.sub.e]/[[sigma].sup.2.sub.ref]) in the assumed range of (-25 dB, 25 dB), where [[sigma].sup.2.sub.ref] is the reference variance.

In logarithmic domain, its thresholds are determined as.

[l.sub.i] [dB] = -25 + [[DELTA].sub.L]i, i = 0,.., L (9)

and levels are determined as.

[[??].sub.i] [dB] = -25 + [[DELTA].sub.L] (i - 1/2), i = 1,...,L (10)

where [[DELTA].sub.L] = 50/L is the step size.

In the linear domain, thresholds and levels are given by (11) and (12), respectively.



The quantization rule is given as [Q.sub.LU]([[sigma].sup.2.sub.e]) = [[??].sub.i] when [[sigma].sup.2.sub.e] [member of] ([r.sub.i-1], [r.sub.i]).

The representative level [y.sup.a.sub.3] and the decision threshold [t.sup.a.sub.2], for [[sigma].sup.2.sub.e] [member of] ([r.sub.i-1], [r.sub.i]) are determined as: [y.sup.a.sub.3] = [g.sub.i] x [y.sup.f.sub.3] and [t.sup.a.sub.2] = [g.sub.i] x [t.sup.f.sub.2], where [g.sub.i] = [square root of [[??].sub.i]] and [y.sup.f.sub.3] and [t.sup.f.sub.2] denote the level and the threshold of fixed three-level quantizer (designed for unit variance), respectively.

On the encoder output we have two digital signals I and J that are transferred to the decoder for each frame, as illustrated in Fig. 3b. Index I represents code-words for M samples in the current frame. Index J consists of [log.sub.2]L bits per frame and it contains information about the level of [Q.sub.LU] that has been used for frame variance quantization. Note that index J carries additional or side information.

Then, the bit rate is given by.

[R.sup.I] = [R.sup.f] + [log.sub.2]L/M (13)

where [R.sup.f] is a bit rate of fixed three-level quantizer.

In addition, Fig. 3b shows that the reconstructed signal y[n] can be obtained as.

y[n] = a x y [n - 1] + [y.sup.a] [n] (14)

where [y.sup.a][n] is the output of the adaptive quantizer.

B. Delta Modulation with a Switched Predictor and an Adaptive Three-Level Quantizer

In order to further improve the performance of the solution presented in Section 3. A, we have introduced a model that involves switched first-order predictor and adaptive three-level quantization, as shown in Fig. 4. The switched predictor employs one of two available coefficients a1 and a2 based on the correlation coefficient p estimated for each frame of the input signal.

[rho] = [M-1.summation over (i = 1)][x.sub.i][x.sub.i+1]/[M.summation over (i = 1)][x.sup.2.sub.i] (15)

where M is the frame size.

The predictor makes a selection between two possible values: if [rho] < 0.8 the input frame is classified as weakly correlated and the switched predictor takes the coefficient a1, otherwise the frame is considered as highly correlated and the coefficient a2 is selected. The values a1 and a2 should be optimal for weakly correlated and highly correlated frames, respectively. By using the predictor coefficients that are optimal for the available signal one can get performance improvements.

After selecting the appropriate value of the predictor coefficient, the same procedure is conducted as in the solution given in Section 3. A. Note that adaptation to the signal variance [[sigma].sup.2.sub.e] = [[sigma].sup.2.sub.e] (1 - [[rho].sup.2]) is performed for each frame, where [[sigma].sup.2.sub.x] is the variance of input signal and [[sigma].sup.2.sub.e] is the variance of prediction error.

The bit rate for the proposed scheme is.

[R.sup.II] = [R.sub.f] + 1 + [log.sub.2]L/M (16)

The number of bits per frame (side information) is increased by one in regard to the equation (13), due to the fact that information about the selected predictor coefficient (index K in Fig. 4) needs to be transferred to the receiving part as well.

C. Application in Speech Coding and Evaluation Metrics

It has been found that the Laplacian distribution accurately models speech signal [1], hence we test the proposed configurations with speech. As an objective measure of performance, we use Signal to Noise Ratio (SNR) which is composed of two components: the prediction gain G and SQNR. Let us assume a speech signal involving F frames, where the particular frame contains M samples. SQNR for the particular frame can be calculated as.

[SQNR.sup.j.sub.e] = 10[log.sub.10]([([[sigma].sup.j.sub.e]).sup.2]/[D.sup.j.sub.e]), j = 1,...F (17)

where [([[sigma].sup.j.sub.e]).sup.2] is the average variance of the j-th frame, as given in (8), and [D.sup.j.sub.e] is the average distortion of the j-th frame.

[D.sup.j.sub.e] = 1/M[M.summation over (i = 1)][([e.sub.ji] - [y.sup.a.sub.ji]).sup.2], j = 1,...,F (18)

where [y.sup.a.sub.ji] are the outputs of adaptive quantizer.

Prediction gain G for the particular frame can be determined as.

[G.sup.j] = 10[log.sub.10](1/M[M.summation over (i = 1)][x.sup.2.sub.ji]/1/M[M.summation over (i = 1)][e.sup.2.sub.ij]), j = 1,...,F (19)

where [x.sub.ji] are the samples of input speech signal.

By averaging SQNR and G given in (17) and (19) over all frames we obtain.

[SNR.sub.DM] = 1/F[F.summation over (j = 1)][SQNR.sup.j.sub.e] + 1/F[F.summation over (j = 1)][G.sup.j] (20)

Let us assume that P out of F frames are classified as weakly correlated. Then, using the equation (20) we obtain.

[SNR.sup.(1).sub.DM] = 1/P[P.summation over (j = 1)][SQNR.sup.j.sub.e] + 1/P[P.summation over (j = 1)][G.sup.j] (21)

[SNR.sup.(2).sub.DM] = 1/F - P [F - P.summation over (j = 1)][SQNR.sup.j.sub.e] + 1/F - P[F - P.summation over (j = 1)] [G.sup.j] (22)

Finally, [SNR.sub.DM] of the solution presented in Section 3.B has the form.

[SNR.sub.DM] = [wSNR.sup.(1).sub.DM] + (1 - w) [SNR.sup.(2).sub.DM] (23)

where w = P/F is the probability of occurrence of weakly correlated frame.

In addition to SNR we use PESQ, which is an objective perceptual speech quality measure that assesses how speech quality is perceived by people. PESQ uses a perceptual model to convert the original and the degraded speech into an internal representation. The degraded speech is time-aligned with the original signal to compensate for the delay that may be caused by degradation. The difference in the representations of two signals is used by the cognitive model to estimate the Mean Opinion Score (MOS). PESQ was standardized by the ITU-T P.862 standard [20] with extensions to wideband speech [24].

D. Selection of Optimal Predictor Coefficients

An analysis of the influence of predictor coefficients on performances of the proposed delta modulation configurations with an adaptive three-level quantization is carried out in order to determine the optimal coefficients which maximize [SNR.sub.DM]. We analyze both the configurations with fixed predictor (see Section 3. A) and switched predictor (see Section 3.B).

The impact of the predictor coefficient choice on SNRDM for frame size M=80 samples is illustrated in Fig. 5 for AM with a fixed predictor and an adaptive three-level quantization implemented according to the criterion 1 and criterion 2. The results are obtained using (20) when the prediction error signal e[n] is formed using the fixed predictor coefficient a ranging from 0 to 1. Note that the maximum value of SNRDM in this example was found for a = 0.98. We observe that the proposed three-level AM attains significantly higher SNRDM value compared to the one achieved by two-level AM (Lloyd-Max quantization with N = 2 levels), and very close to the one achieved by four-level [DELTA]M (Lloyd-Max quantization with N = 4 levels).

Regarding the configuration depicted in Fig. 4, we have established two methods for choosing optimal coefficients [alpha]1 and [alpha]2 of the switched predictor. The first method assumes that weakly correlated frames ([rho] < 0.8) and highly correlated frames ([rho] [greater than or equal to] 0.8) are processed separately using scheme in Fig. 3, and optimal coefficients are determined according to the criterion of maximal [SNR.sub.DM] (using (20)) in the specific region (see also Fig. 6). Note that coefficients [a.sub.1] = 0.12 and [a.sub.2] = 0.99 satisfy the required criterion for frame size M = 80 samples.

The second method for determination of switched predictor coefficients is based on the correlation coefficient of the input speech. Thus, the value [a.sub.1] is obtained as the average correlation coefficient calculated over all weakly correlated frames in the available speech signal. Vice-versa, the value [a.sub.2] corresponds to the average correlation coefficient calculated over all highly correlated frames. Optimal predictor coefficients using this criterion, for M=80 samples, are [a.sub.1] = 0.23 and [a.sub.2] = 0.95.


All experiments were performed using the speech signal that consists of 66500 samples, sampled at 16 kHz. Training sequence of approximately 3 minutes of speech was used to obtain predictor coefficients for configuration with the switched predictor. The coder was then applied to speech that was not included in the training sequence. Log-uniform quantization with L=32 levels was used for variance quantization, whereas 50 dB range of the variances is assumed and the reference variance is fixed at [[sigma].sup.2.sub.ref] = 4 x [10.sup.-4].

Fig. 7 illustrates the estimated correlation coefficient using (15) for each frame of the considered speech. Note that in the area of active speech values of p are close to 1, indicating the high predictability of the signal. in Fig. 8, we present the signal to noise ratio over all signal frames (M=80) for configurations with the three-level quantizer designed according to the criterion of minimal distortion (criterion 1) with the fixed and the switched predictor. it can be seen in both that for the active speech higher values of the SNR are obtained (up to 30 dB), while in inactive speech frames SNR decreases up to 0 dB.

in Table I we present the average values of signal to noise ratio along with the bit rate, for various frame lengths (i.e. M=80, 160, 200, 240 and 320 samples). SNRDM denotes the average signal to noise ratio obtained using a coding scheme with fixed predictor (see Section 3. A) obtained using (20). [SNR.sub.DM.sup.1] and [SNR.sub.DM.sup.2] denote the average values for the configuration with switched predictor (see Section 3.B) obtained according to (23), when the predictor coefficients are optimally selected according to the first and second method for each frame length, as given in Section 3.D. The last column [SNR.sub.DM.sup.a] serves as an upper bound and presents results obtained in an ideal case, when an adaptive first-order predictor was used.

The results in a given table indicate that the maximal value in terms of SNRDM in all considered scenarios is obtained for the frame length M=80, which is expected as the quantizer parameters are adjusted more often.

Comparing results to a delta modulation system with two-level and four-level Lloyd-Max quantization we see that the proposed scheme with fixed predictor attains almost the same SNRDM value as four-level delta modulation, with reduction in the bit rate of 0.63 bit/sample, whereas outperforming two-level delta modulation for nearly 4.2 dB (see Table III). The reason of such benefit lies in applying the variable length coding.

Furthermore, schemes with the switched predictor (see [SNR.sup.1.sub.DM] and [SNR.sup.2.sub.DM]) offer better performance than the one with fixed predictor ([SNR.sub.DM]), at the cost of slightly increased bit rate. Slightly higher performance is obtained when the switched predictor coefficients are chosen according to the first optimization method ([SNR.sup.1.sub.DM]). All proposed schemes have a performance that is very close to upper bound results, as given by [SNR.sup.a.sub.DM].

The results for a three-level delta modulation schemes when a three-level quantizer designed according to the simultaneous criterion of minimal distortion and minimal bit rate (criterion 2) is used are summarized in Table II. The table is organized in the same way as Table I, and similar conclusions can be drawn. We obtain slightly lower SNRDM values compared to the results in Table I, but with savings in bit rate. This model is preferred when higher compression is desired.

Moreover, we compare performance of the proposed schemes with the wideband rate distortion bound, which represents the lower bound of performance of several standardized ADPCM speech codecs for wideband speech, including the G.722 standard [22] and its extensions, as shown in [21]. Since distortion is not the best evaluation measure of speech quality, we generate a mapping of distortion-to-PESQ using the mapping function [21].

PESQ (D) = a[e.sup.-bxD] + 4.5 - a (24)

where a and b are estimated by the least squares fit of the distortion D and PESQ pairs of ADPCM waveform codecs.

The results for the proposed configurations (with threelevel quantization satisfying criterion 1), as well as baselines, for M=80 are given in Fig. 9, along with the rate distortion bound (solid line). We can observe that delta modulation with the switched predictor provides PESQ scores slightly better than the one of four-level delta modulation baseline, and it outperforms configurations with fixed predictor and two-level Lloyd-Max's quantization, at the same time satisfying the rate distortion lower bound.

The complexity of the proposed algorithm remains unchanged compared to the baselines, it is equal to O([N.sup.2]).


Two configurations based on delta modulation with adaptive three-level scalar quantization are proposed in this paper, one with a simple fixed first-order predictor and one with a switched predictor utilizing correlation. Switched predictor divides frames into weakly and highly correlated and chooses an appropriate, optimal coefficient for selected frame. PESQ, SNR and bit rate are used as a measure of performance. Predictor coefficients are optimized frame wise, with maximal SNR being used as criterion for optimization. It has been demonstrated in this paper that both proposed solutions with fixed and switched predictor provide over 4 dB higher SNR compared to the traditional adaptive two-level delta modulation, while comparable performance in terms of SNR, with substantial savings in bit rate, was achieved in comparison to four-level delta modulation baseline. Moreover, all proposed configurations have a performance that is very close to upper bound results, obtained using delta modulation with an adaptive three-level quantizer and an adaptive fixed first-order predictor. The obtained performances in terms of PESQ of considered solutions are further compared to rate distortion bound, showing that it provides satisfying perceived quality of speech.

This work was supported in part by the by the Ministry of Education and Science of the Republic of Serbia, grant no. TR32051 and TR32035, within the Technological Development Program.

Digital Object Identifier 10.4316/AECE.2017.01014


This work was supported in part by the by the Ministry of Education and Science of the Republic of Serbia, grant no. TR32051 and TR32035, within the Technological Development Program.


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Bojan Denic (1), Zoran Peric (1), Vladimir Despotovic (2)

(1) University of Nis, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Nis, Serbia

(2) University of Belgrade, Technical Faculty in Bor, Vojske Jugoslavije 12, 19210 Bor, Serbia

Caption: Figure 1. The dependence of bit rate R on distortion D

Caption: Figure 2. Performance of the proposed three-level quantizer compared to the Lloyd-Max quantizer with N=2 and N=4 levels

Caption: Figure 3. Delta modulation scheme with a fixed predictor and adaptive three-level quantizer: a) encoder, b) decoder

Caption: Figure 4. Delta modulation scheme with a switched predictor and adaptive three-level quantizer: a) encoder, b) decoder

Caption: Figure 5. Selection of optimal predictor coefficient for various quantizers implemented in a delta modulation scheme with fixed predictor

Caption: Figure 6. Selection of optimal values of switched predictor coefficients when three-level quantizer is implemented in AM with switched predictor

Caption: Figure 7. The correlation coefficient for different signal frames

Caption: Figure 8. SQNR for different signal frames (M=80) for N=3 levels quantizer satisfying criterion 1impelmented in the scheme with the fixed predictor and the scheme with the switched predictor (the coefficients are chosen according to the method 1)

Caption: Figure 9. The wideband rate distortion bound, as defined in [21]

M     [SNR.sub    [R.sup.I]      [SNR.sub.DM    [SNR.sub.DM
      .DM] [dB]   [bit/sample]   .sup.1] [dB]   .sup.2] [dB]

80    15.135      1.429          15.382         15.274
160   14.888      1.398          15.152         15.091
200   14.881      1.391          15.168         15.114
240   14.843      1.387          15.110         15.075
320   14.739      1.382          15.015         14.958

M     [R.sup.1]      [SNR.sub.DM
      [bit/sample]   .sup.a] [dB]

80    1.441          15.516
160   1.404          15.290
200   1.396          15.297
240   1.391          15.268
320   1.385          15.142


M     [SNR.sub    [R.sup.I]      [SNR.sub.DM    [SNR.sub.DM
      .DM] [dB]   [bit/sample]   .sup.1] [dB]   .sup.2] [dB]

80    14.652      1.342          14.891         14.747
160   14.481      1.311          14.733         14.616
200   14.487      1.305          14.756         14.679
240   14.468      1.301          14.745         14.667
320   14.378      1.296          14.638         14.551

M     [R.sup.1]      [SNR.sub.DM
      [bit/sample]   .sup.a] [dB]

80    1.355          15.017
160   1.317          14.872
200   1.310          14.886
240   1.305          14.880
320   1.299          14.763


M    [SNR.sub.n=2]   [R.sub.N=2]    [SNR.sub.N=4]   [R.sub.N=4]
     [dB]            [bit/sample]   [dB]            [bit/sample]

80   10.978          1.062          15.529          2.062
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Author:Denic, Bojan; Peric, Zoran; Despotovic, Vladimir
Publication:Advances in Electrical and Computer Engineering
Article Type:Report
Date:Feb 1, 2017
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