INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 2010, VOL. 56, NO. 2, PP. 125-128 Manuscript received April 8, 2010; revised June, 2010. 10.2478/v10177-010-0016-1

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larger signals may get considerably compressed, unlike µlaw percentage of data, pi, that fall in the subinterval i in algorithm, it is guaranteed that these signals after companding n compressed space can be found by p = i , where is the will definitely not be smaller than originally small but i expanded signals. The more uniform the source signal is, the total number of signal data points. That is, less quantization error and thus a higher signal to noise ratio is achieved. ),0[ → pr This paper is organized as follows. In sections 2, µlaw 1 1 algorithm is briefly discussed. Section 3 describes the 21 ),[ → prr 2 proposed companding algorithm, followed by quantization 32 ),[ → prr 3 error analysis in Section 4. Section 5 presents the simulation ]1,[ → pr results, and finally, conclusion is drawn in Section 6. 3 4

4) Calculation of compressing function, II. µLAW The following functions are employed for compressing law is a logarithmic conversion algorithm used in North function, America and Japan that is defined by CCITT G.711. It compresses 16−bit linear PCM data down to eight bits of  − 1 1 0)exp(1 <≤ rxxab 1 logarithmic data. The compressing and expanding functions of  )exp( <≤ rxrxab this algorithm are defined by the following equations,  2 2 1 2 (3) xF )( =  respectively.  3 3 )exp( 2 <≤ rxrxab 3

 )exp( xrxab ≤≤ 1 (1ln + x )  4 4 3 = xxF )sgn()( x ≤≤− 11 (1) ()1ln + As shown in Fig. 2, in order to map signal data from the where is the compression parameter ( in the U.S. and original space to the compressed space, by applying the Japan) and x is the normalized integer to be compressed. above function, we shall have,

−1 y 1 yF = y ( )[( ) ] y ≤≤−−+ 1111/1)sgn()( (2) , b1 is set to 1, a1 − =p1)1ln( r1 1 r2 , − rarb = )exp( a2 = )ln( 2 2 22 III. PROPOSED COMPANDING ALGORITHM 2 − r1 r1 1 r To compand the signal, more than one function is utilized: 3 , = − rarb )exp( a3 = )ln( 33 33 an exponential function for compressing stage and a 3 − r2 r2 logarithmic function for expanding. In each exponential 1 1 , a = ln( ) 4 = −ab 3 )exp( function, to make the output signal more uniformly distributed, 4 1− rr a set of companding parameters are properly adjusted. This 33 algorithm is described below. A. Compression Stage To compress the original signal, the whole signal range is divided into several subintervals and each set of these subs is mapped to a new interval. In this work, we consider 4 4 %100 ofp p3 subintervals and assign a different exponential function as a Samples compressor function for each subinterval. To do so, the p2 following steps are necessary: 3 %100 ofp r3 p1 1) ormalizing signal to [0, 1]: for simplicity, the Samples r2 amplitude of original signal is normalized to [0, 1]. %100 ofp 2) Choosing the range of each subinterval: in this work, 2 the whole range of original signal is divided into 4 Samples

subintervals, r1),0[ , rr 21 ),[ , ,[ rr 32 ) , and r3 ]1,[ . r1 1%100 ofp 3) Counting the number of signal data that fall into each Samples subinterval. In this step, it is necessary to determine the number of data points in each subinterval. Let n1, n2, n3 and n4 be the number of data points falling into subintervals

r1),0[ , rr 21 ),[ , rr 32 ),[ and r3 ]1,[ , respectively. The Fig. 2. A schematic of the technique.

A NOVEL MULTI-EXPONENTIAL FUNCTION-BASED COMPANDING TECHNIQUE FOR UNIFORM SIGNAL COMPRESSION OVER CHANNELS... 127

B. Expansion Stage verification, we constructed an audio signal using MATLAB The decompanding function is the inverse function of that audio recording command (Fig. 3). The amplitude of the can be found as, signals are normalized into [0, 1], and the whole dynamic range is divided into 128 (7 bits) and 256 (8 bits) quantization  1 − y 0)1ln( <≤ py levels, respectively. a 1  1 From Tables 1 and 2, it can be seen that in terms of  1 y quantization error, in good agreement with the theoretical  )ln( 1 <≤ pyp 2 (4) a b analyses shown in Section 4, the proposed companing −1  22 yF )( =  algorithm shows less quantization errors than that obtained 1 y  )ln( yp <≤ p from law at a modest increase of computation cost. It is also  2 3 a b33  0.8  1 y ln( ) yp ≤≤ 1 0.6 a b 3  4 4 0.4

0.2 UANTIZATION RROR NALYSIS IV. Q E A 0

As described in previous section, the original signal is Amplitude -0.2 mapped to a compressed space in such a way that the number -0.4 of data in each subinterval is proportional to the length of that subinterval. By doing so, the compressed signal exhibits a -0.6 -0.8 uniform distribution and it can be quantized using a uniform 0 500 1000 1500 2000 2500 3000 3500 4000 Time (Milliseconds) quantizer. For uniform quantization of a uniform distributed signal, since the quantization error is also uniform over each Fig. 3. A real audio signal for data processing. quantization interval, we have,

TABLE I + 2/ 2 2 2 1 2 1 2xmax 2 (xmax ) (5) RESULTS FOR SYNTHETIC DATA σ q = ∫ q dq = = ( ) = 2 − 2/ 12 12 M 3M No Multi Theoretical Criteria law where σ 2 is the mean square quantization error, ∆ is the Companding Exponential Results q Quantization 8 Bits 0.00018 0.00010 0.00006 0.00005 length of quantization interval, xmax is the maximum signal Error 7 Bits 0.00034 0.00019 0.00010 0.00008 amplitude level, and M is the number of quantization levels. Relative CPU Time 1 1.06 1.92 In general, it can be shown that the mean square quantization error is obtained by [7], [8], 22 TABLE II 2 σα x σ = (6) RESULTS FOR REAL DATA q 3M 2 No Multi Theoretical x Criteria law α = max Companding Exponential Results ' xCx )( Quantization 8 Bits 0.00098 0.00075 0.00042 0.00035 Error 7 Bits 0.00190 0.00140 0.00110 0.00088 2 where σ x is the variance of the signal, and C’(x) is the Relative CPU Time 1 1.07 2 derivative of the companding function with respect to x. For the law companding, α is close to the companding seen that the proposed algorithm would require smaller parameter which is typical equal to 255. Assuming that the number of code words for the same quantization error level number of quantization level, M is 256, Eq. (6) becomes, than when law is applied. σ 2 σ 2 = x (7) q 3 2 VI. CONCLUSIONS 2  xmax  Since σ x ≥   , from Eqs. (5), (7), one can arrive at In this work, quantization error in companding techniques  M  due to nonuniformly distributed companded signal was 2 2 ( q ) ≤ (σσ q ) investigated. To address this problem, a new algorithm that U (8) uses multiexponential companding was proposed, and it was verified using real data. The results showed that the proposed V. SIMULATION RESULTS technique has a few distinct advantages: it has lower To verify the performance of the proposed technique in quantization error than popular law, or it needs smaller terms of quantization error, which is given in Eq. (5) and number of code words for the same quantization error than that computation efficiency given in CPU time, we used both of the case when law is applied. These advantages are synthetic and real data as source signals. In the case where achieved at a cost of a slight increase in computation synthetic data are used for verification, we generated 4000 requirement, which is less of a concern, given modern fast random numbers with a Gaussian distribution; for real data computing platforms. 128 T. MOAZZENI, H. SELVARAJ, Y. JIANG

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