Chapter-3 Waveform Coding Techniques

Transcription

1 Chapter-3 Waveform Coding Technique PCM [Pule Code Modulation] PCM i an important method of analog to-digital converion. In thi modulation the analog ignal i converted into an electrical waveform of two or more level. A imple two level waveform i hown in fig 3.1. Fig:3.1 A imple binary PCM waveform The PCM ytem block diagram i hown in fig 3.. The eential operation in the tranmitter of a PCM ytem are Sampling, Quantizing and Coding. The Quantizing and encoding operation are uually performed by the ame circuit, normally referred to a analog to digital converter. The eential operation in the receiver are regeneration, decoding and demodulation of the quantized ample. Regenerative repeater are ued to recontruct the tranmitted equence of coded pule in order to combat the accumulated effect of ignal ditortion and noie. PCM Tranmitter: Baic Block: 1. Anti aliaing Filter. Sampler 3. Quantizer 4. Encoder An anti-aliaing filter i baically a filter ued to enure that the input ignal to ampler i free from the unwanted frequency component. For mot of the application thee are low-pa filter. It remove the frequency component of the ignal which are above the cutoff frequency of the filter. The cutoff frequency of the filter i choen uch it i very cloe to the highet frequency component of the ignal.

2 Sampler unit ample the input ignal and thee ample are then fed to the Quantizer which output the quantized value for each of the ample. The quantizer output i fed to an encoder which generate the binary code for every ample. The quantizer and encoder together i called a analog to digital converter. Continuou time meage ignal PCM Wave LPF Sampler Quantizer Encoder (a) TRANSMITTER Ditorted PCM wave Regenerative Repeater Regenerative Repeater (b) Tranmiion Path Input Regeneration Circuit Decoder Recontruction Filter Detination Uer (c) RECEIVER Fig: 3. - PCM Sytem : Baic Block Diagram

3 REGENERATIVE REPEATER REGENERATION: The feature of the PCM ytem lie in the ability to control the effect of ditortion and noie produced by tranmitting a PCM wave through a channel. Thi i accomplihed by recontructing the PCM wave by mean of regenerative repeater. Three baic function: Equalization Timing and Deciion Making Ditorted PCM Wave Amplifier - Equalizer Deciion Making Device Regenerated PCM wave Timing Circuit Fig: Block diagram of a regenerative repeater. The equalizer hape the received pule o a to compenate for the effect of amplitude and phae ditortion produced by the tranmiion characteritic of the channel. The timing circuit provide a periodic pule train, derived from the received pule, for ampling the equalized pule at the intant of time where the ignal to noie ratio i maximum. The deciion device i enabled at the ampling time determined by the timing circuit. It make it deciion baed on whether the amplitude of the quantized pule plu noie exceed a predetermined voltage level.

4 Quantization Proce: The proce of tranforming Sampled amplitude value of a meage ignal into a dicrete amplitude value i referred to a Quantization. The quantization Proce ha a two-fold effect: 1. the peak-to-peak range of the input ample value i ubdivided into a finite et of deciion level or deciion threhold that are aligned with the rier of the taircae, and. the output i aigned a dicrete value elected from a finite et of repreentation level that are aligned with the tread of the taircae.. A quantizer i memory le in that the quantizer output i determined only by the value of a correponding input ample, independently of earlier analog ample applied to the input. Analog Signal Dicrete Sample ( Quantized ) 0 T T 3T Time Fig:3.4 Typical Quantization proce. Type of Quantizer: 1. Uniform Quantizer. Non- Uniform Quantizer

5 In Uniform type, the quantization level are uniformly paced, wherea in nonuniform type the pacing between the level will be unequal and motly the relation i logarithmic. Type of Uniform Quantizer: ( baed on I/P - O/P Characteritic) 1. Mid-Rie type Quantizer. Mid-Tread type Quantizer In the tair cae like graph, the origin lie the middle of the tread portion in Mid Tread type where a the origin lie in the middle of the rie portion in the Mid-Rie type. Mid tread type: Quantization level odd number. Mid Rie type: Quantization level even number. Output 7Δ/ 5Δ/ 3Δ/ Δ/ Δ Δ 3Δ 4Δ Input Fig:3.5 Input-Output Characteritic of a Mid-Rie type Quantizer

6 Output Δ Δ Δ/ 3Δ/ Input Fig:3.6 Input-Output Characteritic of a Mid-Tread type Quantizer Quantization Noie and Signal-to-Noie: The Quantization proce introduce an error defined a the difference between the input ignal, x(t) and the output ignal, yt). Thi error i called the Quantization Noie. q(t) = x(t) y(t) Quantization noie i produced in the tranmitter end of a PCM ytem by rounding off ample value of an analog bae-band ignal to the nearet permiible repreentation level of the quantizer. A uch quantization noie differ from channel noie in that it i ignal dependent. Let Δ be the tep ize of a quantizer and L be the total number of quantization level. Quantization level are 0, ± Δ., ± Δ., ±3 Δ The Quantization error, Q i a random variable and will have it ample value bounded by [-(Δ/) < q < (Δ/)]. If Δ i mall, the quantization error can be aumed to a uniformly ditributed random variable. Conider a memory le quantizer that i both uniform and ymmetric. L = Number of quantization level X = Quantizer input Y = Quantizer output

7 The output y i given by Y=Q(x) (3.1) which i a taircae function that befit the type of mid tread or mid rier quantizer of interet. Suppoe that the input x lie inide the interval I k = {x k < x x k+1 } k = 1,, L ( 3.) where x k and x k+1 are deciion threhold of the interval I k a hown in figure 3.7. I k-1 I k y k-1 y k X k-1 X k X k+1 Fig:3.7 Deciion threhold of the equalizer Correpondingly, the quantizer output y take on a dicrete value Y = y k if x lie in the interval I k Let q = quantization error with value in the range q then Y k = x+q if n lie in the interval I k Auming that the quantizer input n i the ample value of a random variable X of zero mean with variance x. The quantization noie uniformly ditributed through out the ignal band, it interfering effect on a ignal i imilar to that of thermal noie. Expreion for Quantization Noie and SNR in PCM:- Let Q = Random Variable denote the Quantization error q = Sampled value of Q Auming that the random variable Q i uniformly ditributed over the poible range (-Δ/ to Δ/), a f Q (q) = 1/Δ - Δ/ q Δ/ (3.3) 0 otherwie

8 where f Q (q) = probability denity function of the Quantization error. If the ignal doe not overload the Quantizer, then the mean of Quantization error i zero and it variance σ Q. f Q (q) 1/Δ - Δ/ 0 Δ/ q Fig:3.8 PDF for Quantization error. Therefore E { Q Q } Q q ) q f ( q dq ---- ( 3.4) 1 Q q dq --- (3.5) 1 Thu the variance of the Quantization noie produced by a Uniform Quantizer, grow a the quare of the tep ize. Equation (3.5) give an expreion for Quantization noie in PCM ytem. X Let = Variance of the bae band ignal x(t) at the input of Quantizer. When the bae band ignal i recontructed at the receiver output, we obtain original ignal plu Quantization noie. Therefore output ignal to Quantization noie ration (SNR) i given by ( Signal Power X X SNR) O Noie Power /1 Q (3.6) Smaller the tep ize Δ, larger will be the SNR. Signal to Quantization Noie Ratio:- [ Mid Tread Type ] Let x = Quantizer input, ampled value of random variable X with mean X, variance X. The Quantizer i aumed to be uniform, ymmetric and mid tread type. x max = abolute value of the overload level of the Quantizer. Δ = Step ize

9 L = No. of Quantization level given by L x max (3.7) Let n = No. of bit ued to repreent each level. In general n = L, but in the mid tread Quantizer, ince the number of repreentation level i odd, L = n (Mid tread only) ---- (3.8) From the equation 3.7 and 3.8, n Or 1 xmax 1 The ratio x max x x max n (3.9) i called the loading factor. To avoid ignificant overload ditortion, the amplitude of the Quantizer input x extend from loading factor of 4. Thu with 4 x n 1 1 x max 4 x 4 x to 4 x, which correpond to we can write equation (3.9) a (3.10) X 3 n1 ( SNR) O [ /1 4 1] (3.11) For larger value of n (typically n>6), we may approximate the reult a 3 n1 3 n ( SNR) O [ 1] ( ) (3.1) 4 16 Hence expreing SNR in db 10 log 10 (SNR) O = 6n (3.13) Thi formula tate that each bit in codeword of a PCM ytem contribute 6db to the ignal to noie ratio. For loading factor of 4, the problem of overload i.e. the problem that the ampled value of ignal fall outide the total amplitude range of Quantizer, 8σ x i le than 10-4.

10 The equation 3.11 give a good decription of the noie performance of a PCM ytem provided that the following condition are atified. 1. The Quantization error i uniformly ditributed. The ytem operate with an average ignal power above the error threhold o that the effect of channel noie i made negligible and performance i there by limited eentially by Quantization noie alone. 3. The Quantization i fine enough (ay n>6) to prevent ignal correlated pattern in the Quantization error waveform 4. The Quantizer i aligned with input for a loading factor of 4 Note: 1. Error uniformly ditributed. Average ignal power 3. n > 6 4. Loading factor = 4 From (3.13): 10 log 10 (SNR) O = 6n 7. In a PCM ytem, Bandwidth B = nw or [n=b/w] ubtituting the value of n we get 10 log 10 (SNR) O = 6(B/W) (3.14) Signal to Quantization Noie Ratio:- [ Mid Rie Type ] Let x = Quantizer input, ampled value of random variable X with mean X variance The Quantizer i aumed to be uniform, ymmetric and mid rie type. Let x max = abolute value of the overload level of the Quantizer. X. x max L (3.15) Since the number of repreentation level i even, L = n (Mid rie only) ---- (3.16) From (3.15) and (3.16) xmax n (3.17) X ( SNR) O / (3.18) where X repreent the variance or the ignal power.

11 Conider a pecial cae of Sinuoidal ignal: Let the ignal power be P, then P = 0.5 x max. P 1 P n ( SNR) O 1.5L (3.19) /1 In decibel, ( SNR ) 0 = n (3.0) Improvement of SNR can be achieved by increaing the number of bit, n. Thu for n number of bit / ample the SNR i given by the above equation For every increae of one bit / ample the tep ize reduce by half. Thu for (n+1) bit the SNR i given by (SNR) (n+1) bit = (SNR) (n) bit + 6dB Therefore addition of each bit increae the SNR by 6dB Problem-1: An analog ignal i ampled at the Nyquit rate f = 0K and quantized into L=104 level. Find Bit-rate and the time duration Tb of one bit of the binary encoded ignal. Solution: Aume Mid-rie type, n = log L = 10 Bit-rate = Rb = nf = 00K bit/ec Bit duration Tb = 1/ Rb = 5µec. Problem-: A PCM ytem ue a uniform quantizer followed by a 7-bit binary encoder. The bit rate of the ytem i 56Mega bit/ec. Find the output ignal-to-quantization noie ratio when a inuoidal wave of 1MHz frequency i applied to the input. Solution: Given n = 7 and bit rate Rb = 56 Mega bit per econd. Sampling frequency = Rb/n = 8MHz Meage bandwidth = 4MHz. For Mid-rie type (SNR) 0 = 43.9 db

12 CLASSIFICATION OF QUANTIZATION NOISE: The Quantizing noie at the output of the PCM decoder can be categorized into four type depending on the operating condition: Overload noie, Random noie, Granular Noie and Hunting noie OVER LOAD NOISE:- The level of the analog waveform at the input of the PCM encoder need to be et o that it peak value doe not exceed the deign peak of Vmax volt. If the peak input doe exceed Vmax, then the recovered analog waveform at the output of the PCM ytem will have flat top near the peak value. Thi produce overload noie. GRANULAR NOISE:- If the input level i reduced to a relatively mall value w.r.t to the deign level (quantization level), the error value are not ame from ample to ample and the noie ha a harh ound reembling gravel being poured into a barrel. Thi i granular noie. Thi noie can be randomized (noie power decreaed) by increaing the number of quantization level i.e.. increaing the PCM bit rate. HUNTING NOISE:- Thi occur when the input analog waveform i nearly contant. For thee condition, the ample value at the Quantizer output can ocillate between two adjacent quantization level, cauing an undeired inuoidal type tone of frequency (0.5f) at the output of the PCM ytem Thi noie can be reduced by deigning the quantizer o that there i no vertical tep at contant value of the input.

13 ROBUST QUANTIZATION Feature of an uniform Quantizer Variance i valid only if the input ignal doe not overload Quantizer SNR Decreae with a decreae in the input power level. A Quantizer whoe SNR remain eentially contant for a wide range of input power level. A quantizer that atifie thi requirement i aid to be robut. The proviion for uch robut performance neceitate the ue of a non-uniform quantizer. In a nonuniform quantizer the tep ize varie. For maller amplitude range the tep ize i mall and larger amplitude range the tep ize i large. In Non Uniform Quantizer the tep ize varie. The ue of a non uniform quantizer i equivalent to paing the baeband ignal through a compreor and then applying the compreed ignal to a uniform quantizer. The reultant ignal i then tranmitted. COMPRESSOR UNIFORM QUANTIZER EXPANDER Fig: 3.9 MODEL OF NON UNIFORM QUANTIZER At the receiver, a device with a characteritic complementary to the compreor called Expander i ued to retore the ignal ample to their correct relative level. The Compreor and expander take together contitute a Compander. Compander = Compreor + Expander Advantage of Non Uniform Quantization : 1. Higher average ignal to quantization noie power ratio than the uniform quantizer when the ignal pdf i non uniform which i the cae in many practical ituation.. RMS value of the quantizer noie power of a non uniform quantizer i ubtantially proportional to the ampled value and hence the effect of the quantizer noie i reduced.

14 Expreion for quantization error in non-uniform quantizer: The Tranfer Characteritic of the compreor and expander are denoted by C(x) and C -1 (x) repectively, which are related by, C(x). C -1 (x) = ( 3.1 ) The Compreor Characteritic for large L and x inide the interval I k : dc( x) xmax for k 0,1,... L 1 dx L k where Δ k = Width in the interval I k. Let f X (x) be the PDF of X ( 3. ) Conider the two aumption: f X (x) i Symmetric f X (x) i approximately contant in each interval. ie.. f X (x) = f X (y k ) Δk = x k+1 - x k for k = 0, 1, L (3.3) Let p k = Probability of variable X lie in the interval I k, then p k = P (x k < X < x k+1 ) = f X (x) Δk = f X (y k ) Δk (3.4) with the contraint L 1 p k k 0 1 Let the random variable Q denote the quantization error, then Q = yk X for xk < X < xk+1 Variance of Q i σ Q = E ( Q ) = E [( X y k ) ] ---- (3.5) x Q ( x yk ) f X ( x) dx ---- ( 3.6) x max max Dividing the region of integration into L interval and uing (3.4) L1 xk 1 pk Q ( x yk ) dx k (3.7) k x k

15 Uing y k = 0.5 ( x k + x k+1 ) in 3.7 and carrying out the integration w.r.t x, we obtain that 1 1 L Q p k k (3.8) 1 k 0 Compreion Law. Two Commonly ued logarithmic compreion law are called µ - law and A law. μ-law: In thi companding, the compreor characteritic i defined by equation 3.9. The normalized form of compreor characteritic i hown in the figure The μ- law i ued for PCM telephone ytem in the USA, Canada and Japan. A practical value for μ i 55. c( x ) x max ln(1 x / x ln(1 ) max ) 0 x x max ( 3.9) Fig: 3.10 Compreion characteritic of μ-law

16 A-law: In A-law companding the compreor characteritic i defined by equation The normalized form of A-law compreor characteritic i hown in the figure The A-law i ued for PCM telephone ytem in Europe. A practical value for A i 100. c ( x x max ) 1 A x / x max x ln A x A ln A x 1 l / x A ma max ) 1 A x x ma ( 3.30) Fig. 3.11: A-law compreion Characteritic. Advantage of Non Uniform Quantizer Reduced Quantization noie High average SNR

17 Differential Pule Code Modulation (DPCM) For the ignal which doe not change rapidly from one ample to next ample, the PCM cheme i not preferred. When uch highly correlated ample are encoded the reulting encoded ignal contain redundant information. By removing thi redundancy before encoding an efficient coded ignal can be obtained. One of uch cheme i the DPCM technique. By knowing the pat behavior of a ignal up to a certain point in time, it i poible to make ome inference about the future value. The tranmitter and receiver of the DPCM cheme i hown in the fig3.1 and fig 3.13 repectively. Tranmitter: Let x(t) be the ignal to be ampled and x(nt) be it ample. In thi cheme the input to the quantizer i a ignal e(nt) = x(nt) - x^(nt) (3.31) where x^(nt) i the prediction for unquantized ample x(nt). Thi predicted value i produced by uing a predictor whoe input, conit of a quantized verion of the input ignal x(nt). The ignal e(nt) i called the prediction error. By encoding the quantizer output, in thi method, we obtain a modified verion of the PCM called differential pule code modulation (DPCM). Quantizer output, v(nt) = Q[e(nT)] = e(nt) + q(nt) ---- (3.3) where q(nt) i the quantization error. Predictor input i the um of quantizer output and predictor output, u(nt) = x^(nt) + v(nt) ---- (3.33) Uing 3.3 in 3.33, u(nt) = x^(nt) + e(nt) + q(nt) ----(3.34) u(nt) = x(nt) + q(nt) ----(3.35) The receiver conit of a decoder to recontruct the quantized error ignal. The quantized verion of the original input i recontructed from the decoder output uing the ame predictor a ued in the tranmitter. In the abence of noie the encoded ignal at the receiver input i identical to the encoded ignal at the tranmitter output. Correpondingly the receive output i equal to u(nt), which differ from the input x(nt) only by the quantizing error q(nt).

18 Sampled Input x(nt ) e(nt ) v(nt ) Σ Quantizer + Output ^ x(nt ) Σ Predictor u(nt ) Fig:3.1 - Block diagram of DPCM Tranmitter Input b(nt) Decoder v(nt) Σ u(nt) Output x^(nt ) Predictor Fig: Block diagram of DPCM Receiver.

19 Prediction Gain ( Gp): The output ignal-to-quantization noie ratio of a ignal coder i defined a ( SNR ) 0 X Q ( 3.36) where σ x i the variance of the ignal x(nt) and σ Q i the variance of the quantization error q(nt). Then SNR GP ( SNR) ( X E ) 0 E Q P (3.37) where σ E i the variance of the prediction error e(nt) and (SNR) P i the prediction error-to-quantization noie ratio, defined by ( SNR) The Prediction gain Gp i defined a P G P E Q (3.38) X E (3.39) The prediction gain i maximized by minimizing the variance of the prediction error. Hence the main objective of the predictor deign i to minimize the variance of the prediction error. 1 The prediction gain i defined by GP ---- (3.40) (1 1 ) and E X (1 1 ) ----(3.41) where ρ 1 Autocorrelation function of the meage ignal PROBLEM: Conider a DPCM ytem whoe tranmitter ue a firt-order predictor optimized in the minimum mean-quare ene. Calculate the prediction gain of the ytem for the following value of correlation coefficient for the meage ignal: Rx(1) Rx(1) ( i) ( ii) R (0) R (0) Solution: Uing (3.40) x (i) For ρ1= 0.85, Gp = 3.13 In db, Gp = 5dB (ii) For ρ = 0.95, Gp = 10.6 In db, Gp = 10.1dB x

20 Delta Modulation (DM) Delta Modulation i a pecial cae of DPCM. In DPCM cheme if the bae band ignal i ampled at a rate much higher than the Nyquit rate purpoely to increae the correlation between adjacent ample of the ignal, o a to permit the ue of a imple quantizing trategy for contructing the encoded ignal, Delta modulation (DM ) i preciely uch a cheme. Delta Modulation i the one-bit (or two -level) verion of DPCM. DM provide a taircae approximation to the over ampled verion of an input bae band ignal. The difference between the input and the approximation i quantized into only two level, namely, ±δ correponding to poitive and negative difference, repectively, Thu, if the approximation fall below the ignal at any ampling epoch, it i increaed by δ. Provided that the ignal doe not change too rapidly from ample to ample, we find that the tair cae approximation remain within ±δ of the input ignal. The ymbol δ denote the abolute value of the two repreentation level of the one-bit quantizer ued in the DM. Thee two level are indicated in the tranfer characteritic of Fig The tep ize of the quantizer i related to δ by = δ (3.4) Output +δ 0 Input -δ Fig-3.14: Input-Output characteritic of the delta modulator. Let the input ignal be x(t) and the taircae approximation to it i u(t). Then, the baic principle of delta modulation may be formalized in the following et of relation:

21 e( nt e( nt) x( nt) b( nt u( nt ) ) gn[ e( nt )] and (3.43) ) x( nt ) u( nt T u( nt T) x ) ^ ( nt ) b( nt ) where T i the ampling period; e(nt ) i a prediction error repreenting the difference between the preent ample value x(nt ) of the input ignal and the latet approximation ^ to it, namely x( nt ) u( nt T ).The binary quantity, b( nt ) tranmitted by the DM ytem. i the one-bit word The tranmitter of DM ytem i hown in the figure3.15. It conit of a ummer, a twolevel quantizer, and an accumulator. Then, from the equation of (3.43) we obtain the output a, n u( nt) gn[ e( it)] b( it) (3.44) i1 i1 At each ampling intant, the accumulator increment the approximation to the input ignal by ±δ, depending on the binary output of the modulator. n Sampled Input x(nt ) e(nt ) b(nt ) Σ One - Bit + Quantizer Output ^ x(nt ) Σ Delay T u(nt ) Fig Block diagram for Tranmitter of a DM ytem

22 In the receiver, hown in fig.3.16, the tair cae approximation u(t) i recontructed by paing the incoming equence of poitive and negative pule through an accumulator in a manner imilar to that ued in the tranmitter. The out-of band quantization noie in the high frequency taircae waveform u(t) i rejected by paing it through a low-pa filter with a band-width equal to the original ignal bandwidth. Delta modulation offer two unique feature: 1. No need for Word Framing becaue of one-bit code word.. Simple deign for both Tranmitter and Receiver Input b(nt) Σ u(nt) Low pa Filter u(nt-t) Delay T Fig Block diagram for Receiver of a DM ytem QUANTIZATION NOISE Delta modulation ytem are ubject to two type of quantization error: (1) lope overload ditortion, and () granular noie. If we conider the maximum lope of the original input waveform x(t), it i clear that in order for the equence of ample{u(nt )} to increae a fat a the input equence of ample {x(nt )} in a region of maximum lope of x(t), we require that the condition in equation 3.45 be atified. dx( t) max T dt ( 3.45 ) Otherwie, we find that the tep ize = δ i too mall for the tair cae approximation u(t) to follow a teep egment of the input waveform x(t), with the reult that u(t) fall behind x(t). Thi condition i called lope-overload, and the reulting quantization error i called lope-overload ditortion(noie). Since the maximum lope of the taircae approximation u(t) i fixed by the tep ize, increae and decreae in u(t) tend to occur along traight line. For thi reaon, a delta modulator uing a fixed tep ize i often referred ton a linear delta modulation (LDM).

23 The granular noie occur when the tep ize i too large relative to the local lope characteritic of the input wave form x(t), thereby cauing the taircae approximation u(t) to hunt around a relatively flat egment of the input waveform; The granular noie i analogou to quantization noie in a PCM ytem. The e choice of the optimum tep ize that minimize the mean-quare value of the quantizing error in a linear delta modulator will be the reult of a compromie between lope overload ditortion and granular noie. Output SNR for Sinuoidal Modulation. Conider the inuoidal ignal, x(t) = A co(µfot) The maximum lope of the ignal x(t) i given by dx( t) max dt f0 A (3.46) The ue of Eq.5.81 contrain the choice of tep ize = δ, o a to avoid lopeoverload. In particular, it impoe the following condition on the value of δ: dx( t) max f A (3. 47) dt T 0 Hence for no lope overload error the condition i given by equation 3.48 and A f T (3.48) 0 f 0 AT (3.49) Hence, the maximum permiible value of the output ignal power equal A Pmax ---- (3.50) 8 f0 T When there i no lope-overload, the maximum quantization error ±δ. Auming that the quantizing error i uniformly ditributed (which i a reaonable approximation for mall δ). Conidering the probability denity function of the quantization error,( defined in equation 3.51 ),

24 f Q 1 ( q) for q 0 otherwie The variance of the quantization error i Q. 1 q dq (3.51) Q (3.5) The receiver contain (at it output end) a low-pa filter whoe bandwidth i et equal to the meage bandwidth (i.e., highet poible frequency component of the meage ignal), denoted a W uch that f 0 W. Auming that the average power of the quantization error i uniformly ditributed over a frequency interval extending from -1/T to 1/T, we get the reult: Average output noie power 3 WT 3 fc No ( 3.53) f Correpondingly, the maximum value of the output ignal-to-noie ratio equal ( SNR) O P 3 max (3.54) No 8 Wf0 T Equation 3.54 how that, under the aumption of no lope-overload ditortion, the maximum output ignal-to-noie ratio of a delta modulator i proportional to the ampling rate cubed. Thi indicate a 9db improvement with doubling of the ampling rate. Problem 1. Determine the output SNR in a DM ytem for a 1KHz inuoid ampled at 3KHz without lope overload and followed by a 4KHz pot recontruction filter. Solution: Given W=4KHz, f0 = 1KHz, f = 3KHz Uing equation (3.54) we get (SNR) 0 = or 4.9dB

25 Delta Modulation: Problem. Conider a Speech Signal with maximum frequency of 3.4KHz and maximum amplitude of 1volt. Thi peech ignal i applied to a delta modulator whoe bit rate i et at 60kbit/ec. Explain the choice of an appropriate tep ize for the modulator. Solution: Bandwidth of the ignal = 3.4 KHz. Maximum amplitude = 1 volt Bit Rate = 60Kbit/ec Sampling rate = 60K Sample/ec. STEP SIZE = Volt 3. Conider a Speech Signal with maximum frequency of 3.4KHz and maximum amplitude of 1volt. Thi peech ignal i applied to a delta modulator whoe bit rate i et at 0kbit/ec. Explain the choice of an appropriate tep ize for the modulator. Solution: Bandwidth of the ignal = 3.4 KHz. Maximum amplitude = 1 volt Bit Rate = 0Kbit/ec Sampling rate = 0K Sample/ec. STEP SIZE = Volt 4. Conider a Delta modulator ytem deigned to operate at 4 time the Nyquit rate for a ignal with a 4KHz bandwidth. The tep ize of the quantizer i 400mV. a) Find the maximum amplitude of a 1KHz input inuoid for which the delta modulator doe not how lope overload. b) Find pot-filtered output SNR Solution: Bandwidth of the ignal = f0 =1 KHz. Nyquit Rate = 8K ample/ec Sampling Rate = 3K ample/ec. Step Size = 400 mv a) For 1KHz inuoid, Amax =.037 volt. b) Auming LPF bandwidth = W= 4KHz SNR = = 4.93 db

26 Adaptive Delta Modulation: The performance of a delta modulator can be improved ignificantly by making the tep ize of the modulator aume a time-varying form. In particular, during a teep egment of the input ignal the tep ize i increaed. Converely, when the input ignal i varying lowly, the tep ize i reduced. In thi way, the ize i adapted to the level of the input ignal. The reulting method i called adaptive delta modulation (ADM). There are everal type of ADM, depending on the type of cheme ued for adjuting the tep ize. In thi ADM, a dicrete et of value i provided for the tep ize. Fig.3.17 how the block diagram of the tranmitter and receiver of an ADM Sytem. In practical implementation of the ytem, the tep ize ( nt ) or ( nt ) i contrained to lie between minimum and maximum value. The upper limit, max min, control the amount of lope-overload ditortion. The lower limit,, control the amount of idle channel noie. Inide thee limit, the adaptation rule for nt ) i expreed in the general form ( δ(nt) = g(nt). δ(nt T) (3.55) where the time-varying multiplier g nt ) depend on the preent binary output b nt ) ( of the delta modulator and the M previou value b nt T ),... b( nt MT ). ( ( Thi adaptation algorithm i called a contant factor ADM with one-bit memory, where the term one bit memory refer to the explicit utilization of the ingle perviou bit b nt T ) becaue equation (3.55) can be written a, ( g(nt) = K if b(nt) = b(nt T) g(nt) = K -1 if b(nt) = b(nt T) (3.56) Thi algorithm of equation (3.56), with K=1.5 ha been found to be well matched to typically peech and image input alike, for a wide range of bit rate.

27 Figure: 3.17a) Block Diagram of ADM Tranmitter. Figure: 3.17 b): Block Diagram of ADM Receiver.

28 Coding Speech at Low Bit Rate: The ue of PCM at the tandard rate of 64 kb/ demand a high channel bandwidth for it tranmiion. But channel bandwidth i at a premium, in which cae there i a definite need for peech coding at low bit rate, while maintaining acceptable fidelity or quality of reproduction. The fundamental limit on bit rate uggeted by peech perception and information theory how that high quality peech coding i poible at rate coniderably le that 64 kb/ (the rate may actually be a low a kb/). For coding peech at low bit rate, a waveform coder of precribed configuration i optimized by exploiting both tatitical characterization of peech waveform and propertie of hearing. The deign philoophy ha two aim in mind: 1. To remove redundancie from the peech ignal a far a poible.. To aign the available bit to code the non-redundant part of the peech ignal in a perceptually efficient manner. To reduce the bit rate from 64 kb/ (ued in tandard PCM) to 3, 16, 8 and 4 kb/, the algorithm for redundancy removal and bit aignment become increaingly more ophiticated. There are two cheme for coding peech: 1. Adaptive Differential Pule code Modulation (ADPCM) kb/. Adaptive Sub-band Coding kb/ 1. Adaptive Differential Pule Code Modulation A digital coding cheme that ue both adaptive quantization and adaptive prediction i called adaptive differential pule code modulation (ADPCM). The term adaptive mean being reponive to changing level and pectrum of the input peech ignal. The variation of performance with peaker and peech material, together with variation in ignal level inherent in the peech communication proce, make the combined ue of adaptive quantization and adaptive prediction neceary to achieve bet performance. The term adaptive quantization refer to a quantizer that operate with a time-varying tep ize nt ), where T i the ampling period. The tep ize nt ) i varied o a ( ( to match the variance x of the input ignal x nt ). In particular, we write ( Δ(nT) = Φ. σ^x(nt) (3.57) where Φ Contant σ^x(nt) etimate of the σ x (nt) Thu the problem of adaptive quantization, according to (3.57) i one of etimating nt ) continuouly. x (

29 ^ The computation of the etimate x ( nt ) in done by one of two way: 1. Unquantized ample of the input ignal are ued to derive forward etimate of x ( nt ) - adaptive quantization with forward etimation (AQF). Sample of the quantizer output are ued to derive backward etimate of nt ) - adaptive quantization with backward etimation (AQB) x ( The ue of adaptive prediction in ADPCM i required becaue peech ignal are inherently nontationary, a phenomenon that manifet itelf in the fact that autocorrection function and power pectral denity of peech ignal are time-varying function of their repective variable. Thi implie that the deign of predictor for uch input hould likewie be time-varying, that i, adaptive. A with adaptive quantization, there are two cheme for performing adaptive prediction: 1. Adaptive prediction with forward etimation (APF), in which unquantized ample of the input ignal are ued to derive etimate of the predictor coefficient.. Adaptive prediction with backward etimation (APB), in which ample of the quantizer output and the prediction error are ued to derive etimate of the prediction error are ued to derive etimate of the predictor coefficient. () Adaptive Sub-band Coding: PCM and ADPCM are both time-domain coder in that the peech ignal i proceed in the time-domain a a ingle full band ignal. Adaptive ub-band coding i a frequency domain coder, in which the peech ignal i divided into a number of ub-band and each one i encoded eparately. The coder i capable of digitizing peech at a rate of 16 kb/ with a quality comparable to that of 64 kb/ PCM. To accomplih thi performance, it exploit the quai-periodic nature of voiced peech and a characteritic of the hearing mechanim known a noie making. Periodicity of voiced peech manifet itelf in the fact that people peak with a characteritic pitch frequency. Thi periodicity permit pitch prediction, and therefore a further reduction in the level of the prediction error that require quantization, compared to differential pule code modulation without pitch prediction. The number of bit per ample that need to be tranmitted i thereby greatly reduced, without a eriou degradation in peech quality. In adaptive ub band coding (ASBC), noie haping i accomplihed by adaptive bit aignment. In particular, the number of bit ued to encode each ub-band i varied dynamically and hared with other ub-band, uch that the encoding accuracy i alway placed where it i needed in the frequency domain characterization of the ignal. Indeed, ub-band with little or no energy may not be encoded at all.

30 Application 1. Hierarchy of Digital Multiplexer. Light wave Tranmiion Link (1) Digital Multiplexer: Digital Multiplexer are ued to combine digitized voice and video ignal a well a digital data into one data tream. The digitized voice ignal, digitized facimile and televiion ignal and computer output are of different rate but uing multiplexer it combined into a ingle data tream. 1 : : Multiplex er High-Speed Tranmiio n DeMux : : 1 N N Fig. 3.18: Conceptual diagram of Multiplexing and Demultiplexing. Two Major group of Digital Multiplexer: 1. To combine relatively Low-Speed Digital ignal ued for voice-grade channel. Modem are required for the implementation of thi cheme.. Operate at higher bit rate for communication carrier. Baic Problem aociated with Multiplexer: 1. Synchronization.. Multiplexed ignal hould include Framing. 3. Multiplexer Should be capable handling Small variation

31 Digital Hierarchy baed on T1 carrier: Thi wa developed by Bell ytem. The T1 carrier i deigned to operate at mega bit per econd, the T at 6.31 megabit per econd, the T3 at megabit per econd, and the T4 at mega bit per econd. Thi ytem i made up of variou combination of lower order T-carrier ubytem. Thi ytem i deigned to accommodate the tranmiion of voice ignal, Picture phone ervice and televiion ignal by uing PCM and digital ignal from data terminal equipment. The tructure i hown in the figure Fig. 3.19: Digital hierarchy of a 4 channel ytem. The T1 carrier ytem ha been adopted in USA, Canada and Japan. It i deigned to accommodate 4 voice ignal. The voice ignal are filtered with low pa filter having cutoff of 3400 Hz. The filtered ignal are ampled at 8KHz. The µ-law Companding technique i ued with the contant μ = 55. With the ampling rate of 8KHz, each frame of the multiplexed ignal occupie a period of 15μec. It conit of 4 8-bit word plu a ingle bit that i added at the end of the frame for the purpoe of ynchronization. Hence each frame conit of a total 193 bit. Each frame i of duration 15μec, correpondingly, the bit rate i mega bit per econd.

32 Another type of practical ytem, that i ued in Europe i 3 channel ytem which i hown in the figure Mbit/ Mbit/ (3 channel x 64 = 048 channel) Mbit/ Mbit/ x16 x Fig 3.0: 3 channel TDM ytem 3 channel TDM Hierarchy: In the firt level.048 megabit/ec i obtained by multiplexing 3 voice channel. 4 frame of 3 channel = 18 PCM channel, Data rate = 4 x.048 Mbit/ = 8.19 Mbit/, But due to the ynchronization bit the data rate increae to 8.448Mbit/ec. 4 x 18 = 51 channel Data rate = 4 x8.19 Mbit/ (+ ignalling bit) = Mbit/

33 () Light Wave Tranmiion Optical fiber wave guide are very ueful a tranmiion medium. They have a very low tranmiion loe and high bandwidth which i eential for high-peed communication. Other advantage include mall ize, light weight and immunity to electromagnetic interference. The baic optical fiber link i hown in the figure 3.1. The binary data fed into the tranmitter input, which emit the pule of optical power., with each pule being on or off in accordance with the input data. The choice of the light ource determine the optical ignal power available for tranmiion. Fig: 3.1- Optical fiber link. The on-off light pule produced by the tranmitter are launched into the optical fiber wave guide. During the coure of the propagation the light pule uffer lo or attenuation that increae exponentially with the ditance. At the receiver the original input data are regenerated by performing three baic operation which are : 1. Detection the light pule are converted back into pule of electrical current.. Pule Shaping and Timing - Thi involve amplification, filtering and equalization of the electrical pule, a well a the extraction of timing information. 3. Deciion Making: Depending the pule received it hould be decided that the received pule i on or off. --END--