mywbut.com Quantization and Coding

Transcription

1 Quantization and Coding 1

2 Quantization and Preproceing

3 After reading thi leon, you will learn about Need for preproceing before quantization; Introduction In thi module we hall dicu about a few apect of analog to digital (A/D) converion a relevant for the purpoe of coding, multiplexing and tranmiion. The baic principle of analog to digital converion will not be dicued. Subequently we will dicu about everal loy coding and compreion technique uch a the pule code modulation (PCM), differential pule code modulation (DPCM), and delta modulation (DM). The example of telephone grade peech ignal having a narrow bandwidth of 3.1 khz (from 300 Hz to 3.4 khz) will be ued extenively in thi module. Need for Preproceing Before Digital Converion It i eay to appreciate that the electrical equivalent of human voice i ummarily a random ignal, Fig It i alo well known that the bandwidth of an audible ignal (voice, muic etc.) i le than 0 KHz (typical frequency range i between 0 Hz and 0KHz). Interetingly, the typical bandwidth of about 0 KHz i not conidered for deigning a telephone communication ytem. Mot of the voice ignal energy i limited within 3.4 KHz. While a mall amount of energy beyond 3.4 KHz add to the quality of voice, the two important feature of a) meage intelligibility and b) peaker recognition are retained when a voice ignal i band limited to 3.4 KHz. Thi band limited voice ignal i commonly referred a telephone grade peech ignal. volt Local mean (+ve) time Fig Sketch of random peech ignal v. time A very popular ITU-T (International Telecommunication Union) tandard pecifie the bandwidth of telephone grade peech ignal between 300 Hz and 3.4 khz. The lower cut off frequency of 300 Hz ha been choen intead of 0 Hz for multiple practical reaon. The power line frequency (50Hz) i avoided. Further the phyical ize and cot of ignal proceing element uch a tranformer and capacitor are alo uitable for the choen lower cut-off frequency of 300 Hz. A very important purpoe of queezing the bandwidth i to allow a large number of peaker to communicate imultaneouly through a telephone network while haring cotly trunk line uing the 3

4 principle of multiplexing. A tandard rate of ampling for telephone grade peech ignal of one peaker i 8-Kilo ample/ ec (Kp). Uually, an A/D converter quantize an input ignal properly if the ignal i within a pecified range. A peech i alo a random ignal, there i alway a poibility that the amplitude of peech ignal at the input of a quantizer goe beyond thi range. If no protection i taken for thi problem and if the probability of uch event i not negligible, even a good quantizer will lead to unacceptably ditorted verion of the ignal. A poible remedy of thi problem i to (a) tudy and ae the variance of random peech ignal amplitude and (b) to adjut the ignal variance within a reaonable limit. Thi i often enured by uing a variance etimation unit and a variable gain amplifier, Fig Another preproceing which may be neceary i of DC adjutment of the input peech ignal. To explain thi point, let u aume that the input range of a quantizer i ±V volt. Thi quantizer expect an input ignal whoe DC (i.e average) value i zero. However if the input ignal ha an unwanted dc bia ( x in Fig 3.10.) thi alo hould be removed. For precie quantization, a mean etimation unit can be ued to etimate a local mean and ubtract it from the input ignal. If both the procee of mean removal and variance normalization are to be adopted, the mean of the ignal hould be adjuted firt. x(t) +V +V z(t) +V x 0 -V 0 -V ORIGINAL SIGNAL MEAN ADJUSTED GAIN NORMALIZED pdf of y(t) pdf(z) x x 0 y=y 1 0 z x(t) + + y(t)= x(t) -x MEAN - VARIANCE EST. EST. x σ y A z(t) = A.y(t) Fig Scheme for mean removal and variance normalization 4

5 Pule Code Modulation 5

6 After reading thi leon, you will learn about Principle of Pule Code Modulation; Signal to Quantization Noie Ratio for uniform quantizaton; A chematic diagram for Pule Code Modulation i hown in Fig The analog voice input i aumed to have zero mean and uitable variance uch that the ignal ample at the input of A/D converter lie atifactorily within the permitted ingle range. A dicued earlier, the ignal i band limited to 3.4 KHz by the low pa filter. Analog Voice L P F Analog Speech x(t) x (kt ) x q (kt ) Sample & Hold Quantizer Encoder & P/S k= Intance of ample Serial Output x(t) t x ( t ) T LPF Error Noie due to quantization Decoder R b bit/ec x q (kt ) Fig Schematic diagram of a PCM coder decoder Let x (t) denote the filtered telephone-grade peech ignal to be coded. The proce of analog to digital converion primarily involve three operation: (a) Sampling of x (t), (b) Quantization (i.e. approximation) of the dicrete time ample, x (kt ) and (c) Suitable encoding of the quantized time ample x q (kt ). T indicate the ampling interval where R = 1/T i the ampling rate (ample /ec). A tandard ampling rate for peech ignal, band limited to 3.4 khz, i 8 Kilo-ample per econd (T = 15μ ec ), thu, obeying Nyquit ampling theorem. We aume intantaneou ampling for our dicuion. The encoder in Fig generate a group of bit repreenting one quantized ample. A parallel to erial (P/S) converter i optionally ued if a erial bit tream i deired at the output of the PCM coder. The PCM coded bit tream may be taken for further digital ignal proceing and modulation for the purpoe of tranmiion. The PCM decoder at the receiver expect a erial or parallel bit-tream at it input o that it can decode the repective group of bit (a per the encoding operation) to generate quantized ample equence [x' q (kt )]. Following Nyquit ampling theorem for band limited ignal, the low pa filter produce a cloe replica ˆx () t of the original peech ignal x (t). If we conider the proce of ampling to be ideal (i.e. intantaneou ampling) and if we aume that the ame bit-tream a generated by PCM encoder i available at PCM decoder, we hould till expect ˆx ( t ) to be omewhat different from x (t). Thi i 6

7 olely becaue of the proce of quantization. A indicated, quantization i an approximation proce and thu, caue ome ditortion in the recontructed analog ignal. We ay that quantization contribute to noie. The iue of quantization noie, it characterization and technique for retricting it within an acceptable level are of importance in the deign of high quality ignal coding and tranmiion ytem. We focu a bit more on a performance metric called SQNR (Signal to Quantization Noie power Ratio) for a PCM codec. For implicity, we conider uniform quantization proce. The input-output characteritic for a uniform quantizer i hown in Fig 3.11.(a). The input ignal range (± V) of the quantizer ha been divided in eight equal interval. The width of each interval, δ, i known a the tep ize. While the amplitude of a time ample x (kt ) may be any real number between +V and V, the quantizer preent only one of the allowed eight value (±δ, ±3δ/, ) depending on the proximity of x (kt ) to thee level. Fig 3.11.(a) Linear or uniform quantizer The quantizer of Fig 3.11.(a) i known a mid-rier type. For uch a mid-rier quantizer, a lightly poitive and a lightly negative value of the input ignal will have different level at output. Thi may be a problem when the peech ignal i not preent but mall noie i preent at the input of the quantizer. To avoid uch a random fluctuation at the output of the quantizer, the mid-tread type uniform quantizer [Fig 3.11.(b)] may be ued. 7

8 Fig 3.11.(b) Mid-tread type uniform quantizer characteritic SQNR for uniform quantizer In Fig x(kt ) repreent a dicrete time (t = kt ) continuou amplitude ample of x(t) and x q (kt ) repreent the correponding quantized dicrete amplitude value. Let e k repreent the error in quantization of the k th ample i.e. ek = xq( kt) x( kt) Let, M = Number of permiible level at the quantizer output. N = Number of bit ued to repreent each ample. ±V = Permiible range of the input ignal x (t). Hence, M= N and, M.δ.V [Conidering large M and a mid-rier type quantizer] Let u conider a mall amplitude interval dx uch that the probability denity function (pdf) of x(t) within thi interval i p(x). So, p(x)dx i the probability that x(t) lie dx dx in the range ( x ) and ( x + ). Now, an expreion for the mean quare quantization error e can be written a: x1+ δ / x+ δ / x1 δ / x δ / e = p( x)( x x ) dx+ p( x)( x x ) dx... For large M and mall δ we may fairly aume that p(x) i contant within an interval, i.e. p(x) = p 1 in the 1 t interval, p(x) = p in the nd interval,., p(x) = p k in the k th interval. Therefore, the previou equation can be written a δ / ( 1...) δ / e = p + p + y dy Where, y = x-x k for all k. 8

9 So, 3 δ = ( ) 1 e p1 p δ [( p1 p...) δ ] 1 = + + Now, note that (p 1 + p + + p k + )δ = 1.0 δ e = 1 The above mean quare error repreent power aociated with the random error ignal. For convenience, we will alo indicate it a N Q. Calculation of Signal Power (S i ) After getting an etimate of quantization noie power a above, we now have to find the ignal power. In general, the ignal power can be aeed if the ignal tatitic (uch a the amplitude ditribution probability) i known. The power aociated with x (t) can be expreed a + V S i x x V () t () t p( x) = = dx where p (x) i the pdf of x (t). In abence of any pecific amplitude ditribution it i common to aume that the amplitude of ignal x (t) i uniformly ditributed between ±V. In thi cae, it i eay to ee that S i = () t () t x + V = V x 1 V dx 3 x = 3.V + V V ( Mδ ) = = V 3 1 Now the SNR can be expreed a, S N i Q = V 3 1 δ = ( Mδ ) 1 δ 1 = M It may be noted from the above expreion that thi ratio can be increaed by increaing the number of quantizer level N. Alo note that S i i the power of x (t) at input of the ampler and hence, may not repreent the SQNR at the output of the low pa filter in PCM decoder. However, for large N, mall δ and ideal and mooth filtering (e.g. Nyquit filtering) at the PCM 9

10 decoder, the power S o of deired ignal at the output of the PCM decoder can be aumed to be almot the ame a S i i.e., S o S i With thi jutification the SQNR at the output of a PCM codec, can be expreed a, and in db, A few obervation SQNR = S N S N o Q o Q M = N = 4 N ( ) db S o = 10log 6.0NdB 10 N Q (a) Note that if actual ignal excurion range i le than ± V, S o / N o < 6.0NdB. (b) If one quantized ample i repreented by 8 bit after encoding i.e., N = 8, SQNR 48 db. (c) If the amplitude ditribution of x (t) i not uniform, then the above expreion may not be applicable. Problem Q3.11.1) If a inuoid of peak amplitude 1.0V and of frequency 500Hz i ampled at k-ample /ec and quantized by a linear quantizer, determine SQNR in db when each ample i repreented by 6 bit. Q3.11.) How much i the improvement in SQNR of problem if each ample i repreented by 10 bit? Q3.11.3) What happen to SQNR of problem if each ampling rate i changed to 1.5 k-ample/ ec? 10

11 Logarithmic Pule Code Modulation (Log PCM) and Companding 11

12 After reading thi leon, you will learn about: Reaon for logarithmic PCM; A-law and μ law Companding; In a linear or uniform quantizer, a dicued earlier, the quantization error in the k-th ample i e k = x (t) x q (kt ) and the maximum error magnitude in a quantized ample i, δ Max e k = 3.1. So, if x (t) itelf i mall in amplitude and uch mall amplitude are more probable in the input ignal than amplitude cloer to ± V, it may be gueed that the quantization noie of uch an input ignal will be ignificant compared to the power of x (t). Thi implie that SQNR of uually low ignal will be poor and unacceptable. In a practical PCM codec, it i often deired to deign the quantizer uch that the SQNR i almot independent of the amplitude ditribution of the analog input ignal x (t). Thi i achieved by uing a non-uniform quantizer. A non-uniform quantizer enure maller quantization error for mall amplitude of the input ignal and relatively larger tep ize when the input ignal amplitude i large. The tranfer characteritic of a non uniform quantizer ha been hown in Fig A non-uniform quantizer can be conidered to be equivalent to an amplitude pre-ditortion proce [denoted by y = c (x) in Fig 3.1.] followed by a uniform quantizer with a fixed tep ize δ. We now briefly dicu about the characteritic of thi pre-ditortion or compreion function y = c (x). Fig Tranfer characteritic of a non-uniform quantizer 1

13 x (t) Pre-ditortion y = c (x) Uniform Quantizer x (t) Non-uniform Quantizer Fig An equivalent form of a non-uniform quantizer Mathematically, c (x) hould be a monotonically increaing function of x with odd ymmetry Fig The monotonic property enure that c -1 (x) exit over the range of x(t) and i unique with repect to c (x) i.e., c( x) c 1 ( x) = 1. o/p amplitude + V - V + V y=c(x) i/p amplitude - V Fig 3.6 Fig A deired tranfer characteritic for non-linear quantization proce Remember that the operation of c -1 (x) i neceary in the PCM decoder to get back the original ignal unditorted. The property of odd ymmetry i.e., c (-x) = - c (x) imply take care of the full range ± V of x (t). The range ± V of x (t) further implie the following: c (x) = + V, for x = +V; = 0, for x = 0; = - V, for x = - V; Let the k-th tep ize of the equivalent non-linear quantizer be δ k and the number of ignal interval be M. Further let the k-th repreentation level after quantization when the input ignal lie between x k and x k+1 be y k where 1 = +, k = 0,1,..,(M-1) k ( xk x k + 1 y ) The correponding quantization error e k i e k = x y k ; x k < x x k+1 13

14 Now oberve from Fig that δ k hould be mall if y = c (x) i large. dc( x) dx i.e., the lope of In view of thi, let u make the following imple approximation on c (x): dc( x) V 1, k = 0,1,,(M-1) dx M δ k and =, k = 0,1,.,(M-1) Note that, V M δ k xk+ 1 x k i the fixed tep ize of the uniform quantizer Fig Let u now aume that the input ignal i zero mean and it pdf p(x) i ymmetric about zero. Further for large number of interval we may aume that in each interval I k, k = 0,1,..,(M-1), the p(x) i contant. So if the input ignal x (t) i between x k and x k+1, i.e., x k < x x k+1, ( ) py ( ) p x So, the probability that x lie in the k-th interval I k, M 1 P where, ( x ) 0 x k ( < x 1) p + ( ) I = p P x x = y δ k r k k k k r k k x < = 1 Now, the mean quare quantization error e + V V ( ) ( ) x y k = x = k k= 0 xk M p xk + 1 k = k= 0 δ k x p xdx ( x y ) p( y ) M 1 k + 1 ( x y ) 1 p k k k dx dx ( xk+ 1 y ) ( e M 1 k k can be determined a follow: = k 0 3 k xk y δ k = k 3 3 M 1 1 p 1 1 k 1 ( 1) ( ) = k 0 3 xk+ xk xk+ xk xk xk = δ k ) 14

15 p p kδ k δ M 1 M 1 k 3 δ k k= 0 k k= = = Now ubtituting δ k V M ( ) dc x dx in the above expreion, we get an approximate expreion for mean quare error a 1 e = 3 V M ( ) M 1 dc x pk k= 0 dx The above expreion implie that the mean quare error due to non-uniform quantization can be expreed in term of the continuou variable x, -V< x < +V, and having a pdf p (x) a below: e V M + V V ( ) dc x ( ) dx p x dx Now, we can have an expreion of SQNR for a non-uniform quantizer a: V + V ( ) x ( ) p x dx 3 V SQNR M V + V dc( x) p x dx dx The above expreion i important a it give a clue to the deired form of the compreion function y = c(x) uch that the SQNR can be made largely independent of the pdf of x (t). It i eay to ee that a deired condition i: dc( x) K = where V < x < +V and K i a poitive contant. dx x i.e., c( x) V Kln x = + V for x > and c (x) = - c (x)

16 Note: Let u oberve that c (x) ± a x 0 from other ide. Hence the above c(x) i not realizable in practice. Further, a tated earlier, the compreion function c (x) mut pa through the origin, i.e., c (x) = 0, for x = 0. Thi requirement i forced in a compreion function in practical ytem. There are two popular tandard for non-linear quantization known a (a) The µ - law companding (b) The A law companding. The µ - law ha been popular in the US, Japan, Canada and a few other countrie while the A - law i largely followed in Europe and mot other countrie, including India, adopting ITU-T tandard. The compreion function c (x) for µ - law companding i (Fig and Fig ): μ x ln 1 c( x + ) V x =, V ln 1+ μ V ( ) µ i a contant here. The typical value of µ lie between 0 and 55. µ = 0 correpond to linear quantization. Fig μ-law companding characteritic(mu = 100) 16

17 Fig μ-law companding characteritic (mu = 0, 100, 55) The compreion function c (x) for A - law companding i (Fig ): x c( x) A V x 1 =, 0 V 1 + ln A V A x 1+ ln A V = 1 x, lna A V A i a contant here and the typical value ued in practical ytem i Fig A-law companding characteritic (A = 0, 87.5, 100, 55) For telephone grade peech ignal with 8-bit per ample and 8-Kilo ample per econd, a typical SQNR of 38.4 db i achieved in practice. A approximately logarithmic compreion function i ued for linear quantization, a PCM cheme with non-uniform quantization cheme i alo referred a Log PCM or Logarithmic PCM cheme. 17

18 Problem Q3.1.1) Conider Eq and ketch the compreion of c (x) for µ = 50 and V =.0V Q3.1.) Sketch the compreion function c (x) for A - law companding (Eq ) when V = 1V and A = 50. Q3.1.3) Comment on the effectivene of a non-linear quantizer when the peak amplitude of a ignal i known to be coniderably maller than the maximum permiible voltage V. 18

19 Differential Pule Code Modulation (DPCM) 19

20 After reading thi leon, you will learn about Principle of DPCM; DPCM modulation and de-modulation; Calculation of SQNR; One tap predictor; The tandard ampling rate for pule code modulation (PCM) of telephone grade peech ignal i f = 8 Kilo ample per ec with a ampling interval of 15 μ ec. Sample of thi band limited peech ignal are uually correlated a amplitude of peech ignal doe not change much within 15 μ ec. A typical auto correlation function R (τ ) for peech ample at the rate 8 Kilo ample per ec i hown in Fig R (τ = 15 μ ec) i uually between 0.79 and Thi apect of peech ignal i exploited in differential pule code modulation (DPCM) technique. A chematic diagram for the baic DPCM modulator i hown in Fig Note that a predictor block, a umming unit and a ubtraction unit have been trategically added to the chain of block of PCM coder intead of feeding the ampler output x (kt ) directly to a linear quantizer. An error ample e p (kt ) i fed. The error ample i given by the following expreion: = x x e ( ) ( ) p kt kt ( kt ) x ( kt ) i a predicted value for x( kt ) and i uppoed to be cloe to ( ) that ep ( kt ) i very mall in magnitude. ep ( kt ) for the n-th ample x kt uch i called a the prediction error R (τ) 0.87 R (τ)= Correlation coefficient 0 T 3 T Fig Typical normalized auto-correlation coefficient for peech ignal 0

21 x(t) x(nt ) e(nt ) v(nt ) + Sample Σ Quantizer Encoder _ b(nt ) + Σ + x ( nt ) Predictor u(nt ) Fig Schematic diagram of a DPCM modulator If we aume a final encloed bit rate of 64kbp a of a PCM coder, we enviage maller tep ize for the linear quantizer compared to the tep ize of an equivalent PCM quantizer. A a reult, it hould be poible to achieve higher SQNR for DPCM codec delivering bit at the ame rate a that of a PCM codec. There i another poibility of decreaing the coded bit rate compared to a PCM ytem if an SQNR a achievable by a PCM codec with linear equalizer i ufficient. If the predictor output ( ) x kt can be enured ufficiently cloe to ( ) x kt then we can imply encode the quantizer output ample v(kt ) in le than 8 bit. For example, if we chooe to encode each of v(kt ) by 6 bit, we achieve a erial bit rate of 48 kbp, which i coniderably le than 64 Kbp. Thi i an important feature of DPCM, epecially when the coded peech ignal will be tranmitted through wirele propagation channel. We will now develop a imple analytical tructure for a DPCM encoding cheme to bring out the role that nay be played by the prediction unit. A noted earlier, ep ( kt ) = k-th input to quantizer = x( kt ) x ( kt ) x ( kt ) = prediction of the k-th input ample x( kt ). eq( kt ) = quantizer output for k-th prediction error. c e ( ) p kt =, where c[] indicate the tranfer characteritic of the quantizer 1

22 If q ( kt ) indicate the quantization error for the k-th ample, it i eay to ee that ( ) = ( ) + q ( ) Further the input u ( kt eq kt e p kt kt ) to the predictor i, u ( ) x ( ) q( ) kt = kt + e kt = x ( kt ) + ( ) q ( ) e p kt + kt x( ) q( ) kt kt = Thi equation how that u ( kt ) i indeed a quantized verion of x( kt ) prediction ep ( kt ) will uually be mall compared to x( kt ) and q ( kt ). For a good in turn will be very mall compared to. Hence, the predictor unit hould be o deigned that x kt. variance of ( ) q kt < variance of ep ( kt ) << variance of ( ) A block chematic diagram of a DPCM demodulator i hown in Fig uing a predictor unit The cheme i traightforward and it trie to etimate ( ) u kt identical to the one ued in the modulator. We have already oberved that u ( kt ) i x kt within a mall quantization error of ( ). The analog peech very cloe to ( ) ignal i obtained by paing the ample ( ) u kt q kt Thi low pa filter hould have a 3 db cut off frequency at 3.4kHz. through an appropriate low pa filter. bˆ( nt ) i/p Demodulator output vnt ˆ( ) unt ˆ( o/p ) xˆ( t) + Decoder Σ Low pa filter Analog + Output x( nt ) Predictor unt ˆ( ) Fig Schematic diagram of a DPCM demodulator; note that the demodulator i very imilar to a portion of the modulator

23 Calculation of SQNR for DPCM The expreion for ignal to quantization noie power ratio for DPCM coding i: Variance of x( kt ) SQNR = Variance of q ( kt ) Variance of x( kt ) Variance of ep( kt) =. Variance of ep( kt) Variance of q( kt) A in PCM coding, we are auming intantaneou ampling and ideal low pa filtering. The firt term in the above expreion i the predictor gain (G p ). Thi gain viibly increae for better prediction, i.e., maller variance of e p (kt ). The econd term, SNR p i a property of the quantizer. Uually, a linear or uniform quantizer i ued for implicity in a DPCM codec. Good and efficient deign of the predictor play a crucial role in enhancing quality of ignal or effectively reducing the neceary bit rate for tranmiion. In the following we hall briefly take up the example of a ingle-tap predictor. Single-Tap Prediction A ingle-tap predictor predict the next input example x(kt ) from the immediate previou input ample x([k-1]t ). Let, xkt ˆ( ) = xkk ˆ( 1) = the k-th predicted ample, given the (k-1)th input ample = au.( k 1k 1) Here, a i known a the prediction co-efficient and uk ( 1k 1) i the (k-1)-th input to the predictor given the (k-1)-th input peech ample, i.e., x(k-1). Now the k-th prediction error ample at the input of the quantizer may be expreed a e ( kt ) e ( k) p p = xk ( ) xkk ˆ( 1) = xk ( ) auk. ( 1k 1) The mean quare error or variance of thi prediction error ample i the tatitical expectation of e p(k). Now, Ee [ ( k)] = E[{ xk ( ) auk. ( 1 k 1)} ] p = Exk [().() xk..().( axk uk 1k 1) + a uk 1k 1).( uk 1k 1)] = Exk [ ( ). xk ( ) aexk [ ( ). uk ( 1k 1)] + a. Euk [ ( 1k 1). uk ( 1k 1)]

24 Let u note that E[x(k).x(k)]=R(0). Where ( R( τ ) indicate the autocorrelation coefficient. For the econd term, let u aume that uk ( 1k 1) i an unbiaed etimate of x(k-1) and that uk ( 1k 1) i a atifactorily cloe etimate of x(k-1), o that we can ue the following approximation: Exk [ ( ). uk ( 1k 1)] Exk [ ( ). xk ( 1)] = R( τ = 1. T ) R(1), ay The third term in the expanded form of E[e p(k)] can eaily be identified a: a. E [ u ( k 1 k 1). u ( k 1 k 1)] = a. R( τ = 0) = a. R(0) p Ee [ ( k)] = R(0). ar. (1) + a R (0) R(1) = R(0)[1 a. +a ] R (0) The above expreion how that the mean quare error or variance of the prediction error can be minimized if a = R(1)/R(0). Problem Q3.13.1) I there any jutification for DPCM, if the ample of a ignal are known to be uncorrelated with each other? Q3.13.) Determine the value of prediction co-efficient for one tap prediction unit if R(0) = 1.0 and R(1) =

25 Delta Modulation (DM) 5

26 After reading thi leon, you will learn about Principle and feature of Delta Modulation; Advantage and limitation of Delta Modulation; Slope overload ditortion; Granular Noie; Condition for avoiding lope overloading; If the ampling interval T in DPCM i reduced coniderably, i.e. if we ample a band limited ignal at a rate much fater than the Nyquit ampling rate, the adjacent ample hould have higher correlation (Fig ). The ample-to-ample amplitude difference will uually be very mall. So, one may even think of only 1-bit quantization of the difference ignal. The principle of Delta Modulation (DM) i baed on thi premie. Delta modulation i alo viewed a a 1-bit DPCM cheme. The 1-bit quantizer i equivalent to a two-level comparator (alo called a a hard limiter). Fig how the chematic arrangement for generating a delta-modulated ignal. Note that, ekt ( ) = xkt ( ) xkt ˆ( ) = x( kt ) u([ k 1] T ) R xx (τ) T T Fig The correlation increae when the ampling interval i reduced 6

27 Over ampled input x(kt ) e(kt ) e q (kt )=v(kt ) + Σ 1-bit Quantizer _ ^ x(kt ) u(kt ) Delay by T + Σ + Modulator output Accumulation Unit Fig Block diagram of a delta modulator Some intereting feature of Delta Modulation No effective prediction unit the prediction unit of a DPCM coder (Fig ) i eliminated and replaced by a ingle-unit delay element. A 1-bit quantizer with two level i ued. The quantizer output imply indicate whether the preent input ample x(kt ) i more or le compared to it accumulated approximation x ˆ( kt ). Output xˆ ( kt ) of the delay unit change in mall tep. The accumulator unit goe on adding the quantizer output with the previou accumulated verion x ˆ( kt ). u(kt ), i an approximate verion of x(kt ). Performance of the Delta Modulation cheme i dependent on the ampling rate. Mot of the above comment are acceptable only when two conecutive input ample are very cloe to each other. Now, referring back to Fig , we ee that, ekt ( ) = xkt ( ) { xˆ ([ k 1] T) + v([ k 1] T)} Further, vkt ( ) = e( kt) = ignekt. [ ( )] q Here, i half of the tep-ize δ a indicated in Fig

28 + δ = - Fig Thi diagram indicate the output level of 1-bit quantizer. Note that if δ i the tep ize, the two output level are ± Now, auming zero initial condition of the accumulator, it i eay to ee that k ukt ( ) =. igne [ ( jt )] j= 1 k ( ) ( ukt Further, = v jt ) j= 1 k 1 x ˆ( kt ) = u ([ k 1] T ) = v ( jt ) j= 1 Eq how that xˆ ( kt ) i eentially an accumulated verion of the quantizer output for the error ignal e ( kt ). xˆ ( kt ) alo give a clue to the demodulator tructure for DM. Fig how a cheme for demodulation. The input to the demodulator i a binary equence and the demodulator normally tart with no prior information about the incoming equence. ˆ( ) + vkt + x( kt ) Σ Delay T ˆ( ) ukt LPF 3.4 KHZ. Demodulator output Fig Demodulator tructure for DM Now, let u recollect from our dicuion on DPCM in the previou leon ( Eq ) that, u(kt ) cloely repreent the input ignal with mall quantization error q(kt ), i.e. ukt ( ) = xkt ( ) + qkt ( )

29 Next, from the cloe loop including the delay-element in the accumulation unit in the Delta modulator tructure, we can write u([ k 1] T ) = xˆ ( kt) = x( kt) e( kt) = x([ k 1] T) + q([ k 1] T) Hence, we may expre the error ignal a, ekt ( ) = { xkt ( ) x([ k 1] T)} q([ k 1] T ) That i, the error ignal i the difference of two conecutive ample at the input except the quantization error (when quantization error i mall). Advantage of a Delta Modulator over DPCM a) A one ample of x(kt ) i repreented by only one bit after delta modulation, no elaborate word-level ynchronization i neceary at the input of the demodulator. Thi reduce hardware complexity compared to a PCM or DPCM demodulator. Bit-timing ynchronization i, however, neceary if the demodulator in implemented digitally. b) Overall complexity of a delta modulator-demodulator i le compared to DPCM a the predictor unit i abent in DM. However DM alo uffer from a few limitation uch a the following: a) Slope over load ditortion: If the input ignal amplitude change fat, the tepby-tep accumulation proce may not catch up with the rate of change (ee the ketch in Fig ). Thi happen initially when the demodulator tart operation from cold-tart but i uually of negligible effect for peech. However, if thi phenomenon occur frequently (which indirectly implie maller value of auto-correlation co-efficient R xx (τ) over a hort time interval) the quality of the received ignal uffer. The received ignal i aid to uffer from lope-overload ditortion. An intuitive remedy for thi problem i to increae the tep-ize δ but that approach ha another eriou lacuna a noted in b). 9

30 Slope over load ditortion Fig A ketch indicating lope-overload problem. The horizontal axi repreent time. The continuou line repreent the analog input ignal, before ampling and the tair-cae repreent the output xˆ ( kt ) of the delay element. b) Granular noie: If the tep-ize i made arbitrarily large to avoid lope-overload ditortion, it may lead to granular noie. Imagine that the input peech ignal i fluctuating but very cloe to zero over limited time duration. Thi may happen due to paue between entence or ele. During uch moment, our delta modulator i likely to produce a fairly long equence of , reflecting that the accumulator output i cloe but alternating around the input ignal. Thi phenomenon i manifeted at the output of the delta demodulator a a mall but perceptible noiy background. Thi i known a granular noie. An expert litener can recognize the crackling ound. Thi noie hould be kept well within a tolerable limit while deciding the tep-ize. Larger tep-ize increae the granular noie while maller tep ize increae the degree of lope-overload ditortion. In the firt level of deign, more care i given to avoid the lope-overload ditortion. We will briefly dicu about thi approach while keeping the tep-ize fixed. A more efficient approach of adapting the tep-ize, leading to Adaptive Delta Modulation (ADM), i excluded. Condition for avoiding lope overload: From Fig we may oberve that if an input ignal change more than half of the tep ize (i.e. by ) within a ampling interval, there will be lope-overload ditortion. So, the deired limiting condition on the input ignal x(t) for avoiding lope-overloading i, dx() t max dt T 30

31 Quantization Noie Power Let u conider a inuoid repreenting a narrow band ignal x() t = am co( π ft) where f repreent the maximum frequency of the ignal and a m it peak amplitude. There will be no lope-overload error if π am f or am T π ft The above condition effectively limit the power of x(t). The maximum allowable power a of x(t) = P max = m =. 8π f T Once the lope overload ditortion ha been taken care of, one can find an etimate of SQNR max. Auming uniform quantization noie between + and, the quantization noie power i 4 N Q = = 1 3 Let u now recollect that the ampling frequency f = 1/T i much greater than f. The granular noie due to the quantizer can be approximated to be of uniform power pectral denity over a frequency band upto f (Fig ). The low pa filter at the output end of the delta demodulator i deigned a per the bandwidth of x(t) and much of the quantization noie power i filtered off. Hence, we may write, f the in-band quantization noie power. NQ f Therefore, SQNR max = (Maximum ignal power) / (In-band quantization noie power) 3 f 3 = ( ).( ) 8π f The above expreion indicate that we can expect an improvement of about 9dB by doubling the ampling rate and it i not a very impreive feature when compared with a PCM cheme. Typically, when the permiible data rate after quantization and coding of peech ignal i more than 48 Kbp, PCM offer better SQNR compared to linear DM. Noie Power In- band noie: It i uniform up to 1/T f = 1/T f frequency Fig In-band noie power at the output of the low pa filter in a delta demodulator i hown by the haded region 31

32 Problem Q3.14.1) Comment if a delta modulator can alo be called a a 1-bit DPCM cheme. Q3.14.) Mention two difference between DPCM and Delta Modulator Q3.14.3) Sugget a olution for controlling the granular noie at the output of a delta modulator. Q3.14.4) Let x(t) = co (π 100t. If thi ignal i ampled at 1 KHz for delta modulator, what i the maximum achievable SQNR in db? 3