15. Quantisation Noise and Nonuniform Quantisation

Transcription

1 5. Quntistion Noise nd Nonuniform Quntistion In PCM, n nlogue signl is smpled, quntised, nd coded into sequence of digits. Once we hve quntised the smpled signls, the exct vlues of the smpled signls cn never be restored. This gives rise to rndom vritions clled quntistion noise. This noise cn be reduced to ny desirble level by simply incresing the number of quntistion levels M. However, the lrger the vlue of M, the lrger the number of bits required to code the quntised signl, nd the greter the bndwidth required for trnsmission. In prcticl PCM system for speech trnsmission, we use 8-bit quntistion. Uniform Quntistion Consider the 8-level uniform quntiser (equl spcing) shown in Figure 5.. Figure 5. Messge nd quntised signl. Let M be the totl number of quntistion (mplitude) levels, nd be the spcing between dcent levels. Suppose the messge cn hve mximum swing of + V volts, then V M (5.) The output pek signl voltge is V M/ (5.) nd the quntiser covers rnge of (M - ) (volts) (5.3) Let A be the voltge ssocited with the -th quntistion level. Any smples tht lie between A +.5 volts re quntised to A volts. The quntistion error (noise) is therefore limited to +.5 volts. This region of uncertinty is shown in Figure 5.. Figure 5. Uncertinty region t quntiser output. The men-squred quntistion noise is E(ε ) / ε dε / 5.

2 / (5.4) nd the root-men-squred (rms) quntistion noise is. The output pek signl-to-rms quntistion noise rtio is V (5.5) E( ε ) The corresponding output pek signl-to-rms quntistion noise power rtio (SNR) is ( ) (5.6) log M (db) (5.7) We cn normlise the input signl swing to + volt. In this cse, the spcing becomes /M, nd the output pek signl-to-rms quntistion noise rtio nd the corresponding output pek signl-to-rms quntistion noise power rtio remin unchnged. M Output SNR Reltive bndwidth Tble 5. Output SNR improvement with number of quntistion levels. We cn improve the output SNR by incresing the vlue of M. It cn lso be seen from Figure 5. tht the input signl is directly proportionl to the quntised output signl. For fixed vlue of M, smll input signl will give lower output SNR thn lrge input signl. Therefore, good reproduction for ll types of telephone users would not be possible with uniform spcing. Some users shout while others whisper. To cter for ll types of users, non-uniform quntistion is employed. It gives finer levels for smll input signls nd corser levels for lrge input signls. 5.

3 Nonuniform Quntistion The most common form of nonuniform quntistion is known s compnding. The nme is derived from the words compressing nd expnding. A block digrm of PCM system with compnding is shown in Figure 5.3. The originl signl is compressed nd uniformly quntised. At the receiving end, the decoded signl is expnded. Figure 5.3 A PCM system with compnding. µ-lw Compnder In the United Sttes, Cnd, nd Jpn, µ-lw compnder is used. The compression chrcteristic is given by y(x) ln( + µ x ), < x < (5.8) ln( + µ ) where x is the normlised input signl, nd dy dx µ ln( + µ ) + µ x (5.9) To find the rms quntistion noise of the µ-lw compnder, we consider the spcing centred t x of the compressor, s shown in Figure 5.4. Figure 5.4 Compressor nlysis. For smll, the slope t x is dy /M dx x (5.) nd / M (5.) dy / dx x The men-squred quntistion noise bout the input x is given by 5.3

4 E(ε ) x + x (x - x ) f(x) dx (5.) where f(x) is the probbility density function of the input signl x. Assuming tht f(x) is unchnged over tht rnge, we hve x + E(ε ) f(x ) x 3 f(x ) (x - x ) dx (5.3) Substituting for from eqution (5.) into eqution (5.3), we get E(ε ) f( x ) [ dy / dx x ] (5.4) For,,..., M/ nd including negtive vlues of x, we hve E(ε M/ ) E(ε ) M / f( x ) [ dy / dx x ] For lrge M, becomes dx nd summtion becomes integrtion E(ε ) f( x) dx (5.5) [ dy / dx] x Substituting eqution (5.9) into eqution (5.5), we get E(ε ) ln( + µ ) (+µx) f(x) dx µ x 5.4

5 ln( + µ ) (+ µx+µ x ) f(x) dx µ x ln( + µ ) { µ [f(x) +µxf(x) + µ x f(x)] dx} x ln( + µ ) [.5 + µ µ xf (x) dx + µ σ x ] x ln( + µ ) [ + 4µ µ xf (x) dx + µ σ x ] (5.6) x where σ x x f ( x ) dx (5.7) σ x σ/v is the normlised men input signl power nd σ is the unnormlised input signl power. The output pek signl-to-rms quntistion noise power rtio is E( ε ) 4 xf ( x) dx x ln( + µ ) [ σx + + ] µ µ (5.8) Figure 5.5 shows plot of the output SNR vs normlised input signl when f(x) is Gussin probbility density function. A reltively constnt output SNR over 4 db of input dynmic rnge is chieved. Figure 5.5 Output SNR of 8-bit PCM with nd without compnding. A-Lw Compnder For the rest of the world, n A-lw compnder hs been used, where A. The A- lw compressor chrcteristic is defined by Ax y(x) + ln( A), < x < A (5.9) 5.5

6 nd y(x) + ln( Ax) + ln( A ), A < x < (5.) The A-lw compressor chrcteristic is liner for smll x nd logrithmic for lrge x. The A-lw compressor hs greter dynmic rnge thn the µ-lw compressor, nd hs smller output SNR thn the µ-lw compressor. References [] M. Schwrtz, Informtion Trnsmission, Modultion, nd Noise, 4/e, McGrw Hill, 99. [] H. Tub nd D. L. Schilling, Principles of Communictions Systems, /e, McGrw Hill, 986. [3] L. W. Couch, II, Digitl nd Anlog Communiction Systems, 6/e, Prentice Hll,. 5.6

7 Volts Messge m (t ) / (M -) / V V t (sec.) Figure 5. Messge nd quntised signl. A volts. / / / / ε +-th level Smpled signl -th level } --th level Uncertinty region Figure 5. Uncertinty region t quntiser output. Nonuniform quntiser Messge m (t ) Smpler Compressor Uniform quntiser Encoder ~ ^ m (t ) Low-pss filter Expnder Decoder Figure 5.3 A PCM system with compnding. 5.7

8 Output. -th level.8 y M x Input x -. Figure 5.4 Compressor nlysis. Output SNR (db) 4 µ 55 -lw compnding 3 Uniform quntiser (No compnding) µ - Reltive input level (db) log ( x ) Figure 5.5 Output SNR of 8-bit PCM with nd without compnding. 5.8