[ ] i insensitive to X / σ over a wide range for large μ. -X max =4σ x +X max =4σ x. large xn ( ) X. yn ( ) log logμ

Transcription

1 Digital Seech Processing Lecture 6 Seech Coing Methos Base on Seech Waveform Reresentations an Seech Moels Aative an Differential Coing Seech Waveform Coing-Summary of Part. Probability ensity function for seech samles Gamma Lalacian ( ) e ( 0) / 3 3 ( ) e ( 0 ) 8 π. Coing araigms uniform -- ivie interval from +X ma to X ma into B intervals of length Δ (X ma / B ) for a B-bit quantizer Δ Δ Δ Δ Δ Δ Δ -X ma 4 +X ma 4 Seech Waveform Coing-Summary of Part (e) n ˆ( ) n ( ) + en ( ) /Δ X ma SNR 6B log0 i sensitivity to X / ( varies a lot!!!) -Δ/ Δ/ ma i not great use of bits for actual seech ensities! X ma Xma 0log 0 SNR (uniform) (B8) B loss as X ma / varies over a 3: range i X (or equivalently ) varies a lot across souns, seaers, environments ma i nee to aat coer ( Δ( n)) to time varying or Xma i ey question is how to aat Δ Δ Δ Δ Δ Δ Δ Seech Waveform Coing-Summary of Part seuo-logarithmic (constant ercentage error) - comress (n) by seuo-logarithmic comaner - quantize the comane (n) uniformly - ean the quantize signal [ ] y(n) F (n) n ( ) log + μ Xma X ma sign[ (n) ] log( + μ) large n ( ) X ma μ n ( ) yn ( ) log logμ Xma -X 3 ma 4 +X ma 4 4 -X ma 4 +X ma 4 Seech Waveform Coing-Summary of Part SNR( B) 6B log0 [ ln( + μ) ] 0log0 X ma X ma + + μ μ i insensitive to X / over a wie range for large μ ma maimum SNR coing match signal quantization intervals to moel robability istribution (Gamma, Lalacian) interesting at least theoretically 5 Aative Quantization linear quantization > SNR eens on being constant (this is clearly not the case) instantaneous comaning > SNR only wealy eenent on X ma / for large μ-law comression (00-500) otimum SNR > minimize e when is nown, non- uniform istribution of quantization levels Quantization ilemma: want to choose quantization ste size large enough to accomoate maimum ea-toea range of (n); at the same time nee to mae the quantization ste size small so as to minimize the quantization error the non-stationary nature of seech (variability across souns, seaers, bacgrouns) comouns this roblem greatly 6

2 Solutions to Quantization Dilemna Aative Quantization: Solution -let Δ vary to match the variance of the inut signal > Δ(n) Solution -use a variable gain, G(n), followe by a fie quantizer ste size, Δ > ee signal variance of y(n)g(n)(n) constant Case : Δ(n) roortional to > quantization levels an ranges woul be linearly scale to match > nee to reliably estimate Case : G(n) roortional to / to give y constant nee reliable estimate of for both tyes of aative quantization 7 Tyes of Aative Quantization instantaneous-amlitue changes reflect samleto-samle variations in [n] > rai aatation syllabic-amlitue changes reflect syllable-tosyllable variations in [n] > slow aatation fee-forwar-aative quantizers that estimate from [n] itself feebac-aative quantizers that aat the ste size, Δ, on the basis of the quantize signal, n ˆ[ ], (or equivalently the coewors, c[n]) 8 Fee Forwar Aatation Fee-Forwar Forwar Quantizer Variable ste size assume uniform quantizer with ste size Δ[n] [n] is quantize using Δ[n] > c[n] an Δ[n] nee to be transmitte to the ecoer if c [n]c[n] an Δ [n]δ[n] > no errors in channel, an ˆ [ n[ ˆ[ n] Don t have [n] at the ecoer to estimate Δ[n] > nee to transmit Δ[n]; this is a major rawbac of fee forwar aatation 9 time varying gain, G[n] > c[n] an G[n] nee to be transmitte to the ecoer Can t estimate G[n] at the ecoer > it has to be transmitte 0 Fee Forwar Quantizers fee forwar systems mae estimates of, then mae Δ or the quantization levels roortional to, or the gain is inversely roortional to assume short-time energy [ n] [ m] h[ n m]/ h[ m] h h[ n] i l filt m m 0 [ n] (this can be shown) n [ ] [ ] [ ]/ [ ] where [ ] is a lowass filter E consier hn [ ] α n 0 otherwise [ n] n n m [ m ] α ( α) ( 0< α < ) m [ n] α [ n ] + [ n ]( α) (recursion) this gives Δ [ n] Δ [ n] an G[ n] G / [ n] 0 0 Slowly Aating Gain Control α 099. yn [ ] Gnn [ ] [ ] Gmin G[ n] G Δ Δ[ n] Δ min ma ma [ n] α [ n ] + [ n ]( α) n n m α [ m]( α) m Gn [ ] G 0 [ n] { } yn ˆ[ ] Q G[ nn ] [ ] Δ Δ Δ [ n] 0 [ n] Δ[ n] { } ˆ[ n] Q [ n] α 0.99 > brings u level in low amlitue regions > time constant of 00 samles (.5 msec for 8 Hz samling rate) > syllabic rate

3 Raily Aating Gain Control α 09. [ n] α [ n ] + [ n ]( α) n n m α [ m]( α) m G0 Gn [ ] [ n] yn [ ] Gnn [ ] [ ] yn ˆ[ ] QΔ { Gnn [ ] [ ] } Δ [ n] Δ [ n] Gmin G[ n] G Δ Δ[ n] Δ min ma ma α 0.9 > system reacts to amlitue variations more raily > rovies better aroimation to y constant > time constant of 9 samles ( msec at 8 Hz) for change > instantaneous rate 3 0 Δ[ n] { } ˆ[ n] Q [ n] Fee Forwar Quantizers Δ[n] an G[n] vary slowly comare to [n] they must be samle an transmitte as art of the waveform coer arameters rate of samling eens on the banwith of the lowass filter, h[n] for α 0.99, the rate is about 3 Hz; for α 0.9, the rate is about 35 Hz it is reasonable to lace limits on the variation of Δ[ n] or G[ n], of the form Gmin G[ n] Gma Δmin Δ[ n] Δma for obtaining y constant over a 40 B range in signal levels > Gma Δma 00 (40B range) G Δ min min 4 Fee Forwar Aatation Gain Feebac Aatation n+ M [ n] [ m] M m n Δ[ n] or G[ n] evaluate every M samles use M 8, 04 samles for estimates aative quantizer achieves u to 5.6 B better SNR than non-aative quantizers can achieve this SNR with low "ile channel noise" an wie seech ynamic range by suitable choice of Δ an Δ min ma fee-forwar aatation gain with B3 less gain for M04 than M8 by 3 B > M04 is too long an interval 5 [n] estimate from quantizer outut (or the coe wors) avantage of feebac aatation is that neither Δ[n] nor G[n] nees to be transmitte to the ecoer since they can be erive from the coe wors isavantage of feebac aatation is increase sensitivity to errors in coewors, 6 since such errors affect Δ[n] an G[n] Feebac Aatation Alternative Aroach to Aatation ˆ m m 0 [ n] [ m] h[ n m] / h[ m] [ n] base only on ast values of ˆ [ n] two tyical winows/filters are n. hn [ ] α n 0 otherwise. hn [ ] / M n M 0 otherwise [ n] [ m] n ˆ M m n M can use very short winow lengths (e.g., M ) to achieve B SNR for a B 3 bit quantizer 7 Δ [ n] P Δ[ n ]; P { P, P, P, P } P c[ n ] [ ] 3 4 Δ[ nsigncn ] [ ] n ˆ( ) +Δ [ n ] cn [ ] Δ[ n] only eens on Δ[ n ] an c[ n ] > only nee to transmit coewors also necessary to imose the limits Δ min Δ[ n] Δ ma min ma the ratio Δ / Δ controls the ynamic range of the quantizer 8 3

4 Aatation Gain ey issue is how shoul P vary with c[n-] if (c[n-[ is either largest ositive or largest negative coewor, then quantizer is overloae an the quantizer ste size is too small > P 4 > if c[n-] is either smallest ositive or negative coewor, then quantization error is too large > P < nee choices for P an P 3 9 Aatation Gain + cn [ ] Q B shae area is variation in range of P values ue to ifferent seech souns or ifferent B values Can see that ste size increases (P>) are more vigorous than ste size ecreases (P<) since signal growth nees to be et within quantizer range to avoi overloas 0 Otimal Ste Size Multiliers Quantization of Seech Moel Parameters otimal values of P for B,3,4,5 imrovements in SNR 4-7 B imrovement over μ-law -4 B imrovement over non-aative otimum quantizers Ecitation an vocal tract (linear system) are characterize by sets of arameters which can be estimate from a seech signal by LP or cestral rocessing. We can use the set of estimate arameters to synthesize an aroimation to the seech signal whose quality eens of a range of factors. Quantization of Seech Moel Parameters Quality an ata rate of synthesis eens on: the ability of the moel to reresent seech the ability to reliably an accurately estimate the arameters of the moel the ability to quantize the arameters in orer to obtain a low ata rate igital reresentation that will yiel a high quality rerouction of the seech signal 3 Close-Loo an Oen-Loo Seech Coers Close-looloo use in a feebac loo where the synthetic seech outut is comare to the inut signal, an the resulting ifference use to etermine the ecitation for the vocal tract moel. Oen-loo the arameters of the moel are estimate irectly from the seech signal with no feebac as to the quality of the resulting synthetic seech. 4 4

5 Scalar Quantization Scalar quantization treat each moel arameter searately an quantize using a fie number of bits nee to measure (estimate) statistics of each arameter, i.e., mean, variance, minimum/maimum value, f, etc. each arameter has a ifferent quantizer with a ifferent number of bits allocate Eamle of scalar quantization itch erio tyically ranges from 0-50 samles (at 8 Hz samling rate) > nee about 8 values (7-bits) uniformly over the range of itch erios, incluing value of zero for unvoice/bacgroun amlitue arameter might be quantize with a μ-law quantizer using 4-5 bits er samle using a frame rate of 00 frames/sec, you woul nee about 700 bs for itch erio an bs for amlitue Scalar Quantization 0-th orer LPC analysis frame Each PARCOR coefficient transforme to range: π/<sin - ( i )<π/ an then quantize with both a 4-bit an a 3-bit uniform quantizer. Total rate of quantize reresentation of seech about 5000 bs. 5 6 Vector Quantization Techniques of Vector Quantization coe bloc of scalars as a vector, rather than iniviually esign an otimal quantization metho base on mean-square istortion metric essential for moel-base an hybri coers 7 8 Vector Quantization Waveform Coing Vector Quantizer VQ coe airs of waveform samles, X[n]([n],[n+]); 9 (b) Single element coeboo with cluster centroi (0-bit coeboo) (c) Two element coeboo with two cluster centers (-bit coeboo) () Four element coeboo with four cluster centers (-bit coeboo) (e) Eight element coeboo with eight cluster centers (3-bit coeboo) 30 5

6 Toy Eamle of VQ Coing -ole moel of the vocal tract > 4 reflection coefficients 4-ossible vocal tract shaes > 4 sets of reflection coefficients. Scalar Quantization -assume 4 values for each reflection coefficient > -bits 4 coefficients 8 bits/frame. Vector Quantization -only 4 ossible vectors > -bits to choose which of the 4 vectors to use for each frame (ointer into a coeboo) this wors because the scalar comonents of each vector are highly correlate if scalar comonents are ineenent > VQ offers no avantage over scalar 3 quantization Elements of a VQ Imlementation. A large training set of analysis vectors; X{X,X,,X L }, L shoul be much larger than the size of the coeboo, M, i.e., 0-00 times the size of M.. A measure of istance, ij (X i,x j ), between a air of analysis vectors, both for clustering the training set as well as for classifying test set vectors into unique coeboo entries. 3. A centroi comutation roceure an a centroi slitting roceure. 4. A classification roceure for arbitrary analysis vectors that chooses the coeboo vector closest in istance to the inut vector, roviing the coeboo ine of the resulting nearest coeboo vector. 3 VQ Imlementation The VQ Training Set The VQ training set of L 0M vectors shoul san the anticiate range of: talers, ranging in age, accent, gener, seaing rate, seaing levels, etc. seaing conitions, range from quiet rooms, to automobiles, to noisy wor laces transucers an transmission systems, incluing a range of microhones, telehone hansets, cellhones, seaerhones, etc. seech, incluing carefully recore material, conversational seech, telehone queries, etc The VQ Distance Measure The VQ istance measure eens critically on the nature of the analysis vector, X. If X is a log sectral vector, then a ossible istance measure woul be an L log sectral istance, of the form: R / ( Xi, X j) i j If X is a cestral vector, then the istance measure might well be a cestral istance of the form: R / ( Xi, X j) ( i j) 35 Clustering Training Vectors Goal is to cluster the set of L training vectors into a set of M coeboo vectors using generalize Lloy algorithm (also nown as the K-means clustering algorithm) with the following stes:. Initialization arbitrarily choose M vectors (initially out of the training set of L vectors) as the initial set of coewors in the coeboo. Nearest Neighbor Search for each training vector, fin the coewor in the current coeboo that is closest (in istance) an assign that vector to the corresoning cell 3. Centroi Uate uate the coewor in each cell to the centroi of all the training vectors assigne to that cell in the current iteration 4. Iteration reeat stes an 3 until the average istance between centrois at successive iterations falls below a reset threshol 36 6

7 Clustering Training Vectors Voronoi regions an centrois 37 Centroi Comutation i Assume we have a set of V vectors, X { X, X C,..., XV } C C C where all V vectors are assigne to cluster C. i The centroi of the set X is efine as the vector C Y that minimizes the average istortion, i.e., V C Y min ( Xi, Y) Y V i i The solution for the centroi is highly eenent C on the choice of istance measure. When both X i an Y are measure in a K-imensional sace with the L norm, the centroi is the mean of the vector set V C Y Xi V i i When using an L istance measure, the centroi is the meian vector of the set of vectors assigne to the given class. 38 Vector Classification Proceure i The classification roceure for arbitrary test set vectors is a full search through the coeboo to fin the "best" (minimum istance) match. i If we enote the coeboo vectors of an M -vector coeboo as CBi for i M, an we enote the vector to be classifie ( an vector quantize) as X, then the ine, i, of the best coeboo entry is: i arg min ( X, CB) i M i 39 Binary Slit Coeboo Design. Design a -vector coeboo; the single vector in the coeboo is the centroi of the entire set of training vectors. Double the size of the coeboo by slitting each current coeboo vector, Y m, accoring to the rule: Y + ( ) m Ym + ε Y ( ) m Ym ε where m varies from to the size of the current coeboo, an esilon is a slitting arameter (0.0 tyically) 3. Use the K-means clustering algorithm to get the best set of centrois for the slit coeboo 4. Iterate stes an 3 until a coeboo of size M is esigne. 40 Binary Slit Algorithm VQ Coeboo of LPC Vectors 64 vectors in a coeboo of sectral shaes 4 4 7

8 VQ Cells VQ Cells VQ Coing for Seech Differential Quantization Theory an Practice istortion in coing comute using a sectral istortion measure relate to the ifference in log sectra between the original an the coeboo vectors 0-bit VQ comarable to 4-bit scalar 45 quantization for these eamles 46 Differential Quantization we have carrie instantaneous quantization of [n] as far as ossible time to consier correlations between seech samles searate in time > ifferential quantization high correlation values > signal oes not change raily in time > ifference between ajacent samles shoul have lower variance than the signal itself ifferential quantization can increase SNR at a given bit rate, or lower bit rate for a given SNR 47 Eamle of Difference Signal Preiction error from LPC analysis using Preiction error is about.5 times smaller than signal 48 8

9 n [ ] Differential Quantization + n [ ] Δ n ˆ[ ] Q[ ] Encoer cn [ ] Differential Quantization Δ c'[ n] ˆ '[ n] Decoer + ˆ '[ n] n [ ] P n ˆ[ ] n [ ] n [ ] n [ ] where n [ ] unquantize inut samle n [ ] estimate or reiction of n [ ] n [ ] is the outut of a reictor system, P, whose inut is n ˆ[ ], a quantize version of n [ ] n [ ] reiction error signal n ˆ[ ] quantize ifference (reiction error) signal + (a) Coer 49 P '[ n ] (b) Decoer n [ ] n [ ] n [ ] where n [ ] unquantize inut samle n [ ] estimate or reiction of n [ ) n [ ] is the outut of a reictor system, P, whose inut is n ˆ[ ], a quantize version of n [ ] n [ ] reiction error signal n ˆ[ ] quantize ifference (reiction error) signal 50 Differential Quantization n [ ] n [ ] n [ ] n ˆ[ ] n [ ] + en [ ] Δ Differential Quantization ifference signal, [n], is quantize - not [n] quantizer can be fie, or aative, uniform or non-uniform quantizer arameters are ajuste to match the variance of [n] n [ ] + n [ ] n [ ] n ˆ[ ] Q[ ] Encoer P n ˆ[ ] This art reconstructs the quantize signal, n ˆ[ ] + Pz ( ) cn [ ] αz α ˆ n [ ] n [ ] n ˆ( ) n ( ) + n ˆ( ) n ˆ( ) n ( ) + en ( ) 5 n ˆ[ ] n [ ] + en [ ] en [ ] quantization error n ˆ[ ] n [ ] + n ˆ[ ] reicte lus quantize n ˆ[ ] n [ ] + en [ ] quantize inut has same quantization error as the ifference signal > if <, error is smaller ineenent of reictor, P, quantize [ n] iffers from unquantize n [ ] by en [ ], the quantization error of the ifference signal! > goo reiction gives lower quantization error than quantizing inut irectly 5 Differential Quantization SNR for Differential Quantization quantize ifference signal is encoe into c(n) for transmission ˆ ˆ ˆ H(z) X ( z) D ( z) + P( z) X ( z) ˆ X ( z) Dˆ ( z) H( z) Dˆ ( z) Pz ( ) Hz ( ) α z first reconstruct the quantize ifference signal from the ecoer coewor, c [n] an the ste size Δ net reconstruct the quantize inut signal using the same reictor, P, as use in the encoer 53 the SNR of the ifferential coing system is E [ n] SNR E e [ n ] e SNR G P SNRQ e where SNRQ signal-to-quantizing-noise ratio of the quantizer e GP gain ue to ifferential quantization 54 9

10 SNR for Differential Quantization SNR Q eens on chosen quantizer an can be maimize using all of the revious quantization methos (uniform, non-uniform, otimal) GP, hoefully, >, is the gain in SNR ue to ifferential coing want to choose the reictor, P, to maimize GP since is fie, then we nee to minimize, i.e., esign the best reictor P 55 Preictor for Differential Quantization consier class of linear reictors, P n [ ] α n ˆ[ ] n [ ] is a linear combination of revious quantize values of n [ ] the reictor z-transform is ( ) α Pz z Az ( ) -- reictor system function with reictor imulse resonse coefficients (FIR filter) n [ ] α 0 otherwise 56 Preictor for Differential Quantization Preictor for Differential Quantization the reconstructe signal is the outut, n ˆ[ ], of a system with system function Xz ˆ ( ) Hz ( ) ˆ ( ) ( ) ( ) α Dz Pz Az z where the inut to the system is the quantize ifference signal, n ˆ[ ] n ˆ[ ] n ˆ[ ] α n ˆ[ ] where α n ˆ[ ] n [ ] 57 to solve for otimum reictor, nee eression for ( ) [ ] [ ] [ ] E n E n n E[ n] α ˆ [ n ] E [ n] α[ n ] αe[ n ] ( n ˆ[ ] n [ ] + en [ ]) 58 Solution for Otimum Preictor want to choose { α j}, j, to minimize > ifferentiate wrt αj, set erivatives to zero, giving α j E [ n] α ( [ n ] + e[ n ] ) ( [ n j] + e[ n j] ) 0 j which can be written in the more comact form E ( [ n] [ n] ) ˆ[ n j] E [ n] ˆ[ n j] 0, j the reictor coefficients that minimize are the ones that mae the ifference signal, n [ ], be uncorrelate with ast values of the reictor inut, n ˆ[ j], j 59 Solution for Alhas ( [ ] [ ]) ˆ[ ] [ ] ˆ[ ] 0, E n n n j E n n j j basic equations of ifferential coing n ˆ[ ] n [ ] + en [ ] quantization of ifference signal n ˆ[ ] n [ ] + en [ ] error same for original signal n ˆ[ ] n [ ]+ + n ˆ[ ] feebac loo for signal n [ ] α n ˆ[ ] reiction loo base on quantize inut n ˆ[ ] n ˆ[ ] α n ˆ[ ] irect substitution n ˆ[ j] n [ j] + en [ j] n [ ] α n ˆ[ ] α n [ ] + en [ ] [ ] 60 0

11 ( ) Solution for Alhas E [ n] [ n] ˆ[ n j] E [ n] ˆ[ n j] 0, j n ˆ[ j] n [ j] + en [ j] [ ] α ˆ [ ] α[ [ ] + [ ]] [ [ ] [ ]] + [ [ ] [ ]] [ [ ] [ ]] [ [ ] [ ]] E α [ [ n ] + e[ n ] ] [ n j] + n n n en E n n j E n e n j E n n j E n e n j Eα [ [ n ] + e[ n ] ] e[ n j] α E [ n ] [ n j] + α E e[ n ] [ n j] + [ ] [ ] α [ [ ] [ ]] + α [ [ ] [ ]] E n en j E en en j 6 Solution for Otimum Preictor solution for α - first ean terms to give [ [ ] [ ]] + [ [ ] [ ]] α [ [ ] [ ]] + αe[ en [ j] n [ ] ] + αe[ n [ j] en [ ] ] + αe[ e[ n j] e[ n ] ], j E n j n E e n j n E n j n assume fine quantization so that en [ ] is uncorrelate with n [ ], an en [ ] is stationary white noise (zero mean), giving E[ [ n j] e[ n ] ] 0 n, j, Een [ j] en [ ] δ[ j ] [ ] e 6 Solution for Otimum Preictor Solution for Otimum Preictor we can now simlify solution to form [ ] [ ] [ ] e, φ j α φ j + δ j j where φ[ j] is the autocorrelation of [ n]. Defining terms φ[ j] e ρ[ j] α ρ[ ] + δ[ ], j j j or in matri form ρ Cα ρ[] α + / SNR ρ[] ρ[ ] ρ[ ] α ρ[ ] / ρ[ ] + SNR.. ρ., α., C ρ[ ] α ρ[ ] ρ[ ] + / SNR 63 ρ Cα ρ[] α + / SNR ρ[] ρ[ ] ρ[ ] α ρ[ ] / ρ[ ] + SNR.. ρ., α., C ρ[ ] α ρ[ [ ρ[ ] + / SNR with matri solution α C ρ (efining SNR / ) where is a Toelitz matri can be comute via well unerstoo C > C numerical methos the roblem here is that C eens on SNR / e, but SNR eens on α coefficients of the reictor, which een on SNR > bit of a ilemma e 64 Solution for Otimum Preictor secial case of, where we can solve irectly for α of this first orer linear reictor, as ρ[] α + / SNR can see that α < ρ[ ] < we will loo further at this secial case later 65 Solution for Otimum Preictor in site of roblems in solving for otimum reictor coefficients, we can solve for the reiction gain, GP, in terms of the α j coefficients, as ( [ ] [ ]) ( [ ] E n n n [ n] ) E ( [ n] [ n] ) [ n ] E ( [ n] [ n] ) [ n] where the term ( n [ ] n [ ]) is the reiction error; we can show that the secon term in the eression above is zero, i.e., the reiction error is uncorrelate with the reiction value; thus ( [ ] [ ]) [ ] E n n n E [ n] Eα ( n [ ] + en [ ]) n [ ]) assuming uncorrelate signal an noise, we get αφ[ ] αρ[ ] ( GP ) ot for otimum values of α αρ [ ] 66

12 First Orer Preictor Solution For the case we can eamine effects of sub-otimum value of α on the quantity GP /. The otimum solution is: ( GP ) ot αρ [] Consier choosing an arbitrary value for α ; then we get αρ[ ] α + + αe Giving the sub-otimum result ( G P ) arb αρ [] + α( + / SNR) Where the term α / SNR reresents the increase in variance of n [ ] ue to the feebac of the error signal en [ ]. 67 First Orer Preictor Solution Can reformulate ( GP ) α arb as SNRQ ( GP ) arb αρ [] + α for any value of α (incluing the otimum value). Consier the case of α ρ( ) ρ [] SNR Q ρ [ ) ( GP ) subot ρ [] ρ [] SNR Q the gain in reiction is a rouct of the reiction gain without the quantizer, reuce by the loss ue to feebac of the error signal. 68 First Orer Preictor Solution We showe before that the otimum value of α was ρ[ ] / [ + / SNR] If we neglect the term in /SNR (usually very small), then α ρ[ ] an the gain ue to reiction is ( G P ) ot ρ [] Thus there is a reiction gain so long as ρ[ ] 0 It is reasonable to assume that for seech, ρ[ ] > 0. 8, giving > 77. (or 4.43 B) ( ) G P ot 69 Differential Quantization Δ ^ [n] [n] [n] + + Q - ~ [n] [n] ^ P + SNR G SNR e e First Orer Preictor: ρ[] α + / SNR ρ [] G ρ [] SNRQ ρ [] Q n [ ] α n ˆ[ ] n [ ] n [ ] n [ ] n ˆ[ ] n [ ] + en [ ] n ˆ[ ] n [ ] + n ˆ [ ] n ˆ[ ] n [ ] + en [ ] The error, e[n], in quantizing [n] is the same as the error in reresenting [n] Preiction gain eenent on ρ[], the first correlation coefficient 70 Long-Term Sectrum an Correlation Measure with 3-oint Hamming winow Comute Preiction Gain

13 Actual Preiction Gains for Seech variation in gain across 4 seaers can get about 6 B imrovement in SNR > etra bit equivalent in quantization but at a rice of increase comleity in quantization Delta Moulation ifferential quantization wors!! gain in SNR eens on signal correlations fie reictor cannot be otimum for all seaers an for all seech material 73 Linear an Aative 74 Delta Moulation Linear Delta Moulation simlest form of ifferential quantization is in elta moulation (DM) samling rate chosen to be many times the Nyquist rate for the inut signal > ajacent samles are highly correlate in the limit as T 0, we eect φ [] as T 0 this leas to a high ability to reict [n] from ast samles, with the variance of the reiction error being very low, leaing to a high reiction gain > can use simle -bit (-level) quantizer > the bit rate for DM systems is just the (high) samling rate of the signal 75 -level quantizer with fie ste size, Δ, with quantizer form n ˆ[ ] Δ ifn [ ] > 0 ( cn [ ] 0 ) Δ if [ n] < 0 ( c[ n] ) using simle first orer reictor with otimum reiction gain ( G P ) ot ρ [] as ρ[], ( G P ) ot (qualitatively only since the assumtions uner which the equation was erive brea own as ρ[] ) 76 Illustration of DM basic equations of DM are n ˆ[ ] α n ˆ[ ] + n ˆ[ ] when α (essentially igital integration or accumulation of increments of ±Δ) n [ ] n [ ] n ˆ[ ] n [ ] n [ ] en [ ] n [ ] is a first bacwar ifference of n [ ], or an aroimation to the erivative of the inut how big o we mae Δ--at maimum sloe of a() t we nee Δ () a t ma T t or else the reconstructe signal will lag the actual signal > calle 'sloe overloa' conition--resulting in quantization error calle 'sloe overloa istortion' since n ˆ[ ] can only increase by fie increments of Δ, fie DM is calle linear DM or LDM sloe overloa conition granular noise 77 DM Granular Noise when a() t has small sloe, Δ etermines the ea error > when a () t 0, quantizer will be alternating sequence of 0's an 's, an n ˆ[ ] will alternate aroun zero with ea variation of Δ > this conition is calle "granular noise" nee large ste size to hanle wie ynamic range nee small ste size to accurately reresent low level signals with LDM we nee to worry about ynamic range an amlitue of the ifference signal > choose Δ to minimize mean-square quantization error (a comromise between sloe overloa an granular noise) 78 3

14 Performance of DM Systems normalize ste size efine as Δ / E ( [ n] [ n ] ) oversamling ine efine as F0 FS /( FN) where FS is the samling rate of the DM an F N is the Nyquist frequency of the signal the total bit rate of the DM is BR FS FN F0 can see that for given value of F 0, there is an otimum value of Δ otimum SNR increases by 9 B for each oubling of F 0 > this is better than the 6 B obtaine by increasing the number of bits/samle by bit curves are very shar aroun otimum value of Δ > SNR is very sensitive to inut level for SNR35 B, for F N 3 Hz > 00 Kbs rate 79 for toll quality nee much higher rates Aative Delta Mo ste size aatation for DM (from coewors) Δ [ n] M Δ[ n ] Δmin Δ[ n] Δma M is a function of c[ n] an cn [ ], since cn [ ] eens only on the sign of n [ ] n [ ] n [ ] α n ˆ [ ] the sign of n [ ] can be etermine before theactual quantize value n ˆ[ ] which nees the new value of Δ[ n] for evaluation the algorithm for choosing the ste size multilier is M P> if cn [ ] cn [ ] M Q< if cn [ ] cn [ ] 80 Aative DM Performance sloe overloa conition granular noise sloe overloa in LDM causes runs of 0 s or s granularity causes runs of alternating 0 s an s figure above shows how aative DM erforms with P, Q/, α uring sloe overloa, ste size increases eonentially to follow increase in waveform sloe uring granularity, ste size ecreases eonentially to Δ min an stays there as long as sloe remains small 8 ADM Parameter Behavior ADM arameters are P, Q, Δ min an Δ ma choose Δ min an Δ ma to rovie esire ynamic range choose Δ ma / Δ min to maintain high SNR over range of inut signal levels Δ min shoul be chosen to minimize ile channel noise PQ shoul satisfy PQ for stability PQ chosen to be ea of SNR at P.5, but for range.5<p<, the SNR varies only slightly 8 Comarison of LDM, ADM an log PCM ADM is 8 B better SNR at 0 Kbs than LDM, an 4 B better SNR at 60 Kbs than LDM ADM gives a 0 B increase in SNR for each oubling of the bit rate; LDM gives about 6 B for bit rate below 40 Kbs, ADM has higher SNR than μ-law PCM; for higher bit rates log PCM has higher SNR 83 Higher Orer Preiction in DM first orer reictor gave [ n] α n ˆ[ ] with reconstructe signal satisfying n ˆ[ ] α n ˆ[ ] + n ˆ[ ] with system function Xz ˆ ( ) H( z) ˆ Dz ( ) αz igital equivalent of a leay integrator. consier a secon orer reictor with [ n] α n ˆ[ ] + α n ˆ[ ] assuming two real oles, we can write H( z) as Xz ˆ ( ) H( z), 0 < a, b< ˆ Dz ( ) ( az )( bz ) better reiction is achieve using this "ouble integration" system with u to 4 B better SNR there are issues of interaction between the aative quantizer an the reictor with reconstructe signal n ˆ[ ] α ˆ ˆ ˆ n [ ] + αn [ ] + n [ ] with system function Xz ˆ ( ) H( z) ˆ Dz ( ) ( αz αz ) 84 4

15 Differential PCM (DPCM) fie reictors can give from 4- B SNR imrovement over irect quantization (PCM) most of the gain occurs with first orer reictor reiction u to 4 th or 5 th orer hels 85 DPCM with Aative Quantization quantizer ste size roortional to variance at quantizer inut can use [n] or [n] to control ste size get 5 B imrovement in SNR over μ-law nonaative PCM get 6 B imrovement in SNR using ifferential configuration with fie reiction > ADPCM is about 0- B SNR better than from a fie quantizer 86 Feebac ADPCM DPCM with Aative Preiction nee aative reiction to hanle non-stationarity of seech can achieve same imrovement in SNR as fee forwar system DPCM with Aative Preiction reiction coefficients assume to be time-eenent of the form n [ ] α [ ] ˆ nn [ ] assume seech roerties remain fie over short time intervals choose α[ n] to minimize the average square reiction error over short intervals the otimum reictor coefficients satisfy the relationshis [ ] Rn j α[ n] Rn[ j ], j,,..., where Rn[] j is the short-time autocorrelation function of the form Rn[] j [ m] w[ n m][ j + m] w[ n m j], 0 j m wn-m [ ] is winow ositione at samle nof inut uate α's every 0-0 msec 89 Preiction Gain for DPCM with Aative Preiction 0log0 [ GP ] 0log0 E [ ] n [ ] E n fie reiction 0.5 B reiction gain for large aative reiction 4 B gain for large aative reiction more robust to seaer, seech material 90 5

16 Comarison of Coers 6 B between curves shar increase in SNR with both fie reiction an aative quantization almost no gain for aating first orer reictor 9 6