Pulse Coded Modulation

Transcription

1 Pulse Coded Modulaton PCM (Pulse Coded Modulaton) s a voce codng technque defned by the ITU-T G.711 standard and t s used n dgtal telephony to encode the voce sgnal. The frst step n the analog to dgtal converson of the voce sgnal s the flterng of the analog sgnal, meanng the lmtaton of the frequency band to [300 Hz, 3400 Hz]. The next step s the samplng, usng a frequency whch fulflls the samplng theorem, f s > *f m, and havng the value f s 8 khz. Have to be mentoned that the flterng has the purpose of avodng the alasng phenomenon. After the samplng process the next step s the compresson of the sgnal, process whch realzes the non-unform quantzaton. Compandng Compandng conssts n compresson of the sgnal to be transmtted at transmsson sde and expanson at recepton (compandng compressng + expandng). Ths operaton s performed accordng to the µ (1) and A () compresson laws. µ law A law ln(1 + µ x ) f ( x) sgn( x),0 x < 1 (1) ln(1 + µ ) - employed n USA and n Japan; - µ55; A x 1,0 x < 1 + ln ( A) A f ( x) () 1+ ln( A x ) 1 sgn( x), x < 1 1+ ln( A) A - employed n Europe; - A 87.6, value whch provdes contnuty of the functon; Problem: deduce the mathematcal expressons of the expanson laws correspondng to the presented compresson laws (varable x functon of varable y). In Fgure 1 t s presented the processng nvolved by the compresson and the expanson process: - the samples of the voce sgnal are quantzed on 16 bts usng unform quantzaton; - t s employed one of the compresson laws (1) and () - the least sgnfcant 8 bts are suppressed (see Fgure ) Fgure 1 Block schematc of the transmsson process usng compandng 1

2 At the recever the 8 bt PCM word s converted back nto a 16 bt word by appendng 8 LSBs havng the value (18 10 ) see Fg., reducng n ths way to half the quantzaton error. In the next step the expanson functon s appled to the obtaned 16bt PCM word. The µ compresson law Fgure The re-quantzaton process In Fgure 3 t s presented the normalzed µ law compresson characterstc for dfferent values of the µ parameter, and n Fgure 4 t s presented the approxmated characterstc of the µ compresson law. Fgure 3 Normalzed µ law compresson characterstc

3 Fgure 4 Approxmated/segmented µ law compresson characterstc postve values only In practce the characterstc presented n Fgure 3 (µ55) s approxmated n 16 segments, 8 for the postve values and 8 for the negatve values (Fgure 4), and each segment s splt n 16 sub segments (Fgure 5). Insde each segment on the X axs t s performed a unform quantzaton on 4 bts. Fgure 5 - Segment 1 (splttng n sub-segments) Encodng of the nput sgnal s performed n the followng way (Fgure 6): 3

4 - the most sgnfcant bt, b 7, ndcates the sgnal polarty; - the next 3 bts, b 6 b 5 b 4 ndcate the segment; - the last 4 bts, b 3 b b 1 b 0, ndcate the sub segment; Fgure 6 PCM encodng wth compresson Questons: 1) It s gven a compresson characterstc approxmated n 4 segments, and each segment s splt nto 8 sub-segments. How many bts are requred for PCM encodng? ) A compresson characterstc s descrbed by the sequence of coordnates (0,0) (0.1,0.5) - (0.3,0.5) - (0.6,0.75) (1,1). Gve the sequence of coordnates for the segmented expanson characterstc. Calculaton of some of the compresson characterstcs parameters: x - Compresson rato for segment : Rc, where x represents the y length of segment at the nput of the characterstc (X axs), and y s the length of segment at the output of the characterstc (Y axs) (see Fgure 4); the length of any segment at the output s When the compresson rato s smaller than 1 means that we have expanson, and when the compresson raton s larger than 1 means that we have compresson. - Input quantzaton step on segment : represents the length of the subsegments of the nput segment and can be computed as: x x q. No. sub segments 16 - Output quantzaton step: represents the length of the sub-segments of the output segments and can be computed as: y 0.15 q o Elementary quantzaton step: represents the quantzaton step used for unform quantzaton provdng the same (or smlar) performances as the non-unform quantzaton (14 bts n the case of µ law): q e. b - Number of elementary quantzaton steps of the nput quantzaton q step: n q e q - The power of the quantzaton nose on segment : Pnq. 1 4

5 - Total (average) power of the quantzaton nose: 8, 8 0 Pnq p Pnq where p represents the probablty that the ampltude of the sample s located n segment. If we have the same occurrence probablty of each sample (unform ampltude dstrbuton) then the probablty that a sample s located n segment s equal wth the length of ths segment, f the characterstc s normalzed, meanng that p x. - Total (average) power of the quantzaton nose (consderng only postve segments): 8 ' 1 Pnq p Pnq, p has the same defnton. Segment () x Rc q n Pnq E E E E E E E E-05 Table 1 Computed parameters for the µ compresson characterstc If we compute the total power of the quantzaton nose, when non-unform quantzaton s employed, we obtan Pnq 47.39*10-6 W and f we compute the total power of the unform quantzaton, on the same number of bts, we get Pnq unform 5*10-6 W. Even f the nose power of the unform quantzaton s smaller than the nose power of the non-unform quantzaton, we can notce n Table 1 that for the frst segments, meanng for low sgnal values where the hearng s more senstve, the nose power s much hgher n the case of unform quantzaton than n the case of non-unform quantzaton. The quantzaton error at hgh sgnal levels s less detectable by the human hearng (havng less mportance n ths case) but nfluences sgnfcantly the total quantzaton nose power, whch s more mportant for data transmssons. Supplementary nformaton at: Questons 1) Specfy the advantages and dsadvantages of the non-unform quantzaton. ) Calculate the parameters gven n Table 1 for the A compresson law. 3) A compresson characterstc s approxmated n 4 segments: (0,0) (1/8,0.5) (1/4,0.5) (1/,0.75) (1,1), and each segment s dvded nto 4 subsegments. Whch s the code generated f the nput sample s ampltude s 0.755? (The coordnates of the characterstc are specfed as (x,y)) 5