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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



dictive-subtractive coding. In an actual system the reduced signal 

 would ordinarily be encoded into Shannon-Fano code groups before 

 transmission over the channel. 



If So is the present sample amplitude, and Si , S2 , S3 • • • s„ are previous 

 sample amplitudes we compute a predicted value, Sp , for the present 

 sample which is given by 



Sp = Ksi , S2 , • • • s„) ± 5 



where 5 < | quantizing level. If the conditional probability distribution 

 for the present sample is Psi--sn(so), then the difference, or output, or 



Fig. 6 — Predictive-subtractive coding. 



"error" signal, e, will have the conditional distribution Psi---s,X^ + Sp) 

 for this particular case. The simple distribution is then the weighted 

 average over all cases, i.e. 



PU) = Z) PiSl ,8-2, • ■ ■ S„)p,i...,„(€ + Sp) 



where the sum is over all combinations of Si , S2 • • • s„ . 



Predictive-subtractive coding has especial merit when a simple func- 

 tion can be used for computing Sp . This is often the case. When the 

 function is simply a weighted sum of the past sample amplitudes, i.e. 

 when 



Sp = (asi + 6s2 + CS3 + • • •) ± 8 



we have what is known as linear prediction. Of course, linear prediction 

 can always be used, but it may not be good enough with some types of 

 messages. 



As Wiener has shown the coefficients a, b, c • • • which minimize 

 e- are readily computed. For simplicity, assume only two message sam- 

 ples, Si and S2 , from the past are to be used. We then have 



€ = So — Sp 



€ = So — asi — 5s2 

 e = So -|- asi -\r b S2 — 2asoSi + 2a6 S1S2 — 26 SoSo 



