EFFICIENT CODING 741 



Now 



"^ ~2 "2 J 



So = Si = §2 = Ao 



soSi = S1S2 = Ai 



S0S2 = A-2, 



where .to , -h , aiul .lo are the values of the auto-covariance of the mes- 

 sage wave at displacements of 0, 1, and 2 sampling periods. Thus 



^ = (1 + ft^ + h')A, + 2{ah - a)A, - 2Mo . 



A ■ 

 The autocorrelation (normalized auto-covariance) is given by 0; = -j^ . 



Ai) is proportional to the average power in the message wave, so the 



ratio p = ^ is the ratio of the power in the error signal to the power 



Ao 



in the original message wave. Thus: 



p = {I -\- a- +b~ ) -{- 2{ah - 0)4,1 - 2602 



^ = 2a -f 2(6 - l)</)i = 

 da 



^ = 26 + 2a0i - 2ct>2 = 

 db 



_ <j>2 — 01 



With these values of a and 6: 



P = 1 — 01 — 



(01 — 02)' 



1 - 0? 



If 02 = 01 , then the expressions simplify to 



a = 01 , 6 = 0, p = 1 — 01 . 



As can easily be shown, if 0(a;) = e""'''' , then all the coefficients except 

 a are zero, and a has the value e~". In other words, if the autocorrela- 

 tion function is of exponential shape, the previous sample alone is needed 



