742 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



for linear prediction. Samples before this add no further information as 

 to the location of the mean of the conditional distributions.* 



It happens that in typical television signals the autocorrelation for 

 small displacements shows a very nearly exponential behavior. Thus 

 linear prediction on the basis of the previous picture element alone is 

 a natural method for television, particularly in view of the simplicity of 

 apparatus required. 



Linear prediction is easily instrumented. Fig. 7 shows in block sche- 

 matic form the essentials of a linear predictor. Samples of the message 

 are applied to a delay line. Taps along this line separated by the inter- 

 symbol time of the message, or multiples thereof, make the desired 

 past symbols available. The signals from these taps are merely attenu- 

 ated by amounts corresponding to the coefficients a,h, c • • • and added. 

 A differential summing amplifier is shown to allow for negative coeffi- 

 cients, and also to accomplish the subtraction of the predicted sample 

 amplitude from the present sample amplitude. 



A complete linear predictor-subtractor is nothing but a transversal 

 (time domain) filter whose impulse response is 



f(t) = 8(t) - a8(t - r) - 1)8(1 - 2r) • • • 



and whose equi\'alent frequency response is therefore 



77!/ \ i — iiiiT 7 — 2ia)T 



F{co) = 1 — ae —he 



where r is the delay between taps. If, for example, simple previous value 

 prediction is used (a = l;h,c--- =0) 



F{co) = 1 - e-'^' = 2i sin ^ e~^ . 



* From the preceding expression for p, we see that p = (i.e., perfect predic- 

 tion is possible) if: 



(<t>l - <{>■{)' = (1 - <t>lr 



<f>l - <t>2 = ±(1 - 4>l) 



f02 = 1 



[4>2 = 2<t>l - 1. 



If 02 = 1, the message samples alternate between two independent but constant 

 values. For this case a = 0, b = 1. If <^2 = 2(t>^ — 1 the autocorrelation is a cosine 

 wave so the message consists of samples of a sinusoid. In this case a = 2<t>i , b = 

 — 1. If <^i is nearly unity, the sinusoid is of low frequency, and the prediction 

 approaches "slope" prediction (i.e. extrapolation of a straight line through the 

 last two samples). 



In any case where perfect prediction is possible the wave is periodic and there- 

 fore ^ = 0. 



