EFFICIENT CODING 745 



Other operations besides simple subtraction of the predicted symbol 

 from the present sj^mbol are of course possible. However, in most cases 

 it would seem that if a more complicated operation were indicated, 

 /i-^ramniin<2; would have provided a better start. 



ILLUSTRATIVE EX.\MPLE 



Let us compare the operation of w-gramming and prediction-subtrac- 

 tion techniques on a hypothetical message. We will assume the message 

 has digram statistics, but that longer range statistical influences either 

 do not exist or are ignored. The statistics are then specified entirely by 

 the joint probability distribution p(i, j) of a pair of symbols. Let us 

 assume that there are ^ quantizing levels, and that 



piij) = /va"''"'l 



where a is a constant > 1, and K is given by 



K is the factor which assures that ^ ij p(i, j) = L 



Thus the most likely level is that of the previous sample. A sample 

 differing by one level is 1/a times as likely, one differing by m levels 

 is aT"" times as likely. Figure 8 shows a plot of the relative values of 

 p(i, j) (neglecting the factor K). For ^ = 4, the total array would be 

 the 4x4 portion enclosed by the dashed line. This sort of distribution 

 is rather similar to those of typical television signals, as shown by pre- 

 liminary measurements, although typical values of a have yet to be 

 determined. With no statistical coding, the required channel capacity is 



Ho = log2 i bits/sample. 



If the simple distribution of individual samples is taken into account, 

 the required channel capacity is reduced to 



Hi = -J^pii) log p{i) 



where 



pii) = Z) p(h j) = Z) p(h j) = p(j) 



i i 



Hi may be computed from the array of relative coefficients by adding 

 the rows to form the sums 



