744 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Power reduction is an index of merit when many reduced signals are 

 to be sent by frequency division over one channel, as we have said. 

 When the object is to reduce the channel capacity required for a single 

 message source, then it is the upper bound entropy of the reduced signal 

 which should be minimized, not the power. That is we want — zJi vij) 

 log p(j) to be minimized. For certain types of signals this requires the 

 modes to be shifted to zero, although this is by no means a general rule. 

 Shifting the modes to zero may actually increase the entropy of the 

 "reduced" signal over that of the original message, by adding too many 

 new symbol levels, as the example in the last section shows. 



If the original message has / quantizing levels, the reduced message 

 after predictive-subtractive coding will in general contain more than ^ 



levels since an error of more than " can be made in either direction. 



An n-gramming operation, on the other hand, never increases the al- 

 phabet. 



2 



>- 4 



PRESENT SYMBOL (j j 

 3 4 5 6 7 



1 a"' a"2 a-3 i ^-^ a"^ a~^ 



a-' 1 a-^ a-2 | a-^ a-^ a-^ 



a-2 a-' 1 a-' | a-2 a-^ 



a"3 a~2 a"^ i | a"' a"^ 



3-4 3-3 3-2 3-1 , 



3-5 3-4 3-3 3-2 1 



a-6 a-s 1 



J_i 1 ! 



FIRST ORDER ROWS THEN 



ADD COLUMNS FOR 



DIGRAMMER DISTRIBUTION 



-^ 



ADD THIS WAY 

 FOR SIMPLE 



DISTRIBUTION 



OF ORIGINAL 



MESSAGE 



- V / 



^w 



^^r 



/ 



^ 



Fig. 8 — Joint probability distribution (divide all coefficients by the sum over 

 each array). 



