EFFICIENT CODING 739 



tricks it should be possible to keep the growth of equipment down to 

 something approaching 2"" rather than /"'. 



The output signal from the //-grammer will he, as we have seen, a series 

 of pulses with an amplitude distribution very peaked toward zcm-o and 

 small pulses. If f ,{\\v alpiiabet, is large, these pulses can be efficiently 

 encoded into a Shainion-Fano code. For small alphabets, granularity 

 ti'ouble can be reduced by remapping the output pulses two-by-two 

 into pulses of base t, and tlien encoding these into the Shannon-Fano 

 code. 



The output signal from the w-grammer with English text as the input 

 message is a pulse amplitnde representation of the type of "reduced 

 text" one gets by using running n-gram prediction on English, as de^ 

 scribed by Shannon . 



More efficient encoding would result if the properly matched Shannon- 

 Fano code for each particular conditional distribution were applied to the 

 output pulses, rather than using the same code for all of them. The effi- 

 ciency of the coding operation would then be close to Fn as given by (8) 

 (take A- = n). This would add a great deal to the complexity and with 

 most signals it is felt the gain would be small. If all the conditional dis- 

 tributions were alike after ordering, the improvement w'ould be nil. 



English text was used as the message in describing the n-gramming 

 technique to emphasize the fact that it is a powerful general method 

 which works even when the conditional probability distributions of a 

 message are disorderly, multimodal affairs. It is obviously suited to other 

 types of messages as well. Its main drawback is the complexity of ap- 

 paratus required. 



PREDICTIVE-SUBTRACTIVE CODING 



When the conditional probability distributions of a message are uni- 

 modal (or merely strongly peaked as a rule in the vicinity of a particular 

 sample amplitude) it is not necessary to re-order the distributions in 

 order to obtain a reduced message for coding. The distributions may then 

 merely be shifted along the amplitude scale until their modes are near 

 zero (or their second moments about zero are nearly minimum). This 

 shifting can be accomplished by computing from the preceding (n — 1) 

 gram the amplitude at which this mode or mean is located, and then sub- 

 tracting this computed amplitude (or the nearest quantizing level) from 

 the actual amplitude of the i:)resent sample. The difference in each case 

 is a symbol whose amplitude distribution is peaked in the vicinity of 

 zero amplitude. Fig. 6 shows a block schematic of a system using pre- 



