SPECTRA OF QUANTIZED SIGNALS 447 



is to be replaced by a wave constructed of quantized values selected on a 

 minimum error basis from the discrete set available. Clearly if we assign 

 the quantum values with sufficiently close spacing we may make the quan- 

 tized wave indistinguishable by the ear from the original. The purpose of 

 quantization of magnitudes is to suppress the effects of interference in the 

 transmission medium. By the use of precise receiving instruments we can 

 restore the received quanta without any effect from superposed interference 

 provided the interference does not exceed half the difference between ad- 

 jacent steps. 



By combining quantization of magnitude and time, we make it possible 

 to code the speech signals, since transmission now consists of sending one of 

 a discrete set of magnitudes for each distinct time interval.^ '^ -^ '^ •'' ' ' The 

 maximum advantage over interference is obtained by expressing each dis- 

 crete signal magnitude in binary notation in which the only symbols used 

 are and 1. The number which is written as 4 in decimal notation is then 

 represented by 100, 8 by 1000, 16 by 10,000; etc. In general, if we have N 

 digit positions in the binary system, we can construct 2^^ different numbers. 

 If we need no more than 2^ different discrete magnitudes for speech trans- 

 mission, complete information can be sent by a sequence of N on-or-off 

 pulses during each sampling interval. Actually a total of 2^! different 

 coding plans (sets of one-to-one correspondences between signal magnitudes 

 and on-or-off sequences) is possible. The straightforward binary number 

 system is taken as a representative example convenient for either theoretical 

 discussion or practical instrumentation. We assume that absence of a pulse 

 represents the symbol and presence of a pulse represents the symbol 1. 

 The receiver then need only distinguish between two conditions: no trans- 

 mitted signal and full strength transmitted signal. By spacing the re- 

 peaters at intervals such that interference does not reach half the full 

 strength signal at the receiver, we can transmit the signal an indefinitely 

 great distance without any increment in distortion over that originally 

 introduced by the quantizing itself. The latter can be made negligible by 

 using a sufficient number of steps. 



To determine the number of quantized steps required to transmit specific 

 signals, we require a knowledge of the relation between distortion and step 

 size. This problem is the subject of the present paper.* We divide the 

 problem into two parts: (1) quantizing the magnitude only and (2) combined 

 quantizing of magnitude and time. The first part can be treated by a simple 

 model: the ''staircase transducer", which is a device having the instantane- 

 ous ouput vs. input curve shown by Fig. 1. Signals impressed on the stair- 



* Other features of the quantizing and coding theor>' are discussed in forthcoming 

 papers by Messrs. C. E. Shannon, J. R. Pierce, and B. M. Oliver. 



