580 BELL SYSTEM TECHNICAL JOURNAL 



In the above expression, the primes represent derivatives with respect to 

 time. If Q/P < 1, we expand in Fourier series, obtaining 



■)|(-9" 



^ = e' ^ {e' - ^') 1, ( -M cos m{e - ^). 



When Q/P is small, we retain only the term proportional to Q/P as the error, 

 which may be written in the compact form: 



If the waves are urmiodulated, 6 = pi and v? = qt, giving 

 n- d' = ^(q- p) cos {q - p)t 



APPENDIX III 



PCM FOR BANDWIDTH REDUCTION 



We have treated PCM as a means of increasing bandwidth beyond the value 

 corresponding to one pulse per sample per channel (quantized PAM) and 

 have studied the transmission advantages that accrue therefrom. The PCM 

 method can, in principle, serve to reduce bandwidth. An example of band- 

 width reduction,^' ^^ suggested to the writers by C. E. Shannon, is as follows: 



Any number, say N, of 4 kc telephone channels can be transmitted in the 

 form of one quantized pulse per 125 microseconds, by sampling all channels 

 in the usual way, encoding each sample into a code symbol having, say, 64 

 possible values, assembling all code pulses into one new group and decoding 

 this group at the transmitter. If only one channel were to be transmitted 

 the decoded signal would have 64 possible amplitudes; for two channels it 

 would have 64^ possible ampUtudes, and for A^ channels, 64 . Now, if a 

 single quantized pulse conveying these amplitudes could be transmitted 

 without an error as large as one step, the receiver could encode the quantized 

 pulse, disassemble the resulting code pulses into groups according to channels 

 and decode the groups to obtain the N channel samples. The requirements 

 on transmission circuits capable of the precision required to transmit even 

 two channels in place of one are very severe, however. 



In the event the signals to be transmitted were not speech signals but a 

 very elemental kind of signal such as a black and white pattern requiring for 



** A paper "Reducing Transmission Bandwidtii" by Bailey and Singleton Electronics, 

 Aug. 1948 gives a somewhat different example of reduction. 



■** An early disclosure of a system theoretically capable of any desired amount of band- 

 width reduction is contained in U. S. Patent No. 2,056,284, Oct. 6, 1936, issued to L. A. 

 MacColl. As in the current proposals, the decreased band is obtained at the expense of a 

 vastly greater signal-to-noise ratio requirement and the necessity for precise synchronism 

 between transmitter and receiver. 



