SPECTRA OF QUANTIZED SIGNALS 471 



and the response of the network to (II-3) is therefore: 

 /(/) = I I YSq) I cos f^^ + ph \\{iq)] 



+ ^ Z) (I Fo(2wco« + iq) I cos [{mo3s + q)t (TI— 5) 



+ ph Yoiimcos + ^'9)] + I Yoiimojs — iq) \ cos [(/mo^ — 9)/ 

 + ph Yoiimojs — iq)]). 



But /(/) evidently represents a train of pulses in which the pulse occurring 

 at / = iiT is equal to the nth sample multiplied by g{t — nT) . We have thus 

 obtained the spectrum of a set of samples in which the pulse representing a 

 unit sample is the generalized wave form g{t). Furthermore if the signal 

 frequency q is less than 00^2, an ideal low-pass filter with cutoff at 03s/2 

 responds only to the first component of (II — 5). 



The ''aperture effect" or variation of transfer admittance with signal 

 frequency is thus given by 



Y{iq) = ^ YSq) = fs YoUq). (11-^) 



This is Theorem 11. Since the system is linear when the signal frequency 

 does not exceed half the sampling frequency, the principle of superposition 

 may be applied to composite signals. In the case of distortion from quantiz- 

 ing errors the aperture effect applies to the error component delivered by the 

 low-pass output filter. For an imperfect low-pass filter in the output we 

 multiply the aperture admittance function by the actual transfer admittance 

 of the filter. 



A theorem equivalent to the above has been derived by a different method 

 in a recent paper^* published after completion of the above work. 



REFERENCES 



1. W. R. Bennett, Time Division Multiplex Systems, Bell Sys. Tech. Jmir., Vol. 18, pp. 



1-31; Jan. 1939. 



2. H. S. Black, Pulse Code Modulation, Bell Lab. Record, Vol. 25, pp. 265-269; July, 1947. 



3. VV. M. Goodall, Telephony by Pulse Code Modulation, Bell Sys. Tech. Jour., Vol. 



26, pp. 395-409; July, 1947. 



4. D. D. Grieg, Pulse Count Modulation System, Tele-Tech., Vol. 6, pp. 48-50, 98; Sept. 



1947; also Elect. Comm., Vol. 24, pp. 287-296; Sept. 1947. 



5. A. G. Clavier, P. F. Panter, and D. D. Grieg, PCM Distortion Analysis, Elec. Engg., 



Vol. 66, pp. 1110-1122; Nov. 1947. 



6. H. S. Black and J. O. Edson, PCM Equipment, Elec. Engg., Vol. 66, pp. 1123-1125; 



Nov. 1947. 



7. L. A. Meacham and E. Peterson, An Experimental Pulse Code Modulation System of 



Toll Quality, Bell Sys. Tech. Jour., Vol. 27, pp. 1-43; Jan., 1948. 



8. H. Nyquist, Certain Topics in Telegraph Transmission Theory, A. I. E. E. Trans , 



pp. 617-644; April, 1928. 



9. E. Peterson, Gas Tube Noise Generator for Circuit Testing, Bdl Lab. Record, Vol. 



18, pp. 81-83; Nov. 1939. 



