SPECTRA OF QUANTIZED SIGNALS 455 



We shall call this Theorem II and give the proof in Appendix II. This 

 relation is similar to that found in television and telephotography for the 

 ''aperture effect", or variation of transmission with frequency caused by the 

 finite size of the scanning aperture. The pulse shape g{t) is analogous to a 

 variation in aperture height g{x), where x is distance along the line of scan- 

 ning. Hence it has become customary to use the term ''aperture effect" in 

 the theory of restoring signals from samples. The aperture effect asso- 

 ciated with rectangular pulses lasting from one sample to the next amounts 

 to an amplitude reduction of ir/l or 3.9 db at the top signal frequency (one 

 half the sampling rate) compared to a signal of zero frequency. There is 

 also a constant delay introduced equal to half the sampling period. The 

 latter does not cause any distortion and the amplitude effect can be corrected 

 by properly designed equalizing networks. 



The fact that many pulse spectra can be simply expressed in terms of a 

 flat spectrum associated with sharp pulses and an aperture effect caused by 

 the particular shape of pulse used does not appear to have been recognized 

 in the recent literature, although applications were made by Nyquist in a 

 fundamental paper^ of 1928. Premature introduction of a specific finite 

 pulse not only complicates the work, but also restricts the generality of the 

 results. 



Distortion caused by quantizing errors produces much the same sort of 

 effects as an independent source of noise. The reason for this is that the 

 spectrum of the distortion in the receiving filter output is practically inde- 

 pendent of that of the signal over a wide range of signal magnitudes. Even 

 when the signal is weak so that only a few quantizing steps are operated, 

 there is usually enough residual noise on actual systems to determine the 

 quantizing noise and mask the relation betw^een it and the signal. Eq. 

 (1.1) yields a simple rule enabUng one to estimate the magnitude of the 

 quantizing noise with respect to a full load sine wave test tone. Let the full 

 load test tone have peak voltage E; its mean square value is then E^/2. 

 The total range of the quantizer must be 2E because the test signal swings 

 between —E and -\-E. The ratio 2E/Eq = r is a convenient one to use in 

 specifying the quantizing; it is the ratio of the total voltage range to the 

 range occupied by one step. The ratio of mean square signal to mean square 

 quantizing noise voltage is 



£2/2 6£2 3r2 



El/12 4£V^' 



(1.3) 



Actual systems fail to reproduce the full band /.,/2 because of the finite 

 frequency range needed for transition from pass-band to cutoff. If we in- 

 troduce a factor k to represent the ratio of equivalent rectangular noise band 



