454 BELL SYSTEM TECHNICAL JOURNAL 



The initial points at the minimum sampUng rate are determined on the other 

 hand by the total area under the curves of Fig. 4, since the accepted side- 

 bands on the harmonics in this case exactly fill out the entire error spectrum. 

 These initial points are therefore given quite accurately by Eq. (1.1), which, 

 as pointed out before, is a good approximation for the total areas. We can 

 also give a direct demonstration of the applicability of Eq. (1.1) to the 

 initial points of the curves of Fig. 5 by means of the following theorem: 



Theorem I. The mean square value of the response of an ideal low-pass 

 filter to a train of unit impulses multiplied by instantaneous samples occur- 

 ring at double the cutoff frequency is equal to the mean square value of the 

 samples provided no harmonic of the sampling frequency is equal to twice 

 the frequency of one component or equal to the sum or difference of two 

 component frequencies of the sampled signal. Proof of the theorem is given 

 in Appendix I. To apply it here we resolve the input into two components: 

 the true signal and the error. The former is reproduced with fidelity in the 

 output because it contains only frequencies below half the sampling rate. 

 The error component in the output represents the response of the low-pass 

 filter to the error samples. Except for very special types of signals, the error 

 samples are uniformly distributed throughout the range from minus half a 

 step to plus half a step. Calculation of the mean square value of such a 

 distribution gives Eq. (1.1). 



We have tacitly assumed above that the sampled values applied to the 

 filter in the output of the system are infinitesimally narrow pulses of height 

 proportional to the samples. In actual systems it is found advantageous to 

 hold the sampled values constant in the individual receiving channels until 

 the next sample is received. This means that the input to the channel filter 

 is a succession of rectangular pulses of heights proportional to the samples. 

 The resulting magnitude of recovered signal is much larger than would be 

 obtained if very short pulses of the same heights were used; stretching the 

 pulses in time produces in effect an amplification. The amplification is 

 obtained, however, at the expense of a variation of channel transmission 

 with signal frequency. Infinitesimally short pulses have a flat frequency 

 spectrum, while pulses of finite duration do not. The frequency character- 

 istic introduced by lengthening the pulses is easily calculated by determining 

 the steady state admittance function of a network which converts impulses 

 to the actual pulses used. The general formula for this admittance when a 

 unit impulse input is converted into an output pulse g{t) is easily shown to 

 be: 



F(ia,) =/. r giOe-''-' dt (1.2) 



J_oo 



where/, is the repetition frequency and co is the angular signal frequency. 



