SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 257 
The analysis of the sampled signal shown in Fig. 10.2 gives a physical 
picture of the results of sampling but leads to an unnecessarily awkward 
mathematical formulation of the problem. As in the earlier chapters on 
the treatment of deterministic signals, it is convenient to let the modulat- 
ing-pulse width approach zero while the modulator gain approaches 


NANRRueennneecee ee 
AULLTLUSS ESSE 
Fig. 10.3. Autocorrelation function of random pulse sequence. 
infinity, with the result that the sampling process is approximated by 
impulse-area modulation. By inspection of Fig. 10.3 or (10.22) it is obvi- 
ous that the autocorrelation of the impulse sampled signal reduces to a 
train of impulses which can be expressed as 
ie) 
ee > = &,.(kT)5(r — kT) (10.23) 
k=—o 
Following the treatment of deterministic signals, the sampled autocorrela- 
tion is defined as the sequence of values 
BA (kT) = 7 ®.(ET) (10.24) 
where the 1/T factor is arbitrarily introduced in the definition (10.24) to 
simplify later expressions. 
The spectral density of the signal whose autocorrelation function is 
given by (10.23) is the transform of this equation and has the form 
S7(8) 
k=—o2 
ie » Be (T)6(r — kT)e-*" dr 
Oe (RIM oe (10.25) 
Se 
This function is called the sampled spectral density, or the sampled power 
spectrum, of the signal z(t). If e*” is replaced by z, it is recognized that 
the sampled power spectrum of the signal is simply the two-sided z trans- 
form of the sampled autocorrelation function, &*,(kT). 
