3.2. A random process 𝑥(𝑡) consists of an ensemble of sample functions, each of which
is a square wave of amplitude ±𝑎 and period 𝑇, Fig. 3.8(𝑎). The "phase" of each sample
is defined as the time 𝑡 = 𝜙 at which the sample first switches from +𝑎 to −𝑎 (for 𝑡 >
0 ). 𝜙 varies randomly from sample to sample with a uniform probability distribution
of the form shown in Fig. 3.8(b).
Show that the time history of a single sample function may be represented by the
Fourier series expansion
𝑥(𝑡) =
4𝑎
𝜋
∑
𝑛=1.3,5,…
1
2𝜋𝑛
sin⁡
(𝜙 − 𝑡)
𝑛
𝑇
and hence calculate the ensemble average 𝐸[𝑥(𝑡)𝑥(𝑡 + 𝜏)] to find the autocorrelation
function for the 𝑥(𝑡) process. Verify that this agrees with the result shown in Fig. 3.9
Hint. The Fourier series "recipe" is given by equations (4.1) and (4.2
1