B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

Problems 503
by first finding its trigonometric Fourier series when b = 0.
(b) Show that
2 1 2
Rx (r) = Co +
2 e n cos n woT
n=l
2n
wo = ­ To
Figure P.9.3-4
t --
9.4-1 A simple RC circuit has two resistors R1 and R 2 in parallel (Fig. P9.4-la). Calculate the rms
value of the thermal noise voltage v O across the capacitor in two ways:
(a) Consider resistors R 1 and R 2 as two separate resistors, with respective thermal noise voltages
of PSD 2kTR 1 and 2kTR 2 (Fig. P9.4-lb). Note that the two sources are independent.
(b) Consider the parallel combination of R1 and R 2 as a single resistor of value Rt R 2 / (Ri + R2),
with its thermal-noise voltage source of PSD 2kTR 1 R 2 / (R1 + Rz) (Fig. P9.4-lc). Comment.
Figure P.9.4-1
R,[I[.
(a)
R1 R 2
R1 +R 2
[[ C Vo
p
(c)
9.4-2 Show that Rxy ( r ), the cross-correlation function of the input process x(t) and the output process
y(t) in Fig. 9.12, is
Rxy (r) = h(r) * Rx (r) and Sxy (j) = H (f )Sx (f)
Hence, show that for the thermal noise n(t) and the output v 0 (t) in Fig. 9.13 (Example 9.9),
2kTR
Snva lf) = 1 + j2nfRC
and
9.4-3 A shot noise is similar to impulse noise described in Prob. 9.2-8 except that instead of random
impulses, we have pulses of finite width. If we replace each impulse in Fig. P9.2-8 by a pulse
h(t) whose width is large in comparison to 1/a, so that there is a considerable overlapping of
pulses, we get shot noise. The result of pulse overlapping is that the signal looks like a continuous
random signal, as shown in Fig. P9.4-3.
(a) Derive the autocorrelation function and the PSD of such a random process.
Hint: Shot noise results from passing impulse noise through a suitable filter. First derive
the PSD of the shot noise and then obtain the autocorrelation function from the PSD. The
answers will be in terms of a, h(t), or H(f) .