Description
Exercise 1.
Let Y (t) = X(t) − X(t − d).
(a) Find RX,Y (τ ) and SX,Y (ω).
(b) Find RY (τ ).
(c) Find SY (ω).
c 2020 Stanley Chan. All Rights Reserved. 1
Exercise 2.
Let X(t) be a zero-mean WSS process with autocorrelation function RX(τ ). Let Y (t) = X(t) cos(ωt + Θ),
where Θ ∼ uniform(−π, π) and Θ is independent of the process X(t).
(a) Find the autocorrelation function RY (τ ).
(b) Find the cross-correlation function of X(t) and Y (t).
(c) Is Y (t) WSS? Why?
c 2020 Stanley Chan. All Rights Reserved. 2
Exercise 3.
Consider the system
Y (t) = e
−t
Z t
−∞
e
τX(τ )dτ.
Assume that X(t) is zero mean white noise with power spectral density SX(ω) = N0/2. Find
(a) SXY (ω)
(b) RXY (τ )
(c) SY (ω)
(d) RY (τ )
c 2020 Stanley Chan. All Rights Reserved. 3
Exercise 4.
Consider the random process
X(t) = 2A cos(t) + (B − 1) sin(t),
where A and B are two independent random variables with E[A] = E[B] = 0, and E[A2
] = E[B2
] = 1.
(a) Find µX(t)
(b) Find RX(t1, t2)
(c) Find CX(t1, t2)
c 2020 Stanley Chan. All Rights Reserved. 4
Exercise 5.
Find the autocorrelation function RX(τ ) corresponding to each of the following power spectral densities:
(a) δ(ω − ω0) + δ(ω + ω0)
(b) e
−ω
2/2
(c) e
−|ω|
c 2020 Stanley Chan. All Rights Reserved. 5
Exercise 6.
A WSS process X(t) with autocorrelation function RX(τ ) = e
−τ
2/(2σ
2
T )
is passed through an LTI system
with transfer function H(ω) = e
−ω
2/(2σ
2
H)
. Denote the system output by Y (t). Find
(a) SXY (ω)
(b) RXY (τ )
(c) SY (ω)
(d) RY (τ )
c 2020 Stanley Chan. All Rights Reserved. 6
Exercise 7.
A WSS process X(t) with autocorrelation function
RX(τ ) = 1/(1 + τ
2
)
is passed through an LTI system with impulse response
h(t) = 3 sin(πt)/(πt).
Let Y (t) be the system output. Find SY (ω). Sketch SY (ω)
c 2020 Stanley Chan. All Rights Reserved. 7
Exercise 8.
Consider a WSS process X(t) with autocorrelation function
RX(τ ) = sinc(πτ ).
The process is sent to an LTI system, with input-output relationship
2
d
2
dt2
Y (t) + 2 d
dtY (t) + 4Y (t) = 3 d
2
dt2 X(t) − 3
d
dtX(t) + 6X(t).
Find the autocorrelation function RY (τ ).
c 2020 Stanley Chan. All Rights Reserved. 8
Exercise 9.
Let X(t) be a WSS process with correlation function
RX(τ ) = (
1 − |τ |, if − 1 ≤ τ ≤ 1
0, otherwise.
(1)
It is known that when X(t) is input to a system with transfer function H(ω), the system output Y (t) has a
correlation function
RY (τ ) = sin πτ
πτ
. (2)
Find the transfer function H(ω).
c 2020 Stanley Chan. All Rights Reserved. 9