EE 160 Communication System Fundamentals
draft of April 6, 2000
Homework Assignment 2
Due (in class) 1pm April 12, 2000
Reading:
R. J. McEliece
162 Moore
Haykin, Chapter 3, Sec. 3.11, p. 158–180.
Chapter 4, Secs. 4.1–4.5 (skim), Secs. 4.6–4.7.
Problems to Hand In:
Problem 1. In class on April 3, in my analysis of wideband FM when the message signal
is m(t) = cos ωm t, I asserted that the integral
ωm
2π
+π/ωm
ejβ sin ωm t e−jnωm t dt
−π/ωm
could be simplified to the form
1
π
π
cos(nθ − β sin θ)dθ.
0
Verify that this simplification is valid.
Problem 2. We saw in class on April 3 that in principle, if the received FM signal is of
the form
t
r(t) = A cos(ωc t + k
m(τ )dτ ),
−∞
then the message signal (or rather the signal m̂(t) = ωc + km(t)) can be obtained by
differentiation followed by envelope detection. However, this assumes that the received
amplitute A is constant. In reality A is often a (positive) function of time, say A(t). In
this case the signal
t
r(t) = A(t) cos(ωc t + k
m(τ )dτ ),
−∞
must first be processed by a bandpass limiter.
A bandpass limiter consists of a hard limiter followed by a bandpass filter. The hard
limiter maps a signal x(t) to y(t) defined by
y(t) =
+1 if x(t) ≥ 0
−1 if x(t) < 0.
and the bandpass filter has transfer function
H(ω) =
where is a small number.
1 if |ω − ωc | < 0 if |ω − ωc | > ,
(a) Suppose a signal of the form x(t) = A(t) cos(ωc t + θ), where A(t) > 0, is passed
through a bandpass limiter. What is the resulting output signal?
(b) What is the relevance of part (a) to the problem of FM demodulation?
Problem 3. Haykin, Problem 4.7 (p. 303), except replace part (b) with
(b) Is the process Z(t) wide-sense stationary? Is so, calculate the process mean and
autocorrelation function. If not, explain clearly why not.
Problem 4. In class on April 5, I computed the mean µ and autocorrelation function
R(τ ) for a “sinusoidal wave with random phase” defined as
x(t, θ) = A cos(ωc t + θ),
where A is a constant and θ is a randomly distributed in the interval [0, 2π]. (See also
Example 7 on p. 245.) Now consider a similar random process
x(t, θ, A) = A cos(ωc t + θ),
in which everthing is the same but now A is a Gaussian random variable (independent of
θ) with mean zero and variance σ 2 .
Find the mean and autocorrelation function for the process x(t, θ, A).