EE 3640 Communication Systems I Chi-chao Chao
Spring 2023
Homework Assignment No. 5
Due 1:20pm, May 29, 2023
Reading: Haykin & Moher, Chapter 6.
Problems for Solution:
1. The sample function
x(t) = Ac cos(2πfc t) + w(t)
is applied to the low-pass RC filter shown below. The amplitude Ac and frequency fc
of the sinusoidal component are constants, and w(t) is a white Gaussian noise of zero
mean and power spectral density N0 /2. Find an expression for the output signal-to
noise ratio with the sinusoidal component of x(t) regarded as the signal of interest.
2. A DSB-SC modulated signal is transmitted over a noisy channel, with the power spec-
tral density of the noise being as shown below. The message bandwidth is 4 kHz and
the carrier frequency is 200 kHz. Assuming that the average power of the modulated
wave is 10 watts, determine the output signal-to-noise ratio of the receiver.
3. The power spectral density of the white noise measured at the front end of an AM
receiver is 10−3 watt per Hertz. The modulating wave is sinusoidal, with a carrier
power of 80 kilowatts, and a sideband power of 10 kilowatts per sideband. The message
bandwidth is 4 kHz.
(a) Assuming the use of an envelope detector in the receiver, determine the output
signal-to-noise ratio of the system when the carrier-to-noise ratio is high.
(b) By how many decibels is this system inferior to a DSB-SC modulation system?
4. An unmodulated carrier of amplitude Ac and frequency fc and band-limited white
noise are summed and then passed through an ideal envelop detector. Assume the
noise power spectral density to be of height N0 /2 and bandwidth 2W , centered about
the carrier frequency fc . Determine the output signal-to-noise ratio for the case when
the carrier-to-noise ratio is high. (Hint: The unmodulated carrier is regarded as the
signal of interest.)
5. Consider a phase modulation (PM) system, with the modulated wave defined by
s(t) = Ac cos[2πfc t + kp m(t)]
where kp is the phase sentivity and m(t) is the message signal with bandwidth W
and average power P . Consider the receiver model discussed in class, where the phase
demodulator consists of a phase detector followed by a baseband low-pass filter. The
additive noise w(t) is white Gaussian of zero mean and power spectral density N0 /2.
The phase detector is assumed ideal, i.e., v(t) = θ(t) if its input x(t) is R(t) cos[2πfc t +
θ(t)]. Also assume that the carrier-to-noise ratio at the detector input is high.
(a) Determine the output signal-to-noise ratio.
(b) Determine the figure of merit of the system.
6. Suppose that the transfer functions of the pre-emphasis and de-emphasis filters of an
FM system are given as follows:
jf
Hpe (f ) = k 1 +
f0
1 1
Hde (f ) = .
k 1 + jf /f0
The scaling factor k is to be chosen so that the average power of the emphasized
message signal is the same as that of the original message signal m(t).
(a) Find the value of k that satisfies the above requirement for the case when the
power spectral density of the message signal m(t) is
S0 /[1 + (f /f0 )2 ], −W ≤ f ≤ W
SM (f ) =
0, elsewhere.
(Hint: Note the indefinite integral 1/(a2 + b2 x2 ) dx = (1/(ab)) tan−1 (bx/a).)
R
(b) What is the corresponding value of the improvement factor I produced by using
this pairR of pre-emphasis and de-emphasis filters? (Hint: Note the indefinite
integral x2 /(a2 + b2 x2 ) dx = (x/b2 ) − (a/b3 ) tan−1 (bx/a).)
Homework Collaboration Policy: I allow and encourage discussion or collaboration on
the homework. However, you are expected to write up your own solution and understand
what you turn in. Late homework is subject to a penalty of 5% to 40% of your total points.
