Digital Communications

Mathematical work should be reported in a document. Computer programs supporting the
scientific work should be written in C language, under Unix-like operating systems. Your
report has to include the source of all C programs and scientific plots under gnuplot. It is
highly recommended that your report be written in LaTeX.


Part I – Lowpass and Bandpass Signals
Let Λ(t) be the triangular signal defined as

Λ(t) =

t + 1, 0 t 1,
(1)

0,
otherwise.

t + 1, 1 t 0,
Now we consider three real quantities f0, A, B R+ representing a carrier frequency, an
amplitude, and a frequency band width respectively. Let
x(t) be a lowpass deterministic
signal defined by its Fourier transform,

X(f) = F{x(t)} = AΛ
Its carrier-modulated bandpass version is
2Bf , f R. (2)
x(t) = F1{X(f)} = x(t)ej2πf0t , t R. (3)

I.1 Plot X(f). Take the parameters value A = B = 1 in your plot. Include the .eps
image file in your .tex report.
I.2 Find the mathematical expression of x(t). Plot x(t) for A = B = 1. Include the
expression and the .eps plot of
x(t) in your .tex report.
I.3 Using (2) and (3) find the mathematical expression of X(f). Plot X(f). Take
A = B = 1 and f0 = 20 in your plot. Include the expression and the .eps plot of X(f)
in your .tex report. In practice we have
f0 B, e.g. B = 4 MHz and f0 = 2 GHz.
I.4 Using (3) find the mathematical expression of x(t) = F1{X(f)}. Plot x(t). Take
A = B = 1 and f0 = 20 in your plot. Include the expression and the .eps plot of x(t) in
your .tex report. Any comment about the envelope of the bandpass signal
x(t)?
The analytic signal
x+(t) includes a non-zero spectrum at positive frequencies only,
x+(t) = F1{X+(f)} = F1{X(f)u(f)}, (4)
1

where u(f) is the unit-step function. It can be easily shown that
x+(t) = 1
2
(
x(t) + jxˆ(t)), (5)
where ˆ
x(t) = x(t) πt 1 = F1{-jsgn(f)X(f)}, with the symbol representing linear
continuous-time convolution and sgn(
f) is the sign function.
I.5 Explain why the signal ˆ x(t) is always real and bandpass.
I.6 Write a C program to compute ˆ x(t) and plot this Hilbert transform signal in the time
domain. Include the .c source file and the .eps plot in your .tex report file.
Part II – The Gaussian Tail Function
Let φ(x) = 12πex2/2 be the probability density function of a normal (Gaussian) distribution of zero mean and unit variance. Let X be a random variable having density φ(x),
i.e.
X ∼ N (0, 1). The Gaussian tail function is
Q(x) = P(X > x) = Zxφ(t) dt. (6)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-4 -3 -2 -1 0 1 2 3 4
Probability Density Function
x

Uniform[-1,1]
Normal(0,1)
Normal(0,2)

Figure 1: Probability density function of a normal (Gaussian) distribution N (0, 1) and
N (0, 2) versus a uniform distribution in the range [1, 1].
The density of both
N (0, 1) and N (0, 2) distributions is shown on Figure 1. The
figure also shows a uniform distribution. The function
Q(x) gives the area under the red
curve from
x up to +.
2

II.1 Let η ∼ N(0, σ2) be an additive Gaussian noise. Let Y = A + η, where A R+.
Find the expression of
P(Y < 0) using the Gaussian tail function Q(x). Write this expression in your .tex report.
II.2 Write a C program to compute the probability P(Y < 0) (via numerical integration), given the parameters A and σ2. Plot the numerical value of P(Y < 0) versus the
ratio
A2
2σ2 . This ratio is known in Communication Theory as the signal-to-noise ratio. For
a binary modulation it becomes
NEs0 = NEb 0 = 222 , with a baseband energy per bit Eb = A2
and the noise parameter N0/2 = σ2, where σ2 is the baseband noise variance per real
component. We will see in our course that
N0/2 is the double-sided spectral density of
the additive white noise. In your plot consider a decibel (dB) value for the signal-to-noise
ratio. Plot
P(Y < 0) for a signal-to-noise ratio between 3 dB and 12 dB (a step of 1 dB
is fine). Use a logarithmic scale for the vertical axis. Include the C program and the plot
in your TeX report.
II.3 For Eb
N0 = 9 dB, run 1 million samples of a random variable Y ∼ N(1, σ2) and estimate P(Y < 0). Compare the result to the value obtained by integration in the previous
question. Include the C program of this Monte Carlo simulation and the numerical values
in your TeX report.
II.4 Let η Rn be a vector in n-dimensional real space where each component is N(0, σ2).
We write
η = (η1, η2, . . . , ηn). The components ηi are assumed to be independent (i.i.d.
noise components). Consider two points
S1 and S2 in Rn,
S1 = (1, 1, . . . , 1), S1 = (2, 2, . . . , 2). (7)
Find the mathematical expression of
P(kS2 ηk < kS1 ηk), i.e., this is the probability
of a noisy point 0 +
η being closer to S2 than S1. Use the Gaussian tail function Q(x).
Write the formula of
P(kS2 ηk < kS1 ηk) in your TeX report.
II.5 Explain why Q(x) 12ex2/2, x 0. Plot Q(x) and 1 2ex2/2 on the same graph
where
x = p2Eb/N0. The graph should have Eb/N0 expressed in decibels in the
range [0
, 12]. Use a logarithmic scale for the vertical axis. Include your explanation
and the .eps of the graph plot in your TeX report.

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