Mathematical work should be reported in a document. Computer programs supporting the
scientific work should be written in C language, under Unixlike 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 xℓ(t) be a lowpass deterministic
signal defined by its Fourier transform,
Xℓ(f) = F{xℓ(t)} = AΛ Its carriermodulated bandpass version is 

2Bf , f ∈ R.  (2) 
x(t) = F–1{X(f)} = ℜ xℓ(t)ej2πf0t , t ∈ R.  (3) 
I.1 Plot Xℓ(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 xℓ(t). Plot xℓ(t) for A = B = 1. Include the
expression and the .eps plot of xℓ(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) = F–1{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 nonzero spectrum at positive frequencies only,
x+(t) = F–1{X+(f)} = F–1{X(f)u(f)}, (4)
1
where u(f) is the unitstep function. It can be easily shown that
x+(t) = 1
2
(x(t) + jxˆ(t)), (5)
where ˆ x(t) = x(t) ∗ πt 1 = F–1{jsgn(f)X(f)}, with the symbol ∗ representing linear
continuoustime 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) = √12πe–x2/2 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) = Zx∞ φ(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 signaltonoise ratio. For
a binary modulation it becomes NEs0 = NEb 0 = 2Aσ22 , 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 doublesided spectral density of
the additive white noise. In your plot consider a decibel (dB) value for the signaltonoise
ratio. Plot P(Y < 0) for a signaltonoise 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 ndimensional 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) ≤ 12e–x2/2, ∀x ≥ 0. Plot Q(x) and 1 2e–x2/2 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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