# 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,

 Xℓ(f) = F{xℓ(t)} = AΛ Its carrier-modulated bandpass version is 2Bf , f ∈ R. (2) x(t) = F–1{X(f)} = ℜ xℓ(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
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
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
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.

Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

We value our customers and so we ensure that what we do is 100% original..
At Custom Writing, we believe in exemplary services that are fully geared toward customer satisfaction. That is why we don’t shy away from giving you the following guarantees;

### Money-back guarantee

Trusting us with your work is the best decision you have made, our pleasure lies in seeing you satisfied at 100%. If in the rear chance it happens that you are not satisfied, then know that we will equally not be satisfied. But worry not, our 30 days- Money back guarantee is all you need and that is what we promise you..

### Zero-plagiarism guarantee

We utilize profoundly equipped and gifted writers who produce unique papers liberated from any form of plagiarism. To guarantee this, we run all papers finished by our scholars through a Plagiarism checker to ensure uniqueness and originality. In any case, on the off chance that you have vulnerabilities about the originality or falsification of any paper we have finished and conveyed to you, please get in touch with us straight away. We will quickly investigate, and if the paper is seen as counterfeited, we will take suitable actions including but not limited to, revising the paper for free and in extreme cases we will activate the money back guarantee.

### Free-revision policy

We have an obligation deliver great and specially composed assignments. Our revision strategy endeavors to ensure total client satisfaction, comfort, and a genuine feelings of serenity. We make minor updates and corrections to the underlying request as part of our continuous assistance. However, revisions should just incorporate changes and alterations that were not effectively met, in the underlying request and that are inside the rules as per the current request structure..

Our client's Data is an Integral part of our business but clearly, we are not in the business of offering our clients' very own data to others. We realize that you care how your online data is utilized and shared, we equally value your trust that we will do so cautiously and sensibly. We Promise to ensure the security of your own data during transmission by utilizing encryption conventions and programming. Likewise assist us with securing your information by not sharing your passwords and usernames.

### Fair-cooperation guarantee

In submitting a request with us, you consent to the services we give. We will strive to take the necessary steps to convey a far reaching paper according to your prerequisites. Equally we depend on your cooperation to guarantee that we convey on this order.

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors: