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Foundation of Digital Communications Examples of typical exam-like exercises
Comments
The exam exercises will cover the following three main areas of the course



Probability theory and random variables
Deterministic signals, Fourier transforms, linear systems, power and energy
Random processes
The following is a collection of typical exercises, divided on the three main topics
Exercises on Probability theory and random variables
Exercise 1
A random variable
f ( x) 
 has the following probability density function:
5  5 x
e
2
1.
Draw a qualitative plot of the probability density function f (x )
2.
Evaluate the mean of the random variable E[ ]
3.
Evaluate the probability P (  5)
4.
Evaluate the probability
P [5,5]
Exercise 2
A random variable
f ( x) 
 has the following probability density function:
5  5 x
e
2
The random variable is sent through a nonlinear device, and the output is the random variable
relation of the device is the following:
 . The input-output
if   0 then   1

if   0 then   1
1.
Evaluate the probability P (  1) and P (  1)
2.
3.
Evaluate the cumulative distribution function of the random variable
Evaluate the probability density function of the random variable 

Exercise 3
 is a gaussian random variable with mean equal to 2 and variance equal to 1.
1.
Write the analytical expression for the probability P (  6)
2.
Estimate the numerical value of the probability P (  6)
Note: it is possible to use the following numerical approximation for the erfc() function
erfc( x) 
e x
2
x
1
The random variable
 is sent to a nonlinear device, generating at its output the random variable    2 .
1.
Evaluate the mean E[ ]
2.
Evaluate the variance
 2
Exercise 4
Evaluate the exact error probability of the cascade of two binary symmetric channels (BSC). The first BSC has
4
has a transition probability p  10 , while the second has a transition probability
be assumed to be statistically independent.
p  106 . The two BSC can
Exercises on deterministic signal
Exercise 1
Consider the following deterministic signal x(t )
x(t )
2A
A
T
0
1.
2.
3.
4.
2 T
t
Evaluate its Fourier transform
Evaluate its energy and its average power
Classify the signal (finite energy or finite power)
Evaluate the corresponding spectral density (energy or power, considering that one of the two is meaningless)
Exercise 2
Consider the same deterministic signal x(t ) introduced in the previous exercise. This signal is sent through a linear
and time invariant filter with transfer function:
 A for t  [0, T ]
h(t )  
 0 outside
1.
Evaluate and draw the output y (t ) of the filter
2.
Evaluate the spectral density (energy or power, considering that one of the two is meaningless) of y (t )
Suggestion: try to use the properties of convolution and linear systems
Exercise 3
Evaluate the convolution of the following two signals:
x(t )  e 4t
h(t )  A for t  0,4 and zero outside
Exercise 4
Let’s consider the signal
y(t )  x(t ) cos(2f 0t ) , where the Fourier transform of x(t ) is equal to 1 in the interval
[-B,+B] and zero outside. The frequency f 0 is much greater than B.
z (t )  A y 2 (t )  B . The signal z(t) is then
sent to an ideal low pass filter with bandwidth equal to 3 B, generating the signal g (t ) .
The signal y(t) is sent to a nonlinear device which generates the output
2
1.
Estimate the spectrum of g (t ) .
2.
Write the time-domain expression of the signal g (t ) as a function of x(t )
Exercises on random processes
Exercise 1
Consider the following random process:
y (t )  a0  a1 x(t ) cos 2 2f 0t   
where:

a0

, a1 , f 0 are numeric constants;
x(t ) is a WSS and ergodic random process with zero mean and autocorrelation Rx ( ) ;
 is a random variable uniformly distributed in the interval [0,2 ] and statistically independent from x(t ) .
1.
Evaluate the mean of y (t )
2.
3.
4.
Evaluate the autocorrelation of y (t )
Determine if the process is ergodic for its mean
Evaluate the power spectral density of y (t )

Exercise 2
Consider a digital transmission signal, expressed as:
y (t ) 
i  
 r (t  iT
i  
i
B
 )
where:
  i are statistically independent random variables that can assume the two values  A with equal probability

r (t ) is a deterministic function, shown below
r (t )
A
t
 TB
 TB


1.
Evaluate the mean of y (t )
2.
Evaluate the power spectral density of y (t )
is a random variable uniformly distributed in the interval [0,2 ] and statistically independent from
i .
Exercise 3
We have a LTI filter characterized by the following transfer function.
H( f )
2
1
f
 f1
f1
3
f1 of the filter is equal to 10 GHz. The input of the system is a white gaussian noise with
N0
12
power spectral density equal to
, where N 0  10 [W/Hz].
2
The cut-off frequency
Evaluate:
1. The average power of
2.
nout (t ) in Watt;
Evaluate the analytical and numerical value of the probability
Pnout (t )  A , where A 2  0.4 Watt.
Note: it is possible to use the following numerical approximation for the erfc() function
erfc( x) 
e x
2
x
4