XJEL3430 Digital Communications
PROBLEM SET 1
Problem 1: Calculus
a. Plot and label the following functions on the same graph:
b. Suppose f(t) = 1, for t ∈ [ —1, 0), and f(t) = ∈ 1, for t ∈ [0,1]; It is zero otherwise. Plot
c. Calculate
for T = 1, 10, 100. What is
Problem 2: Complex Numbers and Two-dimensional Vectors
In this problem we reviewsome of the mathematical techniques used often when dealing with complex numbers and functions.
If the setof real numbers corresponds to the pointsonthe x axis, complex numbers can be represented by the points on the two-dimensional xy plane. In fact, any complex number a + ib , where a and b are real numbers and i is the imaginary unit number, can be represented by a point with x-component a and y- component b on the xy plane; see Fig. 1. An alternative way to write a + ib is to use its polar representation , where = cosθ+ isinθ .
Fig. 1: complex numbers on the plane
a. Specify the following complex numbers on an x-y plane and find their corresponding r and θ :
4 + 3 i
(1 - i )* + i 2 , * represents complex conjugate
(1 + i ) e -iπ/2
b. Another way to represent a complex number is by assigning a vector to it, which is originated from the origin of the xy plane, and it ends at the corresponding point to the complex number of interest on the xy plane; see Fig. 1. Using vector representation, show how , where t denotes the time, changes with time. Is this function periodic or nonperiodic? How about the more general function , where ω>0 is the angular frequency? Is a periodic function, or nonperiodic? If periodic, what is its period?
Problem 3: Fourier Transform
Find the Fourier transform of the following functions. Try to use the properties of Fourier transform, alongside the existing tables for well-known functions. In each case, plot both functions.
a. A rectangular pulse with time width 1, centered at time zero, and amplitude 1.
b. A triangle pulse with time width 2, centered at zero, and peak value 1.
c. The function in part (a) multiplied by sin(t), where t denotes the time parameter. Hint: It may be easier if you write sin(t) in terms of exponential functions.
d.
Problem 4: Gaussian Random Variables
Gaussian random variables (RVs) are continuous RVs whose probability density functions (PDFs) are fully described by their mean and variance. If an RV X is Gaussian with mean μ and variance σ2 , it is denoted by and its PDF is given by
For various reasons, we are interested in finding the cumulative distribution function (CDF) of X ,
x
There is no closed form solution forthe CDF of a Gaussian RV. There are, however,
various lookup tables, from which FX (x) can be numerically found. These lookup tables are commonly given in terms of the following three functions:
1. Q function:
2. Error function:
3. Complementary error function:
(a) Show that Q(x) = 1 — FX (x) = Pr{X > x} , where X ~ N(0,1) . Find the numerical values for Q(0), Q(—∞) , and Q(∞) .
(b) Show that if X ~ N( μ, σ2 ), then
Hint: One way to show this isto write the left-hand side in terms of fX (x) , and the right-hand side using the definition of Q function. Try to change variable in one of your integralsto get the other
one. Do not forget to change the interval over which you are integrating once you change the variable.
(c) Suppose X ~ N(1,4) . Calculate Pr(X > 2) and Pr(X ≤ —2) . You may use the MATLAB function qfunc to find the numeric values for these probabilities.