XJEL3430 Digital Communications
PROBLEM SET 2
Problem 1: Vector Spaces
The vector-space concept introduced in class is applicable to many mathematical structures. It will help us visualize these structures by thinking of each member of them as a vector. You have seen the example of 2-dimensional vectors in Problem 2 of PSET 1. Another example is the set of periodic functions of period T, or, those of finite energy. More generally, a set of elements is a vector space if two conditions are satisfied:
- First, we must be able to add these to-be-called vectors according to a proper “addition” operation. That is, if v and w belong to a vector space V, then v + w also belongs to V ; and
- second, to scale them, let’s say for now, by complex numbers. That is if v ∈ V, then for any complex number c, cv is also a member of V.
These are the two main properties that we can easily associate with geometric vectors in the xy plane.
a) Consider the setV = {all periodic complex functions of time, t, with period 1} . Give/draw two example functions that belong to V.
b) Show that if f(t) and g(t) are members of V, then so is f(t) + g(t) .
c) Suppose f(t) belongs to V. For a complex number c, show that h(t) = c f(t) also belongs to V.
d) Is V with the above addition and scaling operations a vector space?
Problem 2: Inner Products and Projection
We saw in the lecture that one valid choice of an inner-product operation for the vector space V = {all periodic complex functions of time, t, with period 1}, is given by
where vf and vg , respectively, represent the vectors corresponding to functions f(t) and g(t) in V.
(a) Verify that the functions φn (t) = e2πint , n ∈ Z , all belong to V.
(b) Verify that φn (t) = e2πint , n∈ Z , are normalized, that is, they have unity length, according to the inner product in (1) .
(c) Verify that φn (t) = e2πint , n ∈ Z , are orthogonal to each other, that is, for n ≠ m ∈Z , the inner product of φn (t) and φm (t) is zero.
(d) Consider functions f (t) = cos(2πt) and g(t) = sin(4πt) . Can you expand these functions in terms of φn (t) = e2πint , n ∈ Z (that is, to write f(t) and g(t) as a linear combination, or, weighted sum of φn (t) functions) .
(e) Find the inner product of f(t) and g(t) in part (d) .
Problem 3: Sampling Theorem
An analogue signal, x(t) , with Fourier transform as shown in the figure below, is the input to an ideal sampler followed by an ideal low pass filter. The ideal sampler takes consecutive samples by a train of Dirac delta functions, and, at its output, it generates We can control and change TS at our wish. The ideal low pass filter has a bandwidth WLP , which is also controllable. We denote the output of the low pass filter by y(t) , and its Fourier transform is denoted by Y(ω) .
(a) Is x(t) a bandlimited signal? What is its bandwidth in rad/sec? Denote the bandwidth by W for the rest of this problem.
(b) What is the Nyquist ratefor x(t) ?
(c) Suppose TS = π/(2W) . Sketch the Fourier transform of xS (t) for —3W ≤ ω ≤ 3W .
(d) Suppose TS = π/(2W) and WLP = W . Find y(t) in terms of x(t) .
(e) Suppose TS = π/W and WLP = W /2 . Sketch Y(ω) for — 2W ≤ ω ≤ 2W . (You do not need to write Y(ω) in an analytical form.)