Homework 5 (Due: March 2)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 5
Due: March 2, 2017

Reading
Madhow: Section 3.2.
Problems
  1. Consider the vectors
         ⌊   ⌋       ⌊    ⌋       ⌊    ⌋
       1            1            1
s0 = ⌈ 1 ⌉   s1 = ⌈ 2 ⌉  s2 = ⌈  4 ⌉ .
       1            3            9

    1. Use the Gram-Schmidt procedure to find orthonormal basis vectors which span the space of these vectors.
    2. What is the dimension of the space spanned by these three vectors?
    3. Compute the representation of the si in terms of the orthonormal basis vectors determined in part (a).
    4. Repeat parts (a) — (c) for the signals
              (                          (
        ||{   2  0 ≤  t < 1          ||{  - 1  0 ≤ t < 1
           - 2 1 ≤  t < 2              3   1 ≤ t < 2
s0(t) = |   2  2 ≤  t < 3   s1(t) = |   1   2 ≤ t < 3
        |(   0  else                |(   0   else
        (                          (
        ||{   1  0 ≤  t < 1          ||{  - 1  0 ≤ t < 1
s (t) =    - 2 1 ≤  t < 2   s (t) =    - 1  1 ≤ t < 2
 2      ||   0  2 ≤  t < 3    3     ||  - 3  2 ≤ t < 3
        (   0  else                (   0   else.

  2. Consider the Hilbert space L2(-1, 1) of square integrable signals on the intervals [-1, 1]. The signals {1,t,t2} form a basis for a subspace L of L2(-1, 1).
    • Apply the Gram-Schmidt procedure to the signals {1,t,t2} to generate an orthonormal basis {en}n=02 of L.
    • It turns out that in general
             ∘  -------
e (t) =   2n-+-1P  (t), for n = 0,1,2,...
 n           2    n

      where Pn(t) are the Legendre polynomials

                   n  n     2 n
Pn (t) = (--1)- d-(1 --t)--
         2nn!     dtn

      Verify that the orthonormal basis signals that you computed via the Gram-Schmidt procedure equal those computed via the Legendre polynomials.

    • Compute the projection of the signal
                       et + e-t
x (t) = cosh (t) = --------
                    2

      onto the subspace L. Express your answer as a second order polynomial.

  3. Karhunen-Loeve Expansion Let the covariance function of a wide-sense stationary process be
             {  1 - |τ | for |τ| ≤ 1
KX (τ) =
              0     otherwise.

    Find the eigenfunctions and eigenvalues associated with the Karhunen-Loeve expansion of Xt over (0,T) with T < 1.

  4. The following signals are used to communicate one of two equally likely messages over a channel perturbed by a zero mean, white Gaussian random process, Nt, with spectral height N0-
 2,
           {                         {  √ --
          1  0 ≤ t < T                2  0 ≤ t < T2-
s0(t) =    0  else          s1(t) =     0   else

    1. Use the Gram-Schmidt procedure to find orthonormal functions Ψ0(t) and Ψ1(t) to represent s0(t) and s1(t).
    2. Sketch the signals si(t) and the basis functions Ψi(t).
    3. Express s0(t) and s1(t) in terms of the basis functions Ψi(t).
    4. Define the random variables
            ∫
        T
Xij =     sj(t)(si(t) + n (t))dt,i,j = 0,1.
       0

      Find the joint density function of Xi0 and Xi1 for i = 0 and i = 1.

    5. Define the random variables
           ∫  T

Yij =  0 Ψj (t)(si(t) + n (t))dt,i,j = 0,1.

      Find the joint density function of Y i0 and Y i1 for i = 0 and i = 1.