EE 5111: Estimation Theory Mini project 1 Solved

EE 5111: Estimation Theory

Mini Project 1
1        Problem
Consider the following OFDM-like system model:

                                                                 y = XFh + n,                                                                                           (1)

where y ∈ C512 are the set of observations, X is a 512 dimensional diagonal matrix with known symbols, h is the L tap time domain channel vector, F is the 512×L matrix performing IDFT1 and n is complex Gaussian noise with variance σ2.

For the following set of experiments, generate a set of random bits and modulate them as QPSK symbols to generate2 X. h is a multipath Rayleigh fading channel vector with an exponentially decaying power-delay profile p where p[k] = e−λ(k−1),k = 1,2...L . That is, each component of h will be ], where . Here, λ

is the decay factor (and choose λ = 0.2 for your simulations). Now, perform the following experiments on the described problem set up.

1.   Estimate h using least squares method of estimation with L = 32.[1]

2.   Now, suppose that h is sparse with just 6 non zero taps. Assuming that you know the non zero locations, estimate h using Least squares with the sparsity information.

3.   Next, introduce guard band of 180 symbols on either side[2], i.e. now we have reduced number of observations. For this case:

a    Repeat (1),(2) for the above set up.

b   Apply regularization and redo least squares. Use various values of α for regularization with αI and compare the estimation results.

 

4.   Perform least squares estimation on h with the following linear constraints :

h[1] = h[2] h[3] = h[4] h[5] = h[6]

For each of the above experiments, you have to compare E[hˆ] and h, theoretical and simulated MSE of estimation, all averaged over 10000 random trials. (Generate different instances of X and n for each trial.) Repeat the experiments for σ2 = {0.1,0.01} for each case. Plot hˆ and h for one trial in each of the above cases.

5.   Next, for the scenarios in question 2 and 3, compare the results with the estimates you get from the following steps :

– Step 1 :



Algorithm 1: To find the non-zero locations of the sparse vector h (support estimate).



Input: Observation y, matrix A = XF, sparsity ko = 6

Initialize  y for k ← 1 to k0 do

Identify the next column as tk = argmax|AHj rk−1|

j

Expand the current support as 

Update residual: rk = [I512 − Pk]y where P .

Increment k → k + 1 end

Output: Support estimate Sˆ = Sompk



– Step 2 : Now that you know the non-zero locations of h, estimate h using least squares.

*In the algorithm Aj is the jth column of matrix A, AS denotes the sub-matrix of A formed using the columns indexed by S and A† = (AHA)−1AH is the Moore-Penrose pseudo inverse of A. Also, IN is the N dimensional identity matrix.

2        Submission guidelines
You need to submit the solutions for this problem no later than March 1, 2021 (11:59 pm). Upload a compressed file containing the program/programs your team has written for the mini project in Moodle. The viva for each team will be conducted jointly on a date and time convenient for all the members.

2


 
[1] Note that you are dealing with complex data now and hence the least squares estimate for the model y = Xb shall now be bˆ = (XHX)−1XHy
[2] Suppress to zero the first and last 180 symbols in X

1