Description

1           Problem

Consider the following OFDM system model:

y = XFh + n,                                                                                (1)

where y ∈ C⁵¹² is 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:
   • Repeat (1),(2) for the above set up.
   • 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 k_o = 6

Initialize y for k ← 1 to k₀ do

Identify the next column as t_k = argmax|A^H_j r^k−1|

j

Expand the current support as

Update residual: y^k = [I₅₁₂ − P_k]y where P.

Increment k → k + 1 end

Output: Support estimate S^ˆ = ^S_omp^k

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

*In the algorithm A_j is the j^th column of matrix A, A_S denotes the sub-matrix of A formed using the columns indexed by S and A^† = (A^HA)⁻¹A^H is the Moore-Penrose pseudo inverse of A. Also, I_N is the N dimensional identity matrix.

[1] Note that you are dealing with complex data now and hence the least squares estimate for the model y = Xb shall now be b^ˆ = (X^HX)⁻¹X^Hy

[2] Suppress to zero the first and last 180 symbols in X

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