18 November, 2021
1 Homework #1, AA2021-2022: MIMO System Identification and De-
convolution
Multiple Input Multiple Output (MIMO) are multichannel systems that model various applications. The Homework’s
objective is to get practice and evaluate the MSE in MIMO estimation and deconvolution by using Matlab. The
general M × N MIMO system is described as a set of MN filters that combine the input signals x1[k], . . . , xN [k]
onto the ith signal as output
yi[k] =
N∑
ℓ=0
hi,ℓ[k] ∗ xℓ[k] + wi[k],
this can be represented as  y1[k]...
yM [k]
 =
 h11[k] . . . h1N [k]... . . . ...
hM1[k] . . . hMN [k]
 ∗
x1[k]...
xN [k]
+
 w1[k]...
wM [k]

or more compactly for the kth sample
y[k] = H[k] ∗ x[k] +w[k]
where H[0], . . . ,H[K − 1] is the length-K MIMO filter response.
The MIMO system considered in this homework exemplifies a digital communication system. The input samples
x[0], ...,x[P − 1] are of length P , where the first Q samples are called pilot or training samples, and they are known
to both the transmitter and the receiver sides. The remaining P − Q samples represent the unknown transmitted
information. In this context, the student is requested to
(a) estimate the MIMO channel Hˆ[k] using the known Q pilots;
(b) estimate the unknown signal x[Q + 1], ...,x[P ] through deconvolution using the previously estimated MIMO
channel Hˆ[k]
The MIMO filter response is modelled as
hi,j [k] = α
|i−j| × βk for k = 0, 1, 2, ...
where 0 ≤ α ≤ 1 is the coupling effect among MIMO lines (α ≈ 0 denotes weak coupling, and α ≈ 1 denotes
strong coupling), and the second term account for the time decaying. The noise is Gaussian, temporally white, i.e.,
E[w[k]w[ℓ]] = Cδ[ℓ− k], and mutually correlated among paired lines, i.e.,
w[k] ∼ N (0,C) where [C]i,j =
{
ρ i ̸= j
1 i = j
where |ρ| ≤ 1 is the correlation coefficient. The input signal x[k] contains both pilot and information samples, and
can be generated from uncorrelated and white random processes:
x[k] ∼ N (0, σ2xI).
1.1 Noise generation
Generate using Matlab code a correlated noise for any arbitrary choice of ρ and compute the sample covariance
Cˆ from a set of L samples {w[k]}Lk=1 and compare with the true covariance C for L = linespace(10, 200, 10) and
ρ = linespace(0, 0.99, 10) to evaluate how close the sample covariance Cˆ is to C after defining a proper metric that
is represented graphically vs ρ for the values of L.
1
1.2 MIMO estimation
Let us consider an M = N = 4 MIMO filter response. Estimate the 4× 4 MIMO filter response under the following
settings:
(i) consider a memoryless filter, i.e., K = 1. After defining the MLE of the MIMO filter and the corresponding
filter estimate Hˆ, evaluate numerically and analytically the MSE of the MIMO filter vs the signal to noise
ration (SNR) defined as SNR = σ2x/σ
2
w = σ
2
x, and plot (in dB-dB
1 scale) the MSE vs SNR = −10 : 2 : 30 dB
for Q = linespace(1, 50, 5) and α = linespace(0, .99, 5). Compare all results with the CRB.
(ii) consider a memory filter, i.e., K = 4. Repeat the same exercise above in point (i) for memory causal filters
with different coefficients βk = {0.9k, 0.5k, 0.1k} for k = [0, 1, 2, 3]T . After defining the problem and its Matlab
solution, the student must include in the final Hw-Report a clear definition of the theoretical structure used to
solve the problem.
Hint: it is possible to reduce the above problem to a simple linear algebraic filtering by a careful design of
convolution-matrices.
1.3 MIMO deconvolution
Let us consider a MIMO filter with no-memory (i.e., hi,j = α
|i−j|) as in the exercise in Sec.1.2(i), with an input
signal x[k] of length P . In a communication system, the first Q samples are known to both the transmitter and
receiver and are used to estimate the MIMO filter response, while the remaining are unknown to the receiver and
represent the transmitted information. For a fixed P , the choice of the number Q is important and must be selected
carefully as tradeoff between accurate MIMO filter response estimation (higher values of Q) and low overhead (lower
values of Q). Therefore, the goal is to estimate the MIMO filter Hˆ using the first Q pilot samples x[k] and use it
to estimate the unknown information samples xˆ[k] from y[k], with k = {Q + 1, Q + 2, ..., P} using both MLE and
MMSE criteria. Evaluate numerically and analytically the MSE vs SNR = σ2x and plot (in dB-dB scale) the MSE
vs SNR = −10 : 2 : 30 for P = 200, Q = linespace(1, P, 20), ρ = .1, and α = .5. Define a proper metric that capture
the tradeoff in the selection of the parameter Q and represent it graphically vs SNR, and discuss the results.
1The decibel scale is a logarithmic scale that can better capture exponentially growing/decreasing values: PdB = 10log(Plinear). To
obtain a dB-dB scale plot on Matlab, you can use the function plot(10log10(SNR), 10log10(MSE))
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