Page 28 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 4 – AI and machine learning solutions in 5G and future networks

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 4

where the dictionary matrices A ̃ R ∈ ℂ × and A ∈ 3. MLGS‑SBL
̃
T
ℂ × contain the Rx and Tx array steering vectors eval‐
In this section, we propose a model‐based approach using
uated on a grid of size for the AoA and a grid of size

for the AoD, respectively. When and are cho‐ the framework of Compressed Sensing (CS), to estimate

sen properly, i.e., much greater than , ∈ ℂ × be‐ the mmWave channel given the received pilot measure‐

ments and the frequency‐ lat transmit vector. We inte‐
comes a sparse matrix containing the channel path gains
grate a fast greedy search procedure and a high perform‐
on the locations that match with the actual AoDs and
ing statistical inference method to estimate the channel.
AoAs. We represent (7) in the frequency domain as
The algorithm consists of the following steps:
̃

̃ ∗
H[ ] ≈ A [ ]A , (8)
R
T
1. Preconditioning
for = 0, … , − 1, and
2. Multi‐level greedy search for dictionary learning
−1

[ ] = ∑ exp (− 2 ) . (9)
3. Noise variance estimation
=0
̃
̃
Note that the dictionary matrices A and A are com‐ 4. Sparse Bayesian learning for channel estimation
T
R
mon to all the subcarriers due to the frequency‐ lat ar‐

ray response vectors. Hence, the sparse matrices [ ] 5. Channel de‐noising
for = 1, … , have the non‐zero elements at the same
We provide a detailed description of each step below.
indices. This means that they share a common sparsity
pattern [4].
3.1 Preconditioning
Now, we vectorize (8) to get
Sparse signal recovery using greedy algorithms, such as
̄
̃
̃
(H[ ]) = (A ⊗ A ) ( [ ]). (10) OMP, are likely to choose the correct support set when
v
T R
the noise covariance matrix is diagonal. In our mmWave
̄
̃
We de ine = A ⊗ A ̃ ∈ ℂ × and h [ ] = channel estimation problem, RF combining by W at the
v
tr
T R front end of the receiver results in correlated noise, which
( [ ]) ∈ ℂ , and substitute (H[ ]) in (2) to
v
needs to be whitened using a preconditioning ilter [4].
get
( )
(y ( ) [ ]) = ( ) h[ ] + n [ ], (11) The scaled noise covariance matrix before whitening is
∗
∗ [n [ ]n [ ]]
( )
where n [ ] = W ( ) n ( ) [ ]. By concatenating the RF C = 2
w
tr
combined signals of training frames, we get ∗ ∗
(1)
=blkdiag{W (1) W , … , W ( ) W ( ) }. (14)
tr
tr
tr
tr
(1)
y (1) [ ] (1) n [ ]
⎡ ⎤ ⎡ ⎤ v ⎡ ⎤
⎢ ⋮ ⎥ = ⎢ ⋮ ⎥ h [ ] + ⎢ ⋮ ⎥ . We get the above by noting that
( )
( )
( )
⎣
⎣ y
[ ] ⎦
⎦
[ ] ⎦
⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟ ( ) ( ) ∗ ( ) ∗ ( )
⎣ n
2
y[ ] n [ ] [n [ ]n [ ]] = W tr W [ − ]. (15)
tr
(12)
Now, by stacking the received signals of subcarriers, we We perform a Cholesky decomposition of C to obtain
w
∗
get the inal system equation C w = D D , where D w ∈ ℂ × is upper triangu‐
w
w
lar. Let us de ine D −∗ to denote the inverse of D . Now,
∗
w
w
Y = [y[1] … y[ ]] we multiply the RF combined received signal (12) by D −∗
w
v
= [h [1] … h [ ]] + [n [1] … n [ ]] to obtain the noise‐whitened received signal:
v

v
= H + N . (13)

−∗
−∗
−∗
v
y [ ] = D y[ ] = D h [ ] + D n [ ]
w w w w
−∗
v
= h [ ] + D n [ ], (16)

w
w
Our goal is to estimate H[ ], for = 0, … , − 1, given Y
−∗
and . As the AoDs and AoAs are the same for all the sub‐ where = D ∈ ℂ × . Concatenating the
w
w
carriers, H ∈ ℂ × has a joint row sparse structure, noise‐whitened received signals of all the subcarriers,
v

i.e., the support set of each column of H are the same. we get
Also, we do not have the knowledge of the sparsifying dic‐ v
tionary and the noise variance, which makes the chan‐ Y = [y [1] … y [ ]] = H + N , (17)
w
w
w
w
w
nel estimation problem more challenging. In the follow‐
−∗
ing sections, we present three different solutions to this where Y w ∈ ℂ × , w = D ∈ ℂ × , and
w
channel estimation problem. N w = D −∗ [n[1] … n[ ]] ∈ ℂ × . Thus, we need
w
v
to estimate the row sparse matrix H given Y and .
w
w
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