Page 35 - 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 we have used ⟨ (A), (B)⟩ = (A B) and the the path is suf iciently signi icant to be included in the
cyclic shift property of the trace. model, or whether to stop the OMP iterations. The value
of is optimized using the dataset information as de‐
The above maximization problem can be solved by scribed later in Section 6.3.
searching over a discretized set of values of the path pa‐
rameters, ( , , ). We start by considering a small set 5.3 Re inement
ℓ
ℓ
ℓ
of values for the path parameters equally spaced in their
domains. We choose a resolution of 4 values for ℓ Once a path has been detected, we proceed to re ine the
and of /2 ( /2) values for ( ). Since we are con‐ path components by iterative projections. We do this by
r t ℓ ℓ
sidering a smaller resolution for the angles than that re‐ freezing two of the variables and increasing the resolution
quired for them to cover the entire angular spectrum, we of the third, in an alternating fashion.
substitute each ( ) by a sector beam‐pattern ̂ ( )
R ℓ R ℓ
of width 4 / . The same manipulation applies to the First steps:
r First, we adapt our estimation to handle a
phase angles at the transmitter, and thus we can de ine higher resolution due to the manipulation we did with the
̄
̂ ( , ) = ̂ ( ) ⊗ ̂ ( ). The sector beam‐pattern
R−T ℓ ℓ T ℓ R ℓ angular resolution.
we consider is the same as the one de ined in [31]. With
this de inition, we can extract a coarse version of the path We start by increasing the time resolution by comput‐
∗
̄
parameters ( , , ) by maximizing ing the maximum of | ̂ ∗ R−T ( , ) ( ))| for ixed
ℓ
ℓ
ℓ
F
ℓ
ℓ
w
ℓ
( , ) and with a resolution of 32 equally spaced
ℓ
ℓ
ℓ
∗
= | ̂ ∗ ( , ) ( )|. (51) points.
̄
ℓ , ℓ , ℓ R−T ℓ ℓ w F ℓ
∗
Then, by ixing and using the identity ( ) =
ℓ
5.2 Detection ̄ ∗
( ⊗ ) ( ),we can simplify the expression to
No w want t kno those parameters con‐
∗
sidered path det T w tak into ∣ ̂ ( ) ( ) ̂ ( )∣ (56)
ℓ
ℓ
ℓ
T
R
account null hypothesis of = 0 that all ∗
̄
ℓ
of ar independent whit noise thus with ( ) such that ( ( )) = ( ).
ℓ
ℓ
w
F

̂ ∗ ( , ) ̄ ( ) is also comprised of white noise
∗
R−T ℓ ℓ w F ℓ We then proceed to re ine the angle components with the
Consequently Rayleigh
ℓ , ℓ , ℓ hig of ant For simplicity let as‐
distribution, which has the cumulative distribution
sume that > By increasing the resolution of to
ℓ
funtion r
32 equally spaced points, we do not need to use the
t
2 sec‐tor beam‐pattern manipulation, thus we can simply
( ) = 1 − exp (− ℓ , ℓ , ℓ ) . (52) maximize
2
ℓ , ℓ , ℓ 2
∗
∣ ̂ ( ) ( ) ( )∣ (57)
Since there is a channel contribution only for a small num‐ R ℓ ℓ T ℓ
ber of path parameters, we have that the median of all
over while the other path parameters are ixed.
ℓ
computed values of should be close to that of the
ℓ , ℓ , ℓ Finally, we re ine the expression with respect to the re‐
Rayleigh distribution √2 ln(2). This insight is key to our maining angle. Again, the manipulation is not required,
algorithm. Then, can be approximated as and we can maximize
∗
≃ ( )/√2 ln(2). (53) ∣ ( ) ( ) ( )∣ (58)
ℓ , ℓ , ℓ R ℓ ℓ T ℓ
Inthiscase, thecumulativefunctionofmax canbecom‐ over while the other path parameters are ixed.

ℓ
puted as ( ) = ∏ ( ). Explicitly, it is given by
max
Iteration steps: No that irst st w re‐
(max( )) =
max ℓ , ℓ , ℓ moved the angle uncertainty caused by the sector beam‐
r t
max( ) 2 patt we can proceed to repeat the same steps itera‐
(1 − exp (− ℓ , ℓ , ℓ )) . tively by substituting all sector beam‐patterns ̂ by array
2 2
responses .
(54)
parameters of path hav estimated,
Using this, we can compute a detection threshold corre‐ ( , , ), we use them to reconstruct the path and sub‐
ℓ
ℓ
ℓ
sponding to a con idence level of as tract it from the received pilots, thereby altering the resid‐
ual. Then, the residual is updated, and the next path is ob‐
1
√
( ) − log (1 − ( ) r t ). (55) tained following the same The residual is updated
ℓ , ℓ , ℓ 2
until a stopping condition is reached, as discussed in
We compare the optimal value of obtained by Section 5.2. How to obtain the best stopping condition for
ℓ , ℓ , ℓ
solving (51) with the above threshold to decide whether our algorithm is discussed in Section 6.3.
© International Telecommunication Union, 2021 19