Presentation at IEEE WCNC 2018 on simple asymptotic bounds on channel estimation and prediction. This work is the presentation of the following paper: https://sfx.aub.aau.dk/sfxaub?sid=pureportal&doi=10.1109/WCNC.2018.8377005.

1. Asymptotic Performance Bound on Estimation
and Prediction of Mobile MIMO-OFDM Wireless
Channels
April 16, 2018
Ramoni Adeogun,PhD
Email: ra@es.aau.dk
Department of Electronic Systems
Aalborg University
Denmark

2. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
1Contributions
Contributions
Derive simple, easily interpretable expressions for the lower
bounds on the performance of channel estimation, interpolation
and prediction.
• Eliminate dependence of performance bound on actual channel
parameters.
• Provide useful insights into the effects of system design
parameters the error performance.
Frequency
Time

3. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
2Channel Model
Ray-based channel model for wideband channels
H(p, q) =
Z
z=1
αz ar(µr
z )aT
t (µt
z )ej(pνz −qηz )
νz = ∆tωz and ηz = 2π∆fτz are the normalized Doppler
frequency and normalized delay.
Array response vector for a ULA
ar(µr
z ) = [1 e−jµrz e−j2µrz · · · e−j(N−1)µrz ]T
; µr
z = 2πδr sin θz
Channel between each transmit-receive antenna pair
h(n, m, p, q) =
Z
z=1
αz ej(pνz −(n−1)µrz −(m−1)µtz −qηz )
Estimated channel equals true channel plus noise
ˆh(n, m, p, q) = h(n, m, p, q) + w(n, m, p, q)

4. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
3Fisher Information and CRB
Problem: estimate deterministic unknown θ from observation y
given statistical model p(y|θ)
The Fisher Information Matrix (FIM)
J(θ) = Eθ[ θ log p(y|θ) T
θ log p(y|θ)]
measures the amount of information that y carries about θ
FIM relates to estimation error covariance via
E[(θ − ˆθ)(θ − ˆθ)T
] ≥ J−1
(θ)
Estimation error on y(θ) relates to FIM via
E[(y − ˆy)(y − ˆy)T
= θyJ−1
(θ) T
θ y]

5. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
4Prediction Error Bounda
aLarsen, Swindlehurst, and Svantesson, “A Performance Bound for MIMO-OFDM
Channel Estimation and Prediction”.
Vectorized form of MIMO channel
h =
Z
z=1
αz (ar(µr
z ) ⊗ at(µt
z ))ej(pνz −qηz )
Mean square error bound computed via
MSEB(p, q) = Tr
∂h
∂Θ
[J(Θ)]−1 ∂h
∂Θ
H
J(Θ) computed via
J =
2
σ2
(PH
5 P5) (PH
4 P4) (PH
3 P3) (PH
2 P2) (PH
1 P1)
P5 = [α
T
α
T
α
T
α
T
1
T
j1
T
] P4 = [Dr Ar Ar Ar Ar Ar]
P3 = [At Dt At At At At] P2 = [Ad Ad Dd Ad Ad Ad]
P1 = [Aτ Aτ Aτ Dτ Aτ Aτ ]
Require averaging over channel realizations.
Computational complexity increases with system size
Expression not readily interpretable.

6. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
5Asymptotic Performance Bound
Channel between each antenna pair:
h(n, m, p, q) =
Z
z=1
αz ej(pνz −(n−1)µrz −(m−1)µtz −qηz )
Parameters: Θ = [θ1, θ2, · · · , θZ ]
θz = [R(αz ) I(αz ) µr
z µt
z νz ηz ]
Mean square error bound
MSEB(p, q) =
N
n=1
M
m=1
∂h
∂Θ
[J(Θ)]−1 ∂h
∂Θ
H
∂h
∂Θ
=
∂h
∂θ1
∂h
∂θ2
· · ·
∂h
∂θZ
With Gaussian noise assumption:
[J(Θ)]ij =
2
σ2
R


Q−1
q=0
P−1
p=0
N
n=1
M
m=1
∂h
∂Θi
∂h
∂Θj
H



7. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
6Asymptotic Performance Bound (2)
FIM submatrix for zth path: [J(θz)] = NMPQ
σ2 K
K =














2 0 0 0 0 0
0 2 0 0 0 0
0 0 2N2
3
NM
2
−
NPUt
2
NQUf
2
0 0 NM
2
2M2
3
−
MPUt
2
MQUf
2
0 0 −
NPUt
2
−
MPUt
2
2P2U2
t
3
−
QPUt Uf
2
0 0
NQUf
2
MQUf
2
−
QPUt Uf
2
2Q2U2
f
3














Assuming uncorrelated scattering, the FIM has a block diagonal
structure [J(Θ)] = blkdiag[J(θ1) J(θ2) · · · J(θZ )]
Asymptotic Mean Squared Error (AMSE):
AMSEB(p, q) =
Z2σ2
13PQ

44 −
36p
PUt
+
60p2
P2U2
t
−
36q
QUf
+
60q2
Q2U2
f
−
36qp
P2U2
t
Q2U2
f


Based on the assumption of normally distributed complex
amplitudes, for a Z-path channel E[||H||2
F ] = NMZ and ANMSEB
becomes
ANMSEB(p, q) =
Zσ2
13NMPQ

44 −
36p
PUt
+
60p2
P2U2
t
−
36q
QUf
+
60q2
Q2U2
f
−
36pq
QUf PUt



8. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
7Asymptotic Performance Bound (3): Inter-
pretations
ANMSEB(p, q) =
Zσ2
13NMPQ



44 −
36p
PUt
+
60p2
P2U2
t
−
36q
QUf
+
60q2
Q2U2
f
−
36pq
QUf PUt



 (1)
The subcarriers near the edge of the frequency band are less
predictable than those near the centre.
The NMSE grows linearly with an increasing noise variance σ2
and number of propagation paths Z.
The NMSE decreases with increasing number of antennas at
either or both ends of the link.
The contribution to the NMSE from the Doppler frequency
estimation and delay estimation lead to a quadratic increase with
prediction horizon and with frequency.
The contributions from the cross correlation of error terms lead to
the negative linear terms, thus reducing the ANMSEB.

9. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
8Simulation Results
Parameters B Nsc Q P N M αz θ
r/tz
Value 20 MHz 2048 64 100 2 2 CN (0, 1) U[−π, π)
Results:
−20 −10 0 10 20
−22
−21
−20
−19
−18
−17
−16
−15
−14
Prediction Horizon (λ)
RNMSE(dB)
BOUND
ASYMPTOTIC BOUND
SNR = 0 dB
SNR = 5 dB
0 200 400 600 800 1000
−35
−30
−25
−20
−15
Number of Time Samples
RNMSE(dB)
BOUND
ASYMPTOTIC BOUND
2 × 2
4 × 4
6 × 6
0 5 10 15 20 25 30
−24
−22
−20
−18
−16
−14
Number of Paths
RNMSE(dB)
BOUND
ASYMPTOTIC BOUND
4 × 4
2 × 2

10. Ramoni Adeogun,PhD Email: ra@es.aau.dk | Asymptotic Performance Bound on Estimation and Prediction of Mobile MIMO-OFDM Wireless Channels
9Conclusion
Derived simple closed-form expression for bound on estimation,
interpolation and prediction of wideband MIMO channels.
Bound relates to system parameters but not actual multipath
parameters.
• Linearly increasing with noise and number of paths
• Linearly decreasing with number of antennas
• Quadratic increase with delay and Doppler estimation errors
• Improved estimation and prediction is achievable with joint
estimation.
• Subcarriers at the center are more predictable than those at the
center of the frequency band.

11. Thanks for your attention!