This document summarizes a study on multipath channel delay estimation using subspace fitting. It presents a signal model that characterizes the multipath channel using clusters of components. It describes the challenges of delay estimation when components are dense and unresolved. It then analyzes the effects of unresolved components on delay estimation bias using approximations of the channel vector and subspace perturbations. Numerical simulations examine delay estimation root mean square error under different scenarios where component delays and powers are varied. The results show increasing error as delays become closer and powers lower relative to the line-of-sight component.

1. Analysis of multipath channel delay estimation
using subspace fitting
Tarik Kazaza
Collaborators: Jac Rommeb
, Gerard J.M. Janssena
and Alle-Jan van der Veena
Email: t.kazaz@tudelft.nl
a
Circuits and Systems, Delft University of Technology, The Netherlands
b
Holst Centre - IMEC-NL, Eindhoven, The Netherlands
Asilomar 2020, Virtual Conference

2. Localization is needed the most in urban and indoor scenarios
Urban canyons Airports and shopping malls
The propagation in these environments is harsh and characterized by a mixture
of specular and dense multipath components.
Tarik Kazaz 1 / 19

3. The problem of channel estimation for ranging
Tx
Rx
h(t)
t
𝛼1
𝜏1 𝜏2 𝜏3 𝜏4
𝛼2
𝛼3
𝛼4
h(t)
What are the goals of channel estimation for ranging?
resolve multipath components (MPCs) present in the channel
detect the line-of-sight (LOS) component
estimate delay of the LOS component
Tarik Kazaz 2 / 19

4. Challenges for delay estimation in the presence of dense MPCs
h(t)
t𝜏1,1 𝜏2,1 𝜏 𝑃,1
𝛼1,1
𝛼1,2
𝛼1,3
t𝜏1,1 𝜏2,1 𝜏 𝑃,1
𝛼1,1
𝛼1,2
𝛼1,3
h(t)
gRx(t)
𝑒−𝑗𝜔𝑡
We observe the continuous-time phenomenon of signal propagation using signals
and transceivers with limited bandwidth. This results in
limited resolution in the delay domain
missed detection of dense MPCs
Tarik Kazaz 3 / 19

5. Challenges for delay estimation in the presence of dense MPCs
h(t)
t𝜏1,1 𝜏2,1 𝜏 𝑃,1
𝛼1,1
𝛼1,2
𝛼1,3
t𝜏1,1 𝜏2,1 𝜏 𝑃,1
𝛼1,1
𝛼1,2
𝛼1,3
h(t)
gRx(t)
𝑒−𝑗𝜔𝑡
Our objective in this work is to answer following question
What are the effects of the dense, i.e., unresolved, MPCs on the bias of the delay
estimation using multiband subspace fittinga
?
aT. Kazaz, G.J.M. Janssen, A.J. van der Veen. Time Delay Estimation from
Multiband Radio Channel Samples in Nonuniform Noise. 2019 53nd Asilomar
Conference on Signals, Systems, and Computers.
Tarik Kazaz 4 / 19

6. Signal model
h(t)
t
𝜏1,1 𝜏2,1 𝜏 𝑃,1
𝛼1,1
𝛼1,2
𝛼1,3
We use the extended Saleh-Valenzuela model to characterize the clustered multipath
channel
h(t) =
P
p=1
Kp
k=1
αp,kδ(t − τp,k) , H(ω) =
P
p=1
Kp
k=1
αp,ke−jωτp,k
, (1)
- P is the number of clusters and Kp is the number of MPCs in the pth cluster
- αp,k ∈ R is the gain of the kth MPC in the pth cluster
- τp,k ∈ R+ is the time-delay of the kth MPC in the pth cluster.
Tarik Kazaz 5 / 19

7. Data model
We assume that the receiver estimates frequency response hi in
i = 0, . . . , L − 1, bands using OFDM signals with N pilot subcarriers.
The multiband channel vector h = [hT
0 , . . . , hT
L−1]T
∈ CNL
satisfies the
model
h = A(φ)α + q :=





M
MΘ1
...
MΘL−1





α +





q0
q1
...
qL−1





, (2)
- A(φ) = [a(Φ1,1), . . . , a(ΦP,K)] ∈ CNL×K
- M ∈ CN×K
is a Vandermonde matrix
M =





1 1 · · · 1
Φ1,1 Φ1,2 · · · ΦP,KP
...
...
...
...
ΦN−1
1,1 ΦN−1
1,2 · · · ΦN−1
P,KP





, (3)
- Φp,k = e−jφp,k
, φp,k = 2πωsτp,k are subcarrier dependent phase shifts
- Θi = diag([Φni
1,1, . . . , Φni
P,K]) ∈ CK×K
are band dependent phase shift
- α collects the gains of MPCs, and q is zero-mean Gaussian noise vector
Tarik Kazaz 6 / 19

8. Multiband delay estimation
h = A(φ)α + q :=





M
MΘ1
...
MΘL−1





α +





q0
q1
...
qL−1





, (4)
A(φ) has multiple shift-invariance structures, and φ can be estimated using
a subspace fitting by solving the following optimization problem
ˆφ = argmin
φ
{J(φ)} = argmin
φ
tr{P⊥
A(φ) ˆUEW ˆUH
E } , (5)
- P⊥
A(φ) = I − PA(φ)
- PA(φ) is the projection matrix onto the column space
- ˆU is the estimated basis that spans column space
- W is weighting matrix
Tarik Kazaz 7 / 19

9. Performance analysis: Assumptions
MPCs within a single cluster have similar delays, i.e., τp,k ≈ τp,1 + ∆τp,k,
where ∆τp,k = τp,k − τp,1, and it is small ∀k.
The steering vectors of the same cluster can be approximated with first-order
Taylor series expansion as
a(φp,k) ≈ a(φp,1) + ∆φp,kd(φp,1) ,
- ∆φp,k = φp,k − φp,1 and d(φp,1) = ∂a(φ)
∂φ |φ=φp,1
Then the multiband channel vector can be approximated as
h ≈ [A( ˜φ) + Ddiag(γ)] ˜α + q , (6)
- A( ˜φ) = [a(φ1,1), a(φ2,1), . . . , a(φP,1)] ∈ CNL×P
- D = [d(φ1,1), . . . , d(φP,1)] ∈ CNL×P
- the elements of vectors ˜α = [α1, . . . , αP ]T
and γ = [γ1, . . . , γP ]T
are
αp =
Kp
k=1
αp,k, and γp =
Kp
k=2 αp,k∆φp,k
αp
.
Tarik Kazaz 8 / 19

10. Performance analysis: First-order subspace perturbations
To form the cost function J(φ) in (5), the first step is to estimate ˆU from
R = E{hhH
}.
R = A( ˜φ)R˜αAH
( ˜φ) + E + σ2
nI , (7)
- R˜α = E{ ˜α ˜αH
} and σ2
n is the noise power
- E = A( ˜φ)diag(e)DH
+ Ddiag(e)AH
( ˜φ) is the perturbation matrix
- e = [e1, . . . , eP ]T
, where ep =
Kp
k=2 σ2
p,k∆φp,k
Further assumptions
The second order terms of ∆φp,k are small and thus can be ignored
The scattering is wide-sense stationary and uncorrelated (WSSUS) and under
these conditions it can be shown that
- R˜α = diag(σ˜α), σ˜α = [σ2
1, . . . , σ2
P ]T
,
- σ2
p =
Kp
k=1 σ2
p,k, ∀p and σ2
p,k = |αp,k|2
.
Tarik Kazaz 9 / 19

11. Performance analysis: First-order subspace perturbations
The basis of the column space can be estimated from R by using eigenvalue
decomposition
When unresolved MPCs are present, the perturbations E perturb the
estimated signal subspace, and we can write
UE = Us + Ue . (8)
- Us is the estimated signal subspace without perturbations
- Ue is the perturbation matrix
The first-order Taylor series expansion of the columns of the perturbed
subspace
uE,i ≈ us,i + ue,i , (9)
ue,i =
P
p=1,i=p
ρi,pus,p +
NL−P
m=1
βi,mun,m ,
ρi,p =
uH
s,pEus,i
λi − λp
and , βi,m =
uH
n,mEus,i
λi
.
- λp, p = 1, . . . , P, are the estimated eigenvalues
Tarik Kazaz 10 / 19

12. Performance analysis: Algorithm
The first order approximation of the cost function J(φ) around the true value
of parameters φ0 is
0 = J (φ0) + J (φ0)( ˆφ − φ0) , (10)
- J (φ0) = ∂J(φ)
∂φ |φ=φ0 is gradient of J(φ)
- J (φ0) = lim∆φ→0 J (φ)|φ=φ0
, and J (φ) is the Hessian of J(φ)1
From (10), the first-order error is
| ˆφ − φ0 |≈ (J (φ0))−1
J (φ0) , (11)
- J (φ) = 2Re diag(A† ˆUEW ˆUH
E P⊥
AD)
- J (φ) = −2Re DH
P⊥
AD A† ˆUEW ˆUH
E (A†
)H
T
1M. Viberg, B. Ottersen, T. Kailath. Detection and estimation in sensor arrays using
weighted subspace fitting. 1991 IEEE Transactions on Signal Processing.
Tarik Kazaz 11 / 19

13. Performance analysis: Algorithm
The first-order error is
| ˆφ − φ0 |≈ (J (φ0))−1
J (φ0) , (12)
Using (9) the expressions for the gradient and Hessian of J(φ) can be
simplified to
J (φ) ≈2Re diag[(AH
A)−1
diag(σ−1
˜α e)DH
P⊥
AD] ,
J (φ) = −2Re DH
P⊥
AD (AH
A)−1 T
.
(13)
- σ−1
˜α = [σ−2
1 , . . . , σ−2
P ]T
, σ2
p =
Kp
k=1 σ2
p,k and σ2
p,k = |αp,k|2
- e = [e1, . . . , eP ]T
, ep =
Kp
k=2 σ2
p,k∆φp,k
From (12) and (13) ⇒ the bias in delay estimation of the LOS is proportional
to the product of the power of the interfering MPCs and their delay
differences compared to the LOS.
At the same time, this bias is inversely proportional to the total power of all
MPCs in the cluster that contains the LOS.
Tarik Kazaz 12 / 19

14. Numerical Simulations
- Receiver estimates hi in L = 4 bands, by using the OFDM probing signal with
N = 12 subcarriers and bandwidth B = 12 MHz.
- The number of snapshots is set to 32, and the band’s central frequencies are
{2410, 2450, 2480, 2560} MHz, respectively.
- The average RMSE is computed using 103
independent Monte-Carlo trials.
Scenarios:
Scenario 1.: Three clusters of MPCs, i.e., P = 3, the clusters have {2, 3, 1}
underlying MPCs with powers set to
{σ2
1,1 = 1, σ2
1,2 = 0.5, σ2
2,1 = 0.85, σ2
2,2 = 0.55, σ2
2,3 = 0.35, σ2
3,1 = 0.55}
- The delay of the LOS component and MPCs in the second and third cluster
are kept fixed and set to
{τ1,1 = 5, τ2,1 = 33, τ2,2 = 33.5, τ2,3 = 34, τ3,1 = 95} ns
- The delay for the MPC in the first cluster τ1,2 is changing during trials and
takes values of {6, 6.5, 7, 8} ns
- SNR is changing during trials
Tarik Kazaz 13 / 19

15. Numerical Simulations
Scenarios:
Scenario 2.: In the first cluster, K1 = 3, with powers of MPCs set to
{σ2
1,1 = 1, σ2
1,2 = 0.5, σ2
1,3 = 0.37}.
- The delays of LOS component and MPCs in the second and third cluster
are kept fixed and are the same as in the previous scenario
- The delay of the 2nd MPC in the first cluster is set to τ1,3 = 8 ns
- The delay of the 1st MPC τ1,2 in the first cluster is changing during trails
and takes values of {5.5, 6, 6.5, 7} ns
Scenario 3.: The power of the second MPC in the first cluster is changing
relative to the LOS component’s power.
- The SNR is set to 10 dB
- The delay of the second MPC is set to τ1,2 = 6 ns
- Other parameters are the same as in the first scenario
Tarik Kazaz 14 / 19

16. Results: Scenario 1.
-10 -5 0 5 10 15 20 25
10 -3
10 -2
10 -1
10 0
10 1
Tarik Kazaz 15 / 19

17. Results: Scenario 2.
-10 -5 0 5 10 15 20 25
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Tarik Kazaz 16 / 19

18. Results: Scenario 3.
0 0.5 1 1.5
10 -1
10 0
Tarik Kazaz 17 / 19

19. Conclusions
The bias in delay estimation of the LOS is proportional to the sum of
products of the powers of the LOS cluster’s MPCs with their delay
differences compared to the LOS.
The bias in delay estimation of the LOS is inversely proportional to the
total power of components in the first cluster, i.e., the sum of MPCs
and LOS powers.
The derived expression for the bias sets a tight bound on the expected
performance of the delay estimation algorithm
In the low-SNR regime, the finite sampling and noise effects are
dominant compared to errors introduced by unresolved MPCs.
When the delay between LOS and interfering MPCs increases, but MPCs stay
unresolvable, the bias will increase.
Tarik Kazaz 18 / 19

20. Thank You!
Tarik Kazaz 19 / 19