1 Multipath Parameter Estimation from OFDM Signals in Mobile Channels Nick Letzepis, Member, IEEE, Alex Grant, Senior Member, IEEE, Paul Alexander Member, IEEE, David Haley, Member, IEEE Abstract We study multipath parameter estimation from orthogonal frequency division multiplex signals transmitted over doubly dispersive mobile radio channels. We are interested in cases where the trans- mission is long enough to suffer time selectivity, but short enough such that the time variation can be accurately modeled as depending only on per-tap linear phase variations due to Doppler effects. We therefore concentrate on the estimation of the complex gain, delay and Doppler offset of each tap of the multipath channel impulse response. We show that the frequency domain channel coefficients for an entire packet can be expressed as the superimposition of two-dimensional complex sinusoids. The maximum likelihood estimate requires solution of a multidimensional non-linear least squares problem, which is computationally infeasible in practice. We therefore propose a low complexity suboptimal solution based on iterative successive and parallel cancellation. First, initial delay/Doppler estimates are obtained via successive cancellation. These estimates are then refined using an iterative parallel cancellation procedure. We demonstrate via Monte Carlo simulations that the root mean squared error statistics of our estimator are very close to the Cramer-Rao lower bound of a single two-dimensional sinusoid in Gaussian noise. I. I NTRODUCTION In wireless communications, reflection and diffraction of the transmitted radio signal results in the superimposition of multiple complex-scaled and delayed copies of the signal at the receiver. N. Letzepis and A. Grant are with the Institute for Telecommunications Research, University of South Australia, e-mail: [email protected], [email protected]. P. Alexander and D. Haley are with Cohda Wireless Pty. Ltd., e-mail: {paul.alexander,david.haley}@cohdawireless.com.au. This work was supported by Cohda Wireless Pty. Ltd. and the Australian Research Council under grant LP0775036. November 16, 2010 DRAFT arXiv:1011.2809v2 [cs.IT] 15 Nov 2010
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1
Multipath Parameter Estimation from OFDM
Signals in Mobile Channels
Nick Letzepis, Member, IEEE, Alex Grant, Senior Member, IEEE,
Paul Alexander Member, IEEE, David Haley, Member, IEEE
Abstract
We study multipath parameter estimation from orthogonal frequency division multiplex signals
transmitted over doubly dispersive mobile radio channels. We are interested in cases where the trans-
mission is long enough to suffer time selectivity, but short enough such that the time variation can be
accurately modeled as depending only on per-tap linear phase variations due to Doppler effects. We
therefore concentrate on the estimation of the complex gain, delay and Doppler offset of each tap of
the multipath channel impulse response. We show that the frequency domain channel coefficients for
an entire packet can be expressed as the superimposition of two-dimensional complex sinusoids. The
maximum likelihood estimate requires solution of a multidimensional non-linear least squares problem,
which is computationally infeasible in practice. We therefore propose a low complexity suboptimal
solution based on iterative successive and parallel cancellation. First, initial delay/Doppler estimates
are obtained via successive cancellation. These estimates are then refined using an iterative parallel
cancellation procedure. We demonstrate via Monte Carlo simulations that the root mean squared error
statistics of our estimator are very close to the Cramer-Rao lower bound of a single two-dimensional
sinusoid in Gaussian noise.
I. INTRODUCTION
In wireless communications, reflection and diffraction of the transmitted radio signal results in
the superimposition of multiple complex-scaled and delayed copies of the signal at the receiver.
N. Letzepis and A. Grant are with the Institute for Telecommunications Research, University of South Australia, e-mail:
The receiver now performs the matched filter to the transmitted sinusoids (less the cyclic
prefix), i.e.
Yl,k =1√KL
∫ lTd
Tcp+(l−1)Td
y(t)w∗(t− lTd)e−j2π(k−1−bK/2c)(t−Tcp)/T dt
=1
KL
∑l′,k′,p
apXl′,k′e−j2πτp
(k′−1T− K
2T
)e−j2π(k′−k)Tcp/T
×∫ lTd
Tcp+(l−1)Td
w(t− τp − (l′ − 1)Td)w∗(t− (l − 1)Td)e
−j2π(νp+ k−k′
T
)tdt+ Zl,k
=1
KL
∑l′,k′,p
apXl′,k′e−j2πτp
(k′−1T− K
2T
)e−j2π(k′−k)Tcp/T e
−j2π(νp+ k−k′
T
)(l−1)Td
× Aw(τp + (l′ − l)Td, νp +
k − k′
T
)+ Zl,k, (35)
where
Aw(τ, ν) =
∫ Td
Tcp
w(t− τ)w∗(t)e−j2πνt dt, (36)
which resembles the ambiguity function of w(t).2 In practical OFDM systems, usually maxp τp <
Tcp and maxp νp 1/T and the windowing function is usually designed such that
Aw(τp + (l′ − l)Td, νp + k−k′
T
)≈ 0 for k 6= k′ or l 6= l′. Hence we may write,
Yl,k =1
KL
∑p
apXl,ke−j2πτp( k−1
T− K
2T )e−j2πνpTd(l−1)Aw(τp, νp) + Zl,k
=1
KL
∑p
apXl,ke−j2π(k−1)τp/T e−j2π(l−1)νpTd + Zl,k, (37)
2The function Aw(τ, ν) is not quite the ambiguity function of w(t) because of the limits of integration.
November 16, 2010 DRAFT
19
where ap = e−jπKτp/T Aw(τp, νp). With some slight abuse of notation, for the remainder of the
paper for brevity of notation (and without loss of generality) we will replace ap with ap. Defining
Hl,k according to (8), we obtain (7).
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November 16, 2010 DRAFT
FIGURES 21
ν (Hz)
τnsec
−1500 −1000 −500 0 500 1000−500
−400
−300
−200
−100
0
100
200
300
400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) K = 52, L = 90
ν (Hz)
τnsec
−1500 −1000 −500 0 500 1000−500
−400
−300
−200
−100
0
100
200
300
400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b) K = 52, L = 242
Fig. 1. Magnitude contour plot of the ambiguity function (18) of an 802.11a OFDM system with PSK modulation, T = 6.4µsec, Td = 8 µsec
November 16, 2010 DRAFT
FIGURES 22
Algorithm 1: Initial estimation via successive cancellation.
1: E(1) = Y2: for p = 1, . . . , P do3: (τp, νp) = arg maxτ,ν
∣∣∣ψ†(ν)(E(p) X∗
)φ(τ)
∣∣∣24: Contruct R(p) and w(p) using (22) and (23) with τ1, . . . , τp and ν1, . . . , νp.5: a(p) = (R(p))−1w(p)
6: E(p+1) = Y −[Ψ(ν(p))diag(a(p))Φ†(τ (p))
]X
7: end for
November 16, 2010 DRAFT
FIGURES 23
Algorithm 2: Estimate refinement algorithm.
1: τ (0) = τ ,ν(0) = ν and a(0) = a2: for i = 1, . . . , N do3: for p = 1, . . . , P do4: E = Y −
[Ψ(ν
(i−1)p )diag(a
(i−1)p )Φ†(τ
(i−1)p )
]X
5: (τ(i)q , ν
(i)q ) = arg maxτ,ν
∣∣ψ†(ν) (E X∗)φ(τ)∣∣2
6: end for7: a(i) = R−1(τ (i), ν(i))w(τ (i), ν(i))8: end for
November 16, 2010 DRAFT
FIGURES 24
Algorithm 3: 2-D Bisection Algorithm.
1: Initialise (τ(0)min, τ
(0)max) = (τmin, τmax) and (ν
(0)min, ν
(0)max) = (νmin, νmax).
2: for i = 1, . . . , Nbisect do3: ∆τ (i) = (τ
(i−1)max − τ (i−1)
min )/M , ∆ν(i) = (ν(i−1)max − ν(i−1)
min )/N4: Construct τ (i) and ν(i) using (27).5: Υ(i) = Ψ†(ν(i)) [Y X∗] Φ(τ (i))
6: (n, m) = arg maxn,m |Υ(i)n,m|2
7: τ(i)max = τ
(i)m + β∆τ (i), τ (i)
min = τ(i)m − β∆τ (i)
8: ν(i)max = ν
(i)n + β∆ν(i), ν(i)
min = ν(i)n − β∆ν(i)
9: end for10: τ = τ
(I)m , ν = ν
(I)n .
November 16, 2010 DRAFT
FIGURES 25
−10 0 10 20 30 40 5010
−3
10−2
10−1
100
SNR (dB)
Pm
iss
L = 128L = 256L = 512
Fig. 2. Probability of miss detecting all taps of a P = 3 tap multipath channel.
November 16, 2010 DRAFT
FIGURES 26
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
SNR (dB)
στ(n
sec)
(a) L = 128
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
103
SNR (dB)
σν(H
z)
(b) L = 128
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
SNR (dB)
στ(n
sec)
No refinement
20 Iterations
(c) L = 256
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
103
SNR (dB)
σν(H
z)
(d) L = 256
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
SNR (dB)
στ(n
sec)
(e) L = 512
0 5 10 15 20 25 30 35 4010
−2
10−1
100
101
102
103
SNR (dB)
σν(H
z)
(f) L = 512
Fig. 3. Three tap estimation root-mean squared (RMS) error. Dashed lines show the CRLB (26) (single-tap estimation). Solidlines with squares show RMS error performance of Algorithm 1 only. Solid lines with circles show the RMS error performanceafter refinement Algorithm 2 with 20 iterations.