arXiv:0802.4344v1 [cs.IT] 29 Feb 2008
An Improved Scheme for Initial Ranging in
OFDMA-based Networks
Luca Sanguinetti* and Michele Morelli
H. Vincent Poor
Department of Information Engineering
University of Pisa
Pisa, Italy
luca.sanguinetti, michele.morelli@iet.unipi.it
Department of Electrical Engineering
Princeton University
Princeton, NJ USA
poor@princeton.edu
Abstract— An efficient scheme for initial ranging has recently
been proposed by X. Fu et al. in the context of orthogonal
frequency-division multiple-access (OFDMA) networks based on
the IEEE 802.16e-2005 standard. The proposed solution aims at
estimating the power levels and timing offsets of the ranging
subscriber stations (RSSs) without taking into account the effect
of possible carrier frequency offsets (CFOs) between the received
signals and the base station local reference. Motivated by the
above problem, in the present work we design a novel ranging
scheme for OFDMA in which the ranging signals are assumed
to be misaligned both in time and frequency. Our goal is to
estimate the timing errors and CFOs of each active RSS. Specifically, CFO estimation is accomplished by resorting to subspacebased methods while a least-squares approach is employed for
timing recovery. Computer simulations are used to assess the
effectiveness of the proposed solution and to make comparisons
with existing alternatives.
I. I NTRODUCTION
The main impairment of an orthogonal frequency-division
multiple-access (OFDMA) network is represented by its remarkable sensitivity to timing errors and carrier frequency
offsets (CFOs) between the uplink signals and the base station
(BS) local references. For this reason, the IEEE 802.16e-2005
standard for OFDMA-based wireless metropolitan area networks (WMANs) specifies a synchronization procedure called
Initial Ranging (IR) where subscriber stations that intend to
establish a link with the BS can use some dedicated subcarriers
to transmit their specific ranging codes [1]. Once the BS has
revealed the presence of ranging subscriber stations (RSSs), it
has to estimate some fundamental parameters including timing
errors, CFOs and power levels.
Two prominent schemes for initial synchronization and
power control in OFDMA were proposed in [2] and [3]. In
these works, a long pseudo-noise sequence is transmitted by
each RSS over the available ranging subcarriers. Timing recovery is then accomplished on the basis of suitable correlations
computed in the frequency- and time-domain, respectively.
The main drawback of these methods is their sensitivity to
*This work was completed while the author was with Princeton University
and it was supported by the U.S. National Science Foundation under Grants
ANI-03-38807 and CNS-06-25637.
multipath distortion, which destroys orthogonality among the
employed codes and gives rise to multiple access interference
(MAI). Better results are obtained in [4] by using a set
of generalized chirp-like (GCL) sequences, which echibits
increased robustness against the channel selectivity. A different approach to managing the IR process has recently been
proposed in [5]. Here, the pilot streams transmitted by RSSs
are spread in the time-domain over adjacent OFDM blocks
using orthogonal codes. In this way, signals of different RSSs
can be easily separated at the BS as they remain orthogonal
after propagating through the channel. Timing information is
eventually acquired in an iterative fashion by exploiting the
autocorrelation properties of the received samples induced by
the use of the cyclic prefix (CP). Unfortunately, this scheme is
derived under the assumption of perfect frequency alignment
between the received signals and the BS local reference.
Actually, the occurrence of residual CFOs results into a loss
of orthogonality among ranging codes and may lead to severe
degradations of the system performance in terms of misdetection probability and estimation accuracy.
In the present work we propose a novel ranging scheme
for OFDMA systems that is robust to time and frequency
misalignments. The goal is to estimate timing errors and
CFOs of all active RSSs. The number of active codes is
found by resorting to the minimum description length (MDL)
principle [6] while the multiple signal classification (MUSIC)
algorithm [7] is employed to detect which codes are actually
active and to determine their corresponding CFOs. Timing
estimation is eventually achieved through least-squares (LS)
methods. Although the proposed solution allows one to estimate the timing errors of each RSS in a decoupled fashion,
it may involve huge computational burden in applications
characterized by large propagation delays. For this reason,
we also present an alternative scheme derived from ad hocreasoning which results into substantial computational saving.
It is worth noting that timing synchronization in OFDMA
uplink transmissions has received little attention so far. A
well-established way to handle timing errors is to design the
CP length large enough to include both the channel delay
spread and the two-way propagation delay between the BS
and the user station [8]. This leads to a quasi-synchronous
system in which timing errors can be viewed as part of the
channel impulse response (CIR) and are compensated for by
the channel equalizer. Unfortunately, this approach poses an
upper limit to the maximum tolerable propagation delay or,
equivalently, to the maximum distance between the BS and
the subscriber stations [9]. For this reason, its application to
a scenario with large cells (as envisioned in next broadband
wireless networks) is hardly viable. In the latter case, accurate
knowledge of the timing errors is required in order to align
the uplink signals to the BS time scale.
II. S YSTEM
DESCRIPTION AND SIGNAL MODEL
A. System description
The investigated OFDMA network employs N subcarriers
with frequency spacing ∆f and indices in the set J =
{0, 1, . . . , N − 1}. Following [5], we denote R the number
of subchannels reserved for the IR process. Each subchannel
is divided into Q subbands uniformly spaced over the signal
bandwidth at a distance (N/Q)∆f from each other. A given
subband is composed of a set of V adjacent subcarriers. The
subcarrier indices within the qth subband (q = 0, 1, . . . , Q−1)
of the rth ranging subchannel (r = 0, 1, . . . , R − 1) are
(r)
(r)
collected into a set Jq = {iq,ν ; ν = 0, 1, . . . , V − 1} with
entries
rN
qN
+
+ ν.
(1)
i(r)
q,ν =
Q
QR
The rth subchannel is thus composed of subcarriers with
Q−1 (r)
indices taken from J (r) = ∪q=0
Jq . Hence, a total of
NR = QV R ranging subcarriers are available in the system
(r)
with indices in the set JR = ∪R−1
. The remaining
r=0 J
N − NR subcarriers are used for data transmission and are
assigned to data subscriber stations (DSSs) which have already
completed their IR process and are assumed to be perfectly
synchronized to the BS time and frequency scales [5].
We denote by M the number of consecutive OFDM blocks
reserved for IR and assume that each ranging subchannel can
be accessed at most by M −1 RSSs. The latter are separated by
means of specific ranging codes selected in a pseudo-random
fashion from a predefined set {c1 , c2 , . . . , cM−1 }, with ck =
[ck (1), ck (2), . . . , ck (M )]T (the superscript T denotes the
transpose operation). As in [5], we assume that different RSSs
employ different codes. Without loss of generality, in what
follows we concentrate on the rth ranging subchannel and
denote by K (r) ≤ M − 1 the number of simultaneously active
RSSs. Also, to simplify the notation, the subchannel index (r)
is dropped in all subsequent derivations.
The waveform transmitted by the kth RSS (1 ≤ k ≤ K)
propagates through a multipath channel characterized by an
impulse response hk = [hk (0), hk (1), . . . , hk (L − 1)]T of
length L (in sampling periods). At the BS, the received
samples are not synchronized with the local references. We
denote by θk the timing error expressed in sampling periods
while εk is the frequency offset normalized to the subcarrier
spacing. As discussed in [8], subscriber stations that intend
to start the ranging process compute initial frequency and
timing estimates on the basis of a downlink control signal
broadcast by the BS. The estimated parameters are then
employed by each RSS as synchronization references for
the uplink ranging transmission. This means that during IR
the CFOs are only due to Doppler shifts and/or estimation
errors and, in consequence, they are assumed to lie within
a small fraction of the subcarrier spacing. Furthermore, in
order to eliminate interblock interference (IBI), we assume
that during the ranging process the CP length comprises
NG ≥ θmax +L sampling periods, where θmax is the maximum
expected timing error [9]. This assumption is not restrictive,
since in many standardized OFDM systems the initialization
blocks are usually preceded by long CPs.
B. Signal model
We denote by Ym the QV -dimensional vector that collects
the DFT outputs corresponding to the considered subchannel
during the mth OFDM block. Since the DSSs are assumed
to be perfectly synchronized to the BS references, their signals will not contribute to Ym . In contrast, the presence of
uncompensated CFOs destroys orthogonality among ranging
signals, thereby leading to some interchannel interference
(ICI). However, as the subchannels are well separated in the
frequency domain, we can reasonably neglect interference on
Ym arising from ranging signals of subchannels other than
the considered one. Under this assumption, we may write
Ym =
K
X
ck (m)ejmωk NT A(ωk )Sk (θk ) + nm
(2)
k=1
where ωk = 2πεk /N , NT = N + NG is the duration of the
cyclically extended block and nm is a Gaussian vector with
zero mean and covariance matrix σ 2 IQV (we denote by IN
the identity matrix of order N ). Also, we have defined
A(ωk ) = FV(ωk )FH
(3)
where V(ωk ) accounts for the CFOs and is given by
V(ωk ) = diag ejnωk ; n = 0, 1, . . . , N − 1
(4)
H H
H
while F = F0 , F1 , . . . , FH
(the superscript H denotes
Q−1
the Hermitian transposition) with Fq (q = 0, 1, . . . , Q − 1)
denoting a V × N matrix with entries
1
[Fq ]ν,n = √ e−j2πniq,ν /N
N
0 ≤ ν ≤ V −1, 0 ≤ n ≤ N −1.
(5)
Vector Sk (θk ) in (2) can be partitioned as Sk (θk ) =
T
T
Sk (θk , i1 ), STk (θk , i2 ), . . . , STk (θk , iQ−1 ) ,
where
Sk (θk , iq ) is a V -dimensional vector with elements
Sk (θk , iq,ν ) = e−j2πθk iq,ν /N Hk (iq,ν ),
0 ≤ ν ≤ V − 1 (6)
while Hk (iq,ν ) denotes the channel frequency response over
the iq,ν th subcarrier and is given by
Hk (iq,ν ) =
L−1
X
hk (ℓ)e−j2πℓiq,ν /N .
(7)
ℓ=0
From (6) we see that θk simply appears as a phase shift across
the DFT outputs. The reason is that the CP length is larger
than the maximal expected propagation delay, thereby making
the ranging signals quasi-synchronous.
In the following sections we show how vectors {Ym }M−1
m=0
can be exploited to compute frequency and timing estimates
for all active ranging codes.
III. E STIMATION
OF THE
K
X
ck (m)ejmωk NT Sk (θk ) + nm .
(8)
This equation indicates that each CFO results only in a phase
shift between contiguous OFDM blocks. Collecting the iq,ν th
DFT output of all M ranging blocks into a vector Y(iq,ν ) =
[Y0 (iq,ν ), Y1 (iq,ν ), . . . , YM−1 (iq,ν )]T , we may write
K
X
Sk (θk , iq,ν )Γ(ωk )ck + n(iq,ν )
(9)
k=1
where n(iq,ν ) is Gaussian distributed with zero-mean and
covariance matrix σ 2 IM , while Γ(ωk ) = diag{ejmωk NT ; m =
0, 1, . . . , M − 1} is a diagonal matrix that accounts for the
phase shifts induced by ωk .
Inspection of (9) reveals that, apart from thermal noise,
vector Y(iq,ν ) is a linear combination of the frequency-rotated
codes {Γ(ωk )ck }. This means that the signal space is spanned
by the K vectors {Γ(ωk )ck } that correspond to the active
RSSs [10]. Then, if we temporarily assume that the number
K of active codes is known at the receiver, an estimate of
ωk (k = 1, 2, . . . , K) can be obtained by resorting to the
MUSIC algorithm [7]. To see how this comes about, we use
the observations {Y(iq,ν )} to obtain the following sample
correlation matrix
R̂Y =
1
QV
V
−1 Q−1
X
X
Y(iq,ν )YH (iq,ν ).
(10)
v=0 q=0
Next, based on the forward-backward (FB) approach [10], we
compute
e Y = 1 (R̂Y + JR̂TY J)
R
(11)
2
where J is the exchange matrix with 1’s on the anti-diagonal
and 0’s elsewhere. We denote by λ1 ≥ λ2 ≥ · · · ≥ λM the
e Y arranged in non-increasing order, and by
eigenvalues of R
{s1 , s2 , . . . , sM } the corresponding eigenvectors. The MUSIC
algorithm relies on the fact that the eigenvectors associated
(12)
ω
e
with
1
Ψk (e
ω ) = PM
m=K+1
k=1
Y(iq,ν ) =
ω̂k = arg max {Ψk (e
ω)} ,
CFO S
To simplify the derivation, we assume that the CFOs are
adequately smaller than the subcarrier spacing, i.e., |ωk | ≪
1. In such a case, matrices A(ωk ) in (3) can reasonably be
replaced by IN to obtain [8]
Ym ≈
with the M −K smallest eigenvalues are an estimated basis of
the noise subspace and, accordingly, they are approximately
orthogonal to all vectors in the signal space [7]. Hence, an
estimate of ωk is obtained by minimizing the projection of
Γ(e
ω )ck onto the noise subspace, i.e.,
H ω )s
cH
m
k Γ (e
2.
(13)
It is worth observing that CFO recovery must be accomplished for any active RSS. However, since the BS has no
prior knowledge as to which codes have been transmitted
in the considered subchannel, it must evaluate the quantities
{ω̂1 , ω̂2 , . . . , ω̂M−1 } for the complete set {c1 , c2 , . . . , cM−1 }.
At this stage the problem arises of identifying which
codes are actually active. The identification algorithm looks
for the K largest values in the set {Ψk (ω̂k )}M−1
k=1 , say
{Ψuk (ω̂uk )}K
,
and
declare
as
active
the
corresponding
k=1
codes {cuk }K
k=1 . The CFO estimates are eventually found as
T
ω̂u = [ω̂u1 , ω̂u2 , . . . , ω̂uK ] .
At this stage we are left with the problem of estimating the
parameter K to be used in (13). For this purpose, we adopt
the MDL approach and obtain [6]
n
o
(14)
K̂ = arg min F (K̃)
K̃
where F (K̃) is the following metric
F (K̃) =
1
K̃(2M − K̃) ln(QV )−QV (M − K̃) ln ρ(K̃) (15)
2
with ρ(K̃) denoting the ratio between the geometric and
arithmetic mean of {λK̃+1 , λK̃+2 , . . . , λM }. Finally, replacing
K by K̂ in (13) leads to the proposed MUSIC-based frequency
estimator (MFE) while the described identification algorithm
is called the MUSIC-based code detector (MCD)
IV. E STIMATION
OF THE TIMING DELAYS
After code detection and CFO recovery, the BS must acquire
information about the timing delays of all ranging signals.
This problem is now addressed by resorting to LS methods.
In doing so we still assume that the number of active codes
has been correctly estimated so that K̂ = K. Also, to simplify
the notation, the indices {uk }K
k=1 of the detected codes are
relabeled following the map uk −→ k for k = 1, 2, . . . , K.
We begin by reformulating (9) in a more compact form.
For this purpose, we collect the CFOs and timing errors
in two K-dimensional vectors ω = [ω1 , ω2 , . . . , ωK ]T and
θ = [θ1 , θ2 , . . . , θK ]T . Then, after defining the matrix
C(ω) = [Γ(ω1 )c1 Γ(ω2 )c2 · · · Γ(ωK )cK ] and the vector
S(θ, iq,ν ) = [S1 (θ1 , iq,ν ), S2 (θ2 , iq,ν ), . . . , SK (θK , iq,ν )]T ,
we may rewrite (9) in the equivalent form
Y(iq,ν ) = C(ω)S(θ, iq,ν ) + n(iq,ν ).
(16)
Omitting for simplicity the functional dependence of S(θ, iq,ν )
on θ and assuming ω̂ ≈ ω, from (16) the maximum likelihood
estimate of S(iq,ν ) is found to be
Ŝ(iq,ν ) = [CH (ω̂)C(ω̂)]−1 CH (ω̂)Y(iq,ν ).
and we have denoted by ŝk (ν, ℓ) the Q-point IDFT of the
sequence {Ŝk (iq,ν ); 0 ≤ q ≤ Q − 1}. In the sequel θ̂k is
termed the LS-based timing estimator (LS-TE).
Once θ̂k has been computed from (23), it is used in (22) to
estimate the CIR of the kth RSS as
ĥk =
(17)
Substituting (16) into (17) yields
Ŝ(iq,ν ) = S(iq,ν ) + ξ(iq,ν )
(18)
Ŝk (iq,ν ) = e
2πθk
N iq,ν
L−1
X
hk (ℓ)e
−j 2πn
N iq,ν
+ ξk (iq,ν ). (19)
ℓ=0
iT
h
On denoting Ŝk (ν) = Ŝk (i0,ν ), Ŝk (i1,ν ), . . . , Ŝk (iQ−1,ν ) ,
2πθk
and Φ(θk , ν) = diag{e−j N
may rewrite (19) as follows
iq,ν
; q = 0, 1, . . . , Q − 1}, we
Ŝk (ν) = Φ(θk , ν)F(ν)hk + ξ k (ν)
(20)
where ξ k (ν) = [ξk (i0,ν ), ξk (i1,ν ), . . . , ξk (iQ−1,ν )]T while
F(ν) is a matrix of dimension Q × L with entries [F(ν)]q,ℓ =
2πℓ
e−j N iq,ν for 0 ≤ q ≤ Q − 1 and 0 ≤ ℓ ≤ L − 1
Equation (20) indicates that, apart from the disturbance term
ξ k (ν), Ŝk (ν) is only contributed by the kth RSS, meaning that
ranging signals have been successfully decoupled at the BS.
We may thus exploit vectors {Ŝk (ν) ; ν = 0, 1, . . . , V − 1}
to get LS estimates of (θk , hk ) separately for each RSS. This
amounts to minimizing the following objective function with
respect to (θ̃k , h̃k )
Λk (θ̃k , h̃k ) =
V
−1
X
ν=0
2
Ŝk (ν) − Φ(θ̃k , ν)F(ν)h̃k
.
(21)
V −1
1 X H
ĥk =
F (ν)ΦH (θ̃k , ν)Ŝk (ν)
QV ν=0
(22)
0≤θ̃k ≤θmax
where Υ(θ̃k ) is given by
θ̃kX
+L−1
ℓ=θ̃k
|ŝk (0, ℓ)|2
(26)
and becomes periodic in θ̃k with period Q. In such a case,
the estimate θ̂k is affected by an ambiguity of multiples of Q.
This ambiguity does not represent a serious problem as long
as Q can be chosen to be greater than θmax . Unfortunately, in
some applications this may not be the case. For example, in
[5] we have Q = 8 while θmax = 102.
A. Reduced-complexity timing estimation
Although separating the RSS signals at the BS considerably
reduces the system complexity, evaluating Υ(θ̃k ) for θ̃k =
0, 1, . . . , θmax may still be computationally demanding, especially in applications where θmax is large. For this reason, we
now develop an ad-hoc reduced complexity timing estimator
(RC-TE).
We begin by decomposing the timing error θk into a
fractional part βk , less than Q, plus an integer part which
is multiple of Q, i.e.,
θ k = βk + p k Q
(27)
where βk ∈ {0, 1, . . . , Q− 1} while pk is an integer parameter
taken from {0, 1, . . . , P − 1} with P = ⌊θmax /Q⌋. Omitting
the details, it is possible to rewrite Υ(θ̃k ) as
(28)
where
Υ1 (β̃k ) =
β̃kX
+L−1 V
−1
X
ν=0
2
|ŝk (ν, ℓ)|
(29)
while Υ2 (β̃k , p̃k ) is shown in (30) at the top of the next page.
The RC-TE is a suboptimal scheme which, starting from
(28), estimates βk and pk in a decoupled fashion. More
precisely, an estimate of βk is first obtained looking for the
maximum of Υ1 (β̃k ), i.e.,
n
o
Υ1 (β̃k ) .
(31)
β̂k = arg max
0≤β̃k ≤Q−1
θ̃kX
+L−1 V
−1
X
ℓ=θ̃k
V =1
=
ℓ=β̃k
where we have used the identity FH (ν)F(ν) = Q · IL . Then,
substituting (22) into (21) and minimizing with respect to θ̃k
yields the timing estimate in the form
n
o
Υ(θ̃k )
(23)
θ̂k = arg max
Υ(θ̃k ) =
Υ(θ̃k )
Υ(θ̃k ) = Υ1 (β̃k ) + Υ2 (β̃k , p̃k )
For a fixed θ̃k , the minimum of Λk (θ̃k , h̃k ) is achieved at
(25)
It is worth noting that for V = 1 the timing metric (24) reduces
to
where ξ(iq,ν ) is a zero-mean disturbance term. From (6) and
(7) it follows that
−j
V −1
1 X H
F (ν)ΦH (θ̂k , ν)Ŝk (ν).
QV ν=0
ν=0
2
ŝk (ν, ℓ)ej2πℓν/N
(24)
Next, replacing βk with β̂k in the right-hand-side of (28) and
maximizing with respect to p̃k , provides an estimate of pk in
the form
Υ2 (β̃k , p̃k ) = 2ℜe
+L−1 V
−2 V −ν−1
β̃kX
X
X
ℓ=β̃k
ν=0
ŝk (ν, ℓ)ŝ∗k (ν + n, ℓ)e−j2πn(ℓ+p̃k Q)/N
n=1
(30)
0
max
0≤p̃k ≤P −1
(32)
FLM
A further reduction of complexity is possible when V =
2. Actually, in this case it can be shown that maximizing
Υ2 (β̂k , p̃k ) is equivalent to maximizing cos(ϕk − 2π p̃k Q/N ),
where
ϕk = arg
+L−1
β̂kX
ŝk (0, ℓ)ŝ∗k (1, ℓ)e−j2πℓ/N
ℓ=β̂k
.
(33)
10
-1
MCD
Pf
p̂k = arg
10
n
o
Υ2 (β̂k , p̃k ) .
10
10
-2
-3
Ω = 0.05
Ω = 0.075
K=2
K=3
The estimate of pk is thus obtained in closed-form as
N ϕk
.
p̂k =
2πQ
V. S IMULATION
10
-4
-2
(34)
0
2
4
6
8
10
12
14
16
18
20
22
SNR, dB
RESULTS
A. System parameters
The simulated system is inspired by [5]. The total number
of subcarriers is N = 1024 while the number of ranging
subchannels is R = 8. Each subchannel is composed by
Q = 16 subbands uniformly spaced at a distance N/Q = 64.
Any subband comprises V = 2 adjacent subcarriers while
the ranging time-slot includes M = 4 OFDM blocks. The
ranging codes are taken from a Fourier set of length 4 and
are randomly selected by the RSSs at each simulation run
T
(expect for the sequence [1, 1, 1, 1] ). The discrete-time CIRs
have L = 12 channel coefficients. The latter are modeled
as circularly symmetric and independent Gaussian random
variables with zero means (Rayleigh fading) and exponential
power delay profiles, i.e., E{|hk (ℓ)|2 } = σh2 ·exp(−ℓ/12) with
2
ℓ = 0, 1, . . . , 11 and σh2 chosen such that E{khk k } = 1.
Channels of different users are assumed to be statistically
independent. They are generated at each new simulation run
and kept fixed over an entire time-slot. The normalized CFOs
are uniformly distributed within the interval [−Ω, Ω] and vary
at each simulation run. We consider a cell radius of 10 km,
corresponding to a maximum transmission delay θmax = 204.
A CP of length NG = 256 is chosen to avoid IBI.
B. Performance evaluation
We begin by investigating the performance of MCD in terms
of probability of making an incorrect detection, say Pf . This
parameter is illustrated in Fig. 1 as a function of SNR = 1/σ 2
under different operating conditions. The number of active
RSSs varies from 2 to 3 while the maximal frequency offset
is either Ω = 0.05 or 0.075. Comparisons are made with the
ranging scheme discussed by Fu, Li and Minn (FLM) in [5],
where the kth ranging code is declared active provided that
the quantity
Fig. 1.
Pf vs. SNR for K = 2 or 3 when Ω is 0.05 or 0.075.
Q−1 V −1
1 XX H
c Y(iq,ν )
Zk = 2
M q=0 ν=0 k
2
(35)
exceeds a suitable threshold η which is proportional to the
estimated noise power σ̂ 2 . The results of Fig. 1 indicates that
the proposed scheme performs remarkably better than FLM
because of its intrinsic robustness against CFOs. As expected,
the system performance deteriorates for large values of K and
Ω. The reason is that increasing K reduces the dimensionality
of the noise subspace, which degrades the accuracy of the
MUSIC estimator. Furthermore, large CFO values result into
significant ICI which is not accounted for in the signal model
(8), where A(ωk ) has been replaced by IN .
Fig. 2 illustrates the root mean-square-error (RMSE) of the
frequency estimates obtained with MFE vs. SNR. Again, we
see that the system performance deteriorates when K and
Ω are relatively large. Nevertheless, the accuracy of MFE is
satisfactory under all investigated conditions.
The performance of the timing estimators is measured in
terms of probability of making a timing error, say P (ǫ), as
defined in [8]. More precisely, an error event is declared
to occur whenever the estimate θ̂k gives rise to IBI during
the data section of the frame. In such a case, the quantity
θ̂k − θk + (−NG,D + L)/2 is larger than zero or smaller
than −NG,D + L − 1, where NG,D is the CP length during
the data transmission phase. Fig. 3 illustrates P (ǫ) vs. SNR
as obtained with RC-TE and FLM when NG,D = 64. The
operating conditions are the same of the previous figures. Since
the performance of LS-TE is virtually identical to that of RCTE, it is not reported in order not to overcrowd the figure. We
10
0
MFE
Frequency RMSE
10
10
10
10
-1
[2]
[3]
-2
[4]
-3
Ω = 0.05
Ω = 0.075
K=2
K=3
[5]
-4
-2
0
2
4
6
8
10
12
14
16
18
20
22
[6]
SNR, dB
Fig. 2. Accuracy of the frequency estimates vs. SNR for K = 2 or 3 when
Ω is 0.05 or 0.075.
[7]
[8]
0
10
[9]
-1
10
P(ε )
FLM
[10]
-2
10
-3
10
Ω = 0.05
Ω = 0.075
K=2
K=3
-4
RC-TE
10
-2
0
2
4
6
8
10
12
14
16
18
20
22
SNR, dB
Fig. 3.
P (ǫ) vs. SNR for K = 2 or 3 when Ω is 0.05 or 0.075.
see that for SNR values larger than 6 dB the proposed scheme
provides much better results than FLM.
VI. C ONCLUSIONS
We have derived a novel timing and frequency synchronization scheme for initial ranging in OFDMA-based networks.
The proposed solution aims at detecting which codes are actually being employed and provides timing and CFO estimates
for all active RSSs. CFO estimation is accomplished by resorting to the MUSIC algorithm while a LS approach is employed
for timing recovery. Compared to the timing synchronization
algorithm discussed in [5], the proposed scheme is more robust
to frequency misalignments and exhibits improved accuracy.
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