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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
Convergence Property of Transmit Time
Pre-Synchronization for Concurrent Cooperative
Communication
Yong Jun Chang and Mary Ann Ingram
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
Email: {yongjun.chang, mai}@gatech.edu
Abstract—This paper proposes a preamble detection-based cooperative transmit time pre-synchronization technique specially
for Concurrent Cooperative Transmission (CCT). CCT enables
a collection of power-constrained radios to transmit as a group
and achieve a transmit range that is much greater than the
range of a single device. In the absence of a GPS system or a
globally synchronized clock, each cooperating node can derive
its transmit time autonomously by detecting the preamble from
a received signal. The paper first compares two methods for
estimating the start of packet (SOP) received in the presence
of a carrier frequency offset (CFO): the peak method, which
is optimal in a single diversity channel without CFO, and a
mean method, which outperforms peak with large CFO. Next,
the SOP is estimated by combining all diversity channels using
an approximated Best Linear Unbiased Estimator (BLUE). Using
theory and measurement, we show that the time error statistics
are convergent as the number of hop increases.
I. I NTRODUCTION
Cooperative transmission (CT) is a physical layer wireless
communication scheme in which spatially separated sensor
nodes collaborate to transmit the same source message so
that the transmission power required for each node can be
reduced or the range can be extended [1][2]. Collaborating
nodes can be chosen in a centralized or distributed manner and
grouped into a cluster. Therefore, we can consider cooperative
communication to be cluster-to-cluster transmission (MIMO)
or cluster-to-single-node transmission (MISO). Concurrent CT
(CCT) is a special form of CT which allows cooperating
nodes transmit the diversity versions of the message at the
approximately same time. Any relative synchronization errors
in the transmit times of the cooperators roughly add to the
delay spread of the multipath channel, so our aim is for the
root mean square (RMS) transmit time spread to be at most
comparable to the channel delay spread.
This paper considers how synchronization time errors evolve
over consecutive CCT hops without single source redistribution within a cluster. We show by both measurement in
a typical indoor office environment as well as theoretical
analysis that the error statistics are convergent with the number
of hops. The measurement data shows that the error spread is
comparable to delay spreads of the indoor environment.
The impairment caused by time synchronization errors
on the performance of cooperative MIMO was theoretically
treated in [3]. The authors also showed that jitters as large as
10% of the bit duration do not have much effect on the BER
performance of the system. Using a globally synchronized
clock by network time protocol or a GPS has been proposed
for CCT [4]. These approaches require additional devices or
a convergence period for accurate time synchronization.
In this paper, we show experimental results for CCT, using
binary frequency shift keying (BFSK), with non-coherent
demodulation. Orthogonality is achieved in the frequency
domain, by having different cooperating radios transmit on
different orthogonal carriers. Because CCT is not supported
by any off-the-shelf radios, we use software-defined radios
(SDRs) to implement our approach.
II. S YSTEM M ODEL AND P REAMBLE D ETECTION
Without using a globally synchronized clock, cooperating
nodes can decide their cooperative transmission time by adding
fixed amount of period to the reception time of the message
to be relayed. It can be achieved by locating the preamble or
sync word embedded in the packet.
Suppose that s(t) is the modulated baseband signal of a
known preamble which has energy |s(t)|2 dt = 1. The
received signal r(t) has a frequency offset f which is introduced by carrier frequency difference between the transmitter
and the receiver. Assuming a quasi-static fading channel with
a channel coefficient α by
r(t) = h · s(t − τ ) · ej2πf t + w(t)
(1)
where τ is a propagation delay and w(t) is zero-mean complex
additive white Gaussian noise (AWGN) with a power spectral
density of N0 .
If the preamble is known a priori at the receiver, the
beginning of the preamble can be found by cross-correlating
the received signal r(t) with a preamble signal s(t) and finding
a threshold or peak. Mathematically, the cross-correlation of
the preamble-bearing signals s(t) and the received signals r(t)
which has encountered a frequency offset f can be described
as
d(t, τ, f ) = h · q(t, τ, f ) + z(t)
(2)
∞
where q(t, τ,
f ) = −∞ s(x − τ )s∗ (t + x)ej2πf (x−τ ) dx
∞
and z(t) = −∞ w(x)s∗ (t + x)dx. z(t) is zero-mean complex
AWGN with a variance N0 . Under the assumption of high
signal-to-noise ratio (SNR) and f = 0 [5] the estimation of
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
Fig. 2. Illustration of the multiple Concurrent Cooperative Transmission
(CCT) hops
Fig. 1. Decision function Λ(t) of a single preamble in the presence of carrier
frequency offset (CFO) and the accuracy of peak and mean estimator in the
presence of CFO
SOP
time can be achieved by choosing t to maximize Λ(t) =
P −1
2
i=0 |d(t − iT )| where P is the length of preamble.
The accuracy of correlation rule is proportional to the SNR
and sidelobe suppression of the correlation output [6]. With
a carrier frequency offset (CFO), however, peak searching
suffers performance degradation because CFO makes the filter
mismatched. Example filter output shown in in-set figure in
Figure 1, which is the decision function Λ(t) over (−T, T )
where h = 1, P = 1, τ = 0, and noise-free z(t) = 0. It is
shown that as CFO increases, signal strength diminishes.
A. Mean Estimation
Instead of searching a peak of correlation output to determine a SOP time, we propose a mean (center-of-mass)
estimation. Without loss of generality, let us assume τ = 0
which implies that the truth time of SOP is 0. We also
assume that the soft-valued signal Λ(t) is sampled with an
oversampling rate 1/Ts . The finding a mean of Λ(t) can be
realized as follows:
K
−K k · Λ[k]
.
(3)
ω = Ts K
−K Λ[k]
In Appendix A, we show that the estimation error ω has an
approximately zero mean Gaussian distribution in high SNR
and its variance σω2 can be approximated by
σω2 =
where
K=
2
i
j
N0 KT s2
|h|2
(4)
ij|q[i, f ]||q[j]|(s s)2 [i − j]
.
( i q 2 [i])2
(5)
Figure 1 shows a MATLAB simulation of the root-meansquare error (RMSE) of peak and mean estimator in the
presence of CFO = {0, 5, 10, 15}ppm. For this simulation,
1MHz sampling rate Ts = 10−6 and a BFSK waveform
with a rectangular shaping pulse are used. It also shows the
theoretical RMSE treated in Equation 4 by using a dotted line.
While peak estimation outperforms mean estimation in the
absence of or relatively small CFO, mean estimation gives
better performance in the presence of relatively large CFO.
Intriguingly, it is shown that RMSE of peak estimator at
15ppm CFO rises rapidly because the output of correlator Λ(t)
becomes blunt and finally it becomes bimodal as shown in the
in-set figure in Figure 1.
III. C OOPERATIVE T RANSMIT T IME S YNCHRONIZATION
In this section, we analyze the Start of Packet (SOP) time
estimation error at each relay in a sequence of relay clusters,
such that each cluster has N relays and does concurrent
cooperative transmission (CCT). We assume that the packet is
originally transmitted by a single-antenna source (S) at time
T , as shown in Figure 2 for N = 2. To simplify the analysis,
we assume that the N nodes in each cluster are co-located
(not shown this way in the figure) and that the clusters are
arranged on a line with equal spacing. This implies that the
propagation distances between nodes in different clusters are
equal and deterministic. We assume that each relay in a cluster
transmits a preamble that is orthogonal to the other N − 1
preambles being transmitted in the cluster. These orthogonal
preambles define the N diversity channels for synchronization,
and are transmitted simultaneously. We assume that each
receiver has a bank of N correlators, with one correlator
for each diversity channel. We assume independent Rayleigh
fading in all diversity channels.
For all terms used in this paper, a superscript and subscript
represent the cluster and node index, respectively. For exam(j)
ple, Tk is a time of transmission of Relay k in Cluster j.
(j)
For example of two subscripts case, h21 is a channel between
(j)
Relay 1 in Cluster j − 1 and Relay 2 in Cluster j. T̂kn is
(j−1)
an estimate of Tn
based only on correlation in channel
(j)
(j)
hkn at the Node k in the Cluster j. ωkn is the “correlator
(j)
estimation error” of T̂kn transmitted from the Node n of
(j)
previous cluster. T̂k is an approximated Best Linear Unbiased
Estimate (BLUE) of T without clock error, which is a linear
(j)
(j)
combination of T̂kn ’s. ξk is a clock compensation error of
the Node k in the Cluster j.
Each relay autonomously estimates the time of the SOP.
(j)
(j)
Let Rk be the kth relay in the jth cluster, and let T̂k be its
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
estimate of the SOP. The details on how that estimate is formed
are explained below. Next, ideally, each relay determines its
transmit time by adding an universally known constant, Tproc ,
(j)
to T̂k , to allow enough time for the signal processing on all
relays in a cluster to be completed [7]. However, because each
relay has a slightly different oscillator frequency from the other
relays, some clocks run fast and some run slow; this is “clock
skew.” Each relay will attempt to compensate its clock skew,
but the compensation is not perfect, so some relays will still
have Tproc expire faster than others. The deviation in Tproc
expiration on Node k of Cluster j from an imaginary universal
(j)
(j)
reference clock is designated as the “clock error,” ξk . ξk is
assumed to be zero mean and Gaussian random variable with
a variance σξ2 . Therefore, the transmit time of the kth relay in
(j)
(j)
(j)
the jth cluster is Tk = T̂k + ξk .
(j)
To compute T̂k , the estimate of the SOP at the kth node in
(j)
the jth cluster, the node first creates N estimates, T̂kn , n =
1, 2, . . . , N , of the SOP in each of its N diversity channels,
respectively, by defining each estimate to be a centroid of the
correlator output as discussed in Section II. Since only one
of the previous cluster nodes transmits in a given diversity
(j)
channel, T̂kn is actually an estimate of the SOP of the packet
(j−1)
(j)
(j−1)
(j)
received from Rn
, i.e. T̂kn = Tn
+ ωkn .
A. Approximated Linear Unbiased Estimation
The performance of cooperative transmit time synchroniza(j)
tion can be assessed by a covariance of transmission time T̂k .
In this subsection, we design an approximated linear unbiased
(j)
estimator to minimize the variance of transmission time T̂k
(j)
(j)
where T̂k is a linear combination of T̂kn for 1 ≤ n ≤ N .
(j)
Let akn denote a coefficient of combiner. By best linear
unbiased estimator (BLUE) and under the assumption that the
average transmission time errors of previous cluster nodes are
(j)
(j) (j)
(j)
same, the coefficient vector ak = [ak1 , ak1 , . . . , akN ]T =
(V−1 1)/(1T V−1 1) where V is a covariance matrix of
(j)
(j)
(j)
(j)
T̂ k = [T̂k1 , T̂k2 , . . . , T̂kN ]T and 1 is the column vector of
(j)
N ones. Since single correlation errors ωkn are uncorrelated
over index n, the covariance matrix becomes a diagonal
matrix V = diag[σω2 1 , σω2 2 , . . . , σω2 N]. Therefore, the
estimator
N
(j)
(j) 2
(j) 2
.
coefficient becomes akn = |hkn | /
i=1 |hki |
To simplify analysis, we assume that all deterministic times
are equal to zero; this includes the propagation time and Tproc .
This allows us to focus only on the behavior of time estimation
errors. Since all the errors are zero mean, we have that all
estimators are trying to estimate the same thing, T , which is
the original transmit time of the source (S).
B. Markov Process Model
(j)
(j)
(j)
Let the vector T (j) = [T1 , T2 , . . . , TN ]T represent the
transmit times of each relay in Cluster j (or “hop j”). From the
arguments above, each element of T(j) is an unbiased estimate
of the original packet transmit time, or E{T(j) } = T · 1.
Let the estimation error vector of T (j) be denoted as e(j) =
(j) (j)
(j)
[e1 , e2 , . . . , eN ]T . Our objective is to get a recursion for
T
the covariance of transmit time error C(j) = E e(j) e(j) ,
from which we derive the sample variance.
Because our estimator coefficients at Cluster j do not
depend on the multi-path channels of previous hops, we
can see that the statistics of the errors at Cluster j are
independent of the errors of past hops, assuming T (j−1) is
given. In other words, the conditional joint probability density
function (PDF) of T (j) , given the entire past transmit times
{T (j−1) , T (j−2) , . . . , T (1) }, is the same as the conditional
joint probability density function (PDF) of T (j) , given only the
latest vector of transmit times T (j−1) . This makes T (j) , j =
1, 2, . . . , N , a vector-valued, continuous-state, discrete-time
(j)
(j)
(j)
Markov Process. ek with substitution of T̂k and T̂kn can
be written as
(j)
(j)
(j)
ek = T̂k + ξk − T =
N
(j) (j) (j)
akn T̂kn − T + ξk
n=1
=
N
(j)
akn e(j−1)
+
n
n=1
N
N
(j)
(j)
(j)
akn ωkn + ξk
n=1
(j)
(j)
noting that
and
n=1 akn = 1 for all j and k. Let A
(j)
be a estimator coefficient and correlator error matrix
ω
(j)
(j)
whose elements of kth column and nth row are akn and ωkn
(j) (j)
(j) T
(j)
respectively, and ξ = [ξ1 , ξ2 , . . . , ξN ] . The covariance
matrix C(j) has an iterative form as
T
T
C(j) = E{e(j) e(j) } = E A(j) C(j−1) A(j)
(j)
(j)
(j)
(j) T
(j)
(j)
[A ◦ ω ]1 + ξ
(6)
+ E [A ◦ ω ]1 + ξ
where ◦ denotes an element-wise product operator. In the
following sections, we show that the expected value of the
(j)
sample variance of estimation error ek is convergent even
though the absolute covariance matrix is not.
C. Convergence Property
The performance of transmit time pre-synchronization can
be assessed by relative transmit time error of relay nodes
within a cluster. The relative transmit time error of jth cluster’s
relay nodes is basically equivalent to the sample variance of
estimation error e(j) . To simplify analysis of a convergence
property of the covariance matrix C(j) , we assume that the
statistics of all channels, clocks and noises do not vary with
relay index within a cluster. These assumptions yield that the
variance and the covariance of the estimation error are not
functions of a node index. Therefore, the diagonal terms of
(j)
the matrix C(j) , denoted as Cd , are identical. Also, the off(j)
diagonal terms, denoted as Co , are identical. In Appendix B,
(j)
(j)
we show that Cd and Co can be written as
2
N − 1 (j−1)
(j−1)
Cd
C
+
+ H + N σξ2
N +1
N +1 o
1 (j−1) N − 1 (j−1)
Co
Co(j) = Cd
+
N
N
N
(j) (j)
where H = n=1 E{agn ωgn }.
(j)
Cd =
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
(a) Transmit Power = 0dBm
Fig. 3. A floor map and photo of the experiment which had conducted on
the fifth floor of the Centergy Building at the Georgia Institute of Technology
In [8], the expected value of sample variance in which the
samples are not mutually independent and pairwise covariances are constant γ, is given by E{σs2 } = σ 2 − γ where
s2 is the sample variance and σ 2 is the variance. From this
relationship, we can get the linear difference equation form of
the expected value of the sample variance at jth cluster as
E{σs2
(j)
}=
N −1
(j−1)
E{σs2
} + H + N σξ2 .
N (N + 1)
The expected value of the sample variance is convergent to
E{σs2 } =
N (N + 1)
(H + N σξ2 ) as j → ∞.
N2 + 1
IV. E XPERIMENTAL S ETUP AND R ESULTS
The synchronization method we designed in Sections II
and III was evaluated in a Software Defined Radio (SDR)
testbed. Each wireless node in this experiment is composed of
a RF-daughterboard (RFX-2400), an Universal Software Radio
Peripheral (USRP1) board, a personal computer (PC), and the
GNU radio software. The USRP1 board has an ADC/DAC
and a FPGA to convert passband signal to baseband signal
and vice versa. All baseband processing is done on the PC.
Binary frequency shift keying (BFSK) with non-coherent
envelope detection was used for the experiments. Orthogonal
preambles were achieved by choosing orthogonal center frequencies. We used 64 kbps bit-rate with 1 Mhz sampling-rate.
The total length of a packet is 24 bytes consisting of 4 bytes
preamble, 6 bytes header, 10 bytes data and 2 bytes CRC. The
source packet is scrambled at every transmission in which the
index of scrambler is included in the header. All baseband processing for non-coherent BFSK modulator/demodulator and
the transmit time synchronization schemes are programmed
by using C++ and Python languages.
(j)
=
The rms transmit time spread (RTTS), σs
N
i
(b) Transmit Power = -5dBm
Fig. 4.
Measured RTTS of “ping-pong” experiment
large, we designed the “ping-pong” experiment, where two
groups of cooperating nodes transmit the source message
back and forth up to 10 hops (or ”CT”s). Figure 3 shows
a floor map and photo of the “ping-pong” experiment was
conducted on the fifth floor of the Centergy Building at
the Georgia Institute of Technology. The experiment was
(j)
repeated 500 times to get 500 trials of σs . USRP1 boards
are programmed to generate a trigger pulse at the time of
each transmission and the trigger pulse was captured by a
customized FPGA board by wire.
Figure 4 shows the empirical RTTS of the cooperative nodes
with two different transmit powers. Each curve represents an
empirical cumulative density function (CDF) of RTTS of each
CT. We note that we measured sample variance of cooperative
nodes so that the CDF does not follow the CDF of the error
statistic we derived in Section III. As shown in the figure, the
CDFs corresponding to −5dBm transmit power are shifted by
approximately 20% compared to the CDFs corresponding to
0dBm. This was predicted in the simulation result in Figure 1.
We also observe that RTTS tends to be converged after the
third hop. This provides strong evidence that our analysis of
convergence property is correct. Moreover 90% of the RTTS is
less than 300ns in both transmit powers, which indicates that
CCT can support up to 300 kbps data rate in narrow-band
waveforms without significant ISI degradation [3].
V. C ONCLUSION
(j)
(Ti −T̄ (j) )2
N −1
of cooperative relay nodes in jth CT
was used to evaluate the performance where T̄ (j) is the
sample mean of transmission time of all relay nodes at
Cluster j. To simulate multiple consecutive cluster hops in a
In this paper, we proposed a preamble detection-based
transmit time synchronization scheme for CCT. We designed a
mean estimation method for non-coherent BFSK demodulation
to estimate the start of packet (SOP) time. We showed that
978-1-4244-5637-6/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
the mean estimation method outperforms the peak searching method in the presence of CFO by both analysis and
simulation. Diversity channels are formed by having each
cooperating node transmit on an orthogonal carrier frequency.
The estimates of the SOP times in each diversity channel are
combined using an approximate BLUE method, to produce
an overall estimate of the SOP, which is the reference for
the retransmit times. We show that the transmit time error
statistics are convergent over consecutive CCT hops based
on measurement in an SDR testbed in a typical indoor
office environment, as well as by theoretical analysis. Our
experiment and implementation show that CCT without using
single-source redistribution within clusters and without using
a globally synchronized clock is practical for the indoor
environment.
A PPENDIX A
Let us consider a single bit preamble P = 1. At any
index k, Λ[k] can be considered as an absolute square of
a non-zero mean complex Gaussian random variable whose
normalized value has a non-central chi-square distribution
with two degrees of freedom and non-centrality parameter
λ = μ2 /σ 2 where μ and σ 2 are mean and variance of
unnormalized complex Gaussian random variable. The mean
and variance of normalized non-central chi-square distribution
with two degrees of freedom are 2+λ and 4(1+λ) respectively.
Therefore, the mean and variance of Λ[k] at any index k are
μΛ[k] = N0 + |h · q[k]|2
2
σΛ[k]
= N02 + 2N0 |h · q[k]|2
where q[k] is a sampled signal of q(t). Similarly, the mean
and variance of k · Λ[k] are
2
= k 2 (N02 + 2N0 |h · q[k]|2 )
σk·Λ[k]
By
(CLT), a probability distribution
central limit theorem
of
k · Λ[k] and
Λ[k] can be approximately
Gaussian
distribution. Let X and Y be denoted as k·Λ[k] and Λ[k]
respectively and define Z = X/Y . In Equation 2, z[k], a
sampled version of z(t), is a filtered white Gaussian noise
so called a colored Gaussian noise . Its auto-covariance is
Cov(z[i], z[j]) = N0 · (s s)[i − j]. The mean and variance of
X and Y are
μΛ[i]
μx = 0, μy =
=
i
σy2
=
i
i
σi·Λ[i] σi·Λ[j] (s s)2 [i − j]
j
(j)
(j−1)
Cd = Cd
N
2
(j−1)
E{(a(j)
gn ) } + Co
n=1
+
N
(j−1)
Co(j) = Cd
(j)
E{a(j)
gx agy }
x=1 y=1
x=y
(j)
E{a(j)
gn ωgn } +
n=1
N N
N
E{ξn(j) }
n=1
N
2
(j−1)
(E{a(j)
gn }) + Co
n=1
N
N (j)
E{a(j)
gx }E{agy }
x=1 y=1
x=y
(j)
It can be shown that E{agn } = 1/N regardless any inN
(j)
(j)
dexes [10]. Assuming that X = |hgn |2 and Y = k=1 |hgk |2 ,
(j) 2
E{(agn ) } can be written as E{X 2 }/E{Y 2 }. Since X
and Y have exponential and gamma distribution respectively,
(j)
E{agn } can be simplified to N (N2+1) . Also we can get
N
N (j)
E{a(j)
gx agy } = E{(
x=1 y=1
x=y
N
2
a(j)
gn ) } − E{
n=1
= 1 − E{
N
N
2
(a(j)
gn ) }
n=1
2
(a(j)
gn ) } =
n=1
N −1
N +1
(j)
It is noted that the third and fourth term of Cd do not depend
on the index j because we assume that the statistics of all
channels, clocks and noises do not vary.
ACKNOWLEDGMENT
The authors would like to acknowledge Zhen GAO in Tianjin University, China for his importance review and discussion.
R EFERENCES
μk·Λ[k] = k(N0 + |h · q[k]|2 )
σx2
From Equation 6,
A PPENDIX B
(j)
and Co can be written as
(j)
Cd
σΛ[i] σΛ[j] (s s)2 [i − j].
j
By Geary-Hinkley transformation
√ [9], a random variable ω
can be transformed to ω = (Ts KN0 /|h|)u where u is a
zero-mean unit variance Gaussian random variable.
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