Li2419041913

Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1904-1913
Optimized Joint Time Synchronization And Channel Estimation
Scheme For Block Transmission UWB Systems
SESHU KUMAR .N1, BHARAT KUMAR SAKI2, MANJULA BAMMIDI3,
P SRIHARI4,
1
PG Student, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA.
2
PG Student, Department Of CSIT, JNTUK college of Engineering, Visakhapatnam, AP, INDIA
3
Associate Professor, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA.
4
Associate Professor, HOD, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA.

Abstract
In this paper, timing synchronization in synchronization accuracy is much more stringent
high-rate ultra-wideband (UWB) block than conventional wideband systems. Moreover, the
transmission systems is investigated. A new joint highly dispersive nature of UWB channels presents
timing and channel estimation scheme is proposed additional challenges to timing acquisition. The
for orthogonal frequency division multiplexing literature is abundant with research on preamble
(OFDM) and single carrier block transmission assisted synchronization methods for OFDM in
with frequency domain equalization (SC-FDE) conventional wideband channels [4]-[7]. Since this
UWB systems. The scheme is based on a newly paper is concerned with timing synchronization,
designed preamble for both coarse timing and the
subsequent channel estimation. Despite of the There are only a few publications in the
presence of coarse timing error, the estimated literature on synchronization for multi-band (MB)
channel impulse response (CIR) is simply the OFDM UWB systems [9]—[13], all using the
cyclic shifted version of the real CIR, thanks to preamble defined in the WiMedia standard [1] except
the unique structure of the preamble. The coarse [10]. Cross-correlation with the preamble template is
timing error (CTE) is then determined from the used in [9]—[11] for synchronization, leading to
CIR estimation and used to fine tune the timing high complexity. The methods in [12], [13] employ
position and the frequency domain equalization autocorrelation of the received signal as the timing
coefficients. The proposed scheme saves preamble metric and take the maximum of the metric to be the
overhead by performing joint synchronization timing point, which tend to synchronize to the
and channel estimation, and outperforms existing strongest path but not necessarily the first significant
timing acquisition methods in the literature in path. In this paper, we propose a generic joint timing
dense multipath UWB channels. In addition, the and CE scheme for block transmission systems,
impact of timing error on channel estimation and which begins with an auto-correlation based coarse
the performance of SC-FDE UWB systems are timing estimator using a newly designed preamble,
analyzed, and the bit-error-rate (BER) followed by CE and CE assisted timing adjustment.
degradation with respect to certain timing error is In the coarse timing stage, the coarse FFT window
derived. position is established and the initial CE is
performed. The particular preamble structure makes
Index Terms—Synchronization, UWB, SC-FDE, the CE robust to the coarse timing error, and the
timing error analysis, joint timing and channel proposed technique to accurately estimate the coarse
estimation. timing error allows both fine timing adjustment and
CIR adjustment for later data demodulation. Both
I. Introduction synchronization and CE are acquired from the
For high data rate ultra-wideband (UWB) procedure. This design saves the preamble overhead
communications, two block transmission systems and improves the synchronization performance in
have been proposed: orthogonal frequency division UWB channels.
multiplexing (OFDM) [1] and single carrier with The fine timing estimation is related to the
frequency domain equalization (SC-FDE) [2], both time of arrival (ToA) estimation. For ToA estimation
of which compare favorably to the impulse based in MB-OFDM UWB systems, energy detectors can
UWB systems in terms of equalization complexity be found in [14], [15], and an energy jump detector
and energy collection in dense multipath UWB has been proposed in [16]. In this work, we propose a
channels. SC-FDE has lower peak to average power newly designed energy jump detector and provide its
ratio (PAPR) and lower sensitivity to carrier performance analysis. In addition, the impact of
frequency offset (CFO) than OFDM, but is less timing error on SC-FDE systems has been analyzed
robust to timing error [3].Due to the very high data in this work, which is not available in the literature.
rate of UWB systems, the requirement on the The remainder of the paper is organized as follows.

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Section II presents the system model and the structure for accurate CE whose result will be used in
synchronization problem formulation. The proposed the subsequent timing adjustment and frequency
timing and CE scheme is introduced in Section HI. domain equalization. As shown in Fig. 1, in addition
The timing error impact on SC-FDE systems with to the CP, a cyclic suffix is padded after the
minimum mean square error frequency domain preamble body. Note that the suffix is used only for
equalization (MMSE-FDE) is analyzed in Section the preamble, and data blocks have CP only as in
IV. Section V gives the performance evaluation and conventional OFDM. In the proposed preamble
discussion of the proposed method in UWB shown in Fig. 1, A and B represent the first half and
channels. Finally Section VI concludes the paper. the second half of one block generated by

II. System Model N 1
1
Although the system model adopts cyclic Pt ( n ) 
N
 k 0
Pt (k )e j 2 nk / N o  k  N  1
prefixed single carrier block transmission with
frequency domain equalization over UWB channels, (1)
the synchronization technique developed directly where Pt(k) is a random ±1 sequence in
applies to an OFDM system. The pth transmitted frequency domain (FD), referred to as FD pilot
block is composed of digitally modulated symbols block, which results in flat amplitude in FD for good
xPk (0<k<N-1), where N is the block length. A cyclic CE performance. The [A B] structure can be repeated
prefix of length Ng is used to avoid IBI as in an M times and finally a prefix and a suffix are added to
OFDM system. Denote the symbol-spaced equivalent have [B A B ... B A], where A and B are the first Nx
CIR by h= [h o, hl,..., hL], where L is the maximum samples of A and last Nx samples of B respectively.
channel delay. The UWB channel is assumed to be The length of the prefix and suffix denoted by N x
constant over the data blocks. As long as the needs to be larger than the maximum coarse timing
maximum channel delay is shorter than the CP length error, and Nx should be larger than the channel length
(L < Ng), there is no FBI in data demodulation with L to avoid IBI. By repeating [A B] we can achieve
perfect timing. The task of synchronization is to find noise averaging for more accurate CE, and the
the beginning of the data block, also referred to as adjacent [A B] blocks serve as cyclic extension. For
finding the FFT window position. Perfect the case of M—1, the whole preamble becomes [B A
synchronization in cyclic prefixed block transmission B A], thus the synchronization and CE overhead is
systems can be achieved in the LBI-free region of the 2Nx+N, smaller than the preamble pattern proposed
CP ([-Ng+L,0]) [5]. As long as the FFT window in [1] where the overhead is 2(N+Ng). Following the
starts within the IBI-free region, data demodulation conventional auto-correlation based metric function,
can be carried out perfectly. Otherwise timing error we define our timing metric function as
will cause performance degradation to a system. In M(d) =
Section IV, the effect of timing offset on the 2 N x ( M 1) N 1
performance of SC-FDE UWB will be analyzed
2 C (d )
2  rdx x rd  N 1
.  2 N x ( M 1) N 1
k 0
III. The Proposed Joint Timing and Channel R(d )
  rd  N 1
2 2
Estimation Scheme ( rd 1 )
Timing within the IBI-free region does not k 0

cause extra interference to the detection. However, in (2)
UWB systems, the IBI-free interval given a short CP and coarse timing is established by finding the
is often very small or even zero, depending on maximum of (2) as given by 6C = maxM(d). Note
channel realizations. Thus, it is necessary to locate the range of d to look for the maximal can be defined
the first significant arrival path of the transmitted as [9S, 9S + Q], where 6S is the point where the
signal block, which is defined as the exact timing metric function first exceeds the preset threshold and
point in the following. With the definition of the Q can be set as N. Simply speaking, the metric
exact timing point, we define the timing error as the calculator activates the maximization procedure once
offset between the established timing point and the the arrival of the signal is sensed. To reduce the
exact timing point. computation complexity at the implementation stage,
the metric can be implemented in an iterative manner
A. Preamble Structure and Coarse Timing similar to [4] as
In dense multipath channels, coarse timing
obtained by evaluating a metric function can not C(d + 1) = C(d) - rdrd+N+ rd+2Nx+(M-1)N rd+2Nx+MN
provide sufficient accuracy, since coarse timing often (3) and
falls far away from the exact timing due to the large R(d+l) = R(d)-rd2-rd+N2 + rd+2Nx+(M-1)N2 + 
channel dispersion. We propose to use the estimated rd+2Nx+MN2 (4)
channel information to fine tune the timing position.
A new preamble structure is designed to have the which implies a sliding window implementation. For
periodical property for acquisition and cyclic every received sample, only 3 new multiplication and

1905 | P a g e

6 add operations are needed to compute the iterative  1
timing metric. This sliding window correlator saves and er i    | g i  k  mod N | 2
substantial complexity compared to the metrics k 0
proposed in [7] and [9] which need to perform the  1
whole correlation for every received sample. el i    | g i  k  1 mod N | 2 (9)
k 0
B. Channel Estimation and Timing Adjustment Where (gmax) is the maximum absolute
In this step, CE is carried out using the same value of the elements in g and T) is a preset threshold
preamble after the initial acquisition and the whose determination will be given in Section V.Fig.
estimation accuracy is not affected by the timing 2 is an example of g obtained by simulation using the
error. Denote the coarse timing error (CTE) by ec CM2 channel model. In this example, the block
number of symbols. Consider the low complexity length is N = 128. Due to the coarse timing error, the
least square (LS) CE. If M = 1, the frequency domain actual CIR is cyclic shifted, and thus some strong
CE is given by H(k) = 75$^, where PT(k) and Pt(k) taps at the end of the vector g are observed which
denote the received and transmitted frequency correspond to the channel taps shifted. The first
domain training symbol at the A:th subchannel significant tap is near the end of the vector g,
respectively. The estimation can be improved by meaning that the current timing is several samples to
employing the repetitive [A B] structure to reduce the right of the exact timing point. The first tap of g
the noise effect. In addition, the cyclic structure can be determined by (6)-(9) and explained as
given by B and A provides IBI-free CE as long as the follows. Once an estimated channel tap g(i) is above
coarse timing error |ec| < Nx. In this case, the a normalized threshold ?75max» the energy of f
estimated channel is given by channel taps on the right side of i represented by
Hk = Hke2ck/N + Wk er{i) is compared to the energy of £ channel taps on
(5) the left side of i represented by e/(i), to evaluate the
where Wk is the Gaussian noise in frequency domain. energy jump on the point i. The first channel tap to is
Since the proposed preamble has a cyclic detected at the position where the energy jump is the
structure, the estimated frequency domain channel largest. Having a window of £ channel taps is to
using this preamble does not suffer from the coarse reduce the occurrence of mistakenly choosing a noise
timing error induced IBI that is usually present in tap as the first tap. Since before the first tap there are
conventional schemes. The impact of coarse timing only noise, whereas er(i) contains energy from the
error on the CE with conventional preambles will be channel taps and noise, er(i) — ei(i) cancels the noise
shown in Section IV-C. Performing inverse FFT and can well detect the jump point from the noise
(TFFT) on Hk, the estimated CIR g is the CIR h only region to the starting of the CIR g. The
cyclicly shifted by the timing error of c samples, determination of parameters i] and 4 will be
plus the Gaussian noise. The CTE c can be discussed in Section V. Upon obtaining the coarse
estimated from g and then used to adjust timing for timing error, the synchronizer can adjust timing
the subsequent data blocks, which has no cyclic position by ec samples, and at the same time, the CE
property to use as the preamble does. The coarse result should also be modified corresponding to the
timing error estimation is obtained by an energy timing adjustment by H'k = Hker^^klN. Note that in
jump detector written as (6), we determine the timing error to the left or right
of the exact timing by the search result being in the
first half or the second half of the CIR. Therefore, the
 N
 -T0 if 0  T0  N
 C  N P  T0 2 

fine timing can handle at most coarse timing
N 2
 if  T0  N  error, which is usually more than sufficient. The
 2  block diagram of the entire timing synchronizer for
(6) SC-FDE UWB systems is shown in Fig. 3. When the
where timing metric module detects the coarse timing point,
it passes the received signal to the subsequent
 max E (i) 0  i  N - 1, modules. The LS estimator estimates FD channel
T0 response Hk based on the coarse FFT window, and
(7) then the coarse timing error is estimated by the CTE
estimator based on the coarse CE result. Finally the
CTE estimation is used to adjust the FFT window
if | g i |  n | g max | position and the FDE channel coefficients.
E i   e r i   el i 
0

if | g i |  n | g max | C. Performance of Fine Timing Adjustment
(8) The performance of the fine timing
adjustment is solely dependent on the accuracy of the

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first tap search, given a set of estimated channel taps. 2
The conventional energy detector [6], [14], [15] variance  . Similarly, the t adjacent taps on the
usually measures the energy in a window of channel 1
taps and takes the starting point of the maximal left side of (11) have approximately equal variance
measurement window as the first tap position. 2
However, this method requires the knowledge of the
denoted by  . Then both sides of (11) are scaled
2
exact channel length, which is usually difficult to Chi-square distributed random variables (r.v.s) with
estimate accurately in UWB channels. Setting the 2t degrees of freedom and the component Gaussian's
window too long may lead to an early timing while
2 2 2 2
having the window too short may lead to a late variance  + and  +  respectively,
timing. The proposed energy jump detector 1 w 2 w
overcomes this drawback of the conventional energy 2
detector. Its window length does not need to equal where  is the variance of the noise component
the exact channel length and usually a window length w
less than the channel length is sufficient. With a in the coarse CE result. Then the probability in (11)
moderate window length, the energy jump detector can be written as
can achieve accurate timing. Note that another
 
energy jump detector has been proposed in [16],
PED (t )   f t ( x)  f r ( y )dydx
where the metric is in an energy ratio form while that 0 z
of the proposed detector is in a subtraction form. The (12)
detector in [16] is able to deal with the more realistic where fi(x) and fr(x) represent the
channel with continuously varying delays at the cost probability density function (PDF) of the left side
of higher complexity. That is, CIR is estimated more and right side of (11) respectively. The PDF of a
frequently in [16]. In the following, we study the scaled Chi-square r.v. of 2t degrees of freedom is
timing error probability of the proposed energy jump
f x  
1 x / 2 2
detector in comparison with that of the conventional x t 1e  , where  2 is the
2
energy detector. The energy jump factor at the exact  2
timing point d can be written as component Gaussian’s variance. Substituting the
PDF of the Chi-square r,v. into (12), the error
 1  -1 probability can be written as
E d    | g d  k  mod N | 2  | gd - k - 1 mod N | 2
k 0 k 0 1
PED (t ) 
 1
=
r t 
 | h k   d  k  mod N | 2  |  d  k  1 mod N | 2    12   
 2
2


k 0
t 1   2 
1  1  t  k 
(10)
where Wk is the noise component of the kth  k! 
t k
=
k 0   12     2
element of the estimated CIR vector. We study the 1  2
   2  
case where the window length is less than the
 1  
channel length (£ < L), usually resulting in late
1 t 1 1  0k
 t  k  (13)
timing with the conventional energy detector. For the
case of f > L, the analysis is similar. r t  k 0 k! 1   0 t  k
Consider a late timing event of t samples
where t > 0. For the conventional energy detector, where  . is the Gamma function with
the error probability that a late timing of t samples
 12   
2
occurs is given by k   k - 1! and  0  Indeed 70
 2 
1 2

 t -1
PED (t )  Pr   | hk   d  k  mod N | 2 <
represents the energy ratio of the estimated channel
tap (including noise) at the actual first channel tap
 k 0 position over that at the + 1 actual tap position. This

 | h 
t 1
  d    k  mod N | 2 . …(11)
definition takes into account the power decaying
k over time.
k 0 For the proposed energy jump detector, the
Following [15], we consider the complex event of timing error of t samples happens when E(d)
Gaussian channel taps with variances decaying along < E(d + t). Similar to (10),
the index. Assume the small number of t adjacent
channel taps on the right side of (11) have the same

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 -1  -1
1,t t    1
t  l  2!
E d  t    | g d  t  k  mod N |2  | g d  t  k  1 mod N | 2
t 1

k 0 k 0 t  1! 2  1 t l 1
= (20)
 1
  d  k  t  mod N | 2  2,t t    1
t  l  2!
| h
k 0
k t
t 1

t  1!1   2 t l 1
-
 1t t 1

 |  d  k  1 mod N |2  | hk   d  k  mod N |2 .
(21)
1
0 
 
k 0 k 0 Defining , and substituting (17)
(14) 4  
22
1
Then the probability of the event for the proposed and the PDF of the Chi-square r,v. into (16), we have
energy jump detector can be written as '
 0t  1t  2t 2 t t 1
PEJ (t )  Pr E d   E d  t  = PEJ t     
t  k 1 l 1 i 0
 t 1 
Pr 2 | hk   d  k  mod N | 2   k ,l   k  i l t 1

 k 0 
k
  i t -1  e 0 k x
dx
i!l  1! 0

t -1
  (| h   d    k  | 2  |  d -   k  mod N | 2 )). 0t 1t  2t 2 t t 1

k 0 =
t  k 1 l 1
 i 0
(15)
The left side of (15) is a scaled Chi-Square r.v. with  k ,l   k  ki l t 1i  t 
2t degrees of freedom and the component Gaussian's =
i!l  1! 0   k 
i t
2 2
variance 2(  +
 1l 1 t  l  2!i  t 
), while the right side of (15) t t 1
1
1 w
is a sum of two scaled Chi-Square random variables t 

l 1 i  0 t  1!i!l  1!
with It degrees of freedom and the component
2 2 2   1i l 1 2
t

  and  
Gaussian's variances . Then     t l 1 1   i t +
3 w w  2 1 1
the timing error probability can be written as:
    2l 1 1t
i

PEJ (t )   f l ( x)  f r ( y )dydx   f L ( x) Ϋ 
i t 
0 x 0  1   2  1   2  
t  l 1

x dx (16) (22)
Where Ϋ  x 

 T  y dy. According to [17],
The energy ratios defined by are
=  X related to the signal-to-noise-ratio (SNR) and the
Ϋ (x) can be written as UWB channel statistics, and they determine the
Ϋ (x) = performance of the estimator. Fig. 4 shows the
 k ,t   k   k x  e
2 t t -1 i  k z timing error probability comparison of the proposed
1t  2t   t - 1! t 11  energy jump detector and the conventional energy
k 1 t 1 k i 0 i! detector with specific energy ratios. In the proposed
(17) energy jump detector, the first channel tap searching
Where algorithm has slightly higher computation com-
plexity than the conventional energy detector. It
1
1  requires calculating the energy of two windows and

2  
22
2  performing a subtraction while the conventional
detector only computes one. However, the window in
(18) the energy jump detector is usually shorter than the
channel-long window of the conventional energy
1
2  detector.
2 
2

IV. The Impact of Timing Error on the SC-
(19) FDE Performance
The timing error impact on a general
OFDM system and on an MB-OFDM system was

1908 | P a g e

analyzed in [18] and [19] respectively. However, the and BFFT, the demodulated block can be written as
timing error impact on SC-FDE UWB systems has
not been well studied. The sub-symbol level timing x  p =

F H CAFxm  p   F H CF h  p   CFn
offset analysis and timing jitter analysis on SC-FDE m
UWB systems were studied in [3], where it was 1
shown that the sub-symbol level timing offset (26)
determines the sampling position of the equivalent Where F and FH are the N x N FFT and
channel and even the worst sub-symbol level timing IFFT operation matrices respectively, and A is
offset only results in small performance degradation. diagonal representing the estimated frequency
Timing jitter was found to be more detrimental to domain channel response. In the presence of timing
system performance. Here, we study the symbol offset, the quality of CE is degraded as will be shown
level timing offset which will be shown to have more in Section IV-C. However, employing our proposed
adverse effect than sub-symbol level timing offset. preamble and the joint timing and CE algorithm in
Section m, the effect of timing offset on CE is only a
A. Small Delay Spread Channels phase shift in the frequency domain. Therefore, Hk =
Small delay spread channels refer to Hke:>2'*krn/N ignoring the noise effect, where the
channels with the maximum delay shorter than CP, phase shift is directly included in the CE. Let *1>m
i.e. Ng > L. Let m denote the timing error, and = FHCFhy and the (fe,n)th (the matrix index starts
assume m < Ng — L. When m. < 0, the starting from 0) element of *1'm is given by
point of the established FFT window falls in the IBI-
free part of CP, leading to the same performance as 
that of the exact timing point. However, when m > 0, 
the established FFT window is on the right of the 0
 n  N-m
correct window, meaning that the current block m
k1,, n 
window contains m samples of the next block.
 N 1 N 1 2i  m - t  h 
N - m  n  N -1
Assuming carrier frequency is perfectly
 1  C h 
j
N
synchronized, the pth received block as shown in
Appendix I can be written as  N i 0 k 0 i t  n

Y P  h1 x1  p   n
m m (27)
Then the demodulated block can be
rewritten as
(23)
where h is the Toeplitz channel matrix with N 1
x p     x p   I   1,m x 1  p   n
1
 
m
h0 0 ... 0 h L h L-1... h1 1xN as its first row. xm (p) k isi
N k 0
is the N x 1 desired pth data block left cyclic shifted
by m (> 0), n is the Gaussian noise, (28)

x m  p =
2
| Hk |
Where k  , I isi is the
 | H k |  N 0 / 2 Eh 
2

0,... 0, x p 1, - ... N g  Tp 0, x p 1, - N G 1 - Tp,1, ...,

inter-symbol interference from the current block due
 N m
 to the fact that an MMSE receiver is not inter-symbol
interference (ISI) free [2], and n is the Gaussian
x p 1,  N g  m1 - x p,m-1
T

N x1
noise
N
term
N-1
with
N-1
variance

i2lk

(24)
  02
2N
2
n |  C e
I 0 K 0
k
N
| 2 . The ith element
where xpn denotes the nth (—Ng < n < N — 1) N 1
symbol in the pth block, and
 0  N -m  xm 
of I isi is  S i x
n 0
n p ,n Where

 h0 0 ... 0  n i

0 N - m  x  N  m   1 N -1
S n i     k e  j 2 n-i k / N . The third term in
m
h1   h 1 h 0 ... 0 
 0mx n  m   N k 0
  
  (28) is the timing error induced interference. Define

 hm 1 ... h0 
 the index set [0, N—1] as U and the subset [0, m—
1] as U. Then for i e U, the demodulated symbol
The received block in (23) can be seen as
can be written as
the combination of two convolutions plus the
AWGN: one is the circulant convolution between xm  1 N 1 
(p) and the channel h, and the other one is the linear x pmi     k - i, N mi  x p ,i
1, m

convolution between h and xy(p). After FFT, FDE,  N k 0 

1909 | P a g e

+  
in 
 block
 ISI
   
  k -1  k
  
N

Pe x p ,i   Q 
N -1 m 1

 S n i x p,n  S n i   iI, , N mn x p,n +  , i  U / U1
m 0

 n2

 N  s i   E
n m n 0 2
n i

m 1  b 
 iI, , N mn x p1,  N 9  n  nij
m
(33)
n 0 where 5R{-} is taking the real part of a complex
    number. Then the overall average BER can be
IBI N 1

 P x .
1
(29) obtained by Pe 
and for i  U/U 1, the demodulated symbol can be
e p ,i
N i 0

written as B. Large Delay Spread Channels
In the large delay spread channels, there is
N -1 m 1 often no IBI-free region within the CP, and hence
1
 x p ,i   i1,, N m n x p 1,  N g  n
m
x p ,i  k
timing error on both sides of the exact timing point
N k 0 n o
   
may cause performance degradation. When m > 0,
IBI
the established EFT window contains multipath
+ components from both the previous and the next
in  block. Considering the case when timing error is
 block
 ISI
  small, i.e., m < L-Ng, and letting v = L — Ng — m,
N -1 m 1

 Sn i x p,n  S n i   iI, , N mn x p,n  ni
m
 the demodulated block can be written as

 
n m n 0
x p   F H CAFX m  p   F H CF h1 x p  +
m

(30)
The interference includes the in-block ISI H
F CFn, (34)
which comes from the current block and the IBI
x 2  p and h 2 are given by
 
coming from the succeeding block. There are a large Where
number of independent interference symbols in (29)
and (30), therefore Gaussian approximation can be
applied to the sum of ISI according to the 
x 2  p   x p-1, N -   x p , N-L, x p -1, N - 1  x p , N  L  1,


Lyapunov's central limit theorem as shown in the
T
Appendix B. The variance of the interference for the
zth desired symbol is denoted by a2 (i) which can be
…., x p 1, N-1  x p , N- N g 1, 0, ...0

 (35)
Nx1
written as N -

h h L-2
 hL 
N-1 m1  L-1 
 s2 i   | S n i  | 2 | S n i  - i, N mn | 2
m
1,
 0 h L-1 
  h L- v 1 0x  N - 
n m n 0 h2   0 
n i
 0 0   
m 1
 0 
+ | 
n 0
1, m
i, N mn |2 . (31) 
 0 (N- )x  h L-1 0  N -  x  N  


(36)
Without loss of generality, considering the BPSK
The IBI has contribution from the preceding block
symbols, the bit error rate (BER) is given by Pe(xPii)
and the succeeding block. Note that
= P(5R(£Pii) > 0xp_i = — 1). Therefore, the error
h1 x 2  p   h 2 x1  p   0, and then (34) can be
m   m
probability is given by
  rewritten as

Pe x p ,i   Q 
N
0
m

  k -1 p k   i1,, N  m i  

 , i  U1 
x p  F H CAFxm  p  FH CF h1  h 2 x1  p  x 2  p
m 
 m 

 2 
N  s2 i   n
H
+ F CFn. (37)
 
CF h  h 
 Eb  2, m, m 
Let  F H
1 2 and define the
(32)
index subsets U2 = [0, m, -1) and U3 = [N – L + v, N
– Ng]. Following the same procedure in the previous
subsection, the variance of the interference can be
obtained as

1910 | P a g e

 s2 i    | S i    2 , m , 3, v
where   F CF h 2 and U4 is defined as [N -
H
n i , N  mi |2 +
nU z Ng -v,N - Ng]. Thus, the ith symbol error probability
n i
can be derived as
 | S i   
n
2 , m ,
i , N  Lm
2
| +
nU z  
 
  k 0  k   i3,, i  N  L  m  
n i N 1 m, v
 
P x p ,i   Q    ,iU
nU

 | S n i  | 2   | i2nm, | 2 . (38)
,
 4

,
 2

N  s i  
nU 2 U nU U 3 2
n i  Eb 
 
'For large timing error m > L - Ng > 0, the problem (44)
becomes the same as Subsection IV-A.
Then the error probability of the ith symbol of the  
pth block is given by  
  k 01  k  
N

 N -1  P x p ,i   Q   , i  U and i U 4 .
     2,
 k
  

2
N  s2 i   n


P x p ,i   Q  k 0
i , N  m 1
 Eb 
 , i  U2
 2 
N  s2 i   
(45)
 Ek 
  C. The Impact of Timing Error on Channel
(39) Estimation
In the literature, CE and
  synchronization often employ separate preamble
 N 1 
  k 0  k   i2, i  L  N  
m, v blocks instead of the single preamble proposed in
 , 
P x p ,i   Q    ,iU Section IE, for example in the Multi-band OFDM
 3 standard proposal [1]. Conventional preambles
  2

N  s i  
2 usually begin with synchronization blocks, followed
 Ek  by CE pilot blocks and then data symbols. This
  subsection considers timing error induced
(40) degradation in CE with conventional preamble
blocks. Note that channel estimation in SC-FDE is
  identical to OFDM. Consider the least square CE
  with one pilot block p*. Under a timing error of m (>
  k 01  k
N

P x p ,i   Q 
0) samples, the frequency domain received pilot
 , i  U  U 2  U 3 . block can be written as
 2 
 N  s2 i    
P r m  mPt
m m
 Ek   F h1 P t  n.
(41) (46)
When the timing point is to the left of the exact where  and Pt are diagonal matrix with elements
timing point, i.e., m < 0, the interference comes only
from the previous block, and the demodulated block
e j 2im / N and the
frequency domain pilot block respectively, is
is given by
defined as

x p   F CAFx  p   F CF x  p   F CFn
H m H  
2
H 0....,0, , x ,N
0 g  pt , 0, ; x 0,  N g  1  Pt , ...x0 , N g  m  Pt ,m1 
T
Nx1
h
where xo is the data block following the pilot block.
(42)
where   L  N g, resulting in more multipath Define  Nxm as the last m columns of the matrix
m
components from the previous block. The variance of Fh 1 , and its (k, n)th element is given by
the interference can be written as
nU  1
 i   | S n i    3,
|  | S n i  | | i3 n |2
 m-n
 ,n m    hl e  j 2k (n  m) / N ,
2 2 2
 i n N  Lm
k m  0.
nU nU 4 n 0
n i n i l0

(43) (47)
Then the fcth sub-carrier of Pr can be written as

1911 | P a g e

Pr k / m   H k e j 2km / N Pt k  + 3T5,3T;] is used. Since the focus of this paper is
timing synchronization, perfect carrier frequency
m 1

  x  pt ,n   n k ,
 synchronization is assumed in the simulation. The
k ,n 0,n design parameters Nx and £ are determined
n 0
according to the channel statistics. Fig. 5(a) gives the
(48) coarse timing error statistics, which shows the
The estimated channel at the kth sub-carrier is empirical cumulative distribution function (CDF) of
Pr k / m  the coarse timing error at ^ = 10 dB in CM1-CM4
Hk  .
Pt k  UWB channels. It can be seen that the coarse timing
error is most severe in CM4 which is the most
The signal-to-interference-ratio (SIR) on the kth sub- dispersive channel model, indicating that the
carrier can be written as performance of the auto-correlation based coarse

 
timing estimator is highly dependent on the channel
 | H k Pt k  | 2 condition. The more dispersive the channel is, the
SIRk  = larger the coarse timing error. According to Fig. 5(a)
 m 1

 |   ,n x0,n  Pt ,n  | 2 
k
we set Nx = 30. The choice of design parameter £ in
 n 0  the CTE estimator depends on the channel length
| H k | | Pt k  |
2 2 statistics. If £ is too small, the estimation error
(49) probability is higher; Otherwise, the computation
m 1 m 1 complexity is higher. We choose £ to be the average
| 
n 0

k ,n
2
| s  | 
n 0

k ,n pt , n | 2
number of channel taps carrying 50% of the channel
energy for different channel models. Fig. 5(b) shows
where Es is the average energy of the data symbols. the CDF of number of taps carrying 50% of the
For m < 0, since the length of the channel delay channel energy, using the channel sampling rate of 2
spread spans L sampling periods, only q = max(L — ns, and 1000 channel realizations for each channel
(Ng + m),0) block symbols are corrupted by the model. It indicates that in CM4 £ can be chosen as
multipath components from the previous block. 30, in CM3 f = 25, while in CM1 and CM2 £ = 15
Assuming that there is a synchronization block and £ = 20 respectively.
preceding the CE block in a non-joint scheme and
with the same transmitting power, it can be derived
that the SIR is given by

 

 | H k | 2 | Pt k  | 2 q0
SIRk   q 1 ,
 | |  |  P |
q 1
q0
  k ,n  k ,n t , p  n
 2  2
s
 n 0 n 0
(50)
n
for m < 0 where  k , n 

h
i 0
L  n i e  j 2ik / N and p
= N — Ng — q. It can be seen that timing error also
affects the CE accuracy and will further degrade the
system performance derived in Subsections FV-A Fig. 10. (a) Average SINR of the coarse CE in CM1;
and IV-B when the conventional preamble pattern is (b) MSE performanceof the channel estimator, with
used. proposed preamble pattern and with theconventional
preamble pattern.
V. Simulation Results and Discussions
MB-OFDM and SC-FDE block The MAE performance shows that the
transmission schemes are for high-rate UWB proposed CE aided timing method outperforms
systems. Thus, in the simulation, we set the block conventional OFDM timing method significantly in
length N = 128, CP length Ng = 32, following [1]. UWB channels. The timing performance
This block length is determined based on practical improvement of the proposed method over the
implementation complexity. However, the resulting conventional method comes from the more accurate
CP does not guarantee to be always longer than a coarse CE and the first tap searching algorithm. The
UWB channel. The UWB channels simulated follow proposed cyclic extended preamble structure can
CM1-CM4 models proposed by the IEEE 802.15.3a provide an IBI-free coarse stage CE regardless. For
Working Group [1]. A root raised cosine (RRC) example, when the coarse timing position is on the
pulse with a roll-off factor of 0.5 and span of [— right side of the true position, the coarse CE in [6]

1912 | P a g e

will include samples of IBI. According to system has been analyzed. Simulation results have
(49) in Section IV, the SIR of the coarse CE shown that the proposed scheme yields accurate
is given by timing and superior CE performance in UWB
| H k | 2 | Pt k  | 2 channels.
SIRk   1

|  ,n   |2  s  |   ,n  Pt ,n |2
n 0
k k
n 0
REFERENCES
[1] "Multi-band OFDM Physical Layer
Proposal for IEEE 802.15 Task Group 3a,"
Where   n   is defined as in (47). The
k, Mar. 2004.
SINR results of the coarse CE obtained from [2] Y. Wang and X. Dong, "Cyclic prefixed
simulation are plotted in Fig. 10(a) for the proposed single carrier transmission in ultra-
scheme and that of [6] in CM1. In addition, the MSE wideband communications," IEEE Trans.
performance of the channel estimator in CM3 and Wireless Commun., vol. 5, pp. 2017-2021,
CM4 are compared in Fig. 10(b). Furthermore, the Aug. 2006.
first tap searching algorithm in [6] is a conventional [3] "Comparison of frequency offset and timing
energy detector, whose performance is inferior to the offset effects on theperformance of SC-FDE
proposed energy jump detector as described in and OFDM over UWB channels," IEEE
Section III-C. Trans. Veh. Technol, vol. 58, no. 1, pp. 242-
250, Jan. 2009.
[4] T. M. Schmidl and D. C. Cox, "Robust
frequency and timing synchronization for
OFDM," IEEE Trans. Commun., vol. 45,
pp. 1613-1621, Dec. 1997.
[5] M. Speth, F. Classen, and H. Meyr, "Frame
synchronization of OFDM systems in
frequency selective fading channels," in
Proc. IEEE Veh. TechnoLConf., vol. 3,
May 1997, pp. 1807-1811.
[6] H. Minn, V. K. Bhargava, and K. B. Letaief,
"A robust timing and frequency
synchronization for OFDM systems," IEEE
Trans. Wireless Commun., vol. 2, pp. 822-
839, July 2003.
[7] C. L. Wang and H. C. Wang, "An optimized
Fig. 11. MAE performance of the proposed scheme joint time synchronization and channel
with different parameters. estimation scheme for OFDM systems," in
Proc. IEEE Veh. Technol. Conf., May 2008,
Fig. 11 shows the timing performance with pp. 908-912.
different design parameters, such as the number of [8] E. G. Larsson, G. Liu, J. Li, and G. B.
repeated [A,B] (M) and in the first tap search Giannakis, "Joint symbol timing and
algorithm. It can be seen from Fig. 11 that in short channel estimation for OFDM based
channel conditions such as CMI or CM2, increasing WLANs," IEEE Commun. Lett., vol. 5, no.
does not improve performance significantly, due to 8, pp. 325-327, Aug. 2001.
noise averaging (though not shown here), thus
leading to better timing performance. However,
increasing M in long channel conditions such as
CM3 and CM4 cannot improve the performance at
high SNRs whereas increasing is more effective.

VI CONCLUSIOIN
A joint timing and CE scheme that employs
a unique preamble design and CIR assisted fine
timing has been presented for UWB block
transmission systems. The preamble proposed saves
the overhead and has the cycle property resulting in
CE and fine timing robust to coarse timing error.
Based on the CE result, a novel algorithm for
estimating the coarse timing error has been
developed for UWB channels. Furthermore, the
impact of symbol level timing error on an SC-FDE

1913 | P a g e

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