CDMA ... - Semantic Scholar
Blind Self-Synchronized Receivers for DS/CDMA Communications Ioannis N. Psaromiligkos and Stella N. Batalama Department of Electrical Engineering State University of New York at Buffalo Buffalo, NY 14260 E-Mail:{ip2, batalama} Qeng.buffalo.edu short-data-record operations, we design an auxiliary-vector (AV)-type [ 1],[3] L-order scheme. For comparison purposes we also develop corresponding structures of order 2L. In addition, we investigate theoretically the effect of the filter order and the data-record-size on the coarse-synchronization-error-rate and we derive analytic expressions that approximate closely the probability of coarse synchronization error of Matched-Filtertype (MF) and MVDR-type combined schemes. Finite datarecord-size probability of coarse synchronization error analysis for AV-type schemes is prohibitively complex and thus not attempted at this time.
Abstract-We consider blind adaptive linear receivers for the demodulation of DSKDMA signals in asynchronous transmissions. The proposed structures are self-synchronized in the sense that adaptive synchronization and demodulation are viewed and treated as an integrated receiver operation. Two computationally efficient combined synchronizationldemodulation schemes are proposed, developed and analyzed. The first scheme is based on the principles of minimum-variance-distortionlessresponse (MVDR) processing, while the second scheme follows the principles of auxiliary-vector filtering and exhibits enhanced performance in short data record scenarios. The coarse synchronization performance of combined synchronizatioddemodulation receivers under finite data record adaptation is also investigated. Analytic expressions are derived that approximateclosely the probability of coarsesynchronization error of the conventional correlator and the MVDR-type combined synchronizationldemodulation scheme and provide low-cost highly-accurate alternatives to the computationally demanding performance evaluation through simulations.
MODEL 11. SYSTEM We consider an asynchronous DWCDMA system populated by I< active users transmitting over a common additive white Gaussian noise (AWGN) channel. The received signal r ( t ) is given by
I. INTRODUCTION The effectiveness of a receiver designed for rapidly changing wireless direct-sequence code-division-multiple-access (DSKDMA) communication environments depends on establishing a successful tradeoff among the following three design objectives:(i) low computational complexity,(ii) multipleaccess-interference (MAI) near-far resistance, and(iii) system adaptivity with superior performance under limited data support. Adaptive short-data-record designs appear as the natural next step [ I ] to a matured discipline that has extensively addressed the first two design objectives in ideal setups (perfectly known or asymptotically estimated statistical properties). System adaptivity based on short data records is necessary for the development of receivers that exhibit superior bit-error-rate performance when they operate in rapidly changing communication environments that limit substantially the available data support. Combined synchronization and demodulation proposals include the work in [2] where an MVDR Sample Matrix Inversion (SM1)-type receiver is developed of order (length) equal to twice the system processing gain. However, as we shall show, the choice of a high order filter combined with adaptive SMI implementations degrades severely the performance of the algorithms under short data record scenarios. In this paper, we propose two blind single-user combined synchronizatiorddemodulation algorithms that result in linear structures of order L , where L is the system processing gain. First, we develop an MVDR-type L-order algorithm. Then, motivated by the limitations of the MVDR structure under This work was supported in part by the AFOSR under Grant 0035 and by the NSF undergrant CCR-9805359.
K-1
as
a b k ( i ) S k ( t - iT
r(t)=
- r k ) + n(t),
(1)
k = O z=-m
where with respect to the Ic-th user, r k is the signal delay, b k ( i ) E (-1, +l}isthei-thinformationbit, Ek isthetransmitted energy, T is the information bit period, and n ( t )is AWGN. The normalized signature waveform Sk(t) assigned to the Icth user has the form Sk(t) = ~ k ( l ) P ~-~lTc), ( t Ic = 0, . . . , I< - 1, where T, is the chip duration, P T ~is a rectangular pulse with support [0, T,],and L = T/Tcis the system processing gain. s k ( l ) is the l-th element of the signature vec-
E;"=,
A
tor S S = [sk(O), sk(l), . . . , s k ( L - 1)ITthat uniquely identifies the k-th user. Without loss of generality, we assume that q. E [0, T ) , Ic = 0, . . . , I< - 1. Thus, we may write the delay r k as a sum of an integer multiple of chips plus a fraction of achip: r k = ( n k 6 k ) T c , where I l k E ( 0 , . . ., L - 1) and 6 k E [O, 1).
+
111. BACKGROUND After chip-matched filtering and chip-rate sampling, the continuous-time signal r ( t ) in (1) is reduced to a discrete time sequence { r ( i ~ ) } ? = - The ~ . samples can be grouped to form A
the input vector r, = [ r ( i L ) ., . . , r((i the order-L linear receiver as follows:
F49620-99-1-
r,
+ l ) L - 1)ITE RL to
& JEabo(i)sf)+ I S I + M A I + n E R~
949 0-7803-6283-7/00/$10.000 2000 IEEE
(2)
-
where
A
of the generated sequence of AV filters {WdJ\.(t)},offer favorable biadcovariance balance and outperform significantly in .---.. A = (1 - 50)[0,. . . , o , S O ( O ) , . . . so(L - 17k - 1)]T f filter. DR Bit-Error-Rate the W ~ , & ~ V The use of the reference vector d as described above alno Go[so(L- 120 - l),. . . , s o ( L - lQ, . ; . , ? J T , ( 3 ) lows the design of DS/CDMA filters that are distortionless in the vector direction of the user signal of interest. In perfectly L-no-1 synchronous systems, for example, this can be achieved if we and n is an AWGN vector with autocorrelation matrix $1. In choose the reference vector d to be the signature vector SO.In (Z), the terms IS1 and AdAI denote the intersymbol interfer- coarsely synchronized systems ( n o = 0) we choose d to be ence and multiple access interference components of r,, respcc- equal to S : , given by the following expression: tively. The decision on the bit 60(i) that modulates Sr)in the s; = (1 - 50)So + (i,Sb+l', (9) vector r, is set to be
sp
where S r 1 ) fi [O, so(O), . . . , so(L - 2)lT denotes the onechip right-shifted zero-filled version of SO. The distortionless property of the designed filters implies that any reduction of the where W E RL is the filter tap-weight vector. In this paper we develop combined synchronization/ demod- filter output energy will be due to interference and noise supulation algorithms that are based on either the MVDR filtering pression. In the event of a coarse synchronization error, howapproach or the AV processing strategy [ 1],[3]. We recall that ever, a receiver distortionless in the S; direction is not necesan MVDR-type filter is a tap-weight vector W that minimizes sarily distortionless in the Sr' direction, which implies that the the output variance/energy E{ IWTr2I?} ( E { . }denotes the sta- desired transmission is no longer protected by the constraint tistical expectation operation) and at the same time is distortion- W& SE = 1 and a portion of it may be suppressed. The lower less in a normalized reference direction d, i.e. WTd = 1. If the cross-correlation between S: and S t ' is the more severe A R = E{r,rT} denotes the autocorrelation matrix of the re- the suppression of the desired signal can be. Therefore, it is exceived vector r,, then the MVDR tap-weight vector is given hy pected that the output energy of a mismatched receiver will be less than the output energy of a receiver designed with perfect R-ld knowledge of S t . In other words, a receiver attains maximum w d , h l V D R = ____ dTR- Id ' output power when it is in perfect synchronization with the deFor the AV-type linear receiver [ 11, [3] the tap weight vector is sired transmission.
&o(i) = sgn(WTr,),
(4)
a member of an infinite sequence of vectors that converges to the MVDR solution and can be obtained by the following rlzcursion:
DESCRIPTION IV. ALGORITHMIC
Our goal is to integrate synchronization and demodulation into a "combined" scheme so that no re-alignment of the received data and re-evaluation of a demodulation filter is needed. To have a receiver readily available by the time synchronization is achieved we propose an L-order combined synchronization/demodulation algorithm that consists of the following two general steps. In the first step, the [O,T)timing uncerGi+i = Rwd,Av(i) - (dTRWd,Av(i))dl and (:I) tainty interval is quantized to a finite number of hypotheses. For each hypothesis, a linear receiver is evaluated. The reGT+iRWd,AV(i) . Pit1 = (W ceiver with the highest output energy over all hypotheses proG:+ 1 RGi +1 vides a coarsely synchronized receiver that operates on a comWe make a parenthesis here to note that the first subscript plete information symbol. In the rest of this paper we refer to of the filter parameter W denotes the reference vector whille the first step of the proposed algorithm as coarse synchronizatthe second subscript denotes the receiver type. When the lat- ioiddemodulation step. In the second step (which we refer to ter is not present then either type is applicable. The MVDF.- as the refined synchronizatiotddenzodulatioristep), we obtain a type and AV-type algorithms outlined above require knowl!- refined structure by maximizing the output energy of a linear edge of the covariance matrix R which is unknown, in prac- combination of two receivers that have been already evaluated tice, and estimated by sample averaging over a$riite set of in the first step. Detailed description of the second step can be data ri, i = 0 , . . . , N - 1. Using the sample-average estx- found in [4]. A A in (5) and (6)-(8) we obtain the mator R = & The coarse synchronization algorithm is summarized below: MVDR and AV filter estimates i?d,,VDR and { w d , 4 V ( i ) ) : : . L-order coarse synchronization algorithm The AV filter was investigated in [3] where it was shown that 1. For 1 = 0, . . . , L - 1, group the received samples { T ( 7 ? ) } , 1 for short data records N , the early, non-asymptotic, elements into L sequences ofvectors, { r : . ' ) >El R L , 1 = 0, . . . , L - 1,
' ; : E
-
950
where
where -(J) A
Sll"ll'
r l r ) = [ l ' ( i L + l ) , l . ( i L + l + l ) ,... , 1 . ( ( i + l ) L + l - l ) ] ? (IO) We note that superscripts in parentheses denote indices (not to be confused with powers). 2. For1 = 0 , . . . L-1, estimate thel-thautocorrelationmatrix R(i) = E { , ~ r ) r ~ i ~ T 1 by
13 / 'J -((J)A
C T
defined by
A G(13/'~)TR(r~~(lj/~J) = o,,..,2L-l, d(J)
d(J)
11o11
(13)
11011
= argmax(OE(3)lj = O , . . . , ~ L -1)
(19)
The probability of coarse synchronization error is defined by A
P,,, = 1 - P
[
li0
-
where TO is the timing estimate given by (15). We recall that , in (15) is the index of the filter with the highest output energy. If we define % ( T O ) to be the set of all possible filter indices that yield a timing estimate TO within $Tr about T O ,then (20)becomes
jmnr
1
5. Let j,,,,
1
( j = 0,. ..,2L - 1 .
-(3'
v . SHORT- DATA-RECOR D PERFORMANCE ANALYSIS
(12)
We note that the superscript of the filter identifies the index of the autocorrelation matrix used for the evaluation of the filter. 4. Calculate the output energy of the receivers
OEW
I
= argiiias OE
The delay TO can be determined as in (15). MF-type algorithmscan be constructed in an analogous manner.
if j is even if j is odd.
L-lj/?l
dllq
=
0 , . . . 2L- 1,that are distortionless in the vector direction d(J lloll)
(18)
' 1
ceiver, j = 0 , . . . , 2 L - 1 then acoarsely synchronized receiver for the demodulation of the transmission of user 0 is given by
3
11011
0 , . . 0IT.
2 2
dlloll
j,,,,,
GL\(2'),j
-
= W-cJ)RW-(J) is the output energy of the j-th re-
If O E
2=0
where the vector is given by (IO). 3. Calculate 2~ AV (or MVDR) receivers
p,.: , o/ s:,
(14)
3
Then a coarsely synchronized receiver for the demodulation of the transmission of user 0 is given by %$;;;('l). A coarse estimate on the delay
Fo =
{
TO is
11o11
given by
LL,la,/2JTc, if j,,,, ([j,,,x/2J 0.5)Tc, if j,,,,,
+
is even is odd.
( 15)
Pm, = 1 - P [ j , , m E 3t(nl)l.
(21)
We emphasize that different definitions of P,,, can be accomodated by modifying appropriately the definition of X ( r 0 ) . A. MF-type receivers ( 2 L-order)
The robability of coarse synchronization error of the 2Lorder d F - t y p e coarse synchronization algorithm is
In the case of the 2L-order algorithms the received samples { ~ ( n ) }are , grouped into a sequence of overlapping vectors, F ~ ER":
#;tifF= 1 - p
r,= [ r ( i L ) , r ( i L + l ) ,. . . , r ( ( i + 2 ) L - 1 f .
- ( 3 A - ( 3 IT2- ( j
h
)
(16) where O E h f ~ d l l Rdlloll. oll We recall that is the sample avOverlined variables always refer to 2L-order processing to erage estimate of the autocorrelation matrix of the received distinguish themselves from the corresponding variables _used 1;2.~ l t h 1;?~ are ~ distributed ~ h according to a mixture by L-order algorithms. The sample average estimate.-. of of 23h- Gaussian distributions and they are, by construction, -A = statistically dependent, in this work we make the assumption the covariance matrix R= E{F,FT} is formed by that the received vectors are uncorrelated and identically Gaus1 N-1 - -T Ciz0 r;ri . As in the case of L-order algorithms, we cal- sian distributed R ) ) (thus i.i.d.). Under the above asculate 2L AV (or MVDR) receivers WZi,, ,j = 0 , . . . ,215 - 1, sumptions, is distributed according to the Wishart distribuh
A
(.u(O,
11011
that are distortionless in the vector direction
defined by
tion W 2 ~ ( R / NN; ) with h
95 1
-0)= N degrees of freedom [ 5 ] ,and dlloll
- I, 1 A - 1 ~ ) 2~- l -11) -1 . degrees of freedom. Since ( X J/ N ) l I 3 is approximately GausThe random quantity OE,,, DR= (dlloll R dlloll1 , J = [ 6 ] we , may appIoxsian withmean (1 - &)and variance 0,.. . , 2 L - 1, is a multiple of a 2’ random variable with -(,IT 4 3 ) A -(.?)T--(J) j = 0,. . . , 2 6- N - 2L degrees of freedom [ 5 ] , so it can be approximated by h a t e dpJ11 Rdlloll by y3 =(dlpll Rdpll 1, where 2, is a Gaussian variable with mean (1 - &) and N - ? L -11 )T - ( - 1 ) - ( J 7 (dllo((R dllollj Z:, where Z, is a Gaussian varivariance &. We can show that the covariances E{Z,Zp}, able with mean 1 - -and variance &. As in the j, (1. = 0, . . . , 2 L - 1, become very small even for small values case of the conventional MF-type coarse synchronization algoof the data-record-size N . Thus, we may assume that the (de- rithm we evaluate the probability of coarse synchronization erA ments of Z = [Zo, . , . , Z Z L - ~are] ~ uncorrelated. Therefore, ror by the probability of coarse synchronization error may be approx(’L) (31) P c s e ,N ~1 -~ ~ ~ ~ imated by:
& )z,”,
-
p,CsztLF
Lo
(23)
fY(Y)dYI
A
=
where, now, & ( E ) is the pdf of the random vector
[ z o , . . . , Z ’ L - ~ ] ~ The . i.i.d. Gaussian random variables -(,F-(-l)-(J) Z, , j = 0 , . . . , 2 L - 1,have mean ( d l l oR ll dlloll)
where fy(y)is the pdf of Y and
I
vo
U
=
{ ( d o ,d i , . . . , d z - 1 ) E RzL
with d j 0 > d ] , j = 0,.. . , 2 L - 1, j
# j,}.
Evaluation of fy(y)is tedious but not necessary since 1; is equivalent to
-bIT --(-1)-b)
(1 - &)
JOE‘H(T0)
(24)
> 1:
and variance (dlloll R
-?/a
dlloll)
2 91,v-z~).
D. MVDR-type receivers (L-based) In a similar manner, we can show that Pec.k,!,l-,R proximated by (L) pcse,Aii/DR 2:
1-
can be ap-
J,, fg(z)&.
(32)
The latter observation allows us to evaluate ( 2 3 ) by irite-
-
-
grating over DO the pdf f i ( Z ) of a Gaussian vector [ZO,. . . , Z ~ L - Iwhere ]~
-
A
-(jlT---(j)
113
Zj = (dlloll Rdlloll) Z j , j = 0, . . . , 2 L - 1
5
A =
B. MF-type receivers (L-based) In a similar manner, we can show that P;,fe:MFcan be iipproximated by
J,, fg(z)&
(28) - A
where f i ( Z ) is the pdf of the Gaussian random vector Z = . . . , 2 2 ~ - 1 ]with ~ elements
[ZO,
2.& (d(j)TR([j/z])d(j) 11o11 11o11) 1 / 3 23. ’ 3
-
5
A
=
(26)
and thus obtain
P ! J L ~ ~ ~NI F1 -
where fg(Z) is the pdf of the Gaussian random vector [Zo,. . . , z z ~ - 1with ] ~ elements
(2!9)
The i.i.d. Gaussian variables Zj , j = 0,. . . , 2L - 1,have mean (1 - &) and variance
&.
C. MVDR-type receivers (2L-based) For the MVDR-t pe coarse synchronization algorithm that utilizes filters of orier 2L the probability of coarse synchroiiization error IS
The i.i.d. Gaussian variables 2,, j = 0 , . . . , 2L - 1, have mean (1 - s($-Lj) and variance 9($-L’). Summarizing the results in this section, the probability of coarse synchronization error is calculated as the probability that the index of the largest element in 5 is not contained in X ( T ~ ) . We see that in contrast to the MF-type algorithms, the mean and variance of 2,, j = 0,. . . , 2 L - 1, (and consequently the performance) of the MVDR-type algorithms depend explicitly on the filter order. Moreover, for the MVDR-type algorithms both the data record size N and the filter order appear only in the terms ( N - 2 L ) and ( N- L ) for filters of order 2 L and L , respectively. Although the filter order terms do not affect the performance of the MVDR-type alasymptotic (as N gorithms, they do cause a performance drop in the small datarecord-size operating region since the filter order term is subtracted from the data-record-size term. In other words, the data record is “effectively” reduced by a number of data vector samples equal to the filter order. The longer the employed filter is the greater the “effective” data-record-size reduction is. For the MVDR-type coarse synchronization algorithms this translates to poorer performance under short-data-record conditions as the filter order increases. This is a direct consequence of the inversion of a Wishart matrix. So, algorithms that do not require the inversion of a Wishart matrix do not suffer from any I
-
952
.cl)
“effective” data-record-size reduction. Consequently, it is expected that the MF-type and AV-type coarse synchronization algorithms are fairly insensitive to the choice of filter length. $0.7
VI. SIMULATION RESULTSAND COMPARISONS We consider a 5-user asynchronous DS-CDMA system that utilizes Gold sequences of length L = 31. We compare the 0.4 P proposed L-order coarse synchronization algorithms with the 2L-order algorithms. We also include as reference points the L MF (simulated) - *- MF (analytic) and 2L-order MF-type receivers. The coarse-synchronizationMVDR (simulated) error-rate is evaluated by averaging over 1,000 independent MVDR (analytic) simulation experiments. The coarse-synchronization-errorAV (simulated) rates evaluated using the expressions derived in Section V are 20 40 60 80 100 120 140 160 180 200 also presented. The delays of all users are chosen randomly and Data-Record-Size are kept constant over 1 0 independent experiments. The auto- Fig. I : Coarse-Synchronization-Error-Rateperformance as a function of the correlation matrix of the received signal is estimated by sample- data-record-size for linear receivers of order L. averaging for both the MVDR and the AV scheme. In Fig. 1 and 2 we show the coarse-synchronization-errorrate performance as the data support size ranges from 5 to 200 samples, for linear receivers of order L and 2L, respectively. The energies of the interferers are fixed at 15, 16, 17, and 18dB. The energy of the user of interest is fixed at 8dB. For the L-order AV-type algorithms, six (6) auxiliary vectors are used, while E Os\ for the 2L-order AV-type algorithms, four (4) auxiliary vectors are used. We observe that the AV-type algorithms are not very 0.4 sensitive with respect to the length of the observation interval. On the other hand, the MVDR-type L-order algorithm exhibits MF (simulated) improved performance when compared to an MVDR-type 2L0 MVDR (simulated) order scheme. Regardless of the filter order we see that the exMVDR (analytic) 0.1 pressions for the coarse synchronization error probability deAV (simulated) rived in Section V offer very close performance estimates. 20 40 60 80 100 120 140 160 180 200 To .quantify the insensitivity of the proposed receivers to Data- Record-Size synchronization errors we compare the BER performance of the Fig. 2: Coarse-Synchronization-Error-Rateperformance as a function of the AV and the MVDR receiver to the performance of their per- data-record-size for linear receivers of order 2L. fectly synchronized counterparts. Fig. 3 depicts the performance of the proposed receivers as a function of the data record size for receivers of order L . The energy of the user of interest is fixed at 14dB. The energies of the interferers are fixed at 14, 15,16, and 17dB.For the AV-typealgorithmsthree (3)auxiliary vectors are used.
1
1
REFERENCES [I]
[2] [3]
[4]
[5] [6]
D. A. Pados and S . N. Batalama, “Joint space-time auxiliary-vectorfiltering for DS/CDMA systems with antenna arrays,” IEEE Trans. Co~iiniim., vol. 47, pp. 1406-1415,Sept. 1999. U. Madhow, ”Blind adaptive interference suppression for the near-far resistant acquisition and demodulation of direct-sequence CDMA signals,” IEEE Trans. SigrirrlProc., vol. 45, pp. 124-136,Jan. 1997. D. A. Pados and G. N. Karystinos, “An iterative algorithm for the computation of the MVDR filter,” IEEE Trans.Sigiicil Proc., submitted. I. N. Psaromiligkos and S. N. Batalama, “Rapid Synchronization and Combined Demodulation, Part I : Algorithmic Developments,” IEEE J Trans. Coninzun., submitted. 50 100 150 200 Data-Record-Size R. J. Muirhead, Aspects of‘mirlrivririare statistical r h e o n , J. Wiley and Sons, New York 1982. N.C. Severo and M. Zelen, “Normal approximation to the chi-square and Fig. 3: Bit-Error-Rate as a function of the data-record-size for linear receivers non-central F probability functions.” Biometrika, vol. 47, pp. 41 1-416, of order L. 1960.
95 3
Abstract-We consider blind adaptive linear receivers for the demodulation of DSKDMA signals in asynchronous transmissions. The proposed structures are self-synchronized in the sense that adaptive synchronization and demodulation are viewed and treated as an integrated receiver operation. Two computationally efficient combined synchronizationldemodulation schemes are proposed, developed and analyzed. The first scheme is based on the principles of minimum-variance-distortionlessresponse (MVDR) processing, while the second scheme follows the principles of auxiliary-vector filtering and exhibits enhanced performance in short data record scenarios. The coarse synchronization performance of combined synchronizatioddemodulation receivers under finite data record adaptation is also investigated. Analytic expressions are derived that approximateclosely the probability of coarsesynchronization error of the conventional correlator and the MVDR-type combined synchronizationldemodulation scheme and provide low-cost highly-accurate alternatives to the computationally demanding performance evaluation through simulations.
MODEL 11. SYSTEM We consider an asynchronous DWCDMA system populated by I< active users transmitting over a common additive white Gaussian noise (AWGN) channel. The received signal r ( t ) is given by
I. INTRODUCTION The effectiveness of a receiver designed for rapidly changing wireless direct-sequence code-division-multiple-access (DSKDMA) communication environments depends on establishing a successful tradeoff among the following three design objectives:(i) low computational complexity,(ii) multipleaccess-interference (MAI) near-far resistance, and(iii) system adaptivity with superior performance under limited data support. Adaptive short-data-record designs appear as the natural next step [ I ] to a matured discipline that has extensively addressed the first two design objectives in ideal setups (perfectly known or asymptotically estimated statistical properties). System adaptivity based on short data records is necessary for the development of receivers that exhibit superior bit-error-rate performance when they operate in rapidly changing communication environments that limit substantially the available data support. Combined synchronization and demodulation proposals include the work in [2] where an MVDR Sample Matrix Inversion (SM1)-type receiver is developed of order (length) equal to twice the system processing gain. However, as we shall show, the choice of a high order filter combined with adaptive SMI implementations degrades severely the performance of the algorithms under short data record scenarios. In this paper, we propose two blind single-user combined synchronizatiorddemodulation algorithms that result in linear structures of order L , where L is the system processing gain. First, we develop an MVDR-type L-order algorithm. Then, motivated by the limitations of the MVDR structure under This work was supported in part by the AFOSR under Grant 0035 and by the NSF undergrant CCR-9805359.
K-1
as
a b k ( i ) S k ( t - iT
r(t)=
- r k ) + n(t),
(1)
k = O z=-m
where with respect to the Ic-th user, r k is the signal delay, b k ( i ) E (-1, +l}isthei-thinformationbit, Ek isthetransmitted energy, T is the information bit period, and n ( t )is AWGN. The normalized signature waveform Sk(t) assigned to the Icth user has the form Sk(t) = ~ k ( l ) P ~-~lTc), ( t Ic = 0, . . . , I< - 1, where T, is the chip duration, P T ~is a rectangular pulse with support [0, T,],and L = T/Tcis the system processing gain. s k ( l ) is the l-th element of the signature vec-
E;"=,
A
tor S S = [sk(O), sk(l), . . . , s k ( L - 1)ITthat uniquely identifies the k-th user. Without loss of generality, we assume that q. E [0, T ) , Ic = 0, . . . , I< - 1. Thus, we may write the delay r k as a sum of an integer multiple of chips plus a fraction of achip: r k = ( n k 6 k ) T c , where I l k E ( 0 , . . ., L - 1) and 6 k E [O, 1).
+
111. BACKGROUND After chip-matched filtering and chip-rate sampling, the continuous-time signal r ( t ) in (1) is reduced to a discrete time sequence { r ( i ~ ) } ? = - The ~ . samples can be grouped to form A
the input vector r, = [ r ( i L ) ., . . , r((i the order-L linear receiver as follows:
F49620-99-1-
r,
+ l ) L - 1)ITE RL to
& JEabo(i)sf)+ I S I + M A I + n E R~
949 0-7803-6283-7/00/$10.000 2000 IEEE
(2)
-
where
A
of the generated sequence of AV filters {WdJ\.(t)},offer favorable biadcovariance balance and outperform significantly in .---.. A = (1 - 50)[0,. . . , o , S O ( O ) , . . . so(L - 17k - 1)]T f filter. DR Bit-Error-Rate the W ~ , & ~ V The use of the reference vector d as described above alno Go[so(L- 120 - l),. . . , s o ( L - lQ, . ; . , ? J T , ( 3 ) lows the design of DS/CDMA filters that are distortionless in the vector direction of the user signal of interest. In perfectly L-no-1 synchronous systems, for example, this can be achieved if we and n is an AWGN vector with autocorrelation matrix $1. In choose the reference vector d to be the signature vector SO.In (Z), the terms IS1 and AdAI denote the intersymbol interfer- coarsely synchronized systems ( n o = 0) we choose d to be ence and multiple access interference components of r,, respcc- equal to S : , given by the following expression: tively. The decision on the bit 60(i) that modulates Sr)in the s; = (1 - 50)So + (i,Sb+l', (9) vector r, is set to be
sp
where S r 1 ) fi [O, so(O), . . . , so(L - 2)lT denotes the onechip right-shifted zero-filled version of SO. The distortionless property of the designed filters implies that any reduction of the where W E RL is the filter tap-weight vector. In this paper we develop combined synchronization/ demod- filter output energy will be due to interference and noise supulation algorithms that are based on either the MVDR filtering pression. In the event of a coarse synchronization error, howapproach or the AV processing strategy [ 1],[3]. We recall that ever, a receiver distortionless in the S; direction is not necesan MVDR-type filter is a tap-weight vector W that minimizes sarily distortionless in the Sr' direction, which implies that the the output variance/energy E{ IWTr2I?} ( E { . }denotes the sta- desired transmission is no longer protected by the constraint tistical expectation operation) and at the same time is distortion- W& SE = 1 and a portion of it may be suppressed. The lower less in a normalized reference direction d, i.e. WTd = 1. If the cross-correlation between S: and S t ' is the more severe A R = E{r,rT} denotes the autocorrelation matrix of the re- the suppression of the desired signal can be. Therefore, it is exceived vector r,, then the MVDR tap-weight vector is given hy pected that the output energy of a mismatched receiver will be less than the output energy of a receiver designed with perfect R-ld knowledge of S t . In other words, a receiver attains maximum w d , h l V D R = ____ dTR- Id ' output power when it is in perfect synchronization with the deFor the AV-type linear receiver [ 11, [3] the tap weight vector is sired transmission.
&o(i) = sgn(WTr,),
(4)
a member of an infinite sequence of vectors that converges to the MVDR solution and can be obtained by the following rlzcursion:
DESCRIPTION IV. ALGORITHMIC
Our goal is to integrate synchronization and demodulation into a "combined" scheme so that no re-alignment of the received data and re-evaluation of a demodulation filter is needed. To have a receiver readily available by the time synchronization is achieved we propose an L-order combined synchronization/demodulation algorithm that consists of the following two general steps. In the first step, the [O,T)timing uncerGi+i = Rwd,Av(i) - (dTRWd,Av(i))dl and (:I) tainty interval is quantized to a finite number of hypotheses. For each hypothesis, a linear receiver is evaluated. The reGT+iRWd,AV(i) . Pit1 = (W ceiver with the highest output energy over all hypotheses proG:+ 1 RGi +1 vides a coarsely synchronized receiver that operates on a comWe make a parenthesis here to note that the first subscript plete information symbol. In the rest of this paper we refer to of the filter parameter W denotes the reference vector whille the first step of the proposed algorithm as coarse synchronizatthe second subscript denotes the receiver type. When the lat- ioiddemodulation step. In the second step (which we refer to ter is not present then either type is applicable. The MVDF.- as the refined synchronizatiotddenzodulatioristep), we obtain a type and AV-type algorithms outlined above require knowl!- refined structure by maximizing the output energy of a linear edge of the covariance matrix R which is unknown, in prac- combination of two receivers that have been already evaluated tice, and estimated by sample averaging over a$riite set of in the first step. Detailed description of the second step can be data ri, i = 0 , . . . , N - 1. Using the sample-average estx- found in [4]. A A in (5) and (6)-(8) we obtain the mator R = & The coarse synchronization algorithm is summarized below: MVDR and AV filter estimates i?d,,VDR and { w d , 4 V ( i ) ) : : . L-order coarse synchronization algorithm The AV filter was investigated in [3] where it was shown that 1. For 1 = 0, . . . , L - 1, group the received samples { T ( 7 ? ) } , 1 for short data records N , the early, non-asymptotic, elements into L sequences ofvectors, { r : . ' ) >El R L , 1 = 0, . . . , L - 1,
' ; : E
-
950
where
where -(J) A
Sll"ll'
r l r ) = [ l ' ( i L + l ) , l . ( i L + l + l ) ,... , 1 . ( ( i + l ) L + l - l ) ] ? (IO) We note that superscripts in parentheses denote indices (not to be confused with powers). 2. For1 = 0 , . . . L-1, estimate thel-thautocorrelationmatrix R(i) = E { , ~ r ) r ~ i ~ T 1 by
13 / 'J -((J)A
C T
defined by
A G(13/'~)TR(r~~(lj/~J) = o,,..,2L-l, d(J)
d(J)
11o11
(13)
11011
= argmax(OE(3)lj = O , . . . , ~ L -1)
(19)
The probability of coarse synchronization error is defined by A
P,,, = 1 - P
[
li0
-
where TO is the timing estimate given by (15). We recall that , in (15) is the index of the filter with the highest output energy. If we define % ( T O ) to be the set of all possible filter indices that yield a timing estimate TO within $Tr about T O ,then (20)becomes
jmnr
1
5. Let j,,,,
1
( j = 0,. ..,2L - 1 .
-(3'
v . SHORT- DATA-RECOR D PERFORMANCE ANALYSIS
(12)
We note that the superscript of the filter identifies the index of the autocorrelation matrix used for the evaluation of the filter. 4. Calculate the output energy of the receivers
OEW
I
= argiiias OE
The delay TO can be determined as in (15). MF-type algorithmscan be constructed in an analogous manner.
if j is even if j is odd.
L-lj/?l
dllq
=
0 , . . . 2L- 1,that are distortionless in the vector direction d(J lloll)
(18)
' 1
ceiver, j = 0 , . . . , 2 L - 1 then acoarsely synchronized receiver for the demodulation of the transmission of user 0 is given by
3
11011
0 , . . 0IT.
2 2
dlloll
j,,,,,
GL\(2'),j
-
= W-cJ)RW-(J) is the output energy of the j-th re-
If O E
2=0
where the vector is given by (IO). 3. Calculate 2~ AV (or MVDR) receivers
p,.: , o/ s:,
(14)
3
Then a coarsely synchronized receiver for the demodulation of the transmission of user 0 is given by %$;;;('l). A coarse estimate on the delay
Fo =
{
TO is
11o11
given by
LL,la,/2JTc, if j,,,, ([j,,,x/2J 0.5)Tc, if j,,,,,
+
is even is odd.
( 15)
Pm, = 1 - P [ j , , m E 3t(nl)l.
(21)
We emphasize that different definitions of P,,, can be accomodated by modifying appropriately the definition of X ( r 0 ) . A. MF-type receivers ( 2 L-order)
The robability of coarse synchronization error of the 2Lorder d F - t y p e coarse synchronization algorithm is
In the case of the 2L-order algorithms the received samples { ~ ( n ) }are , grouped into a sequence of overlapping vectors, F ~ ER":
#;tifF= 1 - p
r,= [ r ( i L ) , r ( i L + l ) ,. . . , r ( ( i + 2 ) L - 1 f .
- ( 3 A - ( 3 IT2- ( j
h
)
(16) where O E h f ~ d l l Rdlloll. oll We recall that is the sample avOverlined variables always refer to 2L-order processing to erage estimate of the autocorrelation matrix of the received distinguish themselves from the corresponding variables _used 1;2.~ l t h 1;?~ are ~ distributed ~ h according to a mixture by L-order algorithms. The sample average estimate.-. of of 23h- Gaussian distributions and they are, by construction, -A = statistically dependent, in this work we make the assumption the covariance matrix R= E{F,FT} is formed by that the received vectors are uncorrelated and identically Gaus1 N-1 - -T Ciz0 r;ri . As in the case of L-order algorithms, we cal- sian distributed R ) ) (thus i.i.d.). Under the above asculate 2L AV (or MVDR) receivers WZi,, ,j = 0 , . . . ,215 - 1, sumptions, is distributed according to the Wishart distribuh
A
(.u(O,
11011
that are distortionless in the vector direction
defined by
tion W 2 ~ ( R / NN; ) with h
95 1
-0)= N degrees of freedom [ 5 ] ,and dlloll
- I, 1 A - 1 ~ ) 2~- l -11) -1 . degrees of freedom. Since ( X J/ N ) l I 3 is approximately GausThe random quantity OE,,, DR= (dlloll R dlloll1 , J = [ 6 ] we , may appIoxsian withmean (1 - &)and variance 0,.. . , 2 L - 1, is a multiple of a 2’ random variable with -(,IT 4 3 ) A -(.?)T--(J) j = 0,. . . , 2 6- N - 2L degrees of freedom [ 5 ] , so it can be approximated by h a t e dpJ11 Rdlloll by y3 =(dlpll Rdpll 1, where 2, is a Gaussian variable with mean (1 - &) and N - ? L -11 )T - ( - 1 ) - ( J 7 (dllo((R dllollj Z:, where Z, is a Gaussian varivariance &. We can show that the covariances E{Z,Zp}, able with mean 1 - -and variance &. As in the j, (1. = 0, . . . , 2 L - 1, become very small even for small values case of the conventional MF-type coarse synchronization algoof the data-record-size N . Thus, we may assume that the (de- rithm we evaluate the probability of coarse synchronization erA ments of Z = [Zo, . , . , Z Z L - ~are] ~ uncorrelated. Therefore, ror by the probability of coarse synchronization error may be approx(’L) (31) P c s e ,N ~1 -~ ~ ~ ~ imated by:
& )z,”,
-
p,CsztLF
Lo
(23)
fY(Y)dYI
A
=
where, now, & ( E ) is the pdf of the random vector
[ z o , . . . , Z ’ L - ~ ] ~ The . i.i.d. Gaussian random variables -(,F-(-l)-(J) Z, , j = 0 , . . . , 2 L - 1,have mean ( d l l oR ll dlloll)
where fy(y)is the pdf of Y and
I
vo
U
=
{ ( d o ,d i , . . . , d z - 1 ) E RzL
with d j 0 > d ] , j = 0,.. . , 2 L - 1, j
# j,}.
Evaluation of fy(y)is tedious but not necessary since 1; is equivalent to
-bIT --(-1)-b)
(1 - &)
JOE‘H(T0)
(24)
> 1:
and variance (dlloll R
-?/a
dlloll)
2 91,v-z~).
D. MVDR-type receivers (L-based) In a similar manner, we can show that Pec.k,!,l-,R proximated by (L) pcse,Aii/DR 2:
1-
can be ap-
J,, fg(z)&.
(32)
The latter observation allows us to evaluate ( 2 3 ) by irite-
-
-
grating over DO the pdf f i ( Z ) of a Gaussian vector [ZO,. . . , Z ~ L - Iwhere ]~
-
A
-(jlT---(j)
113
Zj = (dlloll Rdlloll) Z j , j = 0, . . . , 2 L - 1
5
A =
B. MF-type receivers (L-based) In a similar manner, we can show that P;,fe:MFcan be iipproximated by
J,, fg(z)&
(28) - A
where f i ( Z ) is the pdf of the Gaussian random vector Z = . . . , 2 2 ~ - 1 ]with ~ elements
[ZO,
2.& (d(j)TR([j/z])d(j) 11o11 11o11) 1 / 3 23. ’ 3
-
5
A
=
(26)
and thus obtain
P ! J L ~ ~ ~NI F1 -
where fg(Z) is the pdf of the Gaussian random vector [Zo,. . . , z z ~ - 1with ] ~ elements
(2!9)
The i.i.d. Gaussian variables Zj , j = 0,. . . , 2L - 1,have mean (1 - &) and variance
&.
C. MVDR-type receivers (2L-based) For the MVDR-t pe coarse synchronization algorithm that utilizes filters of orier 2L the probability of coarse synchroiiization error IS
The i.i.d. Gaussian variables 2,, j = 0 , . . . , 2L - 1, have mean (1 - s($-Lj) and variance 9($-L’). Summarizing the results in this section, the probability of coarse synchronization error is calculated as the probability that the index of the largest element in 5 is not contained in X ( T ~ ) . We see that in contrast to the MF-type algorithms, the mean and variance of 2,, j = 0,. . . , 2 L - 1, (and consequently the performance) of the MVDR-type algorithms depend explicitly on the filter order. Moreover, for the MVDR-type algorithms both the data record size N and the filter order appear only in the terms ( N - 2 L ) and ( N- L ) for filters of order 2 L and L , respectively. Although the filter order terms do not affect the performance of the MVDR-type alasymptotic (as N gorithms, they do cause a performance drop in the small datarecord-size operating region since the filter order term is subtracted from the data-record-size term. In other words, the data record is “effectively” reduced by a number of data vector samples equal to the filter order. The longer the employed filter is the greater the “effective” data-record-size reduction is. For the MVDR-type coarse synchronization algorithms this translates to poorer performance under short-data-record conditions as the filter order increases. This is a direct consequence of the inversion of a Wishart matrix. So, algorithms that do not require the inversion of a Wishart matrix do not suffer from any I
-
952
.cl)
“effective” data-record-size reduction. Consequently, it is expected that the MF-type and AV-type coarse synchronization algorithms are fairly insensitive to the choice of filter length. $0.7
VI. SIMULATION RESULTSAND COMPARISONS We consider a 5-user asynchronous DS-CDMA system that utilizes Gold sequences of length L = 31. We compare the 0.4 P proposed L-order coarse synchronization algorithms with the 2L-order algorithms. We also include as reference points the L MF (simulated) - *- MF (analytic) and 2L-order MF-type receivers. The coarse-synchronizationMVDR (simulated) error-rate is evaluated by averaging over 1,000 independent MVDR (analytic) simulation experiments. The coarse-synchronization-errorAV (simulated) rates evaluated using the expressions derived in Section V are 20 40 60 80 100 120 140 160 180 200 also presented. The delays of all users are chosen randomly and Data-Record-Size are kept constant over 1 0 independent experiments. The auto- Fig. I : Coarse-Synchronization-Error-Rateperformance as a function of the correlation matrix of the received signal is estimated by sample- data-record-size for linear receivers of order L. averaging for both the MVDR and the AV scheme. In Fig. 1 and 2 we show the coarse-synchronization-errorrate performance as the data support size ranges from 5 to 200 samples, for linear receivers of order L and 2L, respectively. The energies of the interferers are fixed at 15, 16, 17, and 18dB. The energy of the user of interest is fixed at 8dB. For the L-order AV-type algorithms, six (6) auxiliary vectors are used, while E Os\ for the 2L-order AV-type algorithms, four (4) auxiliary vectors are used. We observe that the AV-type algorithms are not very 0.4 sensitive with respect to the length of the observation interval. On the other hand, the MVDR-type L-order algorithm exhibits MF (simulated) improved performance when compared to an MVDR-type 2L0 MVDR (simulated) order scheme. Regardless of the filter order we see that the exMVDR (analytic) 0.1 pressions for the coarse synchronization error probability deAV (simulated) rived in Section V offer very close performance estimates. 20 40 60 80 100 120 140 160 180 200 To .quantify the insensitivity of the proposed receivers to Data- Record-Size synchronization errors we compare the BER performance of the Fig. 2: Coarse-Synchronization-Error-Rateperformance as a function of the AV and the MVDR receiver to the performance of their per- data-record-size for linear receivers of order 2L. fectly synchronized counterparts. Fig. 3 depicts the performance of the proposed receivers as a function of the data record size for receivers of order L . The energy of the user of interest is fixed at 14dB. The energies of the interferers are fixed at 14, 15,16, and 17dB.For the AV-typealgorithmsthree (3)auxiliary vectors are used.
1
1
REFERENCES [I]
[2] [3]
[4]
[5] [6]
D. A. Pados and S . N. Batalama, “Joint space-time auxiliary-vectorfiltering for DS/CDMA systems with antenna arrays,” IEEE Trans. Co~iiniim., vol. 47, pp. 1406-1415,Sept. 1999. U. Madhow, ”Blind adaptive interference suppression for the near-far resistant acquisition and demodulation of direct-sequence CDMA signals,” IEEE Trans. SigrirrlProc., vol. 45, pp. 124-136,Jan. 1997. D. A. Pados and G. N. Karystinos, “An iterative algorithm for the computation of the MVDR filter,” IEEE Trans.Sigiicil Proc., submitted. I. N. Psaromiligkos and S. N. Batalama, “Rapid Synchronization and Combined Demodulation, Part I : Algorithmic Developments,” IEEE J Trans. Coninzun., submitted. 50 100 150 200 Data-Record-Size R. J. Muirhead, Aspects of‘mirlrivririare statistical r h e o n , J. Wiley and Sons, New York 1982. N.C. Severo and M. Zelen, “Normal approximation to the chi-square and Fig. 3: Bit-Error-Rate as a function of the data-record-size for linear receivers non-central F probability functions.” Biometrika, vol. 47, pp. 41 1-416, of order L. 1960.
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