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Performance of low-complexity code acquisition for direct-sequence spread spectrum systems K. Yu and I.B. Collings Abstract: The authors consider a low-complexity technique for direct-sequence spread spectrum code acquisition. They present performance analysis in Rayleigh fading channels, and with carrier frequency offset in the receiver. They derive analytical expressions for acquisition probabilities, and use them to determine the mean and variance of the acquisition time. It is shown that acceptable performance can be achieved with significant reduction in hardware requirements, compared to conventional detectors. Simulations confirm the analytical results. Performance comparisons are presented under a range of scenarios including significant frequency offsets and fading channels.

1

Introduction

Timing recovery is a vital task which fundamentally affects the performance of digital communications receivers. For direct-sequence spread spectrum communications, one of the most important timing tasks is code acquisition. This is especially the case for high mobility communications, with fading channels. Such a problem is of course well known and studied, given the widespread use of CDMA systems [1]. However, recently the development of wireless LANs has raised new problems. In particular, these systems use very short spreading sequences, and experience large frequency offsets in the receiver carrier recovery circuitry. These two factors have sparked renewed interest in problems of code acquisition. In this paper we are interested in examining lowcomplexity acquisition techniques required for practical WLAN systems with low spreading gain. Of particular interest is the effect of fading and frequency offset. Our analysis will show that acceptable performance can be achieved with considerably lower computational complexity than is commonly thought to be required. Traditionally, code acquisition designs are divided into two stages, namely detection (code search) and verification. In a parallel acquisition mode, the detection stage simultaneously evaluates all the possible code offsets and picks the one with the highest correlation [2–5]. In a serial search, the detection stage evaluates the possible code offsets one after another until an acceptable correlation is found [6–11]. Hybrid schemes have also been considered including partial parallel searches [2, 12]. Performance analyses for these acquisition structures have been presented, in the case of fading channels in [8, 9, 11, 12], and in the case of frequency offset in [13–16]. In fact, it is well known that for a conventional correlation detector, a frequency offset of half the data rate causes a performance loss of about 4 dB. In this paper we consider the combined

effect of simultaneous channel fading and receiver frequency offset. Of course, the task of code acquisition is intimately linked to the transmission modulation and coding format, and to the transmission channel characteristics. The IEEE standard (802.11) for WLANs [17] must be able to operate in a wide variety of environments, providing a tough challenge for the code acquisition detector. The standard specifies a direct sequence spreading code of only 11 chips, and the frequency offset can be up to 62.5 kHz. The task of code acquisition is significantly more difficult than for current mobile standards such as IS95 and 3G systems. This is particularly the case with increasing user mobility, and with the pressure to reduce the computational complexity. A key contribution of this paper is that we consider and analyse the use of a low-complexity sign correlator in the code acquisition system, as opposed to conventional correlation. Sign operations (including sign quantisation, sign multiplication, and sign summation), significantly reduce the complexity of hardware implementation. We derive analytical performance measures for both the lowcomplexity and conventional schemes, in the presence of fading and frequency offset. Expressions are derived for mean acquisition times and associated variances, for both serial and parallel search schemes, and are given as a function of the fading, SNR and frequency offset. The results show that the sign-correlator technique can provide acceptable performance within 1.6 dB of the conventional approach, and do so with significantly reduced complexity. Simulated acquisition probabilities are generated which demonstrate the validity of the assumptions made in our analysis. The simulated results are seen to match closely our analytic predictions. 2

Signal model and detector

Consider the following transmitted signal xðtÞ ¼ sðtÞ cos 2pf0 t

r IEE, 2003 IEE Proceedings online no. 20030742 doi:10.1049/ip-com:20030742 Paper first received 22nd August 2002 and in revised form 28th May 2003 The authors are with the Telecommunications Laboratory, School of Electrical and Information Engineering, University of Sydney, NSW 2006, Australia IEE Proc.-Commun., Vol. 150, No. 6, December 2003

where f0 is the carrier frequency and t sðtÞ ¼ s h t Tc btTc c T rem N þ1 c 453

where rem denotes remainder, and 1 si 2 pffiffiffiffi for i ¼ 1; . . . ; N N

binary correlator

is the spreading code. Also, Tc ¼ T/N, where N is the spreading gain, Tc is the chip duration, T is the symbol period, and h(t) ¼ 1/Tc for 0otoTc and 0 otherwise. We further define the spreading code vector s ¼ [s1s2?sN]T, and the cyclic k-position shifted code vector s(k) ¼ [sNk+1?s1?sNk]T, where 77s772 ¼ 77s(k)772 ¼ 1. Also ðkÞ denote si as the ith element of s(k). For the WLAN applications of interest, we assume the usual complex-valued fading Rayleigh channel, constant over a symbol interval but independent from symbol to symbol. Also, in this paper we consider frequency offset in the transmitter (or receiver). The received signal is therefore pffiffiffi rðtÞ ¼ P jajsðt kTc Þ where P is the signal power, 7a7 is the channel amplitude (with E [7a72] ¼ 1), and kTc is the transmission delay with k a random integer uniformly distributed between 1 and kmax (i.e. chip synchronisation). Also y is the random phase (uniformly distributed), Df is the carrier frequency offset, and v(t) is Gaussian noise. Now consider the low-complexity sign-correlator code acquisition detector. This detector has all correlation calculations performed as binary operations, significantly reducing hardware requirements. We do not suggest that this detector is particularly novel, but rather our aim is to provide analysis which we will then use to demonstrate that such a low-complexity scheme can prove successful even in fading channels with high levels of frequency offset, typical of current WLAN systems. The first stage in the detector is shown in Fig. 1, with quadrature baseband (two-level) quantized outputs, where sgn (x) models a simple 1-bit logic gate. The second stage of the detector is shown in Fig. 2, where the binary outputs from Fig. 1 are binary correlated with the local spreading code, with the appropriate test timing offset.

binary offset code

r(t )

lTc t

∫t−T

d c

+

R(i )

−

test variable

2

()

binary offset code

Fig. 2

Code offset detector using binary correlator

and

0 B yI ð‘Þ ¼ sgn @

Z‘Tc

1 C rI ðtÞ dtA

ð‘1ÞTc

Now since f0c1/Tc we can write ! pffiffiffi srem ‘1k þ1 N yI ð‘Þ ¼ sgn P jaj ½I I ð‘Þ þ nI ð‘Þ Tc

ð1Þ

where nI(c) is an independent zero-mean Gaussian variable with variance s2n , and Z‘Tc

1 I I ð‘Þ9 Tc

cos ð2pDft þ fÞ dt

ð2Þ

ð‘1ÞTc

f9 2pðf0 þ Df ÞkTc þ y

ð3Þ

Now, when Df{1/Tc, I I (c) is approximately constant for all c in the same symbol, so let us instead consider the average integration over an entire symbol interval, as in [18]. Consider c0 to be a chip sampling instance which corresponds to the first chip of a symbol. Now, for cA[c0, N+c01], we define þN1 1 ‘0 X I I ð‘Þ N ‘¼‘ 0

2 cos 0t

1 NTc

ð‘0 þN Z1ÞT c

cosð2pDft þ fÞ dt

ð‘0 1ÞTc t

∫t−T

1 ¼ ½sin ð2pDf ð‘0 1 þ N ÞTc þ fÞ 2pNTc Df sin ð2pDf ð‘0 1ÞTc þ fÞ

lTc d c

sgn

yQ (l ) (binary)

¼ a cos c

−2 sin 0t

Fig. 1

2

j =1

¼

rQ(t )

()

zQ(i)

∑

yQ(l )

yI (l ) (binary)

sgn

T

N

J9

binary chip matched filter

zI(i)

j=1

cos ð2pðf0 þ Df Þðt kTc Þ þ yÞ þ uðtÞ

rI(t )

T

N

∑

yI(l )

Binary chip matched filter

ð4Þ

where sin ðpDfNTc Þ pDfNTc c ¼pDfNTc þ 2pDf ð‘0 1ÞTc þ f a¼

Turning to the mathematical details we have [Note 1] pffiffiffi rf ðtÞ ¼2 P jajsðt kTc Þ ½cos ð2pð2f0 þ Df Þt 2pðf0 þ Df ÞkTc þ yÞ þ cos ð2pDft 2pðf0 þ Df ÞkTc þ yÞ þ vI ðtÞ Note 1: We present only the in-phase component for notational simplicity since the quadrature component is independent and therefore can be derived in the same way.

454

By replacing I I(c) in (1), with this average expression (4), we have pffiffiffi P ajaj cos cs ‘1k þ ZI ð‘Þ yI ð‘Þ sgn rem

N

þ1

We can now write an ‘average’ expression for zI(i) (defined in Fig. 2). At the ith symbol, yI(c) correlates with test-offset ðdÞ code s(d) whose jth element is sj , where c take values from IEE Proc.-Commun., Vol. 150, No. 6, December 2003

c0 ¼ ((i1)N+1)+k to iN+k. Therefore

(1−PD 2)z AT ðdÞ

yI ðði 1ÞN þ k þ jÞ sgn ðsj Þ

ð5Þ

j¼1

j¼1

þ ZI ðði 1ÞN þ k þ ¼

N X

ðdÞ jÞÞ sgn ðsj Þ

pffiffiffi ðkÞ ðdÞ ð1 rj Þ sgn ð P ajaj cos csj Þ sgn ðsj Þ

ð6Þ

PD1zT V1

JT

pffiffiffi ðkÞ sgn ð P ajaj cos csj

(1−PF2)z AT

N X

SEL

PD 2z AT

ACQ

z

STA

PF1z T

zI ðiÞ ¼

N X

T = NTc

PF 2z AT

V2

FA

j¼1

Fig. 3

where

8 ðkÞ > < 2 nI ðði 1ÞN þ k þ jÞsj cos co0; and pffiffiffiffiffiffiffiffiffiffi rj ¼ jnI ðði 1ÞN þ k þ jÞj4 P =N ajajj cos cj > : 0 elsewhere

And the conditional probability for rj ¼ 2 is Z1 1 x2 pffiffiffiffiffiffi exp 2 dx P ðrj ¼ 2j jaj; cÞ ¼ 2sn 2psn f ða;cÞ f ða; cÞ ¼Q sn

Flow graph theory gives the mean and variance of acquisition time as follows [1, 6, 12] d 1 ¼ ð1 þ A þ JPF ÞT ð9Þ ln GðzÞ E½TACQ ¼ dz P D z¼1 ð7Þ

2 d d Var½TACQ ¼ 2 ln GðzÞ þ ln GðzÞ dz dz z¼1 1 2 2 2 ¼ ½J PF ðA þ 1Þ T PD 1 þ 2 ½1 þ A þ JPF 2 T 2 PD

where Q(x) is the Q-function and rffiffiffiffi P ajaj cos c f ða; cÞ ¼ N Now (6) can be further written as zI ðiÞ ¼ sgn ðcos cÞ

N X

ðkÞ

ðdÞ

ð1 ri Þsgn ðsj Þsgn ðsj Þ

j¼1

Due to non-coherent detection, we only need consider the magnitude of zI(i) and hence the term ~zI ðiÞ as follows: ~zI ðiÞ ¼

N X

ðkÞ

ðdÞ

ð1 rj Þsgn ðsj Þsgn ðsj Þ

ð8Þ

j¼1

This model will be used in the following Section to derive the detection probabilities and acquisition times in each case studied. 3

3.1

State-transition diagram of the acquisition process

SEL is the parallel selection stage. The verification stages are denoted V. V1 indicates that the correct offset was tentatively selected, V2 indicates that an incorrect offset was tentatively selected. FA indicates false alarm

Parallel search structure

where G(z) is the generating function from Fig. 3, T is the symbol period, PD ¼ PD1PD2 and PF ¼ PF1PF2. Clearly, we need now to obtain these acquisition probabilities for the low-complexity sign-correlation detector which is the focus of this paper.

3.2

Detection probabilities

3.2.1 Sign-correlator detector with frequency offset: For hypothesis H1 (codes in synchronisation) we have, from (8), that ~zI ðiÞ ¼ N

IEE Proc.-Commun., Vol. 150, No. 6, December 2003

N X

rj

j¼1

Mean and variance of acquisition time

Figure 3 shows the state diagram of the parallel code acquisition detector, similar to that in [12]. Let hypothesis H1 denote that the codes are in synchronisation while hypothesis H0 means codes are not in synchronisation. A maximum selection criterion (MSC) is applied in the search (or selection) stage using a parallel code offset search (usually performed by parallel hardware) which simultaneously tests all possible offsets. At verification stage a threshold-crossing criterion is applied to the selected offset A times. The A tests are performed over disjoint NTc second intervals and are thus statistically independent. If the threshold is exceeded at least B times out of the A tests, then the hypothesis is accepted, otherwise the search recommences. It is assumed that the channel-decoder will detect false alarms and in that case the search recommences after a JNTc delay.

ð10Þ

Let m1 and s21 be defined to be the mean and variance of ~zI ðiÞ. Then using (7) m1 ¼N

N X

E½rj

j¼1

f ða; cÞ ¼N 2N E Q sn

ð11Þ

Recall that 7a7 is Rayleigh, and since y is uniform, so is c. Defining pffiffiffi P a cos c gðcÞ ¼ pffiffiffiffi N sn 455

gives f ða; c 1 E Q ¼ sn 2p

Z2p Z1 QðgðcÞxÞ 0

0

2x exp ½x2 dx dc 1 ¼ 2p

Z2p

1 2

Z1

0

ð12Þ

Now let us move on to derive the probabilities of acquisition. At search stage (‘SEL’ in Fig. 3) the detection probability PD1 is given by 2y 3N 1 Z1 Z ð19Þ PD1 ¼ pR ðyjH1 Þ4 pR ðxjH0 Þ dx5 dy

exp ½x2

0

2 1 g ðcÞ 2 x dx dc pffiffiffiffiffiffi gðcÞ exp 2 2p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Z2p 1 g2 ðcÞ dc 1 ¼ 4p 2 þ g2 ðcÞ

where

ð13Þ

Substituting (13) into (11) yields 2N arcsin u p Likewise the variance of ~zI ðiÞ is m1 ¼

N X j¼1

þ

N N X X

E½rj þ

N X

ð14Þ

h y iN 1 dy 1 exp 2N N 1 N 1 1 m2 X ¼ 2 exp 21 ð1Þn 2sn s1c n¼0 n 1 pffiffiffiffiffiffi Z 1 n m1 2y exp þ y I0 dy 2 s21c 2s1c 2N 0

N 1 X ð1Þn N N 1 nm21 ¼ exp N þ ns21c N þ ns21c n n¼0

E½r2j

ð20Þ

j¼1

The incorrect detection probability at search stage can be obtained in terms of PD1:

E½ri E½rj m21

i¼1 j¼1;j6¼1

f ða; c ¼N 2 4N ðN 1ÞE Q sn 2 f ða; c þ 4N ðN 1Þ E Q m21 sn Substituting (13) into (15) produces 4 2 2 s1 ¼ N 1 2 arcsin u p

PF 1 ¼ 1 PD1

ð15Þ

ð21Þ

At verification stage (‘V1’ and ‘V2’ in Fig. 3) it can be shown that A X A P1i ð1 P1 ÞAi ð22Þ PD2 ¼ i i¼B

ð16Þ

Also, note that the quadrature correlator output, ~zQ , has the same mean and variance as this in-phase component. We now propose to model ~zf ðiÞ and ~zQ ðiÞ as two Gaussian random variables with mean m1 and variance s21 . Of course, the actual ranges of ~zI ðiÞ and ~zQ ðiÞ are finite on [N, N], therefore the variance of our Gaussian model must be greater than s21 in order to achieve a good match for spreading gains of practical interest. Based on numerical studies it turns out that a modified variance, s21c ¼ s21 þ 0:75 m21 , provides an accurate analytical expression. This modified variance was found to provide accurate theoretical predictions of performance in all cases studied, and therefore validates our Gaussian assumption. Clearly, under the Gaussian assumptions, the test variable (from Fig. 2) RðiÞ ¼ ~z2I ðiÞ þ ~z2Q ðiÞ has a non-central chi-square distribution with non-centrality parameter m2R ¼ 2m21 . The pdf of R(i) conditioned on hypothesis H1 is pffiffiffi pffiffiffi! 2m1 R 1 2m21 þ R pR ðRjH1 Þ ¼ 2 exp ð17Þ I0 s21c 2s1c 2s21c 456

Substituting (17) and (18) into (19) gives pffiffiffiffiffiffi Z1 1 2m21 þ y m1 2y PD1 ¼ exp I0 2 2 s21 2s1c 2s1c 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P a2 u¼ 2 P a þ 2N s2n

s21 ¼N 2 2N

0

0

0

1 1 ¼ arcsin u 2 p

For hypothesis H0 (codes not in synchronisation), we again model ~zI ðiÞ and ~zQ ðiÞ as two Gaussian random variables. With random spreading, we can easily find that their means are zero in this case, and their variances are N, which gives 1 R=2N e pR ðRjH0 Þ ¼ ð18Þ 2N

PF 2 ¼

A X A P2i ð1 P2 ÞAi i i¼B

ð23Þ

Here P1 and P2 are the probabilities that (respectively) H1 test sample and H0 test sample cross the threshold. Therefore pffiffiffi pffiffiffi! Z1 2m 1 x 1 2m21 þ x exp P1 ¼ I0 dx s21c 2s21c 2s21c R1 sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi! 2m21 R1 ð24Þ ; ¼Q 2 s1c s21c P2 ¼

Z1

h x i 1 R1 exp dx ¼ exp 2N 2N 2N

ð25Þ

R1

where R1 is the threshold value which is selected numerically to minimize the mean acquisition time at each SNR [11, 12], and Q(x1, x2) is the Marcum Q-function [19]. Substituting (20)–(23) into (9) and (10) will now give the required performance measures for the reduced complexity signcorrelation detector with frequency offset. IEE Proc.-Commun., Vol. 150, No. 6, December 2003

3.2.2 Conventional correlator detector with frequency offset: In this Section we consider the conventional correlator which requires high-resolution correlation in contrast to the binary correlator in the previous section. Our analysis here follows the procedures in [12], but here we extend the analysis to include the effect of constant frequency offset. In this case the in-phase output is pffiffiffi ð26Þ zI ðiÞ ¼ P ajaj cos cðsðdÞ ÞT sðkÞ þ wI ðiÞ where wI(i )is an independent zero mean Gaussian random variable with variance s2w . Conditioned on 7a7 and H1, we have " # 1 P a2 jaj2 þ R pR ðRjH1 ; jajÞ ¼ 2 exp 2sn 2s2w pffiffiffiffiffiffi PRajaj I0 s2w Note that this function does not depend on c. Averaging with respect to 7a7 gives 1 R pR ðRjH1 Þ ¼ 2 exp 2 ð27Þ 2s2 2s2

4.1

Mean and variance of acquisition time

A general expression for the mean acquisition time in this case is given by (1.183) in [1], and here we extend it to include the false-alarm penalty time J. We therefore have the following expressions for mean and variance of the acquisition time in this serial-detection case E½TACQ ¼

0

¼

N 1 X

ð1Þ

i

i¼0

¼

N 1 X

ð1Þi

i¼0

N 1 i N 1 i

Z1

0

2 1 sw þ is22 exp x dx 2s2w s22 2s22

s2w : s2w þ is22 P ð29Þ

and PF1 ¼ 1PD1. At verification stage the detection and false alarm probabilities can be respectively computed using (22) and (23), but where in this case Z1 1 x R2 exp 2 dx ¼ exp 2 P1 ¼ ð30Þ 2s2 2s22 2s2 R2

P2 ¼

Z1 R2

1 x R2 exp dx ¼ exp 2s2w 2s2w 2s2n

ð31Þ

where R2 is the detection threshold determined in the same way as R1. 4

ð33Þ where f0 ¼1 þ APF 1 þ JPF f1 ¼AðA þ 1ÞPF 1 þ ð1 þ 2A þ J ÞJPF and where the remaining variables in (32) and (33) have analogous definitions to those in the previous Section.

4.2

Detection probabilities

Since the search stage in this serial case only tests one code offset at a time, the analysis can be viewed in the same way as that for the verification stage previously, but with A ¼ 1 and B ¼ 1. In other words for the sign correlator the searchstage probabilities, PD1 and PF1 are given in this case by P1 and P2 as given in (24) and (25) respectively. For the conventional correlator the search-stage probabilities are likewise given by (30) and (31) respectively. For both correlators, the verification-stage probabilities are the same as in the parallel search case of the previous Section. 5

Complexity comparison

Table 1 shows the computational requirements for the serial and parallel acquisition approaches during one symbol duration, under the assumption that the output of the conventional chip matched filter is quantised into q levels. Generally conventional correlation uses (q ¼ 16)-bit operations (or higher), but sign correlation only needs (q ¼ 1)-bit Table 1: Computational requirements per symbol duration for processing gain N (for sign-correlation q ¼ 1, while for conventional-correlation q is typically 16 or more) Search stage

Serial search structure

In this Section we consider the case when a serial search and threshold-crossing criterion is applied at search stage, in contrast to the parallel search in the previous Section. The verification stage is the same as in the preceding Section. While serial search takes longer to achieve acquisition compared to the parallel search, it requires less hardware to implement. IEE Proc.-Commun., Vol. 150, No. 6, December 2003

ð32Þ

1 1 þ 2A 2ð1 þ APD1 Þ Var½TACQ ¼ þ f0 2 PD PD2 N þ7 N N 1 2 þ þ f0 12 PD PD2 1 1 þ þ f1 ðN 1ÞT 2 2 PD 1 1 2 2 þ 2 ð1 þ APD1 Þ ð1 þ 2A þ A PD1 Þ T 2 PD PD

where s22 ¼ 0:5a2 P þ s2w . For hypothesis H0 we have 1 R pR ðRjH0 Þ ¼ 2 exp 2 ð28Þ 2sw 2sw At search stage, PD1 is again given by (19). Substituting (27) and (28) gives N 1 Z1 1 x x PD1 ¼ exp dx 1 exp 2s2w 2s22 2s22

1 ½1 þ APD1 þ ðN 1Þ PD T 1 1 PD ðAPF 1 þ JPF Þ 2

Serial Parallel

Verification stage

Serial

Parallel

q-bit multiplications:

N+1

q-bit additions:

N+1


N (N+1)

q-bit additions:

N (N+1)


N+1

q-bit additions:

N


N+1

q-bit additions:

N 457

operations. In other words the sign-correlator only requires one-sixteenth of the hardware. 6

104 sign−simu−500kHz sign−analy−500kHz conv−simu−500kHz conv−analy−500kHz sign−simu−0kHz sign−analy−0kHz conv−simu−0kHz conv−analy−0kHz

Numerical and simulation results

1.0

0.6

100 −5

0

5 10 average SNR per symbol, dB

20

104 sign−simu−500kHz sign−analy−500kHz conv−simu−500kHz conv−analy−500kHz sign−simu−0kHz sign−analy−0kHz conv−simu−0kHz conv−analy−0kHz

0.4

0 −5

15

Fig. 5 Parallel-search analytical and simulated mean acquisition time of the sign-correlator and conventional approaches for two values of frequency offset (0 and 500 kHz), with a spreading gain of 31

conv−analy−0Hz conv−simu−0Hz sign−analy−0Hz sign−simu−0Hz conv−analy−500kHz conv−simu−500kHz sign−analy−500kHz sign−simu−500kHz

0.2

0


15

20

102

100 −5

0


15

20

Fig. 6 Parallel-search analytical and simulated standard deviation of acquisition time of the sign-correlator and conventional approaches for two values of frequency offset (0 and 500 kHz), with a spreading gain of 31

1.0

0.8 detection probabilities

Using these simulated and analytical detection probabilities, we then computed the corresponding mean acquisition time and standard deviation. The results are shown in Figs. 5 and 6 (note that the curves for the conventional correlator with 0 kHz frequency offset can be related to the simulation figures in [12], however in that paper they considered a partial parallel structure, and had a slightly different fading model). Here we set the verification-stage parameters to check for B ¼ 2 threshold crossings out of A ¼ 3 tests. We also set the false-alarm penalty variable to J ¼ 100. Actually this is somewhat arbitrary since it depends on the particular equaliser–decoder used further down the processing chain (e.g. J4100 in [8, 12]). However our main interest is to characterise and examine the performance of the low-complexity sign correlator, and as such we are primarily interested in the relative performance compared to the conventional approach. The figures show that the low-complexity sign-correlation-based acquisition scheme achieves close to the performance of the conventional method (with and without the presence of frequency offset). The performance degradation is about 1.6 dB. Figures 7 and 8 show PD1 and the mean acquisition time, for two different spreading gains. The plots are without the presence of carrier frequency offset. As the spreading increases the detection probability decreases since of course there are more code offsets to test. The results show that the performance degradation of the low-complexity signcorrelator technique is slightly less for lower WLAN

103

101

Fig. 4 Parallel-search analytical and simulated selection-stage detection probabilities (PD1) of the sign-correlator and conventional approaches for two values of frequency offset (0 and 500 kHz), with a spreading gain of 31

458

102

101

standard deviation, µs

detection probabilities

0.8

mean, µs

103

Each of the simulation studies in this Section assumes the modified Jakes model was used to generate the Rayleigh fading channel with normalised fading rate 0.02 (normalised to the symbol rate). In each case, the curves relating to the conventional detector are generated assuming infinite precision, (i.e. there is no quantisation). The data symbol rate was 1 Msymbol/s. Figure 4 shows the detection probabilities, PD1, for the conventional correlator and the sign correlator, with and without carrier frequency offset. Clearly the analytic expressions we have derived in this paper accurately match the simulated results for practical SNR.

conventional−SG15 sign−SG15 conventional−SG127 sign−SG127

0.6

0.4

0.2

0 −5

0


15

20

Fig. 7 Parallel-search selection-stage detection probabilities (PD1) of the sign-correlator and conventional approaches for two different spreading gains (SG ¼ 15, 127) without frequency offset IEE Proc.-Commun., Vol. 150, No. 6, December 2003

104

104

sign−SG127 conventional−SG127 sign−SG15 conventional−SG15

103

mean, µs

mean, µs

103

sign−serial conventional−serial sign−parallel conventional−parallel

102

102

101

101

0


15

100 −5

20

Fig. 8 Parallel-search mean acquisition time of the sign-correlator and conventional approaches for two different spreading gains (SG ¼ 15, 127) without frequency offset

spreading gains. We also see acceptable performance for high spreading, which is interesting since the implementation savings increase significantly as the spreading gain increases. Figure 9 shows the mean acquisition time when the serial search structure is applied (note that the curves for the conventional correlator with 0 kHz frequency offset can be related to the simulation figures in [8], however in that paper they considered much longer spreading sequences, and plotted their curves normalised to the chip rate, rather than the symbol rate for fading and SNR). As can be seen from the figures here, the performance degradation of the sign-

0


15

20

Fig. 10 Mean acquisition times for the four different combinations of correlators and search structures without frequency offset, with SG ¼ 31

4

10

sign−serial conventional−serial sign−parallel conventional−parallel 3

10

mean, µs

100 −5

2

10

1

10

104

0

10

sign−simu−500kHz sign−analy−500kHz conv−simu−500kHz conv−analy−500kHz sign−simu−0kHz sign−analy−0kHz conv−simu−0kHz conv−analy−0kHz

0


15

20

Fig. 11 Mean acquisition times for the four different combinations of correlators and search structures with a frequency offset of 500 kHz, and with SG ¼ 31

mean, µs

103

−5

acquisition schemes, in fading channels and with frequency offset. We also presented performance comparisons to verify the analysis, and showed that the low-complexity scheme performs with only a 1.6 dB performance degradation.

102

101

−5

0


15

20

Fig. 9 Serial-search analytical and simulated mean acquisition time of the sign-correlator and conventional approaches for two values of frequency offset (0 and 500 kHz) and SG ¼ 31

8

The first author was supported in part by Southern Poro Communications (SPC). 9

based technique is even smaller than those in Figs. 5 and 6. Figures 10 and 11 compare the four different combinations of correlators and search structures with and without frequency offset respectively. 7

Conclusion

This paper generated accurate closed form analytical expressions for both low-complexity and conventional code IEE Proc.-Commun., Vol. 150, No. 6, December 2003

Acknowledgment

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