Adaptive pseudo-noise code acquisition scheme using automatic censoring for DS/SS communication in frequency-selective Rayleigh fading channel A. Aissaoui, Z. Hammoudi and A. Farrouki Abstract: An adaptive pseudo-noise code acquisition scheme based on automatic multipath signal cancellation for mobile communication systems in frequency-selective Rayleigh fading channel is proposed. The proposed system combines a serial strategy and an automatic censored cell-averaging constant false alarm rate (CFAR) detector based on ordered data variability. This system does not require any a priori information about the number of interferences caused by the presence of multipath signals in the reference channel. The mean acquisition time and the detection performances of the proposed processor are evaluated and compared with those of the conventional adaptive acquisition scheme based on fixed-censoring point detector. It is shown that the considered scheme outperforms significantly the conventional one. The effects of various channel parameters on the acquisition performance, namely the number of resolvable paths, the partial correlation length and the signal-to-noise ratio are also investigated.
1
Introduction
Accurate and fast synchronisation play an important role in the efficient utilisation of any spread spectrum system. Typically, the process of synchronisation between the incoming spreading pseudo-noise (PN) code sequence and the local dispreading code is performed in two steps: acquisition and tracking. The first step achieves coarse alignment between the two code sequences, whereas the second achieves finer alignment that further reduces the synchronisation error to a required limit. Acquisition methods can be classified as parallel search methods [1– 3] and serial search methods [4 –7]. A parallel search scheme inspects all the uncertainty phases simultaneously and determines the most likely one. If the period of the PN sequence is large, the parallel schemes require excessive hardware and hence become impractical. In contrast, a serial search strategy inspects one uncertainty phase at a time and determines whether the phase of the local and the incoming sequences are aligned. In the alignment case, the tracking circuit is initiated, or, the new uncertainty phase is examined. Hardware requirements of such schemes are minimal but the average time to reach acquisition state increases. Various combinations of the serial search and parallel search are also possible and are called hybrid search [8]. In all these conventional systems, the decision process is based on a fixed threshold that is adjusted under specific conditions. Since the received signal levels in mobile communications are unknown and location varying, the acquisition schemes with a fixed threshold may cause a significant raise in the false alarm rate. As a result, these # The Institution of Engineering and Technology 2008 doi:10.1049/iet-com:20070012 Paper first received 10th January and in revised form 24th April 2007 The authors are with Laboratoire Signaux et Syste`mes de Communication, De´partement d’Electronique, Faculte´ des sciences de l’inge´nieur, Universite´ de Constantine, Route Ain El Bey, Constantine 25000, Algeria E-mail:
[email protected] IET Commun., 2008, 2, (2), pp. 359– 365
systems are unable to provide good performances due to their lack of adaptability to various environments. To overcome this problem, many acquisition systems based on adaptive thresholding have been investigated [9– 12]. In such studies, the system uses the correlation results to estimate the background noise level. In the presence of a low-noise environment, this level is adaptively reduced. Conversely, when the noise power increases, the system automatically raises the corresponding threshold. In addition, due to the multipath propagation reflected by a variety of terrains, the received signal consists of an unknown number of transmitted signal replicas. Each one is characterised by a particular attenuation and time delay. To circumvent the masking effect caused by the presence of multipath signals in the reference channel, Kim et al. introduced in [9] the order statistics acquisition processor (OSAP), where the kth ordered sample is used to estimate the noise power. In [10], the same authors proposed the adaptive acquisition processor (AAP) by estimating the noise power from the lowest uncensored samples. Also it has been shown that the fixed-censoring point detector AAP(k) performs robustly in multipath situations as long as the number k of censored samples exceeds the number of interfering signal replicas. However, in this PN acquisition scheme, the censoring point is preset and invariant for all environments, whereas in practical applications, the number of multipath signals is unknown. Consequently, if the number of censored cells is poorly selected, the corresponding system may exhibit significant detection degradation and an intolerable mean acquisition time increases. In this paper, we propose a new adaptive PN code acquisition scheme in frequency-selective Rayleigh fading channel. The key idea is to discard the outlying samples corresponding to multipath replicas using an automatic censored cell-averaging-ordered data variability (ACCA-ODV) detector introduced in [13]. We call it automatic censoring acquisition processor (ACAP). Unlike the conventional fixed-censoring point acquisition scheme, the ACAP does 359
not require any a priori knowledge of the number of multipath interferers. The paper is organised as follows. In Section 2, we formulate the problem, set the main assumptions and describe the proposed acquisition system. In Section 3, we analyse the performance of the proposed acquisition scheme. In Section 4, we present the simulations performed on this new technique. A summary along with conclusion is given in Section 5. 2
Problem formulation and system description
The adaptive acquisition system based on a serial search strategy in the presence of more than one unknown path signal is considered. This section describes the basic assumptions and the system of interest. 2.1
Basic assumptions
A widely accepted model for multipath fading channel is a tapped delay line with a tap spacing of one chip [14]. As shown in Fig. 1, each tap is multiplied by an independent time-varying complex Gaussian random variable. It is assumed that there are Lp taps corresponding to Lp resolvable received paths. The amplitude and phase of the fading for the ‘th resolvable path are represented by a‘ and u‘, respectively, where a‘ is a Rayleigh random variable and u‘ is a random variable uniformly distributed over [0, 2p). Furthermore, the fading is assumed to vary sufficiently slowly so that its amplitude and phase are constant during an observation interval but change independently from one interval to another. It is also assumed that the diffuse power in each path decay exponentially with rate m. When the total fading power in all the resolvable paths is normalised to unity, the average fading power in each resolvable path is given by [7] E[a2‘ ] ¼
1 em ‘m e , 1 emLp
the chip duration, t the code phase to be estimated, vc the carrier frequency and n(t) the additive white Gaussian noise with two-sided power spectral density N0/2. The received signal, after in phase and quadrature (I– Q) downconversion, is correlated with the locally generated PN code. It should be noted that the normalised correlation between the received and the local codes presents a maximum value in the alignment case. Conversely, under the non-alignment hypothesis, the correlation between the received sequence and the local code exhibits a negligible value and can be approximated by an additive white Gaussian noise. In addition, the replicas are received with different lags as shown in Fig. 1. In this case, the correlation properties of the orthogonal PN codes allow to separate efficiently the different multipath components. In fact, these components are individually located in the reference cells as illustrated in Fig. 2. Note that there could be as many synchro cells as the number of signal replicas. The remaining out-of-phase positions correspond to no synchro cells. After basebanding, and assuming that there is no data modulation during the acquisition process, the sampled I and Q components shown in Fig. 2 can be written as
XI,j
pffiffiffið MTc ¼ 2 r(t)c(t jDTc ) cos(vc t) dt, 0
j ¼ 0, 1, 2, . . . , N and
XQ,j ¼
pffiffiffi ð MTc 2 r(t) c(t jDTc ) sin(vc t) dt, 0
j ¼ 0, 1, 2, . . . , N ‘ ¼ 0, 1, . . . , Lp
(3)
(4)
(1)
where E[.] denotes the statistical expectation. Fig. 1 shows, for the code synchronisation stage, that the real part of the total received signals consists of the contribution of all resolvable multipath replicas. That is
where N represents the reference window size and D a value determines how much timing of the local PN generator is updated during the search process. The value of D is usually 1 or 1/2. Throughout this paper, D is set to 1.
p 1 pffiffiffiffiffiffiffiffi LX r(t) ¼ 2P0 [a‘ c(t t ‘Tc ) cos (vc t þ u‘ )] þ n(t)
‘¼0
(2) where P0 is the signal amplitude, c(t) the PN waveform, Tc
Fig. 1 Tapped delay line model of the frequency-selective channel 360
Fig. 2 Block diagram of adaptive PN code acquisition with automatic censoring technique IET Commun., Vol. 2, No. 2, February 2008
Substituting (2) into (3) and (4) and after some mathematical manipulations, we obtain p 1 pffiffiffiffiffi LX (a‘ S‘,j cos u‘ ) þ NI,j XI,j ¼ P0 Tc
(5)
‘¼0
XQ,j ¼
p 1 pffiffiffiffiffi LX P0 Tc (a‘ S‘,j sin u‘ ) þ NQ,j
(6)
‘¼0
[13]. Then, at the kth step (k ¼ 0, 1, . . .) of the algorithm, we form the rank-ordered population EY ¼ fY(1), Y(2), . . . , Y( p), ygjy¼Y(N 2 k) , and compute the ODV statistic Vk as a shape parameter of the population EY , which is defined as [15] mp þ y2 Vk ¼ (12) (sp þ y)2 y¼Y (N k) where
and ð MTc NI,j ¼
n(t)c(t jDTc ) cos(vc t) dt
(7)
n(t)c(t jDTc )sin (vc t) dt
(8)
sp ¼
0
where NI,J and NQ,J are independent Gaussian random variables with mean zero and variance s2n ¼ MN0 Tc =2 and ð 1 MTc S‘,j ¼ c(t t ‘Tc )c(t jDTc ) dt (9) Tc 0 Under the non-alignment hypothesis, the XI,j and XQ,j are independent Gaussian random variables with zero mean and variance s2n. Note that TcS‘,j corresponds to the partial autocorrelation over M chips of the PN waveform. The outputs of the I – Q correlators, XIj and XQj , are then squared and summed to form the decision variables denoted by Yj given by 2 2 Yj ¼ XI,j þ XQ,j ,
2.2
j ¼ 0, 1, 2, . . . , N
(10)
Automatic censoring
The decision variables are sent serially into a shift register of length N þ 1. The first sample, denoted as Y0 , corresponds to the test cell. The following N samples correspond to the number of reference cells Yj ( j ¼ 1, 2, . . . , N). The Yj’s are ranked first to form the ordered samples Y(1) Y(2) . . . Y(N), where Y( j), j ¼ 1, . . . , N represents the order statistics obtained from the independent and identically distributed random variables Yj . Then, the appropriate ranked cells are censored and the remaining ones are combined to yield an estimate of the background level given by X ¼
N ^i X
Y (j)
(11)
j¼1
where ^i denotes the estimated number of samples to be censored from the upper end, provided by the automatic censoring block, as shown in Fig. 2. The cell-by-cell censoring procedure allows acceptance or rejection of the successive ranked cells by performing iterative data variability-based hypothesis test [13]. The estimator ^i is used to yield the statistic X and to select the corresponding scaling factor Tk (iˆ ¼ k), such that the probability of false alarm (Pfa) is maintained constant in a homogeneous background. Finally, the adaptive threshold XTk is compared with the test cell Y0 . If Y0 exceeds XTk , the corresponding phase is assumed to be the correct one (H1 state) of the received signal and the acquisition system moves to code tracking. Otherwise (H0 state), the relative phase of the locally generated PN sequence is re-adjusted and the above process is repeated. The basic idea of the automatic censoring algorithm is to consider that the p lowest cells represent the initial estimation of the background level, provided that p . 12 IET Commun., Vol. 2, No. 2, February 2008
Y (j)
and
mp ¼
j¼1
0
ð MTc NQ,j ¼
p X
p X
Y 2 (j)
j¼1
At each step k, the ordered subset Ey of length ( p þ 1) consists of a fixed component (the p lowest cells) to which we add the moving observation y ¼ Y(N 2 k) which varies in the interval under investigation [Y( p þ 1), Y(N)]. Then, the statistic Vk is compared with the threshold Sk and a decision is made in favour of Hh or Hnh according to the following hypothesis ODV test Hnh , Vk S . k Hh
(13)
where Hh and Hnh denote the homogeneous and nonhomogeneous hypotheses, respectively. The successive ODV tests (13) are repeated for k ¼ 0, 1, . . . as long as the hypothesis Hnh is true. The algorithm stops whenever the subset under investigation is declared homogeneous or when all the (N 2 p) highest cells have been tested. Finally, the estimated number ^i of censored cells is set to the iteration index, k. On the basis of the above description, the sequential form of the ODV algorithm is formulated as follows [13]. 1. y ¼ Y(N 2 k). 2. Form the rank-ordered population Ey ¼ fY(1), Y(2), . . . , Y( p), yg. 3. Compute the ODV statistic Vk as a shape parameter of the population Ey . 4. Perform the ODV-based hypothesis test (13). Repeat tasks 1 – 4 for k ¼ 0, 1, . . . , until Hh ¼ True or k ¼ N 2 p. 5. ^i¼ k (estimated number of censored cells). In order to reduce the substantial processing time induced by this sequential scheme, a two-level architecture based on a parallel approach has also been proposed in [13]. Consequently, it ensures a lower computational time than the one required for an integration MTc 3
Performance analysis
In this section, the performances of the adaptive acquisition scheme are analysed for the channel model under consideration. To this end, the expressions of the detection probability (Pd), false alarm rate and mean acquisition time are derived. 3.1
Detection and false alarm rate
In the previous section, we derived the statistics of the correlator outputs, XI,j and XQ,j . It should be noted that the decision variables Yj may represent an H1 or H0 state. Due to the Gaussian nature of XI,j and XQ,j , and under hypothesis H1 , Yj has an exponential probability density function (pdf) 361
Table 1: N
p
Threshold multipliers Tk of ACAP for Pfa 5 1024 Tk T0
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
16
12
0.778
1.012
1.287
1.634
2.088
–
–
–
–
–
–
–
–
24
16
0.468
0.561
0.659
0.769
0.896
1.045
1.223
1.439
1.703
–
–
–
–
36
24
0.291
0.331
0.370
0.411
0.455
0.502
0.554
0.612
0.677
0.749
0.831
0.924
1.031
with parameter s‘. That is, the pdf corresponding to the ‘th resolvable path can be expressed as 1 y ‘ fY (yjH1 ) ¼ 2 exp 2 (14) 2s‘ 2s‘
fixed k to be [18] 1 NY k N jþ1 N Pfa ¼ Tk þ N k j¼1 N kjþ1
where H ‘1 denotes the synchro hypothesis for the th resolvable path and s 2‘ represents the variance of the decision variable, which is given by s2‘ ¼ 1 þ nE a2‘ (15)
The Pd‘ is then deduced from the Pfa expression by replacing Tk by (Tk/s2‘ ) in (23). Thus 1 NY k Tk N jþ1 N þ (24) Pd‘ ¼ N k j¼1 s2‘ N k j þ 1
where n is the total received signal-to-noise ratio (SNR) representing the SNR/chip times the correlation interval in chips, set as
For a design Pfa , the threshold multipliers Tk are computed off-line from (23) and stored in a look-up table. Then, the automatic censoring block provides the estimated number of censored cells, ^i ¼ k, to select the corresponding multiplier Tk . Table 1 lists the values of Tk obtained for Pfa ¼ 1024. Since k is assumed to be random, the overall Pd‘ of the ACAP should be the expected value of Pd‘ with respect to the a priori distributions Prob(^i ¼ k) in a homogeneous environment [13].
n¼
P0 Tc N0
(16)
Similarly, the pdf of the decision variable corresponding to a non-synchro cell can be expressed as y 1 (17) fY (yjH0 ) ¼ exp 2 2 Since the adaptive threshold XTk is assumed to be random, the probability of detection for the ‘th resolvable path and the false alarm rate are given by Pd‘ ¼ E{P[Y0 . XTk jH1‘ ]}
(18)
Pfa ¼ E{P[Y0 . XTk jH0 ]}
(19)
Using the contour integral and the moment generating functions (MGF), the Pfa is shown to be [16] X Pfa ¼ Res[v1 FY0 jH0 (v)FX (Tk v), vj ] (20) j
3.2
(23)
Mean acquisition time
In the frequency-selective fading channels, many resolvable paths may exist. From the acquisition viewpoint, the existence of multipath signals implies that there exists more than one synchro cell (H1). Hence, the mean acquisition time can be calculated using the flow graph method [19] as illustrated in Fig. 3. The nodes represent the states, the labelled branches represent state transitions and z indicates the unit-delay operator whose power is the time delay. Let us assume that there are a total of q states that includes Lp H1-cells and (q – Lp) H0-cells. The following conditions are used to derive the mean acquisition time.
where Res [.] denotes the residue, FX the MGF of the total noise estimator X, FY0 jH0 the MGF of the test cell Y0 under the non-synchro hypothesis H0 and vj the poles of FY0 jH0 lying in the left half v-plane. The expression of MGF FY0 jH0 in a homogeneous Rayleigh environment is given by [16] FY0 jH0 (v) ¼
1 (1 þ v)
(21)
Since our model consists of an automatic switching to the appropriate fixed-censoring point CMLD(k) (censored mean-level detector with k censored cells), when exactly k cells among N are censored from the upper end, FX becomes [17] FX (v) ¼
N k Y j¼1
N kþ1j 1þ v N þ1j
1 (22)
Substituting (21) and (22) into (20), and evaluating the residue at a single pole v0¼ – 1, we obtain the Pfa for a 362
Fig. 3 Equivalent circular state diagram for a serial search code acquisition system with multiple timing hypothesis IET Commun., Vol. 2, No. 2, February 2008
ODV threshold Sk of ACAP for Pfc 5 1022
Table 2: N
p
Sk S0
S1
S2
S3
S4
S6
S7
S8
S9
S10
S11
S12
16
12
0.356
0.246
0.199
0.173
–
–
–
–
–
–
–
–
24
16
0.332
0.235
0.189
0.162
0.143
0.131
0.122
0.117
–
–
–
–
36
24
0.230
0.160
0.130
0.113
0.102
0.093
0.088
0.083
0.080
0.078
0.076
0.074
† A uniform distribution of the incoming PN code sequence yields the same probability at the starting of each node. † A start at the correct phase node is excluded. † The processing time of the proposed ACAP scheme does not affect the derivation of mean acquisition time.
manipulations, we can show that PLp Q 2 j¼1 jPdj j1 i¼1 Pdi þ 2Lp PM þ (q Lp )(1 þ JPfa )(1 þ PM ) E[Tacq ] ¼ MTc 2(1 PM )
According to Fig. 3, the acquisition state generating function can be shown to be
In order to obtain the asymptotic form of the mean acquisition time for q 1, we divide both sides of (30) by the total number of states q and compute its limit with respect to q ! 1, yielding
qL
H(z) ¼
1 HD (z)[1 H0 p (z)] (q Lp ) [1 H (z)H qLp (z)][1 H (z)] M 0 0
(25) lim
q!1
where HD(z) and HM(z) include all paths leading to successful detection and miss, respectively. HD(z), H0(z) and HM(z) can be expressed as ( " j1 #) Lp X Y HD (z) ¼ Pdj z (1 Pdi )z (26) j¼1
where and
Q0
i¼1
i¼1
(31)
Consequently, if q Lp then the mean acquisition time can be approximated by E[Tacq ] ’ 4
(1 Pdi )z ¼ 1
E[Tacq ] (1 þ PM )(1 þ JPfa ) ¼ MTc q 2(1 PM )
(30)
(1 þ PM )(1 þ JPfa ) qMTc 2(1 PM )
(32)
Simulation results
As z approaches one, the numerator and the denominator of (29) become zero. In this case, Hospital’s rule is needed to overcome such an indetermination. After some algebraic
In this section, the mean acquisition time and detection performances of the proposed system are evaluated and compared with those obtained for the OSAP and the AAP(k). We consider a periodic PN sequence with a rate of 1 Mchip/s and a full length L of 1023. Since the search step size is Tc (D ¼ 1), the length q of the uncertainty region for the serial search mode is 1023. The penalty factor J is set to 1000 (i.e. penalty time JMTc in seconds). The detection probabilities are simulated from 105 Monte Carlo trials. The synchro cells and non-synchro cells have been generated using (14) and (17), whereas the power of the simulated multipath signals is selected according to an exponentially decaying profile given by (1) and (15). The performances are analysed for frequency-selective Rayleigh fading channels with an exponentially decaying multipath intensity profile m ¼ 1. For the OSAP processor, we take the (3N/4)th ordered sample to estimate the noise power. The associated parameters for the considered
Fig. 4 Detection probability of ACAP with various partial correlation lengths in a homogeneous environment
Fig. 5 Mean acquisition time of ACAP with various partial correlation lengths in a homogeneous environment
H0 (z) ¼ (1 Pfa )z þ Pfa zJ þ1 where J is the penalty factor. And Lp Y HM (z) ¼ (1 Pdj )z ¼ PM z
(27)
(28)
j¼1
where PM ¼
QLp
j¼1
(1 Pdj )
The mean acquisition time may be calculated as [19]
dH(z) (29) MTc E[Tacq ] ¼ dz z¼1
IET Commun., Vol. 2, No. 2, February 2008
363
Fig. 6 Comparison of mean acquisition times of ACAP, AAP and OSAP in a homogeneous environment
situations are shown in the corresponding figures. The ODV threshold Sk are computed such that a low Pfc is maintained at each step of the censoring algorithm. Table 2 contains the values of the ODV threshold Sk obtained for a probability of false censoring (Pfc) of 1022 [13]. Note that a low Pfc should be maintained at each iteration to ensure the same sensitivity for detecting all multipath signals that may fall in the reference channel. Fig. 4 shows the Pd of the proposed system against the SNR/chip for different values of the partial correlation length M. As expected, we note that the greater the partial correlation length, the more detection we obtain. Fig. 5 illustrates the mean acquisition time of the ACAP against the SNR/chip, with M as a parameter. We observe that when the SNR/chip is greater than 214 dB, the mean acquisition time increases for small values of M. On the contrary, when the SNR/chip is less than 214 dB, it seems that the mean acquisition time is better for large values of M. In our simulation, we restrict ourselves to M ¼ 128. Let us now consider a comparison of the mean acquisition time against the SNR/ chip of the ACAP ( p ¼ 12), the OSAP and the AAP(k) for Lp ¼ 1 resolvable path. In this situation, Fig. 6 describes the case where the reference channel contains only one path located at the cell under test. We observe that the ACAP exhibits the same performances as the AAP(0) and performs better than OSAP and AAP(k)jk=0 . Note that the AAP(0) reduces the CA-CFAR which is known to be the most appropriate processor in such a homogeneous environment. Fig. 7 shows a set of curves representing the mean acquisition time against the SNR/chip of the proposed system along with the OSAP and the AAP(4), in the presence of
Fig. 8 Effects of the number of resolvable paths on the mean acquisition time
five and eight paths. For Lp ¼ 5, we observe that the ACAP, AAP(4) and OSAP exhibit the same performances. However, as illustrated for Lp ¼ 8, if the number of paths exceeds the fixed-number of censored cells, we observe that the ACAP outperforms the conventional AAP and OSAP detectors. This is due to the fact that the ACAP does not rely upon any a priori knowledge of the number of multipath signals. To illustrate such an advantage, Fig. 8 considers two cases for which the number of resolved paths Lp varies. The first is at SNR/chip ¼ 22 dB, which is relatively a high SNR and the second is at SNR/chip ¼ 2 16 dB, which represents a low SNR. From these curves, we observe that the AAP(2) and AAP(4) exhibit a significant mean acquisition time degradation whenever Lp exceeds their fixed-censoring points. Whereas the ACAP remains robust in terms of mean acquisition time, by accommodating up to Lp ¼ 11 resolvable paths. This result clearly shows the capability of the proposed processor to dynamically switch to the optimal detector in every multipath situation. 5
Conclusion
An adaptive PN code acquisition system using automatic censoring CFAR scheme has been proposed. The mean acquisition time performances have been analysed in frequency-selective multipath Rayleigh fading channels. To show the efficiency of the proposed scheme, a comparison with conventional systems has been made. Our analysis considered realistic assumptions that are used in mobile communication systems using binary phase shift keying (BPSK) modulation. The effects of the number of resolvable paths, the partial correlation length and the SNR have also been investigated. From the obtained results, we may conclude that the detection performances as well as the mean acquisition time of the ACAP, with respect to an unknown number of resolvable paths, are more robust than those of the OSAP and the AAP(k). 6
Acknowledgment
The authors are grateful to the reviewers for their constructive comments and Dr. T. Laroussi for his valuable help. 7 Fig. 7 Comparison of mean acquisition times of ACAP, AAP and OSAP in multipath situations 364
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