Wireless Networks 5 (1999) 11–21
11
Performance of multiple-dwell pseudo-noise code acquisition with I-Q detector on frequency-nonselective multipath fading channels Wern-Ho Sheen and Sauz Chiou Department of Electrical Engineering, National Chung Cheng University, Chia-Yi, 621 Taiwan, ROC
Multiple-dwell pseudo-noise code acquisition with a noncoherent I-Q detector is analyzed for Rayleigh and Rician fading channels that takes into account the detection correlations resulting from multipath fading. Minimum mean acquisition times with optimized dwell times and thresholds are obtained, and the effects of multipath fading and frequency offsets are evaluated. In addition, a detailed comparison between I-Q and square-law detectors is conducted under various channel conditions.
1. Introduction Pseudo-noise (PN) code synchronization is essential for spread spectrum (SS) systems to work effectively. PN code synchronization is often accomplished in two steps: code acquisition followed by code tracking [5,17,23]. In the code acquisition, the phase uncertainty of the incoming code is divided into “cells”, and the locally generated PN code is brought to within one-cell phase difference with the incoming code.1 Code tracking is a fine alignment that maintains a fine synchronism between the incoming and locally generated PN codes at all times. In this study, we are concerned with PN code acquisition for direct-sequence (DS) SS systems. Many code acquisition schemes have been proposed for DS/SS systems. Notables are multiple-dwell acquisition, acquisition with sequential detection, rapid acquisition with matched filters and some others [1,2,5–8,10–13,17– 23]. The trade-off among different schemes is hardware complexity versus acquisition speed. In practical applications, the multiple-dwell acquisition with noncoherent I-Q or square-law (S-L) detectors is popular due to its low hardware complexity [20]. The I-Q detector with continuous integration is known to have a better capability of noise suppression than the S-L detector, but suffers from a more severe performance degradation due to non-zero frequency offsets [12,20].2 In the literature, the performance of the multiple-dwell acquisition has been extensively investigated for additive white Gaussian noise (AWGN) channels [2,5–8,10– 13,17,22,23]. However, not many works have been devoted to the similar analysis for multipath fading channels yet. In [16], the performance of this acquisition method with a noncoherent square-law detector was analyzed for both Rayleigh and Rician fading channels. In [20], the performance analysis was done for the case of using an I-Q 1 2
For simplicity, the frequency uncertainty will not be considered. The repeatedly reset integration may be used for the I-Q detector to mitigate the effect of frequency offsets [19,20], but at the expense of losing the advantage on noise suppression over the S-L detector.
J.C. Baltzer AG, Science Publishers
detector for Rayleigh fading channels. The repeatedly reset integration was considered in [20]. In this paper, the performance of the multiple-dwell acquisition with a noncoherent I-Q detector (continuous integration) will be analyzed for frequency-nonselective Rayleigh and Rician fading channels. The detection correlations resulting from multipath fading that have been neglected in previous analyses [20] are treated explicitly in the analysis. Mean acquisition times with optimized dwell times and thresholds are obtained, and the effects of multipath fading and frequency offsets are evaluated. In addition, a detailed comparison between I-Q and square-law detectors is conducted under various channel conditions. Numerical results are only given for singleand double-dwell systems, because the most significant performance improvement obtained by increasing the dwell numbers is from the single to the double dwells [2,16]. The remainder of this paper is organized as follows. Section 2 describes the multiple-dwell acquisition system. Section 3 derives the bounds on mean acquisition time for frequency nonselective Rayleigh and Rician fading channels. In section 4, the relative performance of single- and double-dwell systems, the effects of multipath fading, frequency offset and penalty time, and the relative performance of I-Q and S-L detectors will be presented. Finally, conclusions are given in section 5.
2. Multiple-dwell acquisition In a multiple-dwell acquisition system, the phase uncertainty that is divided into a finite number of cells is searched on a cell-by-cell basis until the true acquisition is obtained. For simplicity, the cell size is set to be one chip period in this study, and, therefore, only one cell in the phase uncertainty will be considered as the sync cell (the hypothesis H1 ) and all others as the non-sync cells (the hypothesis H0 ). If a cell is determined, based on a single or multiple tests, as the sync cell, then a sync hit is declared and the synchronization is transferred to code tracking for a fine code alignment. Otherwise, a new cell will be tested
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W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
Figure 1. The multiple-dwell serial search acquisition with I-Q detector.
Figure 2. The I-Q detector.
according to some types of searching strategies [12,13]. The straight line search will be used as an example in this study. A typical block diagram for the multiple-dwell acquisition is shown in figure 1. The received signal is despread by the locally generated PN code and detected with an I-Q detector. The threshold comparator compares the output of the I-Q detector yi with a suitable threshold Vti to decide if the cell under test is in sync, where 1 6 i 6 N , and N is the dwell (test) number. The Search/Lock Strategy (SLS) is to control the way of tests being performed and to control the voltage control clock that drives the local PN code generator to generate suitable code phases. The immediate rejection strategy will be considered for SLS. That is, if yi fails to exceed the threshold Vti for some i 6 N , then the cell under test will be rejected immediately, and a new cell will be searched. On the other hand, if yi exceeds the corresponding threshold Vti for all i, then the tested cell will be considered as the sync-cell, and the SLS enters the tracking mode. For the hypothesis H1 , this means that code acquisition has been accomplished, and the acquisition process comes to the end. For the hypothesis H0 , however, entering the tracking mode turns out to be a false alarm. For
this case, a fixed penalty time TP will be modeled for the operation of the tracking mode. This model is somewhat simplified, but it is sufficient for our purpose of studying the performance of code acquisition. Figure 2 shows the structure of the considered I-Q detector, where wc is the received carrier frequency, ∆w is the frequency offset between the received carrier and local oscillator, and θ is the random phase uniformly distributed in 0 6 θ < 2π. As seen, the integration is continuous within one dwell period and the dwell times {Tdi }, defined ∆ as Tdi = ti − ti−1 , are not overlapping. The dwell times will be assumed to be multiple of the period of the PN code, which is denoted as LTc with Tc denoting the chip period (full period correlation). This is possible for PN sequences with a small-to-moderate period LTc . Therefore, by defining ∆ ci = M
i X
Mj
(1)
j=0
with M0 = 0 and Mi being an positive integer, we can ci LTc and Tdi = Mi LTc . write ti = M
W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
3. Mean acquisition time
bol rate, as in the mobile and personal communication environments. Then, equations (3) and (4) become
For the considered multipath fading channels, the received signal is given by √ r(t) = 2P c(t − τ ) A + xc (t) cos(wc t) − xs (t) sin(wc t) + n(t), (2) where P and wc are the carrier power and radian frequency, respectively, c(t) is an m-sequence with nonreturn to zero (NRZ) shaping function, A and xc (t) (xs (t)) are the specular and diffused components of the fading channels, respectively, and n(t) is additive white Gaussian noise with one-side power spectral density (PSD) of N0 Watts/Hz. For Rayleigh and Rician fading characteristics, xc (t) and xs (t) are uncorrelated zero-mean Gaussian processes with the same variance σg2 . If A = 0, then the channel is a Rayleigh fading channel, otherwise the channel is a Rician fading channel. The channel is assumed to be wide-sense stationary. Note that since the dwell time Tdi is multiple of the period of PN code sequence, the effect of data modulation is small and can be neglected (assume that the symbol time Tb = LTc ). Besides, in some applications, it may be no data modulation for PN code synchronization, e.g., in [3]. From figure 2, we have √ P gc (i) = Mi
ci −1 M X ci−1 j=M
1 LTc
13
Z
(j+1)LTc
A + xc (t)
gc (i) =
√ P Mi
ci −1 M X
A + xc (j)
ci−1 j=M
1 LTc
Z
(j+1)LTc
c(t − τ )
jLTc
× c(t − τˆ) cos(∆wt + θ) dt √ M ci −1 Z (j+1)LTc 1 P X xs (j) c(t − τ ) + Mi LTc jLTc ci−1 j=M
× c(t − τˆ) sin(∆wt + θ) dt nc (i) + Mi LTc
(5)
and √ P gs (i) = Mi
ci −1 M X
ci−1 j=M
1 A + xc (j) LTc
Z
(j+1)LTc
c(t − τ )
jLTc
× c(t − τˆ) sin(∆wt + θ) dt ci −1 Z (j+1)LTc √ M p X 1 xs (j) c(t − τ ) − Mi LTc jLTc ci−1 j=M
× c(t − τˆ) cos(∆wt + θ) dt ns (i) + , Mi LTc
jLTc
∆
(6)
∆
where xc (j) = xc (jLTc ) and xs (j) = xs (jLTc ). Recall that the cell size is one chip period. Therefore, we can assume that τ = τˆ and τ = kTc + τˆ for the H1 and H0 , respectively, where k is an integer (chip synchronous model). After some manipulations and using the fact that xc (t) and xs (t) are stationary random processes, we have, for H1 ,
× c(t − τ )c(t − τˆ) cos(∆wt + θ) dt √ M ci −1 Z (j+1)LTc P X 1 + xs (t) Mi LTc jLTc ci−1 j=M
× c(t − τ )c(t − τˆ) sin(∆wt + θ) dt nc (i) + Mi LTc
(3)
and
√ Mi −1 P X gc,1 (i) = A + xc (j) Hc (∆w, θi,j ) Mi j=0
√ P gs (i) = Mi
ci −1 M X ci−1 j=M
1 LTc
Z
(j+1)LTc
nc (i) + xs (j)Hs (∆w, θi,j ) + , (7) Mi LTc
A + xc (t)
jLTc
× c(t − τ )c(t − τˆ) sin(∆wt + θ) dt √ M ci −1 Z (j+1)LTc P X 1 − xs (t) Mi LTc jLTc
gs,1 (i) =
√ Mi −1 P X A + xc (j) Hs (∆w, θi,j ) Mi j=0
ns (i) − xs (j)Hc (∆w, θi,j ) + , (8) Mi LTc
ci−1 j=M
× c(t − τ )c(t − τˆ) cos(∆wt + θ) dt ns (i) + , Mi LTc
(4)
where nc (i) and ns (i) are mutually independent stationary Gaussian random variables with zero mean and the variances σn2 = (N0 /2)Mi LTc . Assume that the channel is slowly time-varying such that the channel remains unchanged during the symbol time Tb , i.e., the maximum Doppler shift of the channel is much smaller than the sym-
and for H0 , √ Mi −1 P X gc,0 (i) = A + xc (j) Kc (q, ∆w, θi,j ) Mi j=0 nc (i) + xs (j)Ks (q, ∆w, θi,j ) + , (9) Mi LTc
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W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
where
and √ Mi −1 P X gs,0 (i) = A + xc (j) Ks (q, ∆w, θi,j ) Mi
s20 (i) =
j=0
ns (i) − xs (j)Kc (q, ∆w, θi,j ) + , (10) Mi LTc
Mi −1 M i −1 X P A2 X Kc (q, ∆w, θi,j )Kc (q, ∆w, θi,k ) 2 Mi j=0 k=0 + Ks (q, ∆w, θi,j )Ks (q, ∆w, θi,k ) (21)
and s21 (i) =
where ci−1 LTc + jLTc + θ, θi,j = M Z LTc 1 Hc (∆w, β) = cos(∆wt + β) dt, LTc 0 Z LTc 1 Hs (∆w, β) = sin(∆wt + β) dt, LTc 0 Z LTc 1 Kc (q, ∆w, β) = c(t − qTc ) LTc 0 × cos(∆wt + β) dt, Z LTc 1 c(t − qTc ) Ks (q, ∆w, β) = LTc 0 × sin(∆wt + β) dt,
(11) (12) (13)
∆
m = 0, 1.
+
Hc (∆w, θi,j )Hc (∆w, θi,k ) + Hs (∆w, θi,j )Hs (∆w, θi,k ) 2 = Hc (∆w, 0) + Hs2 (∆w, 0) cos (j − k)LTc . (24)
(15)
Therefore, equations (17), (21), (18) and (22) become σ02 (i) =
(16)
i −1 M i −1 X P σg2 MX ρj,k Kc (q, ∆w, θi,j )Kc (q, ∆w, θi,k ) 2 Mi j=0 k=0 + Ks (q, ∆w, θi,j )Ks (q, ∆w, θi,k )
σn2 /(MiLTc )2 ,
Kc (q, ∆w, θi,j )Kc (q, ∆w, θi,k ) + Ks (q, ∆w, θi,j )Ks (q, ∆w, θi,k ) 2 = Kc (q, ∆w, 0) + Ks2 (q, ∆w, 0) cos (j − k)LTc (23) and
It is easy to show that for given q and θ, gc,m (i) and gs,m (i) are mutually independent and have the same variances σ02 (i) =
From the appendix, we have
(14)
and q is a positive integer 1 6 q 6 L − 1. Note that the add and shift property of the m-sequence has been used in deriving equations (14) and (15). From figure 2, 2 2 yi = gc,m (i) + gs,m (i),
(17)
P σg2 2 Kc (q, ∆w, q) + Ks2 (q, ∆w, 0) 2 Mi ×
s20 (i) =
+
σn2 /(MiLTc )2 ,
M i −1 M i −1 X X j=0
M i −1 M i −1 X X
ρj,k
= E xc (j)xc (k) /σg2 = E xs (j)xs (k) /σg2 . ∆
ρj,k cos (j − k)LTc +
k=0
σn2 , (27) Mi LTc
PA Hc2 (∆w, 0) + Hs2 (∆w, 0) 2 Mi M i −1 M i −1 X X j=0
for the H0 and H1 , respectively, where
(26)
k=0
2
×
(18)
cos (j − k)LTc ,
P σg2 2 Hc (∆w, 0)) + Hs2 (∆w, 0) 2 Mi
j=0
s21 (i) =
k=0
σn2 , (25) Mi LTc
PA Kc2 (q, ∆w, 0) + Ks2 (q, ∆w, 0) 2 Mi ×
σ12 (i) =
ρj,k cos (j − k)LTc +
2
× i −1 M i −1 X P σg2 MX ρj,k Hc (∆w, θi,j )Hc (∆w, θi,k ) 2 Mi j=0 k=0 + Hs (∆w, θi,j )Hs (∆w, θi,k )
M i −1 M i −1 X X j=0
and σ12 (i) =
Mi −1 M i −1 X P A2 X Hc (∆w, θi,j )Hc (∆w, θi,k ) 2 Mi j=0 k=0 + Hs (∆w, θi,j )Hs (∆w, θi,k ) . (22)
cos (j − k)LTc .
(28)
k=0
Also, from equations (12) and (13), it is can be shown that Hc2 (∆w, 0) + Hs2 (∆w, 0) = sinc2 (η),
(19)
Therefore, yi is conditionally chi-square distributed with two degrees of freedom with PDFs given by 2 1 √ sm (i) −(s2m (i)+yi )/2σm (i) pm (yi ) = e I y , 0 i 2 2 (i) 2σm σm (i) yi > 0, m = 0, 1, (20)
(29)
where ∆
η=
∆w 2πRb
(30)
and sinc(x) =
sin πx . πx
(31)
W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
So, equations (27) and (28) are rewritten as σ12 (i) =
M i −1 M i −1 X X P σg2 2 · sinc (η) ρj,k cos (j − k)LTc 2 Mi j=0 k=0
+
σn2 , Mi LTc
(32)
and s21 (i) =
M i −1 M i −1 X X P A2 2 · sinc (η) cos (j − k)LTc . (33) 2 Mi j=0 k=0
With equations (20), (25), (26), (32) and (33), the detection and false alarm probabilities Pdi and Pfi for ith dwell can be evaluated by Z ∞ Pdi = p1 (yi ) dyi (34) Vti
and
Z
∞
Pfi =
p0 (yi ) dyi,
(35)
Vti
respectively. Since yi is non-central chi-square distributed, the integrations (34) and (35) are the Marcum Q functions and can be evaluated by the method of saddle-point integration very efficiently [4]. As evident from equations (25) and (26), the false alarm probability Pfi in equation (35) depends on the value of q. And, to obtain the unconditional false alarm probability is to average equation (35) over all possible q. This can be quite time consuming for the case of a large L. Alternatively, for the case of large L a lower bound Pfl i and a upper bound Pfui on false alarm probability can be devised by using ∆ Kmin = min Kc2 (q, ∆w, 0) + Ks2 (q, ∆w, 0) (36)
interest (small ∆w and low-to-moderate SNRs). Hence, the dependency between detections associated with H0 cells can be neglected without too much affects on the value of mean acquisition time. For the cases of H1 , since only one cell is considered as the sync cell, the detections between H1 cells can be safely considered independent because the time between H1 cells tests is usually much larger than the coherence time of the channel. Nevertheless, the detections between dwells under the the same H1 cell are indeed correlated and requires to be considered. This correlation will be taken into account in the analysis. Based on the above discussions, it is clear that the Markov model is approximately valid for the considered acquisition system and multipath fading channels. Therefore, for the straight line search considered, the mean acquisition time is given by [2,5,6,12,17,23] 2 − PD T1 + T acq = (Q − 1)T0 , (38) 2PD PD where ∆
• Q is a positive integer defined as Q = Tu /Tc with Tu denoting the phase uncertainty, • T0 is the average dwell time for the hypothesis H0 : ! N i−1 X Y T0 = Pfj + PF TP , (39) Tdi i=1
j=1
• T1 is the average dwell time under the hypothesis H1 : ! N i−1 X Y T1 = Tdi Pdj|j−1 , (40) i=1
j=1
• PF is the overall probability of false alarm:
q
and
PF = ∆ Kmax = max Kc2 (q, ∆w, 0) + Ks2 (q, ∆w, 0) q
15
N Y
Pfi ,
(41)
i=1
(37)
in equations (25), (26), (20) and (35), for easy evaluation of mean acquisition time performance. As to be seen, the corresponding lower and upper bounds on mean acquisition time obtained by using Pfl i and Pfui are very tight for the cases of practical interest. For the multiple-dwell acquisition, the mean acquisition time can be easily evaluated if the acquisition process can be modeled as a Markov chain model [2,5,6,12,17,23]. In the following we first show that the Markov chain model is approximately valid for the acquisition system and the multipath fading channels considered. Then, the formulas on mean acquisition time obtained previously [2,5,6,12,17, 23] can be employed for the evaluation and comparisons. As evident in equations (25)–(28), detections between cells and dwells are correlated because of the inherent memory of the multipath fading channels. For the cases of H0 , as to be shown, AWGN will be the dominant factor in determining the false alarm probability for the cases of practical
• PD is the overall probability of detection: PD =
N Y
Pdi|i−1 .
(42)
i=1
In equation (40),
Q0 j=1
∆
Pdj|j−1 = 1,
∆ Pdi|i−1 = Pr yi > Vti | yi−1 > Vti−1 , . . . , y1 > Vt1 , (43) and ∆
Pd1|0 = Pr{y1 > Vt1 }.
(44)
Also, from (42)–(44), we know that PD are given by PD = Pr{yN > VtN , yN −1 > VtN −1 , . . . , y1 > Vt1 }. (45) Note that the correlation between different dwells for H1 has been taken into account in equation (38). For a large
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W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
enough Q, the mean acquisition time can be approximated as 2 − PD . (46) T acq = QT0 2PD In general, it is quite involved to evaluate the exact value of the probabilities Pdi|i−1 for the more than two-dwell systems. Hence, a simple lower bound on Pdi|i−1 that is similar to the one used in [16,17] will be employed for the performance evaluation. The lower bound is given by i Y
Pdj|j−1 >
i X
j=1
Pdj − (i − 1),
0 < i 6 N.
(47)
j=1
Therefore, PD >
N X
Pdi − (N − 1)
(48)
i=1 ∆
= PD,L .
(49)
The lower bound in (49) will be used for evaluating mean acquisition time. Note that for N = 1, we have PD = PD,L . This bound has been shown to be quite tight for two-dwell systems that are of practical interest [14]. 4. Numerical results For simplicity, the channel model used in this section is characterized by [9] E xc (mTs )xc (nTs ) = E xs (mTs )xs (nTs ) ∆
= ρ|m−n| σg2 ,
(50)
although the analysis is applicable to other types of channels, where 0 < ρ 6 1 is the parameter that characterizes the fading rate of the channel. γc is the received SNR, defined as ∆
γc =
P Tc · 2σg2 [1 + A2 /(2σg2 )] . N0
(51)
For Rician fading channels, χ (dB) is defined as ∆
χ = 10 log10
A2 . 2σg2
(52)
All the results are obtained by numerically optimizing the system parameters of dwell times and thresholds. Two sets of PN sequences as shown in table 1 have been considered. The penalty time is given by TP = KLTc for some positive integer K. Figure 3 illustrates the tightness of the bounds on mean acquisition time obtained by using Pfl i and Pfui , respectively. Only single-dwell examples are shown here. Similar results are observed for double-dwell systems. In the figure, the “best” and “worst” curves are obtained by using the smallest Pfl i and the largest Pfui over the considered set of PN sequences, respectively. Only examples for L = 127 are shown because the bounds are tighter for L = 1023. This is due to the fact that the value of out of phase correlation of an m-sequence is smaller for a larger L. Recall that for a small L, we do not need to use Pfl i and/or Pfui since the exact probability of false alarm can be obtained by averaging (35) over all possible q. As can be seen, the bounds are very tight for a small η, say η < 0.2 and become looser for a larger η. Fortunately, only the cases of small η is of practical interest, because the mean acquisition time performance degrades very rapidly when η gets larger and larger, as to be seen in the following. That is, it is necessary to keep η small for the I-Q detector to be useful. Since the bounds are very tight for the cases of practical interest, only the lower bound will be used for the following comparisons. For all the considered cases, the smallest Kmin over the set of considered PN sequence is very small and, hence, the lower bound is virtually the performance without considering the effect of out of phase correlation for the hypothesis H0 . Also, from the tightness of the bound shown in the figure, one can conclude that AWGN is the dominant factor in determining false alarm probability for small η’s. Hence the correlation between detections associated with H0 (caused by nonzero out of phase correlation) can be safely neglected. Figure 4 shows typical effects of multipath fading on the performance of mean acquisition time. The examples shown are for L = 1023, K = 100000 and η = 0. For Rayleigh fading channels, the degradation ranges from 3–5 dB for single-dwell (SD) cases and 5–10 dB for doubledwell (DD) cases for the SNR of interest, respectively. Note that the degradation depends on ρ and γc . Moreover, a smaller ρ (a faster fading) generally results in a
Table 1 The considered PN codes. L = 1023
L = 127
1 + x3 + x10 1 + x2 + x3 + x8 + x10 1 + x + x2 + x3 + x5 + x6 + x10 1 + x2 + x3 + x6 + x8 + x9 + x10 1 + x + x3 + x4 + x5 + x6 + x7 + x8 + x10 1 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10
1 + x3 + x7 1 + x2 + x3 + x4 + x7 1 + x + x2 + x3 + x7 1 + x + x2 + x4 + x5 + x6 + x7 1 + x + x2 + x3 + x4 + x5 + x7 1 + x2 + x4 + x6 + x7 1 + x + x7 1 + x + x3 + x6 + x7 1 + x2 + x5 + x6 + x7
W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
Figure 3. Example lower and upper bounds on mean acquisition time for single-dwell systems.
Figure 4. Typical examples of the effects of multipath fading.
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W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
Figure 5. Typical examples on the effects of penalty time for the case of ρ = 0.99.
poorer performance. This indicates that the I-Q detector is more vulnerable to fast fading than to slow fading. For Rician fading channels the amount of degradation is generally smaller and depends on the value χ. Similar results are observed for L = 127 and/or with different K’s. Figure 5 shows the typical effects of the value K on the mean acquisition time performance. The examples shown are for the parameters of L = 1023, η = 0.0 and ρ = 0.99. Basically, the performance of double-dwell systems is quite insensitive to the value of K for slow fading channels. However, a smaller K results in a smaller mean acquisition time for single-dwell systems. This is also true for the S-L detector. Figure 5 also compares the performance of I-Q and S-L detectors. As seen, for η = 0 and ρ = 0.99, the I-Q detector outperforms uniformly the S-L detector by a margin of 3–4 dB. However, this is not the case for ρ = 0.5, as shown in figure 6, where the S-L detector may outperform the I-Q detector. This phenomenon may be attributed to the fact that the I-Q detector is more vulnerable to fast fading than the S-L detector. Also note that in a faster fading channel a smaller K may result in a smaller mean acquisition time even for double-dwell systems. In figure 5 we also observered that for η = 0 and ρ = 0.99, 2– 4 dB improvement, depending on K, is obtained by using double-dwell systems for both I-Q and square-law detectors. However, for ρ = 0.5, the improvement obtained by using double dwell systems becomes smaller for the I-Q detector. Similar results are observed for L = 127. Figures 7 and 8 show typical effects of frequency offset on the mean acquisition time performance. As seen, the
performance of the I-Q detector degrades very rapidly to the frequency offset. For Rayleigh fading, it is observed in figure 7 that the performance gain obtained by using double-dwell system becomes diminishing as η gets larger, e.g., η > 0.1. Figure 8 compares the performance of I-Q and square-law detectors for Rayleigh fading channels with K = 10000, L = 127, ρ = 0.99 and η 6= 0. As observed, the I-Q detector is much more vulnerable to frequency offset than the S-L detector. 5. Conclusions Multiple-dwell acquisition with an I-Q detector is analyzed for frequency nonselective Rayleigh and Rician fading channels that takes into account the detection correlations resulting from multipath fading. Mean acquisition times obtained by numerical optimization of the system parameters are used for performance evaluation and comparisons. Numerical examples are given for single- and double-dwell acquisition systems only because the most significant performance improvement is obtained by going from single- to double-dwell systems. In addition, a detailed comparison between I-Q and square-law detectors is conducted under various channel conditions. The obtained results are summarized as follows. • Performance degradation due to multipath fading can be very large, for example, 3–5 dB and 5–10 dB degradations are observed for single- and double-dwell systems for SNR of practical interest, respectively. The amount
W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
Figure 6. Typical examples on the effects of penalty time for the case of ρ = 0.5.
Figure 7. Typical effects of frequency offset on the performance with the I-Q detector.
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W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
Figure 8. Performance comparisons of I-Q and S-L detectors with various frequency offsets.
of degradation depends on the dwell number, the channel fading rate, the operating SNR and the power ratio of the specular and diffuse components for Rician fading channels. Faster fading generally results in a poor performance, i.e., the I-Q detector is more vulnerable to fast fading than to slow fading. • For no frequency offset cases, performance improvement can be obtained by using double-dwell systems. The amount of improvement depends on the value of penalty time (characterized by the parameter K), the operating SNR and the channel fading rate. A 2–4 dB improvement is observed in some cases. However, the performance improvement becomes diminishing when frequency offset η gets larger, say η > 0.1 and/or channel becomes faster fading (ρ 6 0.5). Moreover, the performance of double-dwell systems is quite insensitive to the value of K for slow fading. However, for single-dwell systems and/or for faster fading channels, a smaller K results in a smaller mean acquisition time. • The performance of the I-Q detector degrades very rapidly as the frequency offset gets larger and larger. Also, the I-Q detector is much more vulnerable to frequency offset and fast fading than the S-L detector. The relative performance of these two detectors generally depends on the channel parameters of fading rate and frequency offset. No single general rule can be drawn about the relative performance of these two detectors. The selection between these two detectors requires to be considered on case by case basis.
Appendix. Calculation of (23) and (24) Only (23) is explicitly derived here. (24) can be derived in a similar way. From (23), Kc (q, ∆w, θi,j ) Z LTc 1 ∆ = c(t − qTc ) cos(∆wt + θi,j ) dt LTc 0 = cos θi,j Kc (q, ∆w, 0) − sin θi,j Ks (q, ∆w, 0) (A.1) and Ks (q, ∆w, θi,j ) Z LTc 1 ∆ = c(t − qTc ) sin(∆wt + θi,j ) dt LTc 0 = sin θi,j Kc (q, ∆w, 0) + cos θi,j Ks (q, ∆w, 0). (A.2) ci−1 LTc + jLTc + θ. Defining Recall that θi,j = M " # Kc (q, ∆w, θi,j ) ∆ K(q, ∆w, θi,j ) = Ks (q, ∆w, θi,j ) and
"
Ai,j
cos θi,j = sin θi,j ∆
# − sin θi,j , cos θi,j
(A.3)
(A.4)
we have K(q, ∆w, θi,j ) = Ai,j K(q, ∆w, 0),
(A.5)
W.-H. Sheen, S. Chiou / Performance of multiple-dwell pseudo-noise code acquisition
and equation (23) becomes Kc (q, ∆w, θi,j )Kc (q, ∆w, θi,k ) + Ks (q, ∆w, θi,j )Ks (q, ∆w, θi,k ) = K T (q, ∆w, θi,j )K(q, ∆w, θi,k ) = K T (q, ∆w, 0)ATi,j Ai,k K(q, ∆w, 0),
[15]
(A.6)
[16]
where T denotes vector or matrix transposition. From equation (A.6), it is easy to show that
[17]
Kc (q, ∆w, θi,j )Kc (q, ∆w, θi,k ) + Ks (q, ∆w, θi,j )Ks (q, ∆w, θi,k ) 2 = Kc (q, ∆w, 0) + Ks2 (q, ∆w, 0) cos (j − k)LTc . (A.7) References [1] U. Cheng, Performance of a class of parallel spread spectrum code acquisition schemes in the presence of data modulation, IEEE Trans. Commun. 36 (May 1988) 596–604. [2] D.M. DiCarlo and C.L. Weber, Multiple dwell serial search: performance and application to direct sequence code acquisition IEEE Trans. Commun. 31 (May 1983) 650–659. [3] EIA/TIA-IS95, Mobile station – base station compatibility standard for dual-mode wide-band spread spectrum cellular system (July 1993). [4] C.W. Helstrom, Calculating error probabilities for intersymbol and co-channel interference, IEEE Trans. Commun. 34 (May 1986) 430– 435. [5] J.K. Holmes, Coherent Spread Spectrum Systems (Wiley, New York, 1982). [6] J.K. Holmes and C.C. Chen, Acquisition time performance of PN spread-spectrum systems, IEEE Trans. Commun. 25 (August 1977) 778–783. [7] P.M. Hopkins, A unified analysis of pseudonoise synchronization by envelope correlation, IEEE Trans. Commun. 25 (August 1977) 770–778. [8] V.M. Jovanovic, Analysis of strategies for serial-search spread spectrum code acquisition–direct approach, IEEE Trans. Commun. 36 (November 1988) 1208–1220. [9] I. Kanter, Exact detection probability for partially correlated Rayleigh targets, IEEE Trans. Aero. Electron. Syst. 22 (March 1986) 184–195. [10] H. Meyr and G. Poltzer, Performance analysis for general PN spread spectrum acquisition techniques, IEEE Trans. Commun. 31 (December 1983) 1317–1319. [11] S.M. Pan, D.E. Dodds and S. Kumar, Acquisition time distribution for spread spectrum receivers, IEEE J. Selected Areas Commun. 8 (June 1990) 800–807. [12] A. Polydoros and C.L. Weber, A unified approach to serial search spread-spectrum code acquisition – part I: general theory, IEEE Trans. Commun. 32 (May 1984) 542–549. [13] A. Polydoros and C.L. Weber, A unified approach to serial search spread-spectrum code acquisition – part II: a matched filter receiver, IEEE Trans. Commun. 32 (May 1984) 550–560. [14] W.-H. Sheen and H.-C. Wang, Sequential detections with applications to PN code acquisition on multipath fading channels, Technical
[18] [19]
[20] [21]
[22]
[23]
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Report NSC-86-2213-E-194-010, National Chung Cheng University (1996). W.-H. Sheen and S.-G. Wu, Considerations on the performance of serial search pseudo-noise code acquisition with a square-law detector, in: Proc. of 1995 IEEE Vehicular Technology Conference, Chicago, IL (July 26–28, 1995). W.-H. Sheen and S.-G. Wu, Performance of multiple-dwell directsequence pseudo-noise code acquisition with square-law detector on multipath fading channels, submitted for publication (1996). M.K. Simon, J.K. Omura, R.A. Scholtz and B.K. Levitt, Spread Spectrum Communications, Vols. I–III (Computer Science Press, MD, 1985). Y.T. Su, Rapid code acquisition algorithms employing PN matched filters, IEEE Trans. Commun. 36 (June 1988) 724–733. S. Tantaratana, A.W. Lam and P.J. Vincent, Noncoherent sequential acquisition of PN sequences for DS/SS communications with/without channel fading, IEEE Trans. Commun. 43 (February/March/April 1995) 1738–1745. A.J. Viterbi, CDMA, Principles of Spread Spectrum Communication (Addison-Wesley, New York, 1995) Chapter 3. R.B. Ward and K.P. Yiu, Acquisition of PN signals by recursion aided sequential estimation, IEEE Trans. Commun. 25 (August 1977) 784–794. A. Weinberg, Generalized analysis for the evaluation of search strategy effects on PN acquisition performance, IEEE Trans. Commun. 31 (January 1983) 37–49. R.E. Ziemer and R.L. Peterson, Digital Communications and Spread Spectrum Systems (Macmillan, New York, 1985).
Wern-Ho Sheen received the B.S. degree from the National Taiwan Institute of Technology, Taipei, Taiwan, ROC, in 1982, the M.S. degree from the National Chiao Tung University, Hsinchu, Taiwan, ROC, in 1984, and the Ph.D. degree from the Georgia Institute of Technology, Atlanta, in 1991, all in electrical engineering. From 1984 to 1993 he was with Telecommunication Laboratories, Taiwan, ROC, where he was mainly involved in the projects of personal communications and basic rate ISDN. Since 1993 he has been an Associate Professor with the Department of Electrical Engineering, National Chung Cheng University, Taiwan, ROC. His research interests include adaptive signal processing, cellular mobile/personal radio systems, and spread spectrum communications. E-mail:
[email protected]
Sauz Chiou was born in Ping-Tung, Taiwan, ROC, in 1969. He received the B.S. degree in electronics from Tamkang University and the M.S. degree in electrical engineering from the National Chung Cheng University, Taiwan, ROC, in 1991 and 1993, respectively. His research interests include the design and implementation of cellular mobile and personal radio systems.
