Efficient Acquisition Algorithm
for Long Pseudorandom
Sequence
WAN-HSIN HSIEH
National Chiao Tung University, Taiwan CHIEH-FU CHANG
National Applied Research Laboratory - National Space Organization Taiwan
MING-SENG KAO
National Chiao Tung University Taiwan
In this paper, a novel method termed the phase coherence acquisition (PCA) is proposed for pseudorandom (PN) sequence acquisition. By employing complex phasors, the PCA requires only complex additions in the order of N, the length of the sequence, whereas the conventional method using fast Fourier transform (FFT) requires complex multiplications and additions both in the order of N log2N. To combat noise, the input and local sequences are partitioned and mapped into complex phasors in PCA. The phase differences between pairs of input and local phasors are used for acquisition; thus, complex multiplications are avoided. For more noise-robustness capability, the multilayer PCA is developed to extract the code phase step-by-step. The significant reduction of computational loads makes the PCA an attractive method, especially when the sequence length of N is extremely large, which becomes intractable for the FFT-based acquisition.
Manuscript received July 30, 2012; revised March 9 and July 27, 2013; released for publication September 21, 2013.
DOI. No. 10.1109/TAES.2014.120460.
Refereeing of this contribution was handled by W. Blanding.
Authors’ current addresses: W.-H. Hsieh, Applied Research Department, St. Jude Medical, Inc., Taipei, Taiwan E-mail: (hsieh1979@gmail.com); C.-F. Chang, Electrical Engineering Division, National Applied Research Laboratory - National Space Organization, Hsinchu, Taiwan; M.-S. Kao, Communication Engineering Department, National Chiao Tung University, Hsinchu, Taiwan.
0018-9251/14/$26.00C 2014 IEEE
I. INTRODUCTION
Pseudorandom (PN) sequence matching is widely used in various applications. For example, in
spread-spectrum communications and global navigation satellite system (GNSS) receivers, the matching is implemented to search for the correct code phase so as to identify the transmitter. The serial search acquisition for matching the PN sequence by finding the correlation peak is simple and straightforward, but it is rather time-consuming with exhaustive searches required for each code phase [1, 2]. Because the correlation of the PN sequence in the acquisition can be calculated by using the convolution theorem and implemented with the fast Fourier transform (FFT) algorithm, parallel search acquisition based on the FFT algorithm is proposed to significantly reduce acquisition time, but this results in an increase in complexity [3–7]. Specifically, the convolution of two sequences can be derived from the pointwise product of the corresponding Fourier transforms (i.e., x[n]⊗ y[n] F→ X(ω) · Y (ω), where⊗ denotes the convolution and F represents the Fourier transform).
The previous works regarding the reduction of complexity or computation of PN sequence acquisition using the FFT algorithm mainly focused on reducing the number of input elements either in the FFT stage (by summing over a number of chips or superimposing folded segments of the local code), in the inverse FFT (IFFT) stage (by performing only on the portion of data with significant power), or in combination with the Doppler search [8–14]. The FFT/IFFT structure basically remains and serves as the intermediate component for facilitating the search for the code phase. Hence, the computation load for acquisition is dominated by the computation of FFT/IFFT that involves a number of complex
multiplications [3–5].
In this paper, we propose an approach, termed phase coherence acquisition (PCA), which uses a distinct perspective from the conventional FFT/IFFT structure. That is, the PCA extracts the desired code phase information by using phasors in the complex domain. Owing to the simple phase manipulation of phasors and the elimination of inverse mapping into the time domain, the PCA requires much less computation than FFT-based acquisition to search for the correlation peak of PN sequences. It is noteworthy that PCA requires no multiplication as compared with FFT-based acquisition. This superiority becomes prominent when the applied sequence length N is very large such that the FFT-based approach is difficult to be implemented. This paper is organized as follows. First, we describe the motivation of our work. Next, we develop our approach in the noiseless case and provide the essential idea in our development. To achieve noise robustness, we then incorporate a novel segmentation scheme in our approach and propose the PCA method. The simulation results are provided to verify the analysis and demonstrate the performance of the
proposed method. Finally, the computation of PCA and FFT-based method are discussed.
II. MOTIVATION
The convolution theorem states that under general conditions, the Fourier transform of a convolution between two sequences is the pointwise product of the Fourier transforms of these two sequences. The theorem can be represented by
F{x[n] ⊗ y[n]} = F {x[n]} · F {y[n]} , (1) where F denotes Fourier transform. By applying the inverse Fourier transform F−1, we have
x[n]⊗ y[n] = F−1{F {x[n]} · F {y[n]}} . (2) In many applications, the code phase search between two sequences is usually implemented by FFT and its inverse due to the efficient computation compared with the exhaustive direct serial search method. The computation of FFT of N points involves complex multiplications and additions of order N log2N. Due to the diverse need for applications and the increasing complexity of modern algorithms, a more computationally efficient method is needed when the length of processed sequence becomes so large that implementation using the FFT method becomes difficult. Our idea for code phase acquisition that attains much less computation is developed in the following. III. ACQUISITION BY PHASOR
Let SIN = {x0, x1,· · · , xN−1} and
SLO = {y0, y1,· · · , yN−1} be the input and local PN
sequences of length N , respectively, where
xn, yn∈ {1, −1}. In noiseless condition, the cross
correlation between{xn} and {yn} is denoted by
C(m)= N−1 k=0 xk+myk, (3) where m= 0, 1, · · · N − 1.
Let the code phase shift between SIN and SLObe q,
where q ∈ {0, 1, · · · , N − 1}. We first map the input and local sequences into phasors as given by
X= N−1 n=0 xnγ−n (4) Y = N−1 n=0 ynγ−n, (5) where γ = ej2πN and j =√−1. We then calculate = X∗· Y = N−1 n=0 xnγn · N−1 k=0 ykγ−k = N−1 m=0 xk+mγk+m · N−1 k=0 ykγ−k = N−1 m=0 N−1 k=0 xk+mykγm = N−1 m=0 γmC(m), (6)
where the superscript∗ denotes the complex conjugation and C(m) represents the cross correlation between{xn}
and{yn}. To present our concept in a direct and effective
manner, the maximal-length sequence (MLS) is illustrated for the sequence acquisition. The cross correlation between{xn} and {yn} is given by
C(m)= N, if m= q, −1, if m = q. (7) Hence, (6) becomes = N−1 m=0 γmC(m) = C(q)γq+ N−1 m=0,m=q γmC(m) = (N + 1)γq− N−1 m=0 γm = (N + 1)γq = (N + 1)ej2π Nq, (8)
where the equality
N−1 m=0
γm = 0 is applied in the above
derivation.
Let the phase of be as denoted by
= 2π
N q. (9)
The acquisition of the sequence can then be achieved by
q= N
2π. (10)
Because the input sequence consists of+1 and −1, the complex phasor of (4) is obtained by simply N additions (subtractions). Note that the computations of the phasor regarding the local sequence can be omitted by calculating (5) in advance.
In the above derivation, when the phasor of the input sequence is obtained by (4), very few computations are needed to determine the shift q, that is, much fewer than those required for the FFT-based approach. However, the phase accuracy of the complex phasor is sensitive to noise. The phase resolution is 2π/N according to (9). As shown in Fig. 1, when N is large, the distance between adjacent phases is rather small, which easily leads to erroneous phase estimation under the noisy environment. Hence, it becomes necessary to design an algorithm that permits the
Fig. 1. Schematic plot of phase resolution for phasors on complex domain.
distance between adjacent phases to be increased to resist the effect of noise.
IV. PCA
A. Segmentation
Suppose the input sequence SIN has the length of
N= K · M. In the PCA, SIN is first partitioned into
K-disjointed segments of length M as denoted by
A0 = {x0, xK, x2K,· · · x(M−1)K} A1 = {x1, xK+1, x2K+1,· · · x(M−1)K+1} .. . AK-1= {xK−1, x2K−1, x3K−1,· · · xMK−1} = {xK−1, x2K−1, x3K−1,· · · xN−1}. (11)
Similarly, the local sequence SLOis also partitioned into
K-disjointed segments as below:
B0= {y0, yK, y2K,· · · y(M−1)K} B1= {y1, yK+1, y2K+1,· · · y(M−1)K+1} .. . BK-1 = {yK−1, y2K−1, y3K−1,· · · yMK−1} = {yK−1, y2K−1, y3K−1,· · · yN−1}. (12)
Suppose the code phase shift between the input and the local sequences is q= cK + d, where 0 ≤ c < M and 0≤ d < K. We then have yi = xi+q = xi+cK+d, and the
following relationships B0 = {y0, yK, y2K,· · · y(M−1)K} ={xcK+d, x(c+1)K+d,· · ·, x(M−1)K+d,xd,· · ·, x(c−1)K+d} = Ad(c) B1 = Ad+1(c) .. . BK-d-1= AK−1(c), (13)
where Ad(c) denotes the circular shift of Adwith c chips to left.
The remaining BK-d, BK-d+1,· · · , BK-1can be derived by using the same logics but with adjustments, as
given by BK-d= {yK−d, y2K−d, y3K−d,· · · yMK−d} = {x(K−d)+cK+d, x(2K−d)+cK+d,· · · , xMK+d+cK+d} = {x(c+1)K, x(c+2)K,· · · , x0,· · · , xcK} = A0(c+ 1) BK-d+1= A1(c+ 1) .. . BK-1= Ad−1(c+ 1). (14)
From (13) and (14), the relationships between Aiand Bi can be generalized as follows:
Bi = Ad+i(c), 0≤ i ≤ K − d − 1
= A(d+i)modK(c+ 1), K − d ≤ i ≤ K − 1. (15)
B. Acquisition by Phase
In each segment of (11) and (12), we map the sequences into the complex phasors by
Xi = M−1 n=0 xnK+iα−n (16) Yi = M−1 n=0 ynK+iα−n, (17) where α= ej2π M and i= 0, 1, 2, · · · K − 1.
Let Ai(m) be the segment Aiwith m circular shifts to left, denoted by
Ai(m)= {xmK+i, x(m+1)K+i,· · · , xi,· · · , x(m−1)K+i}. (18)
Accordingly, the complex phasor pertaining to Ai(m) is given by Xi(m)= M−1 n=0 x(m+n)K+iα−n = M−1 n=0 x(m+n)K+iα−(m+n)· αm = αm M−1 u=0 xuK+iα−u = αmX i. (19)
According to (19) and (15), the complex phasors Yi are
derived by Yi = αc· Xd+i = ej2π Mc· X d+i, 0≤ i ≤ K − d − 1 (20) Yi = αc+1· Xd+i = ej2πM(c+1)· X d+i, K− d ≤ i ≤ K − 1. (21)
Furthermore, the complex phasors Xiand Yican be
expressed by
Xi = |Xi|ej θi (22)
where θi and φidenote the phases of Xi and Yi,
respectively.
From (20) to (23), we have the following phase relationship: φi = θi+d+ 2π M · c, 0 ≤ i ≤ K − d − 1 = θi+d+ 2π M · (c + 1), K − d ≤ i ≤ K − 1. (24)
Let Gmbe the sum of the K complex phasors, defined as
Gm= K−m−1 i=0 ej(φi−θi+m)+ K−1 i=K−m ej(φi−θi+m−2πM), (25) where m= 0, 1, 2, · · · K − 1.
According to the relationship of (24), when m= d, we have Gd = K−m−1 i=0 ej(φi−θi+d)+ K−1 i=K−m ej(φi−θi+d−2πM) = K · ej2π Mc. (26) Apparently, |Gd| = K. (27)
Note that we have a peak magnitude given by (27) when the K complex phasors are coherently added for Gd. On
the other hand, when m= d, Gmis the sum of the K
phasors of noncoherent phases, and the resultant
magnitude is expected to be much smaller than K. Hence, the value of d can be obtained by finding the peak magnitude among{|Gm|}. In addition, let be the phase
of Gd. From (26), we have = 2π M · c. (28) Thus, c is given by c= M · 2π. (29)
Let the estimates of (c, d) be ( ˆc, ˆd). Practically, when ˆ
d= d and ˆc is equal to c, the shift q = ˆcK + ˆd is
correctly determined.
In PCA, the input sequence is one-bit quantized, partitioned, and transformed into phasors as given by (16). The phase differences between phasors of the input and local sequences are then used for the acquisition, as given by (25). When the phase differences between phasors are coherently added, we can have a large peak (|Gd|) to
determine the correct segment for code phase acquisition. These processes simply require complex additions and eliminate the need for complex multiplications that are the major advantages of the PCA. The segmentation process confers noise robustness in the PCA method by the high correct probability of 1) ˆd= d, because of the coherent
addition of K components, and 2) ˆc= c, because the noise effect is mitigated in determining phase of Gdwith
an enlarged distance between adjacent phases, i.e., from 2π/N to 2π/M by comparing (9) and (28). Hence, the
ultimate accuracy of code phase acquisition is improved with the correct probability of ˆdand ˆc after the
segmentation process. However, the estimated ( ˆc, ˆd) could be erroneous when the signl-to-noise ratio (SNR) is very low, especially ˆc. In such situations, the multilayer scheme can be applied to enhance the noise resistance in the PCA method.
C. Multilayer PCA
In the multilayer PCA, the first-layer process is identical to the method described above. First, the input and local sequences of length N are partitioned into K1
segments of length M1, where N = K1M1. Assume the
shift is denoted as q= c1K1+ d1. In the first layer, only d1is estimated by finding the peak of|G(1)m| in (25), and c1
is left undetermined owing to the sensitivity to the effect of noise. The superscripts (1) and (2) in Gmindicate the
first layer and the second layer, respectively. After the first layer is completed, we assume ˆd1= d1and
Bi= Aˆd1+i(c1), 0≤ i ≤ K1− ˆd1− 1
= Aˆd1+i(c1+ 1), K1− ˆd1≤ i ≤ K1− 1, (30)
where c1is still undetermined.
We rewrite (30) by Bi= Aˆd1+i(c1), 0≤ i ≤ K1− 1 (31) where Aˆd1+i(c1)= Aˆd1+i(c1), 0≤ i ≤ K1− ˆd1− 1 Aˆd1+i(c1+ 1), K1− ˆd1≤ i ≤ K1− 1.
From (31), all the pairs of (Aˆd1+i, Bi) have the same shift
of c1chips in between, which is the key for the following
derivation in the second layer.
The process of the second layer is introduced next. For simplicity, we take the pair (Aˆd1, B0) as an example, where each Aˆd1and B0contains M1elements and their
relative shift is c1, i.e., B0 = Aˆd1(c1). Let M1= K2· M2
and assume c1= c2K2+ d2, where 0≤ c2≤ M2and
0≤ d2 ≤ K2. First, Aˆd1and B0are partitioned into K2-disjointed segments of length M2as before. Following
the same calculation as (25), the sum of the K2complex
phasors is obtained for (Aˆd
1, B0), given as Hr,0= K2−r−1 s=0 ej(φs−θs+r)+ K2−1 s=K2−r ej(φs−θs+r−M22π), (32)
where r = 0, 1, · · · , K2− 1 and (φs, θs) are the
corresponding phases involved in the calculation. When r = d2, we have Hd2,0= K2· e
jM22πc2
under the noiseless condition, which has the maximum magnitude among{Hr,0}.
When the similar calculation is applied to the other pairs (Aˆd1+i, Bi), i = 1, 2, · · · , K1− 1, their associated Hr,ican then be obtained. Afterwards, all of the Hr,iare
used to calculate G(2)r = K1−1 i=0 Hr,i, (33) where r= 0, 1, · · · , K2− 1.
Because all the pairs (Aˆd1+i, Bi) have the same shift
of c1chips in between, where c1= c2K2+ d2, G(2)d2 will
have a peak magnitude among{G(2)
r }. Specifically, in a
noiseless condition, we have
G(2)d2 = K1−1 i=0 Hd2,i = K1−1 i=0 K2· ejM22πc2 = K1K2· ej 2π M2c2 , (34)
where the peak magnitude is K1K2.
Similar to the first layer, by finding the peak magnitude among{|G(2)
r |}, we can estimate d2, which is
denoted by ˆd2. Let be the phase of G(2)dˆ2 as given by
= 2π
M2
· c2. (35)
Similar to (29), the estimate of c2, denoted by ˆc2, is
obtained by
ˆc2= M2·
2π. (36)
According to (35), the separation between adjacent phases is further enlarged from 2π/M1to 2π/M2, which
significantly increases the resistance to noise. Note that c2
can also be left undetermined after the second layer and determined by the third layer, if necessary. Nevertheless, from our simulation results, two layers appear to be sufficient for most applications. Finally, the estimate of q, denoted as ˆq, is calculated as
ˆ
q= ˆc1K1+ ˆd1
= (ˆc2K2+ ˆd2)K1+ ˆd1. (37)
D. Error Detection Capability
When the segment of the first layer is correctly estimated, i.e., ˆd1 = d1, we obtain a much larger peak in
the second layer for ˆd2= d2. Taking the noiseless case, for
example, we have the peak of K1in|G(1)d1|, according to
(26). In contrast, a much larger peak of K1K2is obtained from|G(2)d2| in (34). As a result, the existence of a significant peak in|G(2)
r | of the second layer can be used to
verify the correctness of ˆd1, which shows the inherent
error detection capability of PCA. Accordingly, correct ˆd1
can be obtained with some recursive algorithms using such an error detection property, and the performance of the multilayer PCA can be further improved. This special feature has been verified in our simulations.
V. PERFORMANCE OF PCA
Let the input PN sequence be{xn}. Assume the
sequence is distorted by zero-mean Gaussian noise ζnwith
variance σζ2and is one-bit quantized, as denoted by
wn= sign(xn+ ζn), (38)
where n= 0, 1, · · · , N − 1, sign(z) = 1, if z ≥ 0, and sign(z)= −1, if z < 0.
In the first layer, the input and local sequences,{wn}
and{yn}, are partitioned into K1segments of length M1, as
denoted by
Ai= {wi, wK1+i, w2K1+i,· · · w(M−1)K1+i} (39)
Bi= {yi, yK1+i, y2K1+i,· · · y(M−1)K1+i}, (40)
where i = 0, 1, · · · , K1− 1.
Similar to (16) and (17), the complex phasors are defined by Wi = M1−1 n=0 wnK1+iα −i = |Wi|ej θi (41) Yi = M1−1 n=0 ynK1+iα −n = |Yi|ej φi. (42)
Moreover, according to (25), the sum of complex phasors is given by G(1)m = K1−m−1 i=0 ej(φi−θi+m)+ K1−1 i=K1−m ej(φi−θi+m−M12π) = K1−1 i=0 ej ψi,m, (43)
where ψi,mdenotes the phase difference between the
complex phasors Wi+mand Yi,−π ≤ ψi,m≤ π, and
m= 0, 1, · · · , K1.
Let the shift between{wn} and {yn} be q = c1K1+ d1.
As derived in the Appendix, the magnitude of|G(1)
m| with
m= d1, i.e., the sidelobe, is a random variable with the
Rayleigh distribution given by
f(rs)=
rs
K1/2
e−rs2/K1, (44)
where f (rs) is the probability density function of|G(1)m|
and rs ≥ 0.
In addition,|G(1)
m| with m = d1, i.e.,|G(1)d1|, has the
Rice distribution given by
f(rp)= rp σp21e −(r2 p+μ2Re1)/2σp21· I 0 rpμRe1 σp21 , (45) where rp≥ 0.
For a given rp, the probability of rp> rsis denoted by Pr(rp> rs)= rp 0 f(rs)drs = 1 − exp −r 2 p K1 . (46)
Because there are K1− 1 sidelobes in the G(1)m, the correct
d1is obtained when|G(1)d1|is greater than all the other
K1− 1 sidelobes. Hence, for a given rp, the correct
probability of detecting d1is given by
Pd(rp)= 1− exp −r 2 p K1 K1−1 . (47)
When the distribution of rpis considered, the correct
probability of d1, i.e. ˆd1= d1, is denoted by
PD1=
∞
0
Pd(rp)f (rp)drp. (48)
Furthermore, the probability of correct c1shall be
considered for the correct acquisition in the one-layer PCA. According to (29), c1is obtained from the phase of G(1)d1; thus, the probability of detecting c1can be derived in
light of the phase distribution of G(1)d1. Specifically, let the phase of G(1)d1 be ϕ. For simplicity, assume c1= 0.
According to the schematic concept shown in Fig. 1, c1is
correct if|ϕ| ≤ Mπ
1. Using the joint magnitude and phase
distribution of G(1)d
1 derived in the Appendix, we have
f(rp, ϕ)= rp 2π σp21 exp −r 2
p+ μ2Re1− 2rpμRe1cos ϕ
2σp21
,
(49) where−π ≤ ϕ < π.
The joint probability of correct d1and c1is then
denoted by PC1= Pr ˆ d1= d1,|ϕ| ≤ π M1 = 2 · π M1 0 ∞ 0 Pd(rp)f (rp, ϕ)drpdϕ. (50)
We use the MLS of length N = 220− 1 to verify the analysis. Let K1= 210+ 1 and M1 = 210− 1. The
mentioned correct probabilities are simulated by the Monte Carlo method with 10 000 trials. The correct probabilities of PD1and PC1are shown in Figs. 2 and 3,
respectively. In both figures, the analytical and simulated results are consistent with each other, which justifies the validity of our analysis. The correct probability of d1
approaches one when SNR >−15 dB and begins to degrade with decreasing SNR. Note that the probability of
d1is critical to the performance of PCA. The acquisition
process will fail if d1, i.e., the correct segment, cannot be
correctly detected. On the other hand, the correct probability of PC1is worse than PD1, which approaches
Fig. 2. Correct probability of d1in first layer of PCA.
Fig. 3. Joint correct probability of d1and c1in first layer of PCA. one when SNR > 10 dB but drops to below 0.1 if SNR < 0 dB.
Besides the correct probability, the standard deviation (STD) of ˆc1is also derived to study the deviation of code
phase shift. We consider the STD of ˆc1with the condition
that ˆd1 = d1, which is denoted by σϕ = 1 PD1 π −π ϕ2· ∞ 0 Pd(rp)f (rp, ϕ)drpdϕ 1/2 . (51)
In Fig. 4, the STD decreases with SNR. Specifically, the STD of ˆc1is about four chips when SNR= 0 dB and
decreases to within one chip for SNR≥ 6 dB. According to the STD of ˆc1, the one-layer PCA performs well only in
the case of the high SNR. For applications with low SNR, the second layer is needed to improve performance in PCA.
Let M1 = K2M2and c1= c2K2+ d2in the second
layer of PCA. According to (32) and (33), the sum of complex phasors is given by
G(2)n = K1−1 i=0 K 2−n−1 t=0 ej(φi,t−θi,t+n)+ K2−1 t=K2−n ej(φi,t−θi,t+n−M22π) = K1−1 i=0 K2−1 t=0 ej ψi,t+n, (52) where n= 0, 1, · · · , K2.
Fig. 4. STD of ˆc1in first layer of PCA, when ˆd1= d1.
For simplicity, we assume ˆd1 = d1for the analysis of
the second layer. According to the Appendix, the magnitude distribution of the sidelobe of|G(2)
n | with n= d2is given by f(ls)= ls K1K2/2 e−l2s/K1K2, (53) where ls ≥ 0.
On the other hand, the magnitude distribution of|G(2)
n | with n= d2is denoted by f(lp)= lp σp22e −(l2 p+μ2Re2)/2σp22· I 0 lpμRe2 σp22 , (54) where lp≥ 0.
Similarly, for a given lp, the correct probability of d2is
the probability that lpis greater than all the other K2− 1
sidelobes, which is denoted by
Pd(lp)= 1− exp − l 2 p K1K2 K2−1 . (55)
Considering the distribution of lp, the correct probability
of d2, i.e., ˆd2= d2, is given by
PD2=
∞ 0
Pd(lp)f (lp)dlp. (56)
Furthermore, the joint distribution of the magnitude and phase of G(2)d 2 is given by f(lp, ϑ)= lp 2π σ2 p2 exp −l 2
p+ μ2Re2− 2lpμRe2cos ϑ
2σ2 p2 , (57) where−π ≤ ϑ ≤ π.
Hence, the joint probability of the correct d2and c2is
denoted by PC2= 2 · π M2 0 ∞ 0 Pd(lp)f (lp, ϑ)dlpdϕ. (58)
Fig. 5. Correct probability of d2in second layer of PCA, when ˆd1= d1.
Fig. 6. Joint correct probability of d2and c2in second layer of PCA,
when ˆd1= d1.
The correct probabilities of PD2and PC2are shown in
Figs. 5 and 6, respectively. We use the same parameters for the first layer and take K2= 25+ 1 and M2 = 25− 1
in the second layer. Still, the analytical results are consistent with the simulated values in both figures. The improvement brought by the second layer is significant, because the correct probability of d2approaches one for
SNR from−20 to 20 dB in Fig. 5. Moreover, the joint correct probability of d2and c2is greater than 0.9 when
SNR≥ −20 dB in Fig. 6.
Similarly, the STD of ˆc2with the condition that
ˆ d2= d2is derived by σϑ= 1 PD2 π −πϑ 2· ∞ 0 Pd(lp)f (lp, ϑ)dlpdϑ 1/2 . (59)
In Fig. 7, the STD is much less than one chip for
SNR≥ −20 dB and approaches zero when SNR ≥ 5 dB. The noise robustness of the multilayer PCA is thus verified, especially in the case of low SNR, as comparing Figs. 4 and 7.
To further explore the acquisition performance of PCA, we use the P code, which is a category of long PN sequences used in GNSS applications [15]. Let the code length be N = 216− 1. Three PCA
Fig. 7. STD of ˆc2in second layer of PCA, when ˆd1= d1and ˆd2= d2.
Fig. 8. Detection probability of P code of length N= 216− 1 with
PCA schemes: two-layer PCA1 with (K1, M1)= (28+ 1, 28− 1) and
(K2, M2)= (24+ 1, 24− 1), two-layer PCA2 with
(K1, M1)= ((28+ 1) · (22+ 1), (24+ 1) · (22− 1)) and
(K2, M2)= (24+ 1, 22− 1), and single-layer PCA3 with
(K, M)= ((28+ 1) · (24+ 1), (22+ 1) · (22− 1)) and N-point FFT-based acquisition.
schemes are applied to the P code acquisition: the two-layer PCA1 with (K1, M1)= (28+ 1, 28− 1) and
(K2, M2)= (24+ 1, 24− 1), the two-layer PCA2 with
(K1, M1)= ((28+ 1) · (22+ 1), (24+ 1) · (22− 1)) and
(K2, M2)= (24+ 1, 22− 1), and the single-layer PCA3
with (K, M)= ((28+ 1) · (24+ 1), (22+ 1) · (22− 1)). The performance is shown in Fig. 8. Note that only the dominant detection probability PD1of PCA schemes is
provided for simplicity. The performance of the N -point FFT-based acquisition is also shown in Fig. 8 for comparison. According to the detection probability in Fig. 8, FFT-based method outperforms PCA schemes in low SNR. The PCA, on the other hand, provides flexibility for applications in different ambient SNR. The reduction of computation of these PCA schemes over the FFT-based method will be discussed in the following section. VI. COMPUTATIONS OF PCA
The computations of PCA are studied and compared with that of FFT-based acquisition. Here, we assume the
TABLE I
Computations of Three PCA Schemes with Performance Illustrated in Fig. 8 for P Code of Length N= 216− 1
Operation
Method Multiplications Additions
PCA1 0 7N
PCA2 0 68N
PCA3 0 585N
FFT-based method 32N 32N
computations regarding the local sequence are omitted, because it can be calculated in advance. For the first-layer process of PCA, the derivation of the phasors of the input sequence, as shown in (16), requires K1· (M1− 1)
additions. Also, the calculation of G(1)
m in (25) requires
K1· K1additions (subtractions) for the phase difference
and K1(K1− 1) additions for the sum of phasors.
Regarding the computing of the complex phase in (22), the coordinate rotation digital computer (CORDIC), for computing using shifts and additions, can be used [16]. In CORDIC, let P1denote the parameter associated with the
required phase resolution in the first layer, i.e., tan−1 12P1 ≤ 2πM
1 for (28). For example, when P1 = 8, the
phase resolution is sufficient for M1= 210− 1. As a
result, 3K1P1additions are needed for computing the
complex phases in (22). Hence, the overall addition in the first layer is K1· (M1+ 2K1− 2 + 3P1). For the
second-layer process, K1K2· (M2− 1) additions are
needed for the input phasor and K1· K22+ K1K2(K2− 1)
additions for computing (32) and (33). In addition, assume that the required phase resolution in the second layer is tan−1 12P2 ≤ 2πM2, we then need 3K1K2P2additions for the
complex phase using the CORDIC computing. Therefore,
K1K2· (M2+ 2K2− 2 + 3P2) additions are required in
the second layer. Note that the parameters regarding phase resolution in CORDIC, i.e., P1and P2should be at least
large enough to distinguish between the adjacent phases in the multilayer PCA. Otherwise, the accuracy of phase estimation and subsequent estimation of c will be bounded by the resolution of CORDIC, which would result in a bias in the ultimate code phase acquisition.
To be more specific, we consider the computations involved in acquisition schemes PCA1, PCA2, and PCA3, which have performance illustrated in Fig. 8, for P code of length N= 216− 1. Here, we use P1= 8 and P2 = 4.
According to the summary of computation complexity in Table I, no multiplication is used in the PCA schemes, but the number of additions increases with the number of partitions K. Computations of the FFT-based method are also provided in Table I for comparison. The FFT-based method substantially requires 2N log2Nmultiplications and additions because of N log2Nmultiplications and additions on both the FFT for the input sequence and the IFFT for the conversion of a real sequence from complex phasors [3–5]. Note that PCA1 uses even fewer additions than the FFT-based acquisition. In particular, when the two-layer PCA scheme with K1=2n/2+1, M1=2n/2−1,
TABLE II
Computations of the Two-Layer PCA with
K1= 2n/2+ 1, M1= 2n/2− 1, K2= 2n/4+ 1, and M2= 2n/4− 1
and the FFT-Based Method for the PN Sequence of an Extremely Large Length N= 2n− 1
Operation
Method Multiplications Additions
PCA 0 6N
FFT-based method 2N log2N 2N log2N
K2=2n/4+1, and M2= 2n/4− 1 for a PN sequence
of an extremely large N = 2n− 1 is implemented, the
computations for the phase become relatively
insignificant, and then approximately 3N additions are required for both the first and the second layer in PCA. The computational burden is significantly reduced to 6N additions in the two-layer PCA as compared with
2N log2Nmultiplications and additions in the FFT-based method, as indicated in Table II. Because of its superior computational efficiency, PCA may be applied to the unique pattern search in a long sequence. Potential applications include the long PN code acquisition, such as P(Y) code and spot beam M code, and revealing an encrypted PN code in GNSS and the satellite communication field [8, 17–20].
VII. CONCLUSIONS
In this paper, the PCA method using the phase difference of complex phasors for the PN sequence acquisition is proposed. The PCA requires only complex additions but no complex multiplications. In addition, the acquisition performance can be improved via the use of the multilayer scheme that also provides the inherent error detection capability. Segmentation, phasor acquisition, and the multilayer scheme for the PCA algorithm are introduced in Section IV, and the analysis is conducted in Section V. In the demonstrated case using MLS of length
N= 220− 1 in the two-layer scheme of PCA, the correct
segment of the first layer is obtained with probability approaching one when SNR >−15 dB, as shown in Fig. 2. As we identify the correct segment of the first layer, the acquisition performance attains the correct probability greater than 0.9 for SNR≥ −20 dB after the second layer, as shown in Fig. 6. It is noteworthy that PCA requires much less computation than the FFT-based approach as discussed in Section VI and demonstrated in Tables I and II. Hence, for applications having high SNR margins, such as the spot beam signaling [18], the secure telemetry, tracking and command link [19], or the processing of denoised signals, the use of PCA will significantly reduce the computation, namely, the complex multiplications are eliminated, as compared with the FFT-based method. Moreover, according to Table I, the PCA method also has flexibility in selecting the number of layers and partitions to reduce the computation based on
SNR margins, as shown in Fig. 8, whereas the FFT-based acquisition requires a fixed number of computations regardless of SNR. The superior performance on the computation grants the PCA an effective method when the length of a sequence is so large that the FFT-based acquisition is infeasible. Finally, it is worth mentioning that the SNR performance of PCA regarding the detection probability can be improved when a one-bit quantized input sequence in (16) is replaced by a multibit quantized version. The improvement is obtained by the reduction of quantization loss. However, the price is the computational complexity because the multiplications are required in (16). Further investigations may be interesting but beyond the scope of this work.
APPENDIX. MAGNITUDE AND PHASE DISTRIBUTION OFGm
The distribution of Gmin the first layer, G(1)m , is
derived first. We rewrite (38) as
wn= xn+ βnx¯n (60)
where βn∈ {0, 2} and ¯xndenotes the inverse of xn.
In (60), when βn= 2, we have wn= ¯xn, indicating
that an error occurs because of noise ζn. The
corresponding error probability is given by
Pe= Pr(βn= 2) = Q(1/σn), (61) where Q(z)= z∞e−x2/2dx. We can represent (41) as Wi = M1−1 n=0 wnK1+iα −n = Xi+ M1−1 n=0 βnK1+ix¯nK1+iα −n, (62) where Xi = M1−1 n=0 xnK1+iα−nis assumed to be fixed.
Let E{z} denote the expected value of z. The mean value of Wiis obtained by E{Wi} = E Xi+ M1−1 n=0 βnK1+ix¯nK1+iα −n = Xi+ M1−1 n=0 E{βnK1+i} ¯xnK1+iα −n = Xi+ 2Pe· M1−1 n=0 ¯ xnK1+iα −n = (1 − 2Pe)· Xi, (63) where E{βnK1+i} = 2Peand M1−1 n=0 ¯ xnK1+iα −n= −X i.
Furthermore, to obtain the variance of Wi, we calculate E{WiWi∗} = E Xi+ M1−1 n=0 βnK1+ix¯nK1+iα −n · Xi+ M1−1 m=0 βmK1+ix¯mK1+iα −m ∗ = XiXi∗+ Xi· M1−1 m=0 E{βmK1+i} ¯xmK1+iα m+ X∗ i · M1−1 n=0 E{βnK1+i} ¯xnK1+iα −n +E M 1−1 n=0 βnK1+ix¯nK1+iα −n · M1−1 m=0 βmK1+ix¯mK1+iα m = |Xi|2− 2Pe· XiX∗i − 2Pe· X∗iXi + M1−1 n=0 m=n E{βnK1+iβmK1+i} ¯xnK1+ix¯mK1+iα m−n +E ⎧ ⎪ ⎨ ⎪ ⎩ M1−1 n=0 βnK1+ix¯nK1+iα −n M1−1 m=0 m=n βmK1+ix¯mK1+iα m ⎫ ⎪ ⎬ ⎪ ⎭ ≈ |Xi|2−4Pe|Xi|2+4Pe M1−1 n=0 m=n ¯ xnK1+ix¯mK1+iα m−n = |Xi|2− 4Pe|Xi|2+ 4PeM1, (64) where E M1−1 n=0 βnK1+ix¯nK1+iα−n M1−1 m=0 m=n βmK1+ix¯mK1+iα m ≈ 0 and E{βnK1+iβmK1+i} = 4Pe, when m= n.
By using (63) and (64), the variance of Wiis derived by
V ar{Wi} = E{|Wi|2} − (|E{Wi}|)2
= E{WiWi∗} − E{Wi} · (E{Wi})∗.
= |Xi|2− 4Pe|Xi|2+ 4PeM1− ((1 − 2Pe)Xi)
·((1 − 2Pe)Xi)∗
= 4PeM1− 4Pe2|Xi|2. (65)
It is reasonable to assume that{xnK1+i}involved in Xiis a
PN sequence of length M1. Similar to (8), we have
|Xi|2≈ M1+ 1. (66)
Let the code phase shift between input and local MLS be q = c1K1+ d1. Considering the sidelobe of|G(1)m |, i.e.,
m= d1, in (43), the phases ψi,mcan be considered to be
uniformly distributed between−π and π. According to [21], the magnitude distribution of|G(1)
m|, denoted by rs,
can be modeled using Rayleigh distribution, given as
f(rs)= rs K1/2e −r2 s/K1, (67) where rs ≥ 0.
On the other hand, for the|G(1)m| with m = d1, (43) is
represented by G(1)d1 = K1−1 i=0 ej ψi,m = K1−1 i=0 ejM12πc1+φi , (68)
where φi denotes the phase error induced by noise.
Without loss of generality, we assume c1 = 0. Then,
(68) becomes G(1)d 1 = K1−1 i=0 ej φi. (69)
The φidenotes the phase difference between the input
phasor Wiand local phasor Yi caused by noise. Because
we have the mean and variance of Wi in (63) and (65),
according to [21, Sec. 4.4], the distribution of φi can be
approximated by f(φi)= 1 2πe −ρ1+G√πexp(G2)(1+erf(G)), (70) where ρ= (E{Wi})2 V ar{Wi}, G= √ρ cos(φi), and−π ≤ φi ≤ π.
Moreover, to obtain the magnitude distribution of G(1)d1, (69) is reformulated by G(1)d1 = K1−1 i=0 ej φi = K1−1 i=0 cos(φi)+ j K1−1 i=0 sin(φi) = Re+ jIm, (71) where Re= K1−1 i=0 cos(φi) and Im= K1−1 i=0 sin(φi).
We assume that each cos(φi) and sin(φi) are
independent and identically distributed random variables. LetN(μ, σ2) denote the normal distribution function with mean μ and variance σ2. By the central limit theorem, Re
and Imcan be approximated by two normal distributions NμRe1, σRe12 andNμIm1, σIm12 , respectively, and the parameters are obtained by
μRe1 = E K1−1 i=0 cos(φi) = K1−1 i=0 E{cos(φi} (72) σRe12 = V ar K 1−1 i=0 cos(φi) = K1−1 i=0 V ar{cos(φi)} (73)
μIm1 = K1−1 i=0 E{sin(φi)} (74) σIm12 = K1−1 i=0 V ar{sin(φi)}. (75)
Numerically, we find μRe1= 0, μIm1= 0, and σ2
Re1∼= σIm12 . For simplicity, let σp21 = (σRe12 + σIm12 )/2.
According to [21], the magnitude of G(1)d1, denoted by rp,
can be modeled by Rice distribution given as
f(rp)= rp σp21e −(r2 p+μ2Re1)/2σp21· I 0 rpμRe1 σp21 , (76)
where rp≥ 0 and I0(·) is the modified Bessel function of
the first kind with order zero.
In addition, the joint magnitude of phase distribution of G(1)d1 is given by f(rp, ϕ)= rp 2π σ2 p1 exp −r 2
p+ μ2Re1− 2rpμRe1cos ϕ
2σ2 p1 , (77) where−π ≤ ϕ < π.
The derivation of magnitude and phase distribution of
Gmcan be applied to other layers. For example, for the
sidelobe of|G(2)
n | with n = d2, the phases ψi,t+nin (52)
can be considered to be uniformly distributed between−π and π . The distribution of sidelobe of|G(2)
n |, denoted by
ls, can then be modeled by
f(ls)= ls K1K2/2e −l2 s/K1K2, (78) where ls ≥ 0. Moreover, let μRe2= K1−1 i=0 K2−1 t=0 E{cos(ψi,t+d2)} (79) σRe22 = K1−1 i=0 K2−1 t=0 V ar{cos(ψi,t+d2)} (80) μIm2 = K1−1 i=0 K2−1 t=0 E{sin(ψi,t+d2)} (81) σIm22 = K1−1 i=0 K2−1 t=0 V ar{sin(ψi,t+d2)}. (82) The magnitude of G(2) n with n= d2, denoted by lp, is modeled by f(lp)= lp σp22e −(l2 p+μ2Re2)/2σp22· I 0 lpμRe2 σp22 , (83)
where lp≥ 0 and σp22= (σRe22 + σIm22 )/2.
In addition, the joint magnitude of phase distribution of G(2)d 2 is denoted by f(lp, ϑ)= lp 2π σ2 p2 exp −l 2
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Wan-Hsin Hsieh is currently a senior research engineer of Applied Research
Department in St. Jude Medical, Inc., based in Taipei, Taiwan. He received his Ph.D. degree in electrical engineering from National Chiao Tung University, Hsinchu, Taiwan. From 2012 to 2013, he was with the Division of Sleep and Circadian Disorders at Brigham and Women’s Hospital, Harvard Medical School, where he was a
postdoctoral research fellow in the Medical Biodynamics Program. His research interests include digital signal processing, biomedical signal processing, and nonlinear dynamics in medicine.
Chieh-Fu Chang received his Ph.D. degree in electrical and computer engineering
from Purdue University, West Lafayette, IN. He is currently an associate researcher at the Electrical Engineering Division of the National Applied Research Laboratory -National Space Organization in Taiwan. His current research interests include digital receivers, signal processing, satellite communications, and remote sensing instrument.
[no photo available]
Ming-Seng Kao (S’89—M’90) was born in Taipei, Taiwan, Republic of China, in 1959. He received a B.S.E.E. degree
from the National Taiwan University in 1982, his M.S. degree in optoelectronics from the National Chiao Tung University in 1986, and a Ph.D. degree in electrical engineering from the National Taiwan University in 1990. From 1986 to 1987, he was an assistant researcher at the Telecommunications Laboratories, Chung-Li, Taiwan. In 1990, he joined the faculty of National Chiao Tung University, Hsinchu, Taiwan, where he is now a professor in the
Communication Engineering Department. Between 1993 and 1994, he was a visiting professor at the Swiss Federal Institute of Technology, Zurich, Switzerland, where he worked in optical communications. In 2003, he was a visiting professor at the Nanyang Technological University, Singapore, where he investigated optical devices and networking. He is currently interested in wireless communications and digital receivers.
