J. Cent. South Univ. (2013) 20: 1285−1291 DOI: 10.1007/s11771-013-1613-9

A fast acquisition method of DSSS signals using differential decoding and fast Fourier transform YANG Wei-jun(杨伟君)1, ZHANG Chao-jie(张朝杰)1, 2, JIN Xiao-jun(金小军)1, 2, JIN Zhong-he(金仲和)1, 2, XU Zhao-bin(徐兆斌)1 1. Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China; 2. Micro-Satellite Research Center, Zhejiang University, Hangzhou 310027, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2013 Abstract: In low earth orbit (LEO) satellite or missile communication scenarios, signals may experience extremely large Doppler shifts and have short visual time. Thus, direct sequence spread spectrum (DSSS) systems should be able to achieve acquisition in a very short time in spite of large Doppler frequencies. However, the traditional methods cannot solve it well. This work describes a new method that uses a differential decoding technique for Doppler mitigation and a batch process of FFT (fast Fourier transform) and IFFT (invert FFT) for the purpose of parallel code phase search by frequency domain correlation. After the code phase is estimated, another FFT process is carried out to search the Doppler frequency. Since both code phase and Doppler frequency domains are searched in parallel, this architecture can provide acquisition fifty times faster than conventional FFT methods. The performance in terms of the probability of detection and false alarm are also analyzed and simulated, showing that a signal-to-noise ratio (SNR) loss of 3 dB is introduced by the differential decoding. The proposed method is an efficient way to shorten the acquisition time with slightly hardware increasing. Key words: low earth orbit (LEO) satellite; spread spectrum; fast acquisition; fast Fourier transform (FFT); Doppler mitigation

1 Introduction In a direct sequence spread spectrum (DSSS) system, a synchronized replica of the received pseudo noise (PN) code is required in the receiver to despread the received signal and recover the data sequence. In order to synchronize the PN code, the receiver needs to search over the code phases, and sometimes over a frequency uncertainty region [1−2]. Once the code has been acquired, the system will track the code by a code tracking loop. This work focuses on the code acquisition aspect of the acquisition-tracking problem. The traditional acquisition approach is based on sequential detection. It sequentially searches all code phases over the range of anticipated frequency offsets. This acquisition process, however, costs long time, especially in cases such as low earth orbit (LEO) satellite channel or missile communication scenarios due to long code and large Doppler frequency. A great amount of works have been carried out on techniques of rapid code acquisition. Digital matched filter [3−4] is used to shorten the code phase estimation time. By making judgment on every code chip time, it realizes parallel detection. Fast Fourier

transform (FFT) is inherently a parallel operation and thus is another technique commonly used due to its flexibility and moderate complexity. FFT can be employed to search code phases in parallel [5−6], or to search Doppler frequencies in parallel [1, 7]. These methods, however, only complete 1-dimensioanal parallel search of the 2-dimensional search problem, and still have to process sequential search in the other dimension. By combining the matched filter method with FFT, two dimensions can be searched simultaneously, but it is complex and costs large quantities of hardware resources [8]. Instead of parallel search, differential decoding technique is used to remove Doppler frequency directly [9−11]. In the differential structures, the product of the received signal and its one-chip delayed phase is constantly correlated with the product of a sliding local code and its one-chip delayed phase until a hit is declared. Its performance in presence of white Gaussian noise has been qualitatively analyzed [12]. In this work, a new acquisition method that combines differential decoding technique with FFT is proposed. Differential decoding is executed firstly to remove the Doppler frequency, and then an FFT-based frequency domain correlation is processed to estimate the

Foundation item: Project(60904090) supported by the National Natural Science Foundation of China Received date: 2012−03−06; Accepted date: 2012−06−03 Corresponding author: ZHANG Chao-jie, PhD, Lecturer; Tel: +86−571−87953857; E-mail: [email protected]

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received code phase. Once the code phase acquisition is claimed, the local code will be used to despread the received signal, and another FFT process will be performed to calculate the removed Doppler frequency. Finally, based on the estimated code phase and Doppler frequency, the receiver begins to track the received signal. In this architecture, two FFT processes are performed. They are mathematically equivalent, so the hardware module can be reused. Therefore, the proposed acquisition method is able to do parallel searching both in code phase and Doppler frequency domains with few additional hardware resources compared to conventional FFT methods.

2 Description of PN acquisition methods 2.1 Signal model As in other DSSS communication systems, the outputs of front-end BPF (band-pass filter) are sampled by an ADC (analog-to-digital converter), and can be expressed as [13] rk  2 Pd (tk   )c[(1   )(tk   )] 

cos(IFtk  d tk   k )  nk

(1)

where P is the average signal power, d(t) denotes a random binary data sequence, and c(t) represents the transmitted PN sequence of +1 and –1 values. The quantities τ and η denote the unknown time delay and the fractional perturbation of the chip rate due to Doppler shift, respectively. θk is a slowly time-varying carrier phase function, which is assumed to be constant over several successive chips. The frequency ωIF is the nominal intermediate frequency, and ωd is the carrier signal’s radian Doppler shift. The Doppler shift and the chip rate perturbation are related as η=ωd/2πfR, where fR is the radio frequency and normally larger than 2 GHz. The chip rate perturbation can cause chip shift, but it is always negligible in one correlation period. Hence, this perturbation will not be considered in this work. The last term nk in the right hand side of Eq. (1) can be modeled as additive Gaussian noise represented by the usual band-pass expansion: nk  2{nck cos(IFtk  d tk ) 

Fig. 1 Structure of conventional FFT acquisition method

nsk sin(IF tk  d tk )}

(2)

where nck and nsk are low-pass Gaussian noise processes, following N(0,σ2/2), and mutually independent. The noise power is σ2, and the total power P+σ2 is kept constant by the analog AGC (auto gain control). 2.2 Conventional FFT method The acquisition algorithm searches for the τ and ωd values that maximizes the correlation between the received signal and a reconstruction of it. Traditionally, serial search algorithm quantizes the time uncertainty τ and Doppler uncertainty ωd into a finite number of 2-dimensional grids. The grids are serially tested until it is determined that a particular grid corresponds to the alignment of code phases as well as Doppler frequencies. This 2-dimensional brute force search causes long acquisition time, which cannot meet the requirement of some communication systems such as LEO satellites, since the time when LEO satellites become visible to ground stations is short [14−15]. For this reason, parallel search schemes based on FFT were presented and now extensively used for code acquisition. A typical FFT acquisition method, processing parallel search for code phase and sequential search for Doppler shift, is illustrated in Fig. 1, which consists of a digital down mixer, two low pass filters, a batch process of FFT and IFFT (invert FFT) calculations, and a lock detection module. The down mixer is used to mix the received signal with the local carrier signal, whose frequency is controlled by the lock detection module. The mixing products, both I and Q channels, are low pass filtered to filter sum frequencies and high frequency noise. The output signals are then compounded to process FFT operation. Complex multiplication is employed for the FFT outputs and the pre-computed, conjugated FFT of the PN code. Then, an IFFT process will be carried out on the result signals. According to the principle that frequency-domain multiplication is equivalent with time-domain correlation, the outputs of IFFT exactly represent the correlation results of the received signal and the local PN sequence. The lock detection module searches the maximal correlation value, which is compared with a threshold. If the correlation

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peak exceeds the threshold, the acquisition algorithm will terminate successfully, and the tracking process will be initialized with the derived code phase. If the threshold is numerically larger, the lock detection module will sweep the carrier frequency to a new Doppler frequency and the acquisition process will start all over again. 2.3 Proposed acquisition method With the FFT method, acquisition of signals with negligible Doppler frequency generally can be completed in one correlation period, while for signals with large Doppler range, acquisition usually needs longer time due to sequential search in frequencies. To address this issue, a new acquisition method is proposed: as shown in Fig. 2, with only a slight increase in circuit compared to the conventional FFT method. The increased part plotted in the left dotted rectangle is a Doppler mitigation module, which is used to remove the Doppler frequency. Doppler mitigation utilizes the differential technique, which is based on the shift-and-add property of m sequences [16]. That is, the product of an m sequence and any phase shift of the same m sequence yields another phase of the same m sequence, i.e., c(tk)c(tk −d1T c)=c(tk−d 2Tc)

(3)

where d1, d2[1,L−1]. This indicates that the differential technique still preserves the pseudo noise attributes at its

Fig. 2 Structure of proposed acquisition method

Fig. 3 Proposed structure of Doppler frequency calculation

output and thus the desirable DSSS characteristics. Aside from the Doppler mitigation module, another difference is that the pre-computed, conjugated FFT of the product PN code is used instead of the original one to element-wise multiply against the FFT outputs in the proposed acquisition method, as shown in Fig. 2. Once the Doppler frequency of the received signal is removed, the module composed of FFT and IFFT computes correlations at all code phases simultaneously without trying any local NCO (numerical controlled oscillator) frequencies. If the desired signal exists with adequate power, its code phase shift can be determined in one correlation period. Then, based on the estimated PN code phase, the FFT process will be carried out again to calculate the removed Doppler frequency. The structure of Doppler frequency calculation is shown in Fig. 3, where a lock detection module will provide the Doppler frequency when an FFT output exceeds the preset threshold. After that, the PN tracking loop and carrier tracking loop can be initialized with the correct code phase as well as the Doppler frequency. The two FFT calculations in Fig. 2 and Fig. 3 can be completed by the same FPGA (field programmable gate array) module, thus this does not burden hardware resources. Checking Figs. 1−3 again, we can see that the proposed acquisition method has only four multipliers and two adders more than the conventional FFT method.

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3 Performance analyses In Fig. 2, according to Eq. (1), the mixer output can be described as

rck   rk 2 cos(IFtk  ˆ d tk )  2 Pd k ck ( ) cos(IFtk  d tk   k )  cos(IFtk  ˆ d tk )  2nk cos(IF tk  ˆ d tk )

(4)

which is then low-pass filtered to the following form: rck  Pd k ck ( ) cos(d tk   k )  vck

| R ( , 0) |2  (1 

(5a)

And similarly the other branch can be expressed as rsk  Pd k ck ( ) sin(d tk   k )  vsk

(5b)

where vck and vsk are noise terms. Thus, the differential decoding outputs are given by Ak=rck·rckTD=PdkdkTDck(τ)ck(τ+TD)· cos(Δωdtk−θk)·cos[Δωd(tk−TD)−θk)]+nAk

(6)

where TD is the delayed time, nAk represents the noise term which will be analyzed later. Making V=dkdkTD, Ck=ck(τ)ck(τ+TD), then Ak=PVCkcos(Δωdtk−θk)cos[Δωd(tk−TD)−θk)]+nAk

(7a)

By the same method, it can be obtained that Bk=PVCksin(Δωdtk−θk)cos[Δωd(tk−TD)−θk)]+nBk

(7b)

Ck=PVCkcos(Δωdtk−θk)sin[Δωd(tk−TD)−θk)]+nCk

(7c)

Dk=PVCksin(Δωdtk−θk) sin[Δωd(tk−TD)−θk)]+nDk

(7d)

From Eq. (7), I and Q channels are calculated to be Ik=Ak+Dk=PVCkcos(ΔωdTD)+NIk

(8a)

Qk=Ck−Bk=PVCksin(ΔωdTD)+NQk

(8b)

where NIK and NQK are the noise terms. Since cos(ΔωdTD) and sin(ΔωdTD) are always constants under the assumption of negligible Doppler change rate, Ik and Qk are no more dependent on the Doppler frequency. Furthermore, TD=Tc is chosen, as a result, dkdkTD is almost “1” because data period is ordinarily much longer than one PN code chip time. So, it can be found that the structure is also robust to data modulation. 3.1 Acquisition time The acquisition time is the amount of time required for an acquisition system. Its mean value is defined as [17] Tacq 

2  (2  PD )(q  1)(1  KPFA ) D 2 PD

PFA is the probability of false alarm. For the convenience of analysis, we consider a practical application that the pseudo noise sequence is an m sequence of length 1 023, chip rate Rc is 1.023 MHz, the largest Doppler frequency range is ±100 kHz, the sampling rate is 2.046 MHz, and the number of correlation samples N is 2 046. The correlation losses due to Doppler frequency and code phase estimation errors (i.e. Δωd and Δτ) are expressed as [18−19]

(9)

where K denotes the false alarm penalty, q represents the total number of grids to be searched and τD is the correlation time. PD is the probability of detection, and

|  | 2 ) , (| | ≤ Tc ) Tc

sin(d Ts N / 2) | R (0, d ) |  N sin(d Ts / 2)

(10)

2

2

(11)

where Ts is the sampling time. Therefore, a code phase search interval of Tc/2, whose maximal phase error is Tc/4, can cause correlation peak loss of 2.5 dB derived by Eq. (10), whilst a Doppler frequency search interval of 1 kHz induces 3.92 dB loss according to Eq. (11). In this application, for serial search algorithm, there are 2 046× 200 grids to be searched, which indicates an average acquisition time of 205 s for PD=1, PFA=0 by Eq. (9). For the conventional FFT method, with 200 search grids in frequency domain, the mean acquisition time is 100 ms. In the proposed acquisition method, the two FFT processes are implemented sequentially, resulting in parallel searches of both code phase and Doppler frequency. So, the acquisition time is theoretically the sum of the correlation time for code phase search and Doppler frequency search. For the above application, the acquisition time is 2 ms if the calculation time is neglected. Now the acquisition time of different methods is listed in Table 1. It can be obtained that the acquisition time of the proposed method is dramatically shortened. It is only 0.001% of the serial search algorithm, and one fiftieth of the conventional FFT method. Table 1 Mean acquisition time for PD=1, PFA=0 Search algorithm Acquisition time/s

Serial conventional FFT search method 205

0.1

Proposed method 0.002

3.2 Noise performance There are, however, still tiny drawbacks of the proposed acquisition algorithm. The differential decoding method introduces extra noise. In Eq. (5), the noise terms vck and vsk are the down converted and low-pass filtered noises, which can be expressed as

vck=nckcosΔωdtk−nsksinΔωdtk

(12a)

vsk=ncksinΔωdtk+nskcosΔωdtk

(12b)

Since nck and nsk are wide band noises compared to

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Δωd, vck and vsk are statistically not sensitive to the Doppler error Δωd. For simplicity, let Δωd=0, then vck=nck, vsk=nsk

(13)

Besides, ck and dk are +1/−1 values exclusively, and do not affect the statistics of the noise terms, therefore, they are neglected hereafter in the noise analysis. From Eq. (13), the noise terms in Eq. (7) can be deducted and given in the following forms n Ak  Pnc1 cos 1  Pnc 2 cos  2  nc1nc 2

(14a)

nBk  Pnc1 sin 1  Pns 2 cos  2  ns 2 nc1

(14b)

nCk  Pns1 cos 1  Pnc 2 sin  2  nc 2 ns1

(14c)

nDk  Pns1 sin 1  Pns 2 sin  2  ns 2 ns1

(14d)

as long as the following definitions are made: nc1=nc(tk−TD), nc2=nc(tk), ns1=ns(tk−TD), ns2=ns(tk), α1=ΔωdTD−θk, α2=Δωd(tk−TD)−θk. Since nc1, nc2, ns1 and ns2 are mutually independent, the last terms in the right hand side of Eq. (14) obey the normal product distribution, and E(nc1nc2)=E(nc1)·E(nc2)=0 2

(15a) 4

Var(nc1nc2)=E[(nc1nc2) ]=σ /4

(15b)

The noise in the I and Q channels displayed in Eq. (8) are NIK=nAk+nDk, NQK=nCk−nBk. Putting together Eqs. (14) and (15), their statistical properties are obtained as E(NIK)=E(NQK)=0

(16a)

Var(NIK)=Var(NQK)=Pσ2+σ4/2

(16b)

From Eqs. (8) and (16), the signal-to-noise ratio (SNR) at point Z in Fig. 2 becomes SNR- Z 

P2 2 P 2   4

input SNR is also plotted for comparison. It can be seen that when the input SNR is greater than 0 dB, the SNR deterioration is approximately a constant equal to 3 dB, and when the input SNR is less than 0 dB, the deterioration increases nearly linearly as the input SNR decreases.

4 Simulation results Computer simulations were carried out in order to verify the effectiveness of the proposed method. Figure 5 depicts the simulation results of probabilities of false detection of both the conventional FFT method and the proposed method. A curve representing the theoretical probability of false detection versus the threshold is also plotted. The theoretical curve is expressed as an exponential function of Pfalse  1  (1  PFA ) N FFT

(18)

where NFFT represents the FFT length, and PFA denotes the probability of false alarm at a single bin of IFFT output. PFA can be described as a Rayleigh random variable [20] PFA  Prob( Pn   )   



2

/ 2 e

2



v



 e2

e  y dy  e 

2



e

v2 2 e 2

dv 

/ 2 e 2

(19)

where η is the detect threshold and 2σe2 is the effective noise variance equal to σ2/NFFT.

(17)

which is deteriorative compared to the input SNR P/σ2. The relationship of the deterioration and the input SNR is illustrated in Fig. 4. The solid line is the SNR of point Z, and the reference line representing the same value with

Fig. 5 Simulation results of probabilities of false detection

Fig. 4 SNR deterioration of Doppler mitigation model

By examining Fig. 5, we can see that the probability of false detection of the proposed method is the same as that of the conventional FFT method, and both of them are consistent with the theoretical curve. Thus, in the proposed acquisition structure, the noise after the Doppler mitigation module can still be treated as Gaussian noise. Figure 5 also tells that a threshold setting

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of η ·NFFT/σ =17 can guarantee the probability of false detection smaller than 10−4. Under this threshold setting, the probability of detection is simulated. The probability of detection PD is always characterized in the form of the probability of miss detection PM=1−PD. The simulation results of the probabilities of miss detection versus the carrier-to-noise ratio (C/N0) are illustrated in Fig. 6. The leftmost line (L1) is the probability of miss detection of the conventional FFT method in the case of no Doppler frequency in the received signal, and the second line (L2) is that in the case of a 500 Hz Doppler frequency in the received signal. Comparing these two lines, we can find that the correlation loss due to a 500 Hz Doppler frequency is almost 3.92 dB, which can be obtained in Eq. (11). The other two lines (L3 and L4) in Fig. 6 are the probabilities of the proposed method. It can be seen that the sensitivity of the proposed method is approximately 5 dB worse compared with that of the conventional FFT method when PM=10−3. It is caused by the deterioration of the effective SNR due to differential Doppler mitigation, as depicted in Fig. 4. It can also be seen that probabilities of miss detection of the proposed method are the same whatever the Doppler frequency is. In other words, the correlation peak is independent of Doppler frequencies. In Fig. 7, signal with data modulation is considered. We model the worst situation that data reversal happens to be in the middle of the correlation period. From the simulation results revealed in Fig. 7, it can be seen that the probability of miss detection with data reversal is identical to that without data reversal. That is, the proposed method is very robust to the data modulation, while the conventional FFT method cannot complete acquisition well when there is data reversal in the correlation period.

Fig. 7 Probabilities of miss detection of proposed method when η2·NFFT/σ2=17

5 Conclusions In this work, we present a code acquisition structure for spread spectrum communication systems in face of large Doppler frequencies. It combines the differential technique with FFT in order to achieve parallel search in both code phase and Doppler frequency domains. The acquisition performance has been analyzed in terms of the mean acquisition time, as well as the detection and false alarm probabilities. Simulation results show that the acquisition time of the proposed method is only one fiftieth of the conventional FFT method, while requiring only slightly additional implementation complexity. Furthermore, its robustness to data modulation also contributes to fast acquisition. With the use of differential decoding, the noise performance is sacrificed by a 3 dB SNR loss. For some LEO satellite communications, acquisition time is the most important consideration and this architecture may be very attractive.

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Fig. 6 Simulation results of probabilities of miss detection when η2·NFFT/σ2=17 (L1: Conventional FFT method without Doppler; L2: Conventional FFT method with 500 Hz Doppler; L3: Proposed method without Doppler; L4: Proposed method with 100 kHz Doppler)

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