Acquisition of Time and Doppler Shift in WFRFT Encrypted Chaotic Direct Sequence Spread Spectrum System

Weighted fractional Fourier transform encrypted chaotic direct sequence spread spectrum (WFRFT-CD3S) signal is aperiodic and complex Gaussian distributed. The aperiodicity and complex Gaussian distribution are conducive to the covert transmission of information but also make it difficult to acquire the WFRFT-CD3S signal. An acquisition method implemented in two steps is proposed to acquire the Doppler-shifted WFRFT-CD3S signal. First, a bidirectional correlation search method is proposed to align the received signal with the local spreading sequence in the time domain. The correlation peak attenuation caused by the Doppler frequency shift is eliminated by differential. Second, the accumulated phase angle caused by the Doppler frequency shift is obtained by the differential correlation value between the aligned received signal and the local spreading sequence. Then, the Doppler frequency shift is estimated by the accumulated phase angle. The detection probability, the false alarm probability, and the root mean square error (RMSE) of the Doppler shift estimation are analyzed theoretically and simulated. The theoretical results are verified by simulation. Theoretical results and simulations show that the detection probability is higher than 0.999, the false alarm probability is less than 0.001, and the RMSE of Doppler shift estimation is lower than 63 Hz when the signal-to-noise ratio is higher than −7.3 dB.

INDEX TERMS Acquisition, chaotic direct sequence spread spectrum, aperiodicity, Doppler frequency shift, detection probability, false alarm probability, root mean square error.

SYMBOL LIST b k
The k-th message bit of the transmitter. c k c k is the chaotic sequence corresponding to the k-th message bit. The m-th chip of c k is c k,m . c 0, 1 System initial value. c k, 1 Chaotic initial value of c k . q Tent mapping parameter.  y k,mỹk,m is the differential received signal in the time acquisition process, and the number of chips contained in the delay is N . x k,mxk,m is the differential WFRFT-chaotic signal in the time acquisition process, and the number of chips contained in the delay is N . a cc Acceleration of vehicle. G (a cc ) The correlation value attenuation of the differential signals caused by vehicle speed variation. y a The received signal segment in the a-th round of correlation search. y a The differential received signal in the a-th round of correlation search. x kxk is the differential WFRFT-chaotic sequence in the time acquisition process, and the m-th chip ofx k isx k,m . g k Output of the matched filter corresponding tox k . Residual Doppler frequency shift. y k,mỹ k,m is the differential received signal in the Doppler shift estimation process, and the number of chips contained in the delay is L. x k,mx k,m is the differential WFRFT-chaotic signal in the Doppler shift estimation process, and the number of chips contained in the delay is L. L The number of chips contained in the delay of differential processing. g fd Correlation value ofỹ k,m andx k,m . µ g fd Mathematical expectation of g fd . σ 2 g fd Variance of g fd . θ Phase angle of g fd . f θ The probability density function of θ. σ 2 θ Variance of θ. f d2 Fine estimation of the Doppler frequency shift. f d The estimation of the Doppler frequency shift. S NR Signal-to-noise ratio (SNR). g u The correlation value when the received signal and the local WFRFT-chaotic sequence are not aligned. u g u Mathematical expectation of g u .
The probability density function of η u . P F The false alarm probability. g s The correlation value when the received signal is aligned with the local WFRFT-chaotic sequence. µ g s Mathematical expectation of g s .
The probability density function of η s . P D The detection probability. ε (L) Root mean square error (RMSE) of Doppler shift estimation. |x| Mathematical expectation of x.

I. INTRODUCTION
The broadcast nature of wireless channels makes legitimate communication signals easy to be intercepted by noncooperative receivers. The chaotic direct sequence spread spectrum (CD3S) communication spreads the message bits through the aperiodic chaotic sequence, which is resistant to the periodicity detection [1], [2], [3]. Since the background noise is usually Gaussian and the chaotic sequence is usually non-Gaussian, the noncooperative receiver can still intercept the CD3S signal by analyzing the statistical characteristics of the received signal [4], [5]. The weighted fractional Fourier transform (WFRFT) can change the statistical distribution of the aperiodic signals into Gaussian distribution [6]. Reference [5] proposed a weighted fractional Fourier transform encrypted chaotic direct sequence spread spectrum (WFRFT-CD3S) system, which makes the transmitted signal aperiodic and complex Gaussian distributed. The WFRFT-CD3S system enables message bits to be transmitted covertly and is suitable for enhancing the security of data links for the unmanned aerial vehicle (UAV). However, in the UAV system, the data link suffers from time-selective fading caused by Doppler frequency shift [7], [8], and the acquisition of WFRFT-CD3S signals in the presence of Doppler frequency shift is a challenging issue. In this paper, we further discuss the acquisition of timing and Doppler frequency shift in the WFRFT-CD3S system based on [5]. The Doppler shift attenuates the correlation between the received signal and the local spreading sequence, thereby increasing the complexity of acquisition [9]. So far, few papers have discussed the acquisition of WFRFT-CD3S signals or similar aperiodic CD3S signals in the presence of Doppler shifts. For the periodic direct sequence spread spectrum (DSSS) signal with Doppler frequency shift, the researchers achieve acquisition through the time-frequency two-dimensional search and focus on improving detection performance or reducing computational complexity through different signal processing methods. The extended replica folding acquisition search technique [10] and the all-phase fast Fourier transform [11] were applied to improve detection performance. The compressed sensing theory [12] and the Doppler frequency estimation algorithms [13], [14] were applied to reduce the computational complexity. Reference [15] proposed a sparse discrete fractional Fourier transform (SDFrFT) algorithm with low computational complexity to acquire high-dynamic DSSS signals, and the SDFrFT algorithm was optimized in [16].
In the DSSS system, the spreading sequence is a fixed periodic binary pseudo-noise (PN) sequence. Different acquisition methods for DSSS signals are based on the periodicity and binary characteristics of PN sequences, and these acquisition methods essentially implement a time-frequency two-dimensional search. In the WFRFT-CD3S system, the spreading sequence is the aperiodic and complex Gaussian distributed WFRFT-chaotic sequence. The acquisition methods for the DSSS signal are not applicable to the WFRFT-CD3S signal for three reasons. First, each bit of the WFRFT-CD3S signal is spread by different WFRFT-chaotic codes. Even if the Doppler frequency shift is not considered, the aperiodic WFRFT-CD3S signal cannot be acquired by a fixed WFRFT-chaotic sequence when the WFRFT-CD3S signal leads the local WFRFT-chaotic sequence [5], [17]. Second, the computational complexity of the timefrequency two-dimensional search is unacceptable since the WFRFT-chaotic sequence is aperiodic. Third, the phase angle of the complex Gaussian distributed WFRFT-CD3S signal is random, making it difficult to estimate the Doppler shift before the synchronization. A possible acquisition method is to periodically insert a pilot signal into the WFRFT-CD3S signal and achieve the acquisition through the pilot assistance. However, the pilot signal destroys the aperiodicity of the transmitted signal and reduces the anti-interception performance of the transmitted signal. When the pilot signal is detected, the noncooperative receivers can implement targeted jamming on the pilot to disrupt the acquisition [18], [19]. Therefore, this study does not use the pilot signal to assist the acquisition.
In this study, we accomplish the acquisition of time and Doppler shifts in two steps instead of directly implementing a time-frequency two-dimensional search. First, we align the received signal with the local spreading sequence by a one-dimensional correlation search in the time domain. This process is named time acquisition. The correlation search is performed on the differential received signal and the local differential spreading sequence. There is only a fixed phase angle offset between the differential received signal and the local differential spreading sequence, but no Doppler frequency shift [20], which will not cause attenuation of the correlation peak. Inspired by [17], the correlation search is implemented bidirectionally to ensure that the correlation peak can be detected regardless of whether the aperiodic received WFRFT-CD3S signal lags or leads the local spreading sequence. After the time acquisition, we coarsely estimate the Doppler frequency shift by analyzing the spectrum of the despread signal to obtain a relatively large estimation range. Then, the residual Doppler shift is finely estimated by the correlation of the differential received signal and the local differential spreading sequence.
The rest of the paper is organized as follows: Section II presents the system model of WFRFT-CD3S. The proposed acquisition method of time and Doppler shift is discussed in detail in Section III. Section IV theoretically analyzes the detection probability, false alarm probability, and root mean square error (RMSE) of Doppler shift estimation. Section V presents the numerical simulation results. Finally, conclusions are given in Section VI.

II. SYSTEM MODEL
In the WFRFT-CD3S system, the message bit b k is spread by the WFRFT-chaotic sequence N is the spreading factor. The aperiodic and complex Gaussian distributed WFRFT-chaotic sequence x k is generated in two steps. First, generate the chaotic sequence c k = c k,1 , c k,2 , · · · , c k,N T by two chaotic maps. Then, encrypt the chaotic sequence by WFRFT. The first chaotic map generates the chaotic initial value c k,1 according to the system initial value c 0,1 . In this study, the first chaotic map is the Tent map, which is given by where q ∈ (0, 1) is the Tent mapping parameter. The system initial value c 0,1 needs to meet the initial value condition of the first chaotic map. When the first chaotic map is the Tent map, the system initial value c 0,1 is in (0, 1). The Tent map can generate different chaotic sequences according to different system initial values in (0, 1), but the characteristics of these chaotic sequences are similar. When the system initial value is not in (0, 1), the Tent map cannot generate chaotic sequences. The second chaotic map generates the chaotic sequence c k according to chaotic initial value c k,1 . In this study, the second chaotic map is the improved Logistic map, which is given by The WFRFT-chaotic sequence x k is obtained by encrypting the chaotic sequence c k by WFRFT. For the chaotic sequence c k , the α-order WFRFT is given by [21] where F α is the α-order WFRFT matrix. When α ∈ [0.8, 1], the real and imaginary parts of x k are independent and obey the same Gaussian distribution [6]. F is the normalized Fourier transform matrix, and its element in the m-th row and n-th column is given by The weighting coefficient ω α m is given by The inverse matrix P is given by The WFRFT-CD3S signal is generated by spreading the message bit b k through the WFRFT-chaotic sequence x k . This process can be expressed as where When there is a Doppler frequency shift, the received WFRFT-CD3S signal can be expressed as where f d is the Doppler frequency shift, φ 0 is the initial phase, T c is the chip duration, τ 0 is the transmission delay, and v n is the white Gaussian noise. The mean and variance of v n are 0 and σ 2 , respectively.

III. ACQUISITION A. PROBLEM DESCRIPTION
To facilitate analysis of the effects of Doppler frequency shift, the Doppler frequency shift is temporarily assumed to be constant. When there is a Doppler shift, the correlation value of the WFRFT-chaotic sequence x k can be calculated as where the average chip energy P c is given by Equation (9) shows that the Doppler frequency shift f d attenuates the correlation peak of the WFRFT-chaotic sequence x k by Inspired by [20], we remove the Doppler shift in the received signal by differential processing. The differential received signal is given bỹ where y k,m represents the received signal corresponding to s k,m . The WFRFT-chaotic sequence is differentiated accordingly as followsx The correlation value between the differential signalx k,m and y k,m is given by Since there is only a phase offset but no frequency offset betweenx k,m andỹ k,m , the magnitude of the correlation value in (14) is not attenuated. Furthermore, the phase angle in (14) contains the Doppler shift f d . Therefore, the Doppler shift f d can be estimated from (14) when the received signal is aligned with the local WFRFT-chaotic sequence.
The above analysis assumes that the Doppler frequency shift is constant. However, the Doppler frequency shift varies with vehicle speed in the engineering. Assuming that the vehicle performs a uniformly accelerated motion with the acceleration of a cc when receiving the message bit b k−1 , the Doppler frequency corresponding to the m-th chip of b k−1 can be expressed as where f d0 represents the initial Doppler frequency shift, f d represents the Doppler frequency variation between adjacent chips, and λ is the carrier wavelength. When the Doppler frequency shift varies, the correlation value between the differential signalx k,m andỹ k,m is given by According to (16) and (17), the acceleration of the vehicle attenuates the correlation peak of the differential signals by Fig. 1 shows the curve of the attenuation of the correlation peak of the differential signal with the acceleration. In Fig. 1, the carrier frequency is 2 GHz, the carrier wavelength λ is 0.15 m, the spreading factor N is 1000, the chip duration T c is 1 µs, and the initial velocity of the vehicle is 300 m/s. In Fig. 1, the theoretical result in (18) is consistent with the simulation results. When the acceleration is 500 m s 2 , the correlation peak of the differential signals is attenuated by 0.99992. The acceleration of UAV is usually much less than 500 m s 2 . Therefore, the Doppler frequency variation can be ignored in the engineering. Since the Doppler frequency variation has little impact on acquisition performance, the Doppler frequency is assumed to be constant in the remainder of this paper to simplify the analysis.
Based on the above analysis, we acquire the WFRFT-CD3S signal with Doppler shift in two steps. First, align the received signal with the local spreading sequence by a one-dimensional correlation search betweenx k,m andỹ k,m . Second, estimate the Doppler shift according to (14).

B. TIME ACQUISITION
The WFRFT-chaotic sequence is aperiodic, and the received signal may lag or lead the local WFRFT-chaotic sequence. Therefore, the direction of the correlation search must be consistent with the delay to achieve acquisition. Considering that the delay is unknown, we propose a time acquisition method based on a bidirectional correlation search, which consists of a forward correlation search channel and a backward correlation search channel.
During the correlation search, the unknown binary message bits limit the coherent integration time. In order to avoid the phase inversion, the correlation search must be performed through the WFRFT-chaotic sequence x k corresponding to the single message bit b k . In [17], the received signal is also divided into segments of N chips to facilitate parallel search. However, this approach results in that the number of aligned chips falls between N 2 and N when detection happens, thereby reducing the detection probability. In this study, to ensure that the number of aligned chips is always N , we divide the received signal into segments with a length of 2N chips and N chips overlap.
The proposed time acquisition method based on bidirectional correlation search is shown in Fig. 2. In order to avoid the attenuation of the correlation peak due to Doppler shift, the received signal and the local WFRFT-chaotic sequence are differentiated. Correlation search is performed by matched filters. Each matched filter performs one round of correlation search every time N chips of y n are received. a represents the rounds of correlation search.
In Fig. 2, the received signal segment y a and the differential received signalỹ a are given by The local WFRFT-chaotic sequence x k still contains only N chips. The differential WFRFT-chaotic sequencex k is given byx To make the two input sequences of the matched filter equal in length, the differential WFRFT-chaotic sequence is zero-padded to 2N chips. The output of the matched filter is given by where F −1 is the inverse Fourier transform matrix, U N = I N , O N ,N T is the zero padding matrix, I N is the N -order identity matrix, and O N ,N is the N × N zero matrix.
In the backward search channel, the correlation search is performed through the fixed differential WFRFT-chaotic sequencex 2 . As the WFRFT-CD3S signal is continuously received, the matched filter constructed by the fixed differential WFRFT-chaotic sequencex 2 is equivalent to backward correlation search. In the forward search channel, the correlation search is performed by dynamically generated differential WFRFT-chaotic sequencesx 2a andx 2a−1 . Every time N chips of y n are received, the receiver generates x 2a and x 2a−1 with a total length of 2N chips. which is equivalent to a forward correlation search. Therefore, the two matched filters constructed byx 2a andx 2a−1 are equivalent to forward correlation search.
In Fig. 2, |g max | is the maximum magnitude of the elements in g 2 , g 2a , and g 2a−1 . Z T is the acquisition threshold. When |g max | > Z T , the acquisition is considered to have been achieved, and the delay estimationτ 0 can be obtained from the position of |g max |. Fig. 3 shows the schematic diagram of the alignment between the received signal and the local WFRFT-chaotic sequence. Since the received signal segment y a contains 2N chips, y a contains at least one complete message bit. As shown in Fig. 3, in a certain round of correlation search, the N chips in one spreading period will be all aligned.

C. DOPPLER SHIFT ESTIMATION
There are two problems with estimating the Doppler shift directly from (14). First, the unknown message bits b k and b k−1 cause phase ambiguity. Second, the estimation range of the Doppler frequency shift is |f d | < (2NT c ) −1 . It can be seen from (11) that it is less meaningful to estimate the Doppler  frequency shift in this range. In this subsection, we first make a coarse estimate of the Doppler shift to obtain a larger estimation range. Then, remove the phase ambiguity in (14) and complete the fine Doppler shift estimation. For the DSSS system with binary PN sequence, the despread signal is a single tone signal superimposed with noise. The coarse estimate of the Doppler frequency shift can be obtained from the spectrum of the despread signal. In the WFRFT-CD3S system, since the WFRFT-chaotic sequence is a complex Gaussian distribution, the despread signal is relatively more complicated. The despread signal is given by It can be seen from equation (19) that the despread signal d k,m is not a superposition of a single tone signal and noise. Therefore, it is not guaranteed that the Doppler frequency shift can be estimated from the spectrum of despread signal d k,m .
The WFRFT-chaotic sequence has good autocorrelation properties in the frequency domain, as shown in Fig. 4(a). The Doppler shift of the WFRFT-chaotic sequence in the time domain results in the correlation peak shift in the frequency domain, as shown in Fig. 4(b). Therefore, in the WFRFT-CD3S system, the Doppler frequency shift can be coarsely estimated by searching the correlation peak between the received signal and the local WFRFT-CD3S sequence in the frequency domain. In this study, the correlation search in the frequency domain is implemented by a matched filter, which is given by VOLUME 10, 2022 where y k contains N chips and is the synchronized received signal corresponding to x k . Equation (24) is equivalent to analyzing the spectrum of the despread signal. Denote the m-th element in g as g m , then the coarse estimate of the Doppler frequency shift can be obtained bŷ Considering the periodicity of the Fourier transform of the discrete signal, the estimation range of (25) is When the chip duration T c is 1µs, the estimation range of Doppler shift is (−500 kHz, 500 kHz). The spectral resolution in (24) is (NT c ) −1 . When the Doppler frequency shift is exactly in the middle of the two frequency points, the residual Doppler frequency shift reaches the maximum. That is, the residual Doppler frequency shift is According to (11), the maximum attenuation caused by the residual Doppler frequency shift is 0.6366, which is still a relatively large attenuation. Therefore, it is necessary to implement fine estimation to remove the residual Doppler frequency shift.
In (12), the differential received signalỹ k,m contains two message bits, causing phase ambiguity. To avoid phase ambiguity caused by unknown message bits, we differentiate the chips in the synchronized received signal y k which contains only one message bit. The Doppler frequency shift is compensated byf d1 in the differential process. The differential received signalỹ k,m is given bỹ where L is the number of chips contained in the delay of differential processing. Correspondingly, the differential WFRFT-chaotic signal x k,m is given byx Denote the correlation value ofỹ k,m andx k,m as g fd . The correlation value g fd is equal to The expectation of g fd can be calculated as The fine estimation of the Doppler frequency shift can be calculated byf where the phase angle θ is given by Since L is less than N , it can be seen from (27) that the phase angle in (33) is in (−π, π). That is, residual Doppler frequency shift f d2 is small enough to not cause phase ambiguity.
In summary, the estimated value of the Doppler shift iŝ

IV. THEORETICAL ANALYSIS OF PERFORMANCE A. FALSE ALARM PROBABILITY
The false alarm occurs when the received signal is misaligned with the local WFRFT-chaotic sequence but the magnitude of the correlation value is greater than the acquisition threshold. For ease of writing, we assume that the delay between the received signal and the local WFRFT-chaotic sequence is N chips. This assumption does not affect the theoretical analysis results. Then the correlation value when the received signal and the local WFRFT-chaotic sequence are not aligned is given by The signal-to-noise ratio (SNR) is defined as The correlation value g u obeys a complex Gaussian distribution according to the central limit theorem. The mathematical expectation of g u can be derived as The variance of g u can be derived as (38) Denote the magnitude of g u as η u . Since the mathematical expectation of g u is 0, and the real and imaginary parts of g u are independent and identically distributed, η u obeys the Rayleigh distribution. The probability density function of η u can be expressed as The false alarm probability of time acquisition can be derived as (40)

B. DETECTION PROBABILITY
Detection occurs when the received signal is aligned with the local WFRFT-chaotic sequence and the magnitude of the correlation value is greater than the acquisition threshold. The correlation value when the received signal is aligned with the local WFRFT-chaotic sequence is given by The correlation value g s obeys a complex Gaussian distribution according to the central limit theorem. According to (14), the mathematical expectation of g s is equal to The variance of g s can be derived as Denote the magnitude of g s as η s and the magnitude of µ g s as µ η s . Since the WFRFT-chaotic sequence x k is a complex Gaussian distribution, the real and imaginary parts of g s are correlated when S NR = +∞. As the SNR decreases, the correlation between the real and imaginary parts of g s decreases. When the SNR is low, η s approximately obeys the Rice distribution. The probability density function of η s can be expressed as VOLUME 10, 2022 where I 0 (·) is the zeroth-order modified Bessel function, one of the definition equations of which being The detection probability of time acquisition can be derived as [22] where the Marcum Q-function is defined as [23] Q

C. ROOT MEAN SQUARE ERROR OF DOPPLER SHIFT ESTIMATION
The residual frequency shift of the coarse Doppler frequency shift estimation is processed by the fine estimation. Therefore, the RMSE of the Doppler frequency shift estimation is determined by the fine estimation. According to (43), the variance of g fd can be easily derived as Similar to g s , it can be known that the phase angle of g fd follows the Rice phase distribution. The probability density function of phase angle θ can be expressed as [24] where erf (x) = 2 √ π x 0 exp −u 2 du is the error function, According to [26], the variance of the phase angle θ is approximately equal to (51) The estimate obtained by (32) is unbiased. Combining (32) and (51), the RMSE of the Doppler shift estimate is approximately equal to According to (52), ε (L) takes the minimum value when

V. SIMULATION ANALYSIS
In all simulations, the system initial value c 0,1 , the Tent mapping parameters q, the WFRFT order α, the spreading factor N and the chip duration T c are set to 0.3, 0.65, 0.94, 1000 and 1µs, respectively.
A. SIMULATION ANALYSIS OF TIME ACQUISITION Fig. 5 shows the output of the matched filters when time acquisition happens. The Doppler frequency shift is set to 5 kHz and the SNR is set to 10 dB in Fig. 5. The correlation peak attenuation caused by Doppler frequency shift is avoided by differencing. When the received signal lags the local WFRFT-chaotic sequence, the correlation peak appears in the backward search channel. When the received signal leads the local WFRFT-chaotic sequence, the correlation peak appears in the forward search channel. In Fig. 5(b), the attenuated correlation peak appearing in g 2a is not caused by Doppler frequency shift but is caused by partial chips alignment between the received signal and the local WFRFTchaotic sequence. Fig. 5 shows that the proposed bidirectional correlation search method can acquire the aperiodic WFRFT-CD3S signal with Doppler frequency shift.    6 shows the curves of false alarm probability and detection probability as a function of the acquisition threshold. It can be seen from Fig. 6 that the theoretical results of detection probability and false alarm probability are basically consistent with the simulation results. For the detection probability when the SNR is 0 dB, there is a deviation between the theoretical value and the simulated value but the deviation is limited. Both the detection probability and the false alarm probability show an overall decreasing trend with the increase of the acquisition threshold. Assume that the system detection probability index P FI is set to 0.001. When S NR = 0dB, there is an acquisition threshold Z T ∈ [41, 211] such that P F ≤ P FI and P D ≥ P DI . When S NR = −10dB there is no acquisition threshold Z T that P DI and P FI can be satisfied at the same time.
It can be inferred from Fig. 6 that there is an optimal acquisition threshold Z * T such that P DI and P FI can be satisfied at the lowest SNR. To analyze the optimal acquisition threshold Z * T of the proposed bidirectional correlation search method, Fig. 7 shows the curves of the acquisition threshold superior limit Z max and inferior limit Z min as a function of SNR. In Fig. 7. P DI and P FI are set to 0.999 and 0.001, respectively. The acquisition threshold superior limit Z max is the highest acquisition threshold at which P DI can be satisfied. The acquisition threshold inferior limit Z min is the lowest acquisition threshold at which P FI can be satisfied. When SNR is greater than −7.3dB, Z min is lower than Z max . When SNR is equal to −7.3 dB, Z min and Z max are equal to 131.8. This means that when SNR is greater than −7.3dB, setting Z T to 131.8 can make P DI and P FI be satisfied at the same time. When SNR is lower than −7.3dB, Z min is greater than Z max . Therefore, P DI and P FI cannot be satisfied at the same time when SNR is lower than −7.3 dB. In summary, the optimal acquisition threshold Z * T is 131.8 and the minimum acquirable SNR is −7.3dB under the given simulation conditions.

B. SIMULATION ANALYSIS OF DOPPLER SHIFT ESTIMATION
To analyze the influence of the delay in the difference process on the Doppler frequency estimation error, Fig. 8 shows the RMSE of the Doppler frequency estimation based on the number of chips contained in the delay. As the SNR decreases and the number of chips L increases, the deviation between the theoretical and simulated results increases. The reason is that as the SNR decreases and L increases, the value of K in (50) decreases, thereby increasing the approximation error in (51). The deviation between the theoretical and simulated results in Fig. 8 is limited, which means that it is still feasible to design the system parameters according to the theoretical results in engineering. To ensure the resolution when the RMSE is small, the vertical axis of Fig. 8 is limited to [0 Hz, 50 Hz] and the simulation results when L is close to N or 0 are not shown in Fig. 8. When L is close to N or 0, the RMSE rises rapidly, and L is not suitable to be taken near N or 0. Equation (53) shows that the RMSE is minimized under the given simulation conditions when L = 2N 3 ≈ 667. However, Fig. 8 shows that the RMSEs are close to each other when L ∈ [550, 750]. Therefore, L can take a value around 2N 3 in engineering. Fig. 9 shows the RMSE of the Doppler shift estimation as a function of SNR for different residual Doppler shifts. The VOLUME 10, 2022 delay between the differentiated chips is set to L = 667. The residual Doppler shift is in [−500Hz, 500Hz] under the given simulation condition. Therefore, the residual Doppler shifts in Fig. 9 are set to 100 Hz, 300 Hz, and 500 Hz. The RMSEs under different residual Doppler frequency shifts are close to each other when S NR ≥ −8 dB. When S NR < −8dB and the residual Doppler shift is 500 Hz, the RMSE of Doppler shift estimation increases significantly. The reason for this phenomenon is the phase ambiguity caused by the residual Doppler shift superimposed noise. It can be inferred from Fig. 9 that the residual Doppler shift does not affect the estimation error in the absence of phase ambiguity. However, in (33), higher residual Doppler shifts are more likely to cause phase ambiguity under the influence of noise. The minimum acquirable SNR in the time acquisition phase is −7.3dB. When S NR > −7.3dB, the RMSE of Doppler shift estimation is less than 63 Hz.

VI. CONCLUSION
In this paper, we acquire the aperiodic WFRFT-CD3S signal with Doppler shift through two steps of time acquisition and Doppler shift estimation. The false alarm probability and detection probability of time acquisition, and the RMSE of Doppler frequency shift estimation are theoretically analyzed. The theoretical results are verified by simulation results. The simulation results show that the proposed bidirectional correlation search method can achieve acquisition regardless of whether the received WFRFT-CD3S signal leads or lags behind the local WFRFT-chaotic sequence. The optimal acquisition threshold is obtained through simulation analysis. The optimal acquisition threshold is 131.8 and the minimum acquirable SNR is −7.3 dB under the simulation conditions given in Section V. When the SNR is greater than −7.3 dB, the detection probability is greater than 0.999, the false alarm probability is less than 0.001, and the RMSE of Doppler frequency shift estimation is less than 63 Hz. The Doppler frequency shift is assumed to be constant in this study but the proposed method is applicable to the data link of UAV where the Doppler frequency varies.
The optimal acquisition threshold was obtained by numerical simulation and was not derived theoretically in this study. In engineering, the optimal capture threshold can also be obtained through the numerical analysis method in Section V-A. It will be our future work to reduce the minimum acquirable SNR and derive a theoretical solution for the optimal acquisition threshold.