A novel fine Doppler frequency acquisition algorithm for BDS ‐ 3 B1C signal based on an adaptive filter

Fine Doppler frequency estimation has an important role in accelerating the convergence of the tracking loop in a global navigation satellite system (GNSS) receiver to achieve short time to first fix. BDS ‐ 3 started broadcasting a civil B1C signal to provide open services for global users, which is beneficial for GNSS ‐ based applications. Therefore, a fine Doppler frequency acquisition algorithm based on an adaptive filter is proposed, whose purpose is to acquire the BDS ‐ 3 B1C signal Doppler frequency accurately after the completion of coarse acquisition. The proposed algorithm is based on a first ‐ order complex ‐ coefficients adaptive filter. The adaptive filter depends on the proposed adaptation algorithm to track the input BDS ‐ 3 B1C signal. An accurate Doppler frequency estimate is extracted. Simulation results show the proposed algorithm has high acquisition sensitivity, high acquisition accuracy, short acquisition time, and few hardware resources consumption, and also works well under many different coarse acquisition strategies


| INTRODUCTION
An acquisition engine is one of the most important digital baseband blocks in a global navigation satellite system (GNSS) receiver. It provides coarse estimates of the code phase and Doppler frequency of input signals [1]. The acquisition engine generally outputs these rough estimates to the tracking stage to assist the tracking loop to track the input signal [2]. However, rough Doppler frequency estimates cause the tracking loop to spend a long time on convergence, which cannot satisfy the requirements of a fast convergence rate and short time to first fix (TTFF) [3]. To improve Doppler frequency estimation accuracy, many algorithms are proposed to address this problem [4][5][6][7][8][9][10][11][12][13][14][15][16]. These algorithms are generally divided into two categories: the one-step algorithm and the two-step algorithm.
The one-step algorithm includes three methods. The first increases the number of frequency bins in the sequential searching process of Doppler frequency [4], but it requires a long acquisition time. The second extends the length of Fast Fourier Transform (FFT) by padding zeros to improve frequency resolution [5], which unfortunately requires a large-size FFT. The last one depends on the non-linear relation involving three FFT samples calculated in the coarse acquisition to obtain an accurate Doppler frequency estimate [6,7], but it is suitable only for the high signal-to-noise ratio (SNR) situation.
In the two-step algorithm, the coarse estimates of code phase and Doppler frequency are found by the first step. Based on these coarse estimates, the second step first demodulates primary code and then obtains the fine estimate around the coarse estimate of Doppler frequency. There are four typical methods. The first carries out a sequential searching process of Doppler frequency, again with small frequency steps [8], but it still requires a long acquisition time. The second conducts a parallel frequency search again by one more FFT [9], which leads to an increase in hardware resources consumption. The third is based on the frequency locked loop (FLL) [10]. Although the FLL consumes less hardware resources, it takes a long time to converge. The last is based on the Gram-Schmidt orthogonal method [11,12], which, however, needs a lot of iterations and has a heavy computational burden.
The Chinese third-generation BeiDou navigation satellite system (BDS-3) started broadcasting a civil B1C signal to provide open services for global users, which is beneficial for GNSSbased applications. The BDS-3 B1C signal has a long primary code that contains a total of 10,230 chips in 10 ms, and then acquisition of the BDS-3 B1C signal requires lots of correlation operations, which implies high computational complexity [17]. To acquire the BDS-3 B1C signal by limited resources, some acquisition algorithms such as partial correlation method are proposed. These algorithms also provide a coarse Doppler frequency estimate [18,19]. Hence, a fine Doppler frequency acquisition algorithm is also necessary for the BDS-3 B1C signal.
To overcome those drawbacks of the fine Doppler frequency acquisition methods and fully use the BDS-3 B1C signal, an adaptive filter-based fine Doppler frequency acquisition algorithm for BDS-3 B1C signal is proposed. The method is based on a first-order complex-coefficients adaptive filter. The adaptive filter depends on the proposed adaptation algorithm to track the BDS-3 B1C signal after removal of the primary code. Finally, the fine Doppler frequency estimate is extracted. The proposed algorithm also belongs to the two-step algorithm category.
The rest of this work is organised as follows. First, the BDS-3 B1C signal is briefly introduced in Section 2. Then, coarse acquisition principles of the BDS-3 B1C signal are described in Section 3. Section 4 presents the proposed fine Doppler frequency acquisition algorithm in detail. Simulation results and performance evaluation are given in Section 5. Finally, some conclusions are drawn in Section 6 to summarise the work.

| BDS-3 B1C SIGNAL
The BDS-3 B1C signal is a modern GNSS signal that includes two components: data and pilot. The data component modulates data bits on the primary code, and the pilot component modulates secondary code on the primary code. The primary code contains 10,230 chips in 10 ms, and the secondary code contains 1800 chips in 18 s.
As Table 1 shows, the pilot component of the BDS-3 B1C signal adopts the modulation mode of QMBOC(6,1,4/33) that is composed of two parts: BOC(1,1) and BOC(6,1). On the one hand, the power of BOC(1,1) accounts for 29/44 of the total power of the BDS-3 B1C signal. On the other hand, the subcarrier frequency of the BOC(1,1) is 1.023 MHz and is less than that of the BOC(6,1), which indicates that the acquisition engine designed for the BOC(1,1) generally has lower complexity than that designed for the BOC(6,1). Therefore, only the BOC(1,1) of the pilot component is used to acquire the BDS-3 B1C signal.

| COARSE ACQUISITION PRINCIPLES
This work presents the proposed fine Doppler frequency acquisition algorithm. To illustrate the proposed algorithm, without a loss of generality, the short-time coherent integration is used plus an FFT method to carry out the coarse acquisition of a BDS-3 B1C signal [20][21][22][23], as shown in Figure 1. Many other methods are also suitable for the coarse acquisition of the BDS-3 B1C signal, such as sequential search [5], parallel code phase search [24], and parallel Doppler frequency search [9]. All of these methods may be combined with the proposed method. Moreover, this work adopts the Binary Phase Shift Keying-like method [25] to remove the side peaks of the BOC (1,1) autocorrelation function (ACF).
In general, at the output of the radio-frequency front end, the discrete-time BOC(1,1) signal of the BDS-3 B1C pilot component is given by: where P represents the signal power, s[n] is the secondary code, c[n] is the primary code, τ represents the code phase, f sc = 1.023 MHz, f IF represents the intermediate frequency, f d is the Doppler frequency, T s is the sampling interval, φ 0 is the initial phase, and η½n� is additive white Gaussian noise with a two-side noise power spectrum density of N 0 /2 W/Hz. The local generated complex exponential carrier exp½−j2πðf IF þ f sc ÞnT s � subsequently mixes with r[n], and then the result after the mixing passes through the lowpass filter (LPF). For the convenience of expression, the LPF is assumed to be an ideal finite impulse response (FIR) filter. Afterwards, the local generated primary code c½n −τ� correlates with the result at the output of the FIR filter, and short-time coherent integration is performed. As a result, the lth short-time coherent integration result is given by: Abbreviation: BDS, BeiDou navigation satellite system.

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where T c is the short coherent integration time and T c = NT s . Also in Equation (2), Δτ ¼ τ −τ, RðΔτÞ denotes the primary code ACF and ξ½l� is the noise term. The total coherent integration time is T p , and T p = LT c . Afterwards, an N FFT -points zero-padding FFT is used to conduct parallel Doppler frequency search, and N FFT = 2 m (m ∈ N þ ). To avoid the scalloping loss, N FFT ≥ 2L, and the number of zero is N FFT − L. As a result, the output of the ith FFT operation is given by: where Δf d represents the estimation deviation of Doppler frequency, According to Equation (3), the coarse Doppler frequency es- when the SNR of the input BDS-3 B1C signal satisfies the acquisition sensitivity. Also in Equation (3), ζ½i� is the noise term. According to the central limit theorem, ζ½i� is a complex Gaussian random variable with zero mean and variance 2σ 2 . Furthermore, the real and imaginary part of the ζ½i� are independent Gaussian variables with zero mean and variance σ 2 . At this moment, the SNR in dB is given by: It is known that non-coherent integration is an effective way to extend total integration time to IT p . Based on Equation (3), the result after I-times non-coherent integration is given by: When RðΔτÞ → 0 or the satellite is out of view, there is hypothesis H 0 , and Sðτ;f d Þ ¼ P I−1 i¼0 |ζ½i�| 2 , which indicate the Sðτ;f d Þ follows the central chi-square (χ 2 ) distribution with 2I degrees of freedom. When RðΔτÞ → 1, there is hypothesis H 1 , and Sðτ;f d Þ follows non-central χ 2 distribution with 2I degrees of freedom and non-centrality parameter λ. The power of input signal is generally assumed to be constant during IT p ; In terms of false-alarm probability P fa and detection probability P d , the critical SNR is given by: where F −1 c ð⋅Þ is the inverse function of the cumulative distribution function of central χ 2 distribution, and F −1 n ð⋅Þ is the inverse function of the cumulative distribution function of non-central χ 2 distribution. According to Equations (3), (4), and (6), critical carrier-to-noise ratio (C/N 0 ) in dB-Hz is given by: C/N 0Th represents the required minimum value of the C/ N 0 of the received signal to satisfy the receiver operation characteristics.

| PROPOSED ALGORITHM
The adaptive notch filter technique is widely used for singlefrequency sinusoidal signal enhancement and extraction, and single-frequency interference mitigation [26][27][28][29][30], because the adaptive notch filter is able to estimate the carrier frequency of the input signal accurately, which implies the adaptive notch filter can be used for the fine Doppler frequency estimation. Therefore, based on the adaptive notch filter, a low-complexity adaptive filter is proposed to conduct fine Doppler frequency acquisition.

| Adaptive filter
The transfer function of the first-order complex-coefficients zero-pole constrained notch filter is generally given by: F I G U R E 1 Block diagram of coarse acquisition algorithm based on short-time coherent integration plus an FFT for BDS-3 B1C signal. BDS, BeiDou navigation satellite system QIU ET AL.
where f represents the central frequency of the notch filter, and γ (0 < γ < 1) is the zero-pole constraint factor and determines the 3-dB attenuation bandwidth of the notch filter. According to Borio et al. [31], Equation (8) is composed of two parts. The first part is called the autoregression (AR) block, corresponding to The second part is called the moving-average (MA) block, corresponding to H MA ðzÞ ¼ 1 − expðj2πf Þ ⋅ z −1 . The directform structure of the notch filter is shown as Figure 2, where u[n] and y[n] are the nth complex input and output samples, respectively. As Figure 2 shows, the cost function is given by: where \!\!\! * at the right up corner represents conjugate operation, and x n represents the x[n]. To find the parameter f that minimises the cost function and equals the carrier frequency of the input signal, the cost function is differentiated as: We can see from Equation (10) that the second derivative of the cost function J( f ) exists.
Furthermore, according to the Newton algorithm, d k is defined as the descent direction at the kth iteration, and then the adaptation equation can be generally written as: where α is the step size. Then, the precise linear search method is applied to determine the value of the α, so the function about α is given by: The derivative of the function ϕðαÞ is deduced as: Let dϕðαÞ dα ¼ 0, and then it can be obtained as: Substituting Equation (14) into Equation (10), it can be obtained as: Considering Equations (14) and (15), if Equation (14) is adopted as the adaptation algorithm, the central frequency of the first-order complex-coefficients adaptive notch filter will converge to the carrier frequency of the input signal by only one iteration in theory, which implies a fast convergence rate.
However, Equation (14) means heavy computation, and expðj2πf kþ1 Þ is the expected value. Therefore, we further simplify Equation (14), and the adaptation algorithm is given by: where Fðx n x n−1 * Þ represents the function of extracting the carrier frequency of the input signal x n , and F(0) = 0. Function Fðx n x n−1 * Þ can be implemented by a frequency discrimination function such as a four-quadrant arctangent function arctan2ð⋅Þ.

Equation (16) is further smoothed by
where λ k (0 ≤ λ k < 1) is the forgetting factor, k ≥ 0, and f 0 = 0. To ensure a fast convergence rate and avoid intense oscillation, λ k is adjusted by: F I G U R E 2 Direct-form structure of first-order complex-coefficients zero-pole constrained notch filter where 0 < λ 0 < 1, and 0 < β ≤ λ 0 . Observing Equation (17), the adaptation algorithm is concerned only with x[n], the output of the AR block shown as Figure 2. Consequently, based on this adaptation algorithm, the AR block of the first-order complex-coefficients notch filter can be taken as an adaptive filter to track the carrier frequency of input signals, shown in Figure 3. The AR block has lower complexity than the notch filter.

| Fine acquisition principles
According to coarse Doppler frequency estimatef d , a local carrier numerically controlled oscillator (NCO) generates complex exponential carrier expð−j2πf NCO nT s Þ to mix with input signal r[n]. Based on Equation (17), f NCO is given by: where m ∈ N þ , k denotes the kth iteration of the adaptive filter, f k is the kth output of the adaptation algorithm, and M (M ≥ 2) means that the carrier NCO updates its frequency control word f NCO every M iterations.
According to the coarse code phase estimateτ, local code NCO generates the product of primary code and subcarrier c½n −τ� ⋅ sign½sinð2πf sc nT s Þ� to correlate with the data samples after mixing. Coherent integration is subsequently carried out. The coherent integration time is T coh , and T coh = N p T s . As a result, the ith coherent integration result is deduced as: where ξ p ½i� is the present noise term. The secondary code s [i] in Equation (20) is a pseudorandom noise sequence; therefore, the E i ðτ;f d Þ is a direct spread spectrum signal. Because the adaptive filter merely tracks the single-frequency signal, the fine Doppler frequency acquisition algorithm has to overcome the secondary code chip-sign transition, and thus it is given by: where T d is the secondary code chip duration, and T d = 10 ms. Also, V ∈ N þ , T u is the update cycle of the adaptive filter. Referring to Ward [10], the frequency lock detection method of the adaptive filter is given by Equation (22). Also in Equation (22), α = 0.05, and X = ceil(M/2) + 1. In addition, the frequency lock detection threshold is L Th , and L Th = 0.97. When L F ½k� ≥ L T h , the adaptive filter has entered the lock state. On the contrary, the adaptive filter has exited the lock state when L F [k] < L Th . To improve the accuracy of the F I G U R E 3 Block diagram of adaptive-filter based fine Doppler frequency acquisition algorithm for BDS-3 B1C signal. BDS, BeiDou navigation satellite system F I G U R E 4 Block diagram of generic second-order frequency locked loop for BDS-3 B1C signal. BDS, BeiDou navigation satellite system QIU ET AL. Doppler frequency estimate, the frequency lock detection block outputs the fine Doppler frequency estimatef d p only when the adaptive filter enters the lock state and lasts at least UT u . Otherwise, the frequency lock detection block outputs nothing. Hence, thef d p is given by: Moreover, the total time from when the adaptive filter started working until the frequency lock detection block first outputs the fine Doppler frequency estimate is fine Doppler frequency acquisition time T acq , which is given by: where T acq = 0 means the frequency lock detection block outputs nothing.

| SIMULATIONS AND PERFORMANCE EVALUATION
Because the proposed algorithm is non-linear, only Monte Carlo simulations are suitable to evaluate its performance. A total of 1000 Monte Carlo simulations are conducted for every evaluation, and each simulation consists of 2000 iterations. The performance of the proposed algorithm is evaluated using three parameters: acquisition success rate, acquisition accuracy, and acquisition time. According to Equation (23), the acquisition success rate is defined as: where i means the ith Monte Carlo simulation, and Based on Equations (3), (23), (25), and (26), acquisition accuracy is defined as: where D is the mean absolute deviation between fine Doppler frequency estimatef d p and real input Doppler frequency f d .
The smaller the D, the higher the acquisition accuracy. Eventually, acquisition time based on Equations (24) and (25) is defined as: where T ave represents the mean acquisition time. Furthermore, the proposed algorithm is compared with a generic second-order (2-ord) FLL shown in Figure 4, because the 2-ord FLL is usually used for fine Doppler frequency acquisition [10]. Based on the Laplace domain, the transfer function of the 2-ord loop filter of the 2-ord FLL is given by: where coefficients k 1 and k 2 are concerned with equivalent noise bandwidth B F of the 2-ord FLL, as Equation (30) shows.
Referring to Ward [10], the frequency lock detection method of the 2-ord FLL is given by Equation (31). Also in Equation (31), α = 0.01, and L Th = 0.8: In general, the error of the Doppler frequency estimation after coarse acquisition is a few hundred hertz [34]. For instance, Δf d ∈ ½−250; 250� in hertz. In terms of the QIU ET AL.
-7 2-ord FLL after a Monte Carlo simulation are shown as Figure 5.
As Figure 5 illustrates, the convergence rate of the proposed algorithm is much faster than that of the 2-ord FLL. Before obtaining a fine Doppler frequency estimatef d p , there should be a delay of 1070 iterations for the 2-ord FLL, because the 2-ord FLL tends to converge steadily after approximately 1070 iterations when the PIT is 1 ms. As a consequence, in  Figure 7 shows, although the acquisition accuracy of the proposed algorithm is slightly lower than that of the 2-ord FLL when C=N 0 ≥ 40 dB − Hz, the acquisition accuracy of the proposed algorithm gradually becomes increasingly higher than that of the 2-ord FLL when C/N 0 < 40 dB-Hz. As Figure 8 shows, the acquisition time of the proposed algorithm is much less than that of the 2-ord FLL. It should be noted that D = 10 Hz and T ave = 2000 ms actually means the non-existence of acquisition accuracy and time, respectively, because the P acq is small and gradually tends towards zero when C/N 0 < 34 dB-Hz.
In addition, P fa = 10 −5 and P d = 90%, and thus the critical carrier-to-noise ratio C/N 0Th correspond to Equations (6) and (7), as shown as Figure 9. Every combination in Figure 9 between coherent integration time T p and the number of noncoherent integration I represents a kind of strategy of the coarse acquisition. Hence, the proposed algorithm is able to work well under many different coarse acquisition strategies.

| CONCLUSIONS
An adaptive filter-based fine Doppler frequency acquisition algorithm is proposed for a BDS-3 B1C signal. The proposed algorithm is a kind of two-step algorithm. Based on coarse acquisition estimates of code phase and Doppler frequency, the proposed algorithm tracks the input BDS-3 B1C signal and gives the final fine Doppler frequency estimate.
First, the proposed algorithm has an acquisition sensitivity of 30 dB-Hz, and it is approximately 4 dB better than that of the 2-ord FLL when considering that the acquisition success rate is greater than 90%, that is P acq ≥ 90%. Thus, the proposed algorithm has high acquisition sensitivity and works well under many different coarse acquisition strategies that represent different combinations between a coherent integration time and the amount of non-coherent integration.
Second, the mean absolute deviation of the fine Doppler frequency estimate is less than 2 Hz and the acquisition success rate is approximately 100%, when C=N 0 ≥ 40 dB − Hz. Even if C/N 0 = 30 dB-Hz, the mean absolute deviation of the fine Doppler frequency estimate is only approximately 6 Hz and the acquisition success rate is approximately 90%. Although the acquisition accuracy of the proposed algorithm is slightly lower than that of the 2-ord FLL when C=N 0 ≥ 40 dB − Hz, it gradually becomes increasingly higher than that of the 2-ord FLL when C/N 0 < 40 dB-Hz. Hence, the proposed algorithm has high acquisition accuracy.
Third, the mean acquisition time of the proposed algorithm is around 250 ms when C=N 0 ≥ 35 dB − Hz, and it is less than 500 ms even if C/N 0 = 30 dB-Hz. The mean acquisition time of the proposed algorithm is much less than that of the 2-ord FLL, because the convergence rate of the proposed algorithm is much faster than that of the 2-ord FLL. Overall, the proposed algorithm is better than the 2-ord FLL.
This work deduces all related equations concerned with the proposed algorithm, including the adaptation algorithm, acquisition principles, and evaluation methods. Only one specific set of parameters is used to verify the performance of the proposed algorithm. There is no doubt that different sets of parameters can be obtained based on those equations to bring about a different performance and then satisfy different requirements and applications, which is an important contribution of this work.
Finally, the proposed algorithm can be used to acquire other GNSS signals accurately by the appropriate configuration of related parameters.