The major purpose of this paper is to employ an evolution method to enhance the performance of GNSS signal acquisition. Under the scenario of ignoring data modulation and interference, the incoming signal is down-converted and then transferred to a digital Intermediate Frequency (IF) signal. The signal at sampling time, t_k, can be described as:

where P is the power of the direct line-of-sight GNSS signal, f_c is the IF and . The variables f_d, θ and τ denote the Doppler frequency, carrier phase, and code delay, respectively. The noise components w(t_k) are referred to as white Gaussian noise distribution, in which the power spectrum density is N₀/2. The N₀ stands for single sideband power spectrum density of noise. G(·) is the filtered code sequence expressed by C/A-code, P-code or binary offset carrier (BOC) signal [12,21,22].

The baseband signal r(t_k) multiplies a locally generated replica and the result is processed through correlation operation. The resultant output is expressed as:

where N_k,c is number of samples per coherent accumulation segment, equal to T_c/T_s, where T_s is the sampling period and T_c is code chip rate. u and v are the guessed value of the Doppler frequency and code delay, respectively, in frequency search space [f_c-u_max,lo, f_c+u_max,up] and code search space [v_max,lo, v_max,up]. The values u_max,up and u_max,lo are the upper and lower boundaries of the Doppler frequency, respectively. Similarly, the values v_max,up and v_max,lo are the upper boundary and lower boundary of code delay, respectively. is an unfiltered local replica. The in-phase I_k and quadrature-phase Q_k are squared respectively and then summed altogether. The result is accumulated after K non-coherent integration time. The correlation output is shown as follows:

where R(·) is the correlation function between the filtered incoming signal and local replica and T_c represents the coherent accumulation interval; the values ε_f,k and ε_θ,k are the differences between incoming and internally generated Doppler frequency estimate, and that between incoming and internally generated carrier phase estimate, respectively. The value ε_τ,k represents the estimated code delay error. Each of R(ε_τ,k) and sinc(ε_f,kT_c) has a maximum amplitude of 1 when ε_τ,k and ε_f,k take the value of zero. If these values are not zero, the result is a decrease in the amplitude of correlations. The value w_I,k and w_Q,k denote the in-phase and quadrature-phase noise samples with variance σ² = N_k,cN₀/2, respectively. The acquisition process is to find maximum y(u,v) through a search on the two-dimensional (frequency and code domain) grid of trial points, frequency point u and code point v. Two hypothesis tests define the test statistic for signal detection [23,24,25]. When the value y(u,v) exceeds the set threshold v, the signal is considered present (hypothesis H₁). Otherwise, the signal is absent (hypothesis H₀). Under hypothesis H₁, the probability of signal detection P_D is given by

where ρ = P/2σ², T_y is the test statistic, Pr(·) refers to probability and I₀(·) is the zero-order modified Bessel function of the first kind. The false alarm rate of hypothesis H₀ to H₁ is as shown in Equation 5

where p₁(·) and p₀(·) are non-central and central chi-square χ² distribution with two degrees of freedom, respectively. Suppose the signal is acquired successfully, the Doppler frequency and code delay can be obtained. The constant threshold v = vσ²ln(P_FA^-1) is often selected in order to achieve a certain probability of false alarm in noise alone by employing the well-known Neyman–Pearson criterion [26].

The above depiction shows that what concerns the designer is to rapidly find the required u and v in a short time so as to have the correlation exceed the set threshold. The following chapters demonstrate that the proposed acquisition method can effectively speed up signal acquisition speed and maintain high estimated parameter precision.

In each iteration, fixed and large numbers of initial populations result in a longer signal search time. As a result, the initial detection of the presence of a desired signal (the signal parameters have not been accurately acquired yet) allows us to decrease the number of initial populations in the next search. The (p+1)-th iteration of initial population size is as follows:

where . When the Q_max^(p) exceeds the noise floor, GC method initiates two procedures. The first procedure is that when Q_max^(p) exceeds the set detection threshold v (depicted in Section 2), it indicates that the desired signal has been acquired and the GC process terminates. The second procedure is when the Q_max^(p) is between the detection threshold and the initial detection threshold. In such case, the adaptive scaling scheme begins to control the number of initial population. The initial detection margin is set at 1.4Q and the definition of Q is the same as that in Equation 7. The only difference is that Q is the fitness value acquired under the scenario of no desired signal. Meanwhile, the scaler also controls and adjusts the search space of Doppler frequency and code delay, which is depicted in the following:

where

where α and β are reduced factors, which mainly control the convergence rate during the signal acquisition process.

The adaptive scaling scheme adjusts the Doppler frequency and code delay boundaries with the increase of acquisition iteration. Thus, this adjustment results in faster convergence and more accurate values of the signal parameter.

The termination criterion of GC normally regulates reproduction generation or detects the difference between generations. If there is no evolution after several generations, the evolution has terminated, which indicates the convergence to optimal value. The Errin Equation 20 approximates zero, which means the GC with adaptive scaling scheme process comes to an end.

where M is the final iteration number after proposed scheme converge. When the control process finishes, the chosen optimal individual is decoded so as to obtain the Doppler frequency and code delay of the desired signal.

This section compares the proposed method with the other published methods in acquiring GNSS signals under the same simulation parameter. Three types of signal are presented for signal acquisition analysis under no consideration for mutual jamming between signals. Suppose signals are down converted to 20 MHz and the sampling frequency is 64 MHz, which fulfills Nyquist’s sampling requirement due to the fact that the P-code has a bandwidth of 20.46 MHz. To verify the proposed method as capable of acquiring GNSS signals, the post-correlation signal-to-noise-ratio (PSNR) is adopted to evaluate the performance of different methods and is calculated as follows:

where C_p is the highest peak of search power during the correlation process. E[C_n] depicts the mean power of noise term C_n. The proposed method was evaluated by using a developed program under the MATLAB environment.

The process of applying GC with an adaptive scaling scheme to signal acquisition is shown in Figure 1. For each parameter, all individuals of the current population are evaluated and a local optimal value is selected. This operation is repeated as many times as required with a diversified population. This diversity is assured by the application of GC operators in binary coding. Once a set of local optimal values is obtained, a global one is selected. Similarly, this process is carried out for all search values to produce the solution per iteration. At this stage, if the acquisition process is not terminated, the maximum search space boundaries (including Doppler frequency and code delay) are adjusted by subjecting them to an evolution cadence (variation of fitness function). This process is repeatedly carried out in this manner as long as it has not entered into offspring stagnation; otherwise, the program is interrupted by a fixed maximum iteration number. The initial population size of C/A-code is set at 1500, P-code, and BOC is set at 1500, 7000, and 5750, respectively, with carrier to noise density ratio (C/No) is 45 dB-Hz, crossover probability 0.85, mutation probability 0.008 and false alarm rate P_FA = 0.01 in control process of proposed scheme. The selection criterion of initial population size is calculated as follows:

where μis a scale factor with a value between 0.8 and 1. The chip length of C/A-code is 1023. Figure 2a illustrates the detection probability as a function of the C/No. It is shown that the adoption of GC under an adaptive and without an adaptive scaling scheme improves roughly 0.2 dB and 0.8 dB, respectively, in detection performance, in contrast to the adoption of the traditional method. Figure 2b demonstrates a receiver operating characteristics (ROC) curve with C/No = 38 dB-Hz. It shows that the proposed method improves the detection performance with adaptive scaling scheme.

A large number of initial populations can speed up signal search time, but is also requires a higher hardware cost. The initial population size, crossover and mutation probability value are set, based on an experience rule to be successfully utilized in the GC process to acquire desired signals. Figure 3 shows the examples for the evolution values of two parameters (Doppler frequency and code delay) for C/A-code. The protruding “○” in Figure 3 is caused by mutation in GC. The two figures ( 3a,b ) show that in utilizing the adaptive scaling scheme, the signal search space will gradually narrow down its space after each iteration to ensure the estimated signal parameter to converge to global optimal value and also reduce iteration times. Contrariwise, a longer acquisition time is required without using the scheme. Figure 4 illustrates the 2D signal search path using the proposed method in application to different signal types, which include C/A-code, P-code and BOC. These diagrams show that the use of GC can effectively reduce search times. On the other hand, Figure 5 analyzes the relation between the number of initial population and search times regarding C/A-code. The result demonstrates that when the number of initial population adopts a roughly 1.5 times code period length of C/A-code (1023 chip), the signal can be accurately acquired within fewer search times under C/No = 45 dB-Hz. When the number of the initial population is below 0.5 times code period length, the proposed method cannot acquire the signal. Such a result is the same with P-code and BOC.

In addition, the system computation time of GC depends on the maximum initial population size Z and iteration number M, which takes 0(ZM). For example, regarding the hardware implementation, the standard GC method takes 2 × 1500 plus 1500 multiplications and 2 × 1500 additions for C/A-code. Therefore, the total required number of operations equals 4500 multiplications and 3000 additions. The adoption of GC with an adaptive scaling scheme takes additional memory for the adaptive adjustment. Table 1 compares the simulated performances of our proposed method with those of the existing methods. The execution time is measured using the tic and toc functions in MATLAB. An average personal computer (PC) is adopted for 20 execution time measurements, with the mean computed as well [28]. The table indicates that the proposed method is superior in time and more precise in parameter than other methods in acquiring three different types of signal. It is noteworthy that because P-code has a longer period time, code length of only that segment (about 2 ms) is utilized for signal acquisition during simulation. Hence, its acquisition time is the longest among different signal types. A tradition (serial search) method is superior in its easy implementation of hardware, but is inferior in parameter precision than other methods. Parallel code delay with fixed code/Doppler search space method [28] is higher in parameter precision rapidity, but is also higher in hardware cost. Although parallel frequency with a fixed code/Doppler search space boundary method is rapid in acquisition time, it is limited by the length of FFT and sampling rate. Thus, this method cannot be efficiently promoted and is seldom applied to realistic circuits. In addition, a parallel code delay with adaptive logic control method is low in hardware implementation complexity, high in estimated parameter precision and parallels the proposed method; the parallel code delay with proposed method is more rapid and efficient in acquisition time. Despite the fact that the proposed method is somewhat higher in hardware implementation complexity (count on number of initial population), it has the potential to be implemented in hardware under the gradual and continuous enhancement of integration circuit design technique.

In this section, experimental results are presented to verify the feasibility of the proposed method to reduce acquisition time. Because P-code is currently a military code, the signal cannot be acquired in the actual environment. Thus, the acquisition is conducted only in terms of GPS signal in this experiment, where the IF is 4.092 MHz and sampling frequency is 16.368 Hz.

The data samples from antennae are stored and, afterwards, post-processed for signal acquisition analysis. The results have been obtained for PRN-8, the location of which is at an elevation of 30 degrees at the time of data collection. Table 2 depicts experimental results of GC, utilizing adaptive and non-adaptive scaling schemes, as well as the results of a traditional method. The table demonstrates that the proposed method can achieve better signal parameter precision within a shorter time, as opposed to traditional methods, where one must narrow frequency search step size to enhance parameter precision. This consumes large amounts of signal search time. The table reveals that the average convergence iteration number of the proposed method is below 17 times (except for PRN-8 and PRN-4), which greatly saves a lot of hardware computation time. In addition, the high PSNR can cause the GC process to converge more rapidly in order to speed up the computation time. Note that the iteration numbers of the proposed scheme in the signal acquisition of PRN-8 and PRN-4 are 29 and 30, respectively, which are higher than that of other satellites. The reason is that the estimate parameter converges to a local optimal solution and thus the signal acquisition engine repeats the search process, which increases the iteration number. The same scenario also happens to the GC method without an adaptive scaling scheme in the case of PRN-8 and PRN-4. Figure 6a,b depicts the acquisition parameter search curve of frequency shift and code delay versus iteration number, respectively. Figure 6a shows that the proposed scheme converges to a local optimal solution in the 18th iteration and thus restarts search process. In the 29th iteration the proposed scheme converges to a global optimal solution and successfully acquires a signal parameter. Table 2 compares the performances of the proposed method with a traditional (serial search) method [1] for PRN-8. This table shows that the proposed method can successfully acquire the satellite signal in the sky. Note that search times for PRN-8 and PRN-4 are longer because the signal power of the satellite is weaker.

Table 3 shows that the proposed method estimates the frequency shift of PRN-8 to be 2.275 KHz, which has better precision than the other method. Although the iteration number of proposed scheme in the signal acquisition of PRN-8 is more than that of other satellites, its computation time is much shorter than that of traditional method (shorter by roughly 2.5 times).