FFT-Based Acquisition of GPS L2 Civilian CM and CL Signals

Block processing Fast Fourier Transform (FFT) algorithms have been developed to do signal acquisition for the new CM and CL civilian codes that will appear on the L2 GPS frequency once the Block IIR-M satellites are launched. These algorithms have been developed to aid in the acquisition of weak GPS signals. FFT block processing can be used to significantly reduce the computational burden caused by the extended lengths of the new codes and by the long integration times needed to acquire weak signals. The target application for this technology is high-altitude spacecraft navigation. The new algorithms must work efficiently when the maximum allowable FFT batch size precludes the use of a single FFT for a full code period. Zero-padding and overlapand-discard techniques are used to generate partial correlation accumulations at a range of pseudo-random number (PRN) code start times, and interpolation techniques are used to map these partial accumulations onto a desired grid of start times before summation into full accumulations. Frequency-domain techniques are used to reduce the number of FFT and inverse FFT (IFFT) operations that are needed if one performs the acquisition search over a range of frequencies. The new algorithms have been used to acquire simulated signals with carrierto-noise ratios as low as 9 dB-Hz. They can reduce execution times in comparison to brute-force algorithms by factors ranging from 1,000 to over 30,000. INTRODUCTION A GPS receiver acquires a signal by determining its PRN code start time and carrier Doppler shift. It does this by searching for the maximum detection statistic over the full range of possible code start times and Doppler shifts. At each code start time and Doppler shift the detection statistic is calculated using baseband mixing, code correlation, coherent integration, squaring, and noncoherent integration. The present paper develops efficient ways to calculate detection statistics at multiple code start times and Doppler shifts when acquiring the new civil-moderate (CM) and civil-long (CL) codes at low received power levels. These codes will begin appearing on the L2 frequency when the first Block IIR-M satellites are launched . Weak CL and CM signals are easier to acquire and track than weak L1 C/A code signals because they have lower cross-correlation between different codes and because the CL code does not carry navigation data. Acquisition of weak signals requires the use of long coherent and non-coherent integration intervals. The required number of calculations grows with the length of these integration intervals, which is why it is important to use efficient acquisition calculations. There are several reasons for wanting to acquire very weak GPS signals. High-altitude spacecraft navigation, i.e., navigation at altitudes above the GPS constellation, can benefit from an ability to use weak side-lobe signals . Use of side-lobe signals can significantly increase the number of available satellites at very high altitudes, but most importantly can reduce or eliminate periods when no GPS signals are available. The improved signal observability will improve the robustness of the navigation solution and may alleviate the requirement for a very stable reference oscillator in the receiver. Another application that would benefit from the ability to acquire weak signals is E911 service. The current work has been motivated by the high-altitude spacecraft navigation application, and it considers acquisition scenarios that are typical of high-altitude orbits. This work builds on a number of previous results. It uses the standard acquisition approach that is described in Ref. 4, but it uses FFT-based block processing methods like those described in Ref. 5 in order to expedite the calculation of multiple acquisition statistics. Reference 6 extends the methods of Ref. 5 to use long coherent and non-coherent integration intervals for the acquisition of weak L1 C/A signals. The present work adapts the methods of Ref. 6 to the L2 CM and CL signals. The most significant change is to introduce the use of zero padding from Ref. 7 in the FFT calculations of Ref. 6. These techniques enable CL code acquisition using FFT blocks that are shorter than the CL code period. It is impractical to use block lengths that equal the CL code length due to memory limitations. In the case of the CM code, zero-padded FFT block processing is used to slide the coherent integration intervals with respect to the incoming data stream as the code start time varies. This approach ensures that the correct code start time produces coherent integration intervals which do not cross navigation data symbol transitions. This paper makes several contributions to the art of GPS signal acquisition via FFT block processing. First, it presents a simple means of dealing with the timemultiplexed nature of the CM and CL codes. Second, it adapts the zero-padding methods described in Ref. 7 to work over long data spans that can include multiple code periods. The principal novelty of this approach is the use of interpolation of correlations onto a grid of code start times. Interpolation is required when the code Doppler shift used in the FFT calculations does not match that of the received signal closely enough, or when the code period does not equal an integer number of RF sample periods. Third, it adapts the frequency-shifting and coarse-frequency-grid/fine-frequency-grid approaches of Ref. 6 to the situation of zero-padding and long codes. These methods help reduce the needed number of computations when calculating acquisition statistics at more than one Doppler shift. Fourth, it shows how to include an a priori estimate of the rate of change of the Doppler shift (i.e., a carrier phase acceleration) in the calculations. Aiding by an a priori acceleration estimate is needed in order to keep the power from leaking between different frequency bins when working with long coherent integration intervals. Fifth, it analyzes the operations counts and memory requirements of the new algorithms. Sixth, it tests these techniques under weak signal conditions using the new civilian L2 CL and CM codes. This paper's algorithms and results are presented in 6 main sections. A model of the L2 civilian signal is presented in Section II along with a review of the signal acquisition calculations. Section III develops the FFT-based block processing algorithms for calculating the acquisition statistic at multiple code start times for a given initial Doppler shift and a given Doppler shift rate. Section IV shows how circular frequency shifting allows one to reuse many of the FFT calculations of Section III in order to calculate acquisition statistics for a set of alternate Doppler shifts. Section V adapts the coarse-frequencygrid/fine-frequency-grid approach of Ref. 6 for use with zero-padded calculations. This approach further reduces the required number of FFT and IFFT calculations when searching over multiple Doppler shifts if a single coherent integration interval is broken up into multiple FFT blocks. Section VI analyzes the operations counts and memory requirements of the CL and CM acquisition algorithms. Section VII presents acquisition test results using data from hardware and software simulators. Section VIII summarizes the paper's results and conclusions. II. SIGNAL MODEL AND REVIEW OF SIGNAL ACQUISITION CALCULATIONS The acquisition calculations are based on the following model of the signal as it comes out of an RF front end: yk = A {d[(1+η0)(k∆t–ts) + 0.5ξ(k∆t–ts) ] × cM0[(1+η0)(k∆t–ts) + 0.5ξ(k∆t–ts) ] + c0L[(1+η0)(k∆t–ts) + 0.5ξ(k∆t–ts) ]} × cos[ωIFk∆t + φ0 + ωD0k∆t + 0.5αD(k∆t) ] + νk (1) The quantities in this model are the RF front end output yk at sample time k∆t, the sample period ∆t, the carrier amplitude A, the GPS navigation data symbol stream d[t], the L2 civilian CM PRN code time history interspersed with zeros cM0[t], the L2 civilian CL PRN code time history interspersed with zeros c0L[t], the start time of the CM and CL PRN codes ts, the initial fractional code Doppler shift η0, the fractional code Doppler shift rate ξ, the nominal intermediate value of the L2 carrier frequency ωIF, the initial carrier phase φ0, the initial carrier Doppler shift ωD0, the carrier Doppler shift rate αD, and the thermal/interference noise νk. The value of ωIF is the result of mixing and intentional in the RF front end. Three of the signals in eq. (1) represent bit or chip values. The navigation data symbol stream d[t] is a sequence of +1 and -1 values that switch randomly every 0.020 sec. The cM0[t] PRN code is a known pseudo-random sequence of +1/0/-1 chip values that chip at a nominal rate of 1.023 MHz. The subscript ()M0 indicates that every other chip is a +1/-1 value from the L2 civilian CM code with a zero value between each code chip. Similarly, the c0L[t] PRN code is a known +1/0/-1 sequence with a nominal chipping rate of 1.023 MHz that has zeros interspersed every other chip between the +1/-1 values of the L2 civilian CL code. The cM0[t] code starts with the first CM chip followed by a zero followed by the second CM chip, and so on. The c0L[t] code starts with a zero followed by the first CL chip followed by another zero, followed by the second CL chip, and so on. This arrangement models the time multiplexing of the CM and CL codes in a way that facilitates the developments of this paper while preserving the code's autocorrelation and cross-correlation properties. The CM code repeats after 10,230 chips, which makes cM0[t] periodic with a period of 20,460 chips or 0.020 sec. Each CM code period is aligned with a d[t] navigation data symbol interval. The CL code repeats after 767,250 chips, which gives c0L[t] a repetition period of 1,534,500 chips or 1.5 sec. The code and carrier Doppler shifts and Doppler shift rates are related if one neglects ionospheric effects. The relationships are: η0 = (ωD0+αDts)/ωL2 and ξ = αD/ωL2 where ωL2 = 2π×1227.6×10 6 rad/sec. Calculation of the signal acquisition statistic involves mixing with a replica signal, coherent integration, and non-coherent integration . The mixing and coherent integration for CL code acquisition takes the following form: I0L(m)[ s t̂ , 0 D ω̂ ] =