Simultaneous Frequency Search with a Randomized

Simultaneous Frequency Search with a Randomized Dirichlet Kernel for Fast GPS Signal Acquisition Chun Yang

Andrey Soloviev

Michael Veth

Erik Blasch

Sigtem Technology, Inc.

QuNav, LLC

Veth Research Associates, LLC

Air Force Research Lab

BIOGRAPHIES Dr. Chun Yang received his Bachelor of Engineering from the Northeastern University (Shenyang, China) and the title of Docteur en Science from the Université de Paris-Sud (Orsay, France). After postdoctoral research at the University of Connecticut, he has been working on adaptive array and baseband signal processing for GNSS receivers and radar systems as well as on nonlinear state estimation with applications in target tracking, integrated inertial navigation, and information fusion. Dr. Yang is the winner of 2009 IEEE NAECON Grand Challenge and the recipient of 2007 ION Samuel Burka Award. Dr. Andrey Soloviev holds B.S. and M.S. degrees in applied mathematics and physics from Moscow Institute of Physics and Technology and a Ph.D. in electrical engineering from Ohio University. Previously he served as a Research Faculty at the University of Florida and as a Senior Research Engineer at the Ohio University Avionics Engineering Center. His research interests focus on all aspects of GNSS signal processing and estimation, as a well as multi-sensor fusion for navigation applications. He is a recipient of the ION Early Achievement Award and the RTCA William E. Jackson Award. Dr. Michael Veth received his Ph.D. and M.S. in Electrical Engineering from the Air Force Institute of Technology and a B.S. in Electrical Engineering from Purdue University. He previously served as an Assistant Professor of Electrical Engineering at the Air Force Institute of Technology. His research focus is on applying advanced estimation theory to combine inertial sensors with nontraditional, bio-inspired sensors for non-GPS navigation and control applications. He has authored over 40 technical articles and book chapters and 1 US patent in areas relating to computer vision, navigation, and control theory. Dr. Erik Blasch received his B.S. in mechanical engineering from the Massachusetts Institute of Technology in 1992 and his M.S. degrees in mechanical engineering (1994), health science (1995), and industrial engineering (human factors) (1995) from Georgia Tech

and M.B.A. (1998), M.S.E.E. (1998), M.S. econ (1999), and a Ph.D. in electrical engineering from Wright State University. Currently, he is with the Air Force Research Laboratory (AFRL) Information Directorate as a principal scientist in information fusion. His research interests include target tracking, information/sensor fusion, pattern recognition, human factors, and communication theory. He is a past President of the International Society of Information Fusion (ISIF), a Member of the IEEE AESS Board of Governors, a SPIE Fellow, and is active in AIAA and ION. ABSTRACT Signal acquisition in a GPS receiver aims at quickly obtaining coarse estimates of a GPS signal’s time and frequency parameters so as to initialize the code and carrier tracking loops for subsequent refined signal estimation. Conventional methods divide the time and frequency uncertainty zone of the signal into a grid of search points and then test each and every search point by correlating the incoming signal samples with those of a local replica generated with the parameters thereof. If several grid points can be checked at the same time per correlation, the uncertainty zone can be swept over quickly, leading to a fast acquisition process. In this paper, we present a fast acquisition search technique (FAST) via simultaneous search of allowable frequency errors (SAFE). FAST is based on judicious combining of a number of carrier replicas at selected frequency search points, leading to a combined carrier replica (CCR) and a multi-frequency modulated code replica (MMCR). As such, it can reduce the total test points of MN, where M is the number of frequency bins and N is the number of code lags, into M+N. That is, the method achieves fast acquisition using two linear time searches. Optimality criteria and practical methods (a randomized Dirichlet kernel) to reduce implementation loss of CCR and MMCR are described. Simulation and experimental data processing results are presented to demonstrate the functionality and performance of FAST.

INTRODUCTION GPS signal acquisition aims at quickly obtaining coarse estimates of a GPS signal’s time and frequency parameters as the initial values to close the code and carrier tracking loops for subsequent refined estimation. A straightforward method yet widely used in many practical receivers consists of dividing the time and frequency uncertainty zone of the signal into a grid of search points and then testing each and every search point sequentially until the signal is detected at one search point or the acquisition fails (the signal is deemed not present). Each test consists of correlating the incoming signal samples with those of a local replica generated with its parameters at the grid test point. Sequential search can be sped up when several search points are tested simultaneously. To the extreme, the use of massive parallel correlators can test all the search grid points at once for “instantaneous” acquisition. Examples of such design include YExpress [Wolfert et al., 1998] and DirAc [Betz et al., 2004]. However, in either sequential or parallel search, hardware correlators operate at the sampling rate, which is much slower than the underlying clock rate; and during signal tracking, a large number of correlators will be idle or put into a sleep mode to save power. When the time and frequency uncertainty zone is large and/or the code replica is a long sequence, it is desired to accelerate the acquisition process beyond using parallel correlators particularly for weak signals. The acquisition of weak signals requires signal accumulation over time. To address these issues, a considerable amount of research efforts have been devoted on several important fronts:  Reduce computation. One way to reduce the amount of arithmetic (addition and multiplication) operations in calculating the correlation function is to use the fast Fourier transform (FFT) [Nee and Coenen, 1991; Davenport, 1991]. FFT-implemented correlation actually performs sequential search in time (unless parallelized in hardware). By eliminating redundancy, it reduces the number of multiplications from O(N2) to O(Nlog2N), thus offering significant saving when the number of samples in integration N (i.e. the number of code phases to test) is very large on the order of thousands. Frequency search in the time domain utilizes phase rotation, which consists of multiplying the incoming signal samples with a carrier replica generated at the search frequency. In the frequency domain, however, frequency search reduces to shifting the spectrum, which can be inexpensively implemented as simple index change and memory access [Yang et al., 2007], thus avoiding time-domain complex multiplications.  Compress data. Reducing the number of data points in correlation can speed up the computation. Since the incoming signal is sampled at a rate higher than the chipping rate, a simple method is to average all samples

taken over the duration of a chip into a new sample and to perform correlation at the reduced rate [Starzyk and Zhu, 2001]. Since the code phase is unknown, the averaging correlation is calculated for each and every consecutive sample over a chip (i.e., to solve the ambiguity within a chip). However, the averaging and decimation operation, when applied to the incoming signal, alters the code characteristics, thus making it very sensitive to noise and Doppler shift [Djebouri, 2015]. When the number of samples per FFT-implemented correlation is a power of two, the spectrum of the correlation function can be decimated by first dividing the spectrum into a number of sections of equal size and then adding up the spectrum values per section before taking IFFT – a data compression of the correlation function in the frequency domain [Sagiraju et al., 2006]. However, assistance information about the approximate peak location is required for spectrum division. With such assistance information, a zoom IFFT can be used, instead of a full IFFT, so as to calculate only a section of the correlation function containing the peak [Yang, 2001]. In recognition of the fact that the correlation function has only one peak, it admits a sparse expression (few dominant among mostly near zero values) [Hassanieh et al., 2012; Kong, 2013]. As such, the inverse FFT (IFFT) operation is sparse in the time domain, which can be performed using a compressed sensing technique [Kong, 2013] or in sub-linear time using the sparse Fourier transform [Hassanieh et al., 2012]. These methods only provide a coarse estimate of code phase, which needs individual tests as a follow-up step in order to solve ambiguity. Finally, it is worth noting the claim in [Fish et al., 2011] that search in time and frequency with a matched filter can be accomplished in linear time using the so-called Heisenberg-Weil flag algorithm.  Simultaneous search. If several grid points are assessed per correlation, the uncertainty zone can be swept over quickly, leading to a fast acquisition process. Testing several grid points at the same time is equivalent to implementing parallel correlators, each at a test point, but using a single correlation, thus called simultaneous search. XFAST is such a technique, which folds the code replica such that each sample of the folded code sequence contains several code phases of the original sequence and then tests them at the same time per correlation. The ambiguity within the folds (from a coarse estimate to a refined one) is then solved by individual checks [Yang et al., 1999; 2000]. The idea of folding the code replica is extended to folding the incoming signal in [Li et al., 2008], where the incoming signal is phase-rotated by several carrier replicas, each at a different frequency. These phaserotated signals are then added up to form a single stream for correlation with a folded code replica, leading to the time-frequency folding technique [Li et al., 2008]. Two

additional steps are then applied to resolve the ambiguity of signal parameters within the detected time and frequency folds, respectively. Recently, a particular frequency folding technique is analyzed in [Zhao et al, 2016] where coherent integration is used to recover its folding loss. Joint search of a group of frequencies is described in [Akopian, 2005]. The frequency uncertainty is divided into groups (a coarse grid) of frequencies (a fine grid). The search for fine frequencies can be performed using either FFT or an approximation (subject to degradation) around a coarse grid point they belong to. This precorrelation frequency compensation is useful for long coherent integration of weak signals.

for code search [Yang et al., 1999; 2000] to that for frequency search [Li et al., 2008]. However, instead of folding the incoming signal as in [Li et al., 2008], it folds, superposes, or combines the carrier replicas. As such, it does not affect the incoming signal or add up noise. Besides, the combined carrier replica (CCR) can be precalculated and stored in memory for easy access in realtime without on-line computation. The paper is organized as follows. The method of fast acquisition with two linear time searches is first introduced. The concept of multi-frequency modulated code replica (MMCR) is then described. Simulation results are next presented with implementation loss analysis. Finally, the paper is concluded.

 Reduce search dimension. The spectrum compression processing (SCP) technique is used in [Mathews et al., 2011] where the incoming signal is multiplied with a copy of itself delayed by a half-chip. In this scheme, the delay and multiply (DAM) operation without complex conjugate first squares off the code chip and data bit and at the same time doubles the residual Doppler frequency of each of the satellite signals present. FFT-analysis of the resulting signal over an extended period of time reveals the chipping rate as well as each satellite’s frequency and phase. This method eliminates the dimensions of satellite ID and code phase from the search process. Other means is then needed to obtain the missing information or a Doppler-only method is used for positioning. The multiplication of an incoming signal with the complex conjugate of a copy of itself delayed by an integer number of chips eliminates the need for Doppler search. The code product sequence resulting from the above DAM operation still has good auto-correlation and cross-correlation properties, which can be used not only for its own acquisition (autocorrelation) but also for other satellites (cross-correlation) [Coenen and van Nee, 1992; Jun and Kee, 2006; Hu et al., 2013]. The last category of methods involves nonlinear operations that tend to introduce squaring loss due to noise amplification. Similarly, other methods reduce the search in one way or another at the expense of incurring implementation loss due to approximation. By consequence, it may require longer integration time to recover the signal, sometime counterproductive. A good signal acquisition design therefore needs to trade the pros and cons for an efficient implementation. In this paper, we present a fast acquisition search technique (FAST) via simultaneous search of allowable frequency errors (SAFE). It can reduce the total test points of MN, where M is the number frequency bins and N is the number of code lags, into M+N with minimized implementation loss. It is based on judicious combining of a number of carrier replicas at selected frequency search points, leading to a multi-frequency modulated code replica (MMCR). It can thus be viewed as an extension of the folding technique

FAST ACQUISITION SEARCH TECHNIQUE Assume that the time and frequency uncertainty zone of a GPS signal of interest to acquire is divided into a grid of MN search points where M is the number frequency bins and N is the number of code lags. Instead of testing each and every search point, which would need to perform MN correlations, the proposed method requires M+N search trials. Fig. 1 shows the block diagram of fast acquisition frequency search technique (FAST). It consists of two steps. In Step 1, the incoming signal is correlated with a multi-frequency modulated code replica (MMCR). Since MMCR allows for simultaneous search of allowable frequency errors (SAFE), that is, over the entire frequency uncertainty of M bins, only the search over the code uncertainty of N lags is required. The correlation between the incoming signal and MMCR can be performed using either the conventional correlators or the FFT-implemented correlation. The MMCR can be constructed either in the time domain or in the frequency domain, which will be detailed in Section 3 of this paper. As shown in Fig. 1, successive correlations may be added in power, that is, non-coherent integration, for detection and code phase estimation. Such non-coherent integration over time is useful in weak signal cases. Step 1: code phase estimation with simultaneous frequency search – N

Peak detection

Correlation

Code lag

MMCR Data buffer

Frequency test points Combined carrier replica

Multifrequency modulation

Non-coherent integration

Code phasing

Code strip-off

Phase & frequency estimation (FFT)

Tracking, demodulation, ranging & positioning Frequency bin

Code replica

Step 2: code-stripped carrier estimation – M

Fig. 1 – Fast Acquisition Search Technique (FAST)

Once the code lag corresponding to the detected correlation peak is obtained, Step 2 is entered in which a code sequence with the correct code phase is first constructed. The code is then stripped off from the incoming signal with sample-to-sample multiplications (a despreading process). What is left in the product sequence is the carrier at a

residual Doppler frequency, which can be estimated using FFT, equivalent to a search over the M frequency bins. The estimated code lag and frequency bin are then used to initialize code and carrier tracking loops for subsequent navigation data demodulation and pseudorange measurement formation. The process is performed for all satellites in view either sequentially or in parallel, leading to position fixing for the ultimate navigation solution. MULTI-FREQUENCY MODULATED CODE REPLICA

The incoming signal at a suitable intermediate frequency (IF), denoted by s(t), can be modeled as: L

s(t )   2 Pl Dl (t   l )cl (t   l )e j ( 2f l t 0 l )  n(t )

(1)

l 1

where Pl is the signal power, Dl(t) is the data bit, cl(t) is the spreading code, l is the time delay, fl is the signal frequency (nominal IF plus Doppler frequency shift), 0l is the initial phase for the l-th satellite, n(t) is the receiver noise (additive white complex Gaussian), and L is the number of visible satellites. For simplicity, (1) does not include multipath and other interference signals. Also, the code Doppler effect is not included. For the purpose of this paper, the data bit Dl(t) will be dropped in the subsequent presentation (sometime the satellite index l is also omitted if without confusion). Fast acquisition aims at finding how many satellites L are visible, which one, its code delay l, carrier frequency fl, and possibly initial phase 0l. The code replica for a satellite of interest locally generated at a search grid point ( ˆn , fˆm ) can be written as: ˆ rl (t;ˆn , fˆm )  cl (t  ˆn )e j 2f mt

the 3 dB-loss due to frequency error is f = 0.6/Ti. It explains why the search grid spacing is typically chosen as a half-chip in time and 1/(2Ti) in frequency, respectively. It is important to note from (3e) that the zero-crossings of the sinc-function occur at the frequency error of an integer multiple of the inverse of integration interval, that is, k/Ti for k = 1, 2, …, and between zero-crossings, the envelope decreases inversely proportional to k (i.e., 1/k). In conventional acquisition schemes, frequency search consists of selecting a test point from the frequency uncertainty grid, generating a carrier replica at that frequency as in (2), and using it to down-convert the incoming signal from an unknown IF to the baseband (carrier wipe-off) prior to correlating it with the code replica as in (3). If the selected test frequency is within a half spacing from the signal frequency, the frequency acquisition is successful. Otherwise, the process repeats with another frequency grid test point (a frequency bin). Instead of converting the incoming signal down toward baseband with one test frequency, the FAST method converts the code replica up by covering the entire frequency uncertainty, thus achieving simultaneous search of allowable frequency errors (SAFE). The search grid is illustrated in Fig. 2(a) where the search points are chosen to cover the whole frequency uncertainty zone with a spacing of 1/(2Ti). Search grid over frequency uncertainty

(2)

-10 kHz

-fs/2

The correlation between s(t) in (1) and rl (t;ˆn , fˆm ) in (2) over an integration interval Ti can be written as: Ti

Cl (  n , f m )   s(t )rl* (t;ˆn , fˆm )dt

0

10 kHz

fs/2

21 frequency bins, Spacing = 500 Hz for Ti = 1 ms (a) Frequency Search Grid

(3a)

fs > 2 MHz (Drawing not to scale)

f = 500 Hz

0 Ti

  2 Pl cl (t   l )e j ( 2f l t 0 l ) cl (t  ˆn )e  j 2f mt dt ˆ

f

(3b)

(b) Frequency Responses at Grid Points Spaced by 500 Hz

0

 2 Pl R(  n )sinc(f mTi )e j (f mTi 0 l )

(3c) f = 1000 Hz

where n = l - ˆ n and fm = fl - ˆf m are code timing error and frequency error, respectively, * stands for complex conjugate, and  |  n | , |  n | Tc 1  R(  n )  Ti  Tc  0, otherwise

(3d)

is the correlation function for an ideal random code with the chip duration of Tc,, and the sinc-function is defined as: sin(f mTi ) (3e) sinc(f mTi )  f mTi From (3d), it is easy to verify that the 3 dB-loss of correlation due to timing error is  = Tc/2 while from (3e),

f = 1000 Hz

f

f

(c) Two Sets of Grid Points Spaced by 1000 Hz Offset by 500 Hz

Fig. 2 – Multi-Frequency Modulated Code Replica

The time-domain generation of a multi-frequency modulated code replica (MMCR) can be formulated as: M ˆ ~ r (t;ˆ )   c (t  ˆ )e j 2fmt l



n

m1 M

m l

n

 cl (t  ˆn ) m e j 2fmt  cl (t  ˆn ) g (t ) ˆ

(4a)

m1

M

g (t )    m e j 2f mt m 1

ˆ

(4b)

and m is a complex weight that can be chosen to minimize the implementation loss. A simple choice is m = 1 for all m. g(t) is a combined carrier replica (CCR) for all frequency errors. Note that if ˆf m is symmetric (with equal positive and negative values) and m = 1, g(t) is in the form of the so-called Dirichlet kernel of M/2-th degree (https://en.wikipedia.org/wiki/Dirichlet_kernel). It can be shown that the power spectrum of g(t) and the aperiodic correlation of {m } is a Fourier pair [Yang et al., 2016], which can be used to design { m}. The correlation between s(t) in (1) and the MMCR in (4) with m = 1 over an integration interval Ti for the code timing error n = l - ˆ n can be written as:

taking the FFT of the code replica, making circular shifts of the replica spectrum, each by a search grid point, and then adding the shifted spectra together. The process is illustrated in Fig. 3(c). To conventional correlator OFMR spectrum

MMCR (a)

Code replica

Phase rotation & adding

To FFTimplemented correlator

FFT

To conventional correlator

Combined carrier replica

MMCR spectrum

MMCR

(b) Code replica

To FFTimplemented correlator

FFT

T

i ~ Cl (  n )   s(t ) ~ rl (t ,ˆn )dt

M

 2 Pl R(  n ) sinc(f mTi )e

j (f m Ti 0 l )

Incoming signal

(5a)

FFT

m 1

  l 2 Pl R(  n )

(5b)

 l  sinc(f mTi )e j (f

mTi

0 l )

MMCR spectrum Code replica

(5c)

m 1

For a combined carrier replica with spacing of 500 Hz as shown in Fig. 2(b), any input frequency is covered by the main lobe of four adjacent sinc-functions (centered at four adjacent test points) with large values and by the sidelobe of all other sinc-functions along the search grid with small values. When the spacing is increased to 1000 Hz, two search grids needed to cover the same uncertainty zone, are offset by 500 Hz for one with respect to the other, as shown in Fig. 2(c). Any input frequency is now covered by the main lobe of two adjacent sinc-functions with large values and by the sidelobe of all other sinc-functions along the search grid with small values. If the input frequency coincides with a frequency search grid point, the orthogonality condition occurs. That is, there is a peak response for the search grid point while none for the others, similar to orthogonal frequency division multiplexing (OFDM) [Chiueh and Tsai, 2007]. With conventional correlators, the multi-frequency modulated code replica (MMCR) can be generated from summing of phase-rotated replicas at desired search frequencies as shown in Fig. 3(a). Equivalently, the summation of phase rotations at desired search frequencies can be pre-calculated into the combined carrier replica (CCR) g(t) (4b) prior to FFT as shown in Fig. 3(b). When the correlation is implemented with FFT, the multifrequency modulated code replica (MMCR) spectrum can be easily generated in the frequency domain. It consists of

Correlation function

IFFT

Complex conjugate

(c)

where M

Correlation spectrum

Signal spectrum

0

FFT

Circular shifting & adding

Pre-calculated & stored in memory for periodic codes

Fig. 3 – FFT-Implemented Correlation with MMCR Generated in Frequency Domain

The fast acquisition process described above can be put into a compact format using vector and matrix representations. Assume that a total of K samples over Ti are used in correlation integration. The signal model in (1) can now be written as: s  c1  ξ1 c 2  ξ 2  c L  ξ L  1L  n T

(6a)

where  stands for Hadamard product (Schur product or entrywise product), the superscript T stands for transpose, 1L is a vector of L ones, and s  s(t1 ) s(t2 )  s(tK )

T

(6b) (6c) (6d)

n  n(t1 ) n(t2 )  n(tK )

T

c l  2 Pl Dl c(t1 ) c(t2 )  c(t K )

T



ξ l  e j ( 2f l t1 0 l )

e j ( 2f l t2 0 l )  e j ( 2f l tK 0 l )



T

(6e)

In (6d), it is assumed that the data bit keeps the same sign, which is true for the correlation among effective samples when K is twice as long as the code length. Note that instead of Hadamard product in (6), the use of diagonal matrix can serve the same purpose to put the problem into a vector-matrix representation. The MATLAB uses “.*” for this element by element (entrywise) operation between two compatible vectors or matrices. The M carrier replicas covering the frequency uncertainty grid can be chosen to construct a set of orthogonal bases for the M-dimensional subspace in the K-dimensional space. The orthogonal frequency basis matrix (a Fourier matrix) is given by:



F  ξˆ 1 ξˆ 2  ξˆ M



ˆ ξˆ m  e j 2f mt1



ˆ

ˆ

e j 2f mt2

 e j 2f mtK

(7a)



T

(7b)

The multi-frequency modulated code replica (MMCR) is generated as: ~ rn  c n  ξˆ 1 c n  ξˆ 2  c n  ξˆ M α  c n  Fα  c n  g (8a)





where c n  cl (t1  ˆn ) cl (t2  ˆn )  cl (tK  ˆn )

T

α  1  2   M 

T

(8b) (8c) (8d)

g  Fα

The correlation between s and ~ rn can be evaluated as: ~ Cl ( n )  {~ rnH }  (c n  Fα) H (c  ξ)  1TK (c  c n  ξ  F*α* (9a) where the superscript H stands for complex conjugate transpose (Hermitian) and the following equation is used: (9b) (a  b) H (c  d)  1T (a*  b*  c  d) Note that the Hadamard product is commutative and associative. Using the fact that c  c n  2 Pl R(  n )1K ,and, 1  a  a

(10a)

Eq. (9a) can be written as: ~ Cl (  n )  2 Pl R(  n )1TK (ξ  F*α* )  2 Pl R(  n )(1TK (ξ  F* )α*

(10b)

 2 Pl R (  n ) η α

(10c)

T

*



η  sinc(f1Ti )e j (f1Ti 0 l )  sinc(f M Ti )e j (f M Ti 0 l )



T

(10d) In (10d), let m ˆ represent the test grid point that is the closet to the incoming signal frequency. Then, (10e) ηmˆ  sinc(f1Ti )e j (f T  )  1 mˆ i

ηm  sinc(f1Ti )e

0l

j (f mTi 0 l )

ˆ  0 for m  m

(10f)

Clearly (10c) is equivalent to (5a). Since |T*| ≥ |m|, the noise floor with MMCR (M carrier replicas at once) is no less than that with a single carrier replica. The equality holds when  mˆ  1 and m = 0 for

m  mˆ . It is therefore expected to see an increase in noise floor, proportional to the number of frequency test points included in the MMCR due to non-coherent integration of individual terms, which is a price to pay for the gained speed. The general expression for l defined in (5c) for arbitrary m is: M

 l   msinc(f mTi )e j (f

mTi 0 l

)

(11a)

m 1

M    e j0 l   mˆ sinc(f mˆ Ti )e jf mˆ Ti   msinc(f mTi )e jf mTi  m 1, m  mˆ   (11b)

M    e j0 l   mˆ   msinc(f mTi )e jf mTi  m 1, m  mˆ  

(11c)

By setting |m| = 1, we may choose the phase of m such that the second term in (11c) is minimized for all m ˆ , that is, a minimal cross-correlation noise floor. A simpler option is to choose the phase of m such that the multifrequency modulated code replica (MMCR) in (4b) has the minimal peak to average power ratio (PAPR). Our method of combined carrier replica (CCR) can be understood as a way of frequency folding, extended from the mirror idea of code folding [Yang et al., 1999] to the incoming signal in [Li et al., 2008] and to the local replica in [Zhao et al., 2016]. However, our method differs from their frequency folding techniques in that CCR incorporates design parameters that can minimize implementation loss via a reduced PAPR and noise floor. SIMULATION AND ANALYSIS Examples are used to show the properties of combined carrier replica (CCR), multi-frequency modulated code replica (MMCR), and simultaneous search of allowable frequency errors (SAFE). Together, they enable the fast acquisitions search technique (FAST). Results for processing of both simulated and real data are presented. CCR and MMCR When all search points in the frequency uncertainty zone (±10 kHz) are considered at once, there are 41 frequency search points including the zero for the spacing of 500 Hz. For a signal sampled at 2.046 MHz over 2 ms, Fig 4(a) shows the amplitude spectrum of all carrier replicas over the frequency uncertainty zone (±1.023 MHz). With m = 1, taking IFFT of Fig 4(a) leads to the combined carrier replica (CCR) waveform as shown in Fig. 4(b). For symmetrically placed test points as in Fig. 4(a), the combined carrier replica (CCR) is real-valued. There are twenty cycles over 2 ms or 10 kHz corresponding to the largest frequency error under test. The lower frequency components add up to define the amplitude envelope. Indeed, the mid-section is a sine wave with the unit amplitude while toward both ends the amplitude rises and reaches 41 (aligned phases), which is exactly the number of total carrier replicas in the combination. In this case with m = 1, the resulting CCR is the so-called Dirichlet kernel. The waveform can be easily understood from the following Dirichlet equality: Dn ( x ) 

n

e

k n

n

jk x

 1  2  cos(kx)  k n

sin((n  1 / 2) x (11) sin( x / 2)

When such a combined carrier replica is correlated with the incoming signal, the resulting frequency error responses are overlapped as shown in Fig. 2(b). When the combined replica carrier is used to modulate the code replica, it produces the multi-frequency modulated code replica (MMCR) as shown in Fig. 4(c). Note that only the first half of 2046 samples are used for correlation over 1 ms.

From Fig. 4(b), it is observed that the waveform has a large peak to average power ratio (PAPR). Large PAPR also appears in communications and broadcast signals with OFDM) It has a detrimental effect on performance due to non-constant envelope waveform that may saturate power amplifier, causing nonlinearity issues [Chiueh and Tsai, 2007]. frequency uncertainty: -10 to 10 kHz, fs = 2.046 MHz, Ti = 2 ms

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4(d) and 4(e), respectively. The corresponding MMCR (the real component) is shown in Fig. 4(f). Another method is to use a CCR that only covers only a portion of the frequency uncertainty zone. Fig. 2(c) is an example of two interleaved grids. Search for optimal phase of MMCR to reach minimal PAPR will be addressed separately. Frequency Errors on Correlation Peak and Noise Floor Before looking into simultaneous search of allowable frequency errors, we first study the effects of frequency errors on correlation peak and noise floor for a single carrier. Table 1 lists the test conditions and results, which are illustrated in Fig. 5. When the frequency error f varies from 0 Hz to 250, 500, 750, and 900 Hz, the correlation peak decreases from 2046 (the full value), to 1842, 1303, 614, and 223, as described by the sinc-function (3e), shown in Figs. 5(a) through 3(d), respectively. At 1000 Hz (zero-crossing) and beyond, there is no discernible correlation peak sticking out above the noise floor as shown in Fig. 5(e). ideal correlation with frequency error = 0 kHz

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Fig. 4 – Spectral Specification, Combined Carrier Replica, and Multi-Frequency Modulated Code for Dirichlet Kernel (a) – (c) and Its Randomized Version (d) – (f)

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In our case, a large PAPR, if developed, would not only require a wider data path when implemented in FPGA but also affect the orthogonality of the code replica by overemphasizing a portion of the pseudo-random code sequence. It has the undesired effect of raising the noise floor. Many methods developed to reduce PAPR for orthogonal frequency-division multiplexing (OFDM) signals can be employed [Lim et al., 2009]. PAPR reduction is more difficult for OFDM signals than our case because the former has given data bits whereas we have no restriction in choosing m. A simple method is used to produce Figs. 4(d) to 4(f) wherein the initial phases of the 41 carriers are picked up uniformly from [0, 2] so as to avoid phase alignment. Since the spectrum profile is no longer symmetric, the corresponding CCR waveform becomes complex-valued with their real and imaginary components shown in Figs.

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Fig. 5 – Noise Floor and Correlation Peak as a Function of Frequency Error Table 1 – Frequency Errors on Correlation Peak and Noise Floor f (Hz) Correlation peak Noise floor peak Peak ratio

0 2046 130 15.74

250 1842 160 14.17

500 1303 160 8.14

750 614 177 3.47

900 223 180 1.25

1000 180 -

With f = 5750 Hz between two test points, the worst case scenario, the correlation peak for a single carrier degrades from 2048 in Fig 6(b) to 1842 in Fig. 7(b). In contrast, the correlation peak for the combined carrier remains about the same. So is the noise floor as shown in Fig. 7(a). Fig. 7(c) shows the FFT applied to the code-stripped signal. From the detail around the peak in Fig. 7(d), the input

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Fig. 6 – SAFE with Dirichlet Kernel forf = 5500 Hz uncertainty interval: +/-10kHz, spacing: 500 Hz, input: 5750 Hz 2200

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From Fig. 6(b), the code starting phase can be detected and used to synthesize a code replica to strip it off the incoming signal. The FFT analysis can be applied to the codestripped signal for frequency estimation as shown in Fig. 6(c) with the detail in Fig. 6(d). Since the frequency resolution is 1000 Hz, the FFT peak is between 5000 and 6000 Hz, which interpolates to the input value of 5500 Hz. In the second example, everything is kept same except for the frequency error f = 5750 Hz. Fig. 7(a) shows the correlation function between the incoming signal and the folded carrier replica and Fig. 7(b) shows the detail around the correlation peak.

frequency can be estimated from the two largest values 614 and 1842 at 5000 and 6000 Hz, respectively. With a quadratic or sinc-function interpolation, the estimate is close the true input of 5750 Hz. Clearly, the raised noise floor with Dirichlet kernel reduces the performance margin as compared to a single carrier replica. A method to minimize the noise floor rising is presented next.

abs(fft)

The noise floor peak value rises slightly from 130 at f = 0 Hz to 180 at f = 1000 Hz and stays at this level for others. The correlation peak to noise floor peak ratio varies from 15.74 to 14.17, 8.14, 3.47, and 1.25, respectively. When the frequency bin width is chosen to be 500 Hz, the maximum frequency error is 250 Hz. The maximum peak loss is 204, the noise floor increase is 30, and the peak to peak ratio loss is 1.57, as shown in Fig. 5(f).The correlation and noise floor peak values and their variations as a function of frequency errors establish the baseline with respect to which the implementation loss of the fast acquisition search technique (FAST) can be assessed, which is presented below. Simulation of FAST with Dirichlet Kernel Two examples are presented for simulation of FAST with the Dirichlet kernel. In the first example, the frequency error f = 5500 Hz, which coincides with a test point, and the initial phase is 35 deg.Fig. 6(a) shows the correlation function between the incoming signal and the folded carrier replica and Fig. 6(b) shows the detail around the correlation peak. As expected, the FFT-implemented correlation with a code modulated on a combined carrier replica (CCR) is calculated only once but covers the whole time-frequency uncertainty zone. It produces a correlation peak similar to the conventional FFT-implemented correlation but calculated forty-one times, one for each frequency test point over the same time-frequency uncertainty zone. As shown in Fig. 6(a), the combined carrier correlation peak (the blue curve) is slightly higher than that of a single carrier (the red curve) because of contributions from adjacent test points. It is this sum of contributions from all other carrier replicas that also elevates the noise floor by a factor of M where M is the number frequency test points included in the combined carrier replica. In Fig. 6, M = 41 and the single carrier noise floor peak is 160. The expected peak for combined carrier noise floor is 160 41  1025.

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Fig. 7 – SAFE with Dirichlet Kernel for f = 5750 Hz

Simulation of FAST with Randomized Dirichlet Kernel In this simulation of FAST, a randomized Dirichlet kernel is used for two examples with f = 5500 Hz in Fig. 8 and f = 5750 Hz in Fig. 9, respectively, wherein the initial phase is 35 deg. As shown in Figs. 8(a) and 9(a), the combined carrier correlation peak (the blue curve) is slightly higher than that of a single carrier (the red curve). However, the

noise floor of the combined carrier correlation peak (the blue curve) is not significantly higher than that of a single carrier (the red curve) as was the case with the Dirichlet kernel in Figs. 6(a) and 7(a). Indeed, the peaks of noise floor of the combined carrier correlation with a randomized Dirichlet kernel are about twice higher than those of a single carrier as shown in Figs. 8(a) and 9(a). However, it was about 6.4 with Dirichlet kernel as shown in Figs. 6(a) and 7(a). The example shows the benefits of a CCR with small PAPR. uncertainty interval: +/-10kHz, spacing: 500 Hz, input: 5500 Hz

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Fig. 8 – SAFE with Randomized Dirichlet Kernelf = 5500 Hz uncertainty interval: +/-10kHz, spacing: 500 Hz, input: 5750 Hz

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FAST with Experimental Data The FAST algorithm is applied to experimental data. A radio-frequency (RF) front-end with 3 MHz bandwidth is used, which down-converts GPS L1 signal at 1575.42 MHz to baseband in a single stage and then samples it at 5 MHz in 8 bits. The conventional correlation via point search over the entire time-frequency uncertainty zone (5000 code phases in 0.2 chips and ±10 kHz frequency bins in 500 Hz) is shown in Fig. 10(a) for SVN32. For a given segment of data, the signal is detected at the 3606-th sample in code and -1 as the frequency search grid point. x 10

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noise floor peak. This is in sharp contrast to the ideal case in Figs. 6(a), 7(a), 8(a), and 9(a) where the correlation peak is more than ten times higher than that of the noise floor. Because of this high SNR margin, it allows for search of all allowable frequency errors simultaneously in the ideal case as shown in Figs. 6 through 9. For practical cases, however, the SNR margin is rather limited as shown in Fig. 10(b). The rising of noise floor is proportional to the increasing of frequency search points included in the combined carrier replica. For a signal of low margin, it is expected to reduce the number of frequency points for simultaneous search. When seven search points (±3 around 0 bins) are included in a uniformly randomized Dirichlet kernel, the correlation function is shown in Fig. 11(a) with the detail around the peak in Fig. 11(b). The correlation peak to noise floor is about 2, from which the signal can be easily detected. When eleven search points (±5 around 0 bins) are included in the combined carrier replica using a randomized Dirichlet kernel, the correlation function is shown in Fig. 11(c) with the detail around the peak in Fig. 11(d). The correlation peak to noise floor decreases to about 1.5. The signal can still be detected but with a higher chance for errors (miss detection or false alarm).

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Fig. 10(b) plots all the correlation functions for 41 frequency bins, respectively. For this segment of data, the correlation peak is only about twice as high as the overall

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Fig. 11 – FAST with Experimental Data: Code Phase Estimation

When ten negative search points (-9 to 0 bins) are included in the combined carrier replica using a randomized Dirichlet kernel, the correlation function is

shown in Fig. 11(e) with the detail around the peak in Fig. 11(f). The correlation peak to noise floor ratio increases to about 1.8, which achieves a detection performance similar to Fig. 11(a) with seven search points but using search points as numerous as Fig. 11(b) of eleven search points. Once the signal is detected with an estimate of the code phase, the code can be stripped off from the signal. The underlying frequency of the signal can be estimated with FFT applied to the code-stripped signal as shown in Fig. 12(a). A refined estimate of the input frequency can then be interpolated from the two largest values as detailed in Fig. 12(b). In this setting, the entire frequency uncertainty zone of ±10 kHz can be covered by four segments with ten bins each, representing rising satellites with low Doppler (0 to 10 bins) and high Doppler (11 to 21 bins) and descending satellites with low Doppler (-1 to -10 bins) and high Doppler (-11 to -21 bins), respectively. Search speed (the number of segments and possible noncoherent integrations) is traded for higher SNR margins, leading to an overall fast detection with high probability of detection and low rate of false alarms. 2.5

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Search speed (the number of segments and possible noncoherent integrations) is traded for higher SNR margins, leading to an overall fast detection with high probability of detection and low rate of false alarms. As compared to the conventional method, the acquisition performance for the three FAST configurations in the experimental test is summarized in Table 2 in terms of speed gained and achieved peak margin. Table 2 – Summary of Acquisition Performance Method

Speed Gained

Detection Margin

Conventional

×1

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From the above experiments with real GPS data, it seems that simultaneous search of 7 to 11 frequency bins is realistic. The results suggest a fast acquisition strategy. That is, the entire frequency uncertainty zone of ±10 kHz can be covered by four segments with ten or eleven bins each. They respectively represent (i) slow rising satellites with low Doppler (0 to 10 bins), (ii) fast rising satellites with high Doppler (11 to 21 bins), (iii) slow descending satellites with low Doppler (-1 to -10 bins), and (iv) fast

descending satellites with high Doppler (-11 to -21 bins). Instead of 41 searches in frequency, four are sufficient, thus speeding up the acquisition process by a factor of ten in average. CONCLUSIONS One way to speed up GPS signal acquisition is to make each calculation of the correlation function (a search over the entire time uncertainty) simultaneously cover several frequency bins, thus reducing the amount of search required for the frequency uncertainty by the same factor. Such an approach to simultaneous search of all frequency errors (SAFE) can be achieved by constructing a combined carrier replica (CCR) and modulating any code of interest onto it, leading to a multi-frequency modulated code replica (MMCR). In our method, the simultaneous frequency bins are chosen such that their frequency responses are orthogonal at the peak while the phases of carriers at the frequency bins are such that the resulting combined replica has a reduced peak to average power ratio (PAPR). Simulation and experimental data processing results are presented to demonstrate the functionality and performance of our fast acquisition search technique (FAST), which reduces the total test points of MN, where M is the number of frequency bins and N is the number of code lags, into M+N, that is, two linear time searches. As shown, due to the noise floor raised by implementation loss, the actual number of frequency bins that can be simultaneously searched depends on the margin of SNR to allow reliable detection. As a result, part of our ongoing work is directed toward the search for optimal phase of MMCR to reach minimal PAPR and, even better, a minimal cross-correlation noise floor. REFERENCES D. Akopian, “Fast FFT Based GPS Satellite Acquisition Methods,” IEE Proc.-Radar Sonar Navig., 152(4), Aug. 2005. J.W. Betz, J.D. Fite, and P.T. Capozza, “DirAc: An Integrated Curcuit for Direct Acquisition of the M-Code Signal,” ION GNSS 2004, Long Beach, CA, September 2004. T.D. Chiueh and P.Y. Tsai, OFDM Baseband Receiver Design for Wireless Communications, Wiley, 2007. A.J.R.M. Coenen and D.J.R. van Nee, “Novel Fast GPS/GLONASS Code-Acquisition Technique Using Low Update Rate FFT,” Electronic Letters, 28, Sept. 1992. R.G. Davenport, “FFT Processing of Direct Sequence Spreading Codes Using Modern DSP Microprocessors,” Proc. of IEEE NAECON, 1991. D. Djebouri, “Averaging Correlation for Fast GPS Satellite Signal Acquisition in Multipath Rayleigh Fading Channel,” Microwave Journal, Oct. 2015. A. Fish, S. Gurevich, R. Hadani, A. Sayeed, and O. Schwartz, “Computing the Matched Filter in Linear Time,” arXiv preprint: 1112.4883v1 [cs.IT], Dec. 2011.

Y. Fong, F. Liu, and G. Pang, Novel Mechanisms of Synchronization for Direct Sequence Spread Spectrum Signals, National Defense Industry Press, Beijing, 2011 (in Chinese). H. Hassanieh, F. Adib, D. Katabi, and P. Indyk, “Faster GPS via the Sparse Fourier Transform,” MobiCom’12, Istanbul, Turkey, Aug. 2012. H. Hu, Y. Yuan, H. Wang, M.H. Gao, and D. Zou, “GPS Receiver C/A Code Rapid Acquisition Technology Research,” J. of Information & Communicational Science, 10(2), 2013. H.Y. Jun and C.D. Kee, “A New Technology for GNSS Signal Fast Acquisition within Three Seconds, Applicable to Current GNSS Receivers,” Proc. of ION NTM, Monterey, Jan. 2006. CA. S.H. Kong, “A Deterministic Compressed GNSS Acquisition Technique,” IEEE Trans. on Vehicular Technology, 62(2), Feb. 2013. H. Li, M.Q. Lu, and Z.M. Feng, “Direct P(Y)/M-Code Acquisition Based on Time-Frequency Folding Technique,” Proc. of ION GNSS 2008, Savannah, GA, Sept. 2008. D.W. Lim, S.J. Heo, and J.S. No, “An Overview of Peakto-Average Power Ratio Reduction Schemes for OFDM Signals,” J. of Communications, 11(3), June 2009, 229239. M.B. Mathews, P.F. Macdoran, and K.L. Gold, “SCP Enabled Navigation Using Signals of Opportunity in GPS Obstructed Environments,” NAVIGATION, Journal of the Institute of Navigation, Vol. 58, No. 2, 2011. P.K. Sagiraju, S. Agaian, and D. Akopian, “Reduced Coplexity Acquisition of GPS Signals for Software Embedded Applications,” IEE Proc.-Radar Sonar Navig. 133(1), Feb. 2006, 69-78. J. Starzyk and Z. Zhu, “Averaging Correlation for C/A Code Acquisition and Tracking in Frequency Domain,” Proc. of IEEE Midwest Symp. on Circuits and Systems, Fairborn, OH, August 2001. X. Tang, S. Yong, and F. Wang, “Performance of XFAST in the Presence of Code Doppler,” J. on Communications, Vol. 31, No. 8, August 2010 (in Chinese). X. Tang, J. Pang, Y. Huang, and F. Wang, “Optimization of XFAST Design” J. of Central South University (Science and Technology), Vol. 45, No. 4, April 2014 (in Chinese). D.J.R. van Nee and A.J.R.M. Coenen, “New Fast GPS Code Acquisition Technique Using FFT,” Electronic Letters, 27, Jan. 1991. R. Wolfert, S. Chen, S. Kohli, D. Leimer, and J. Lascpdy, “Direct P(Y)-Code Acquisition under a Jamming Environment,” Proc. of IEEE PLANS, April 1999. C. Yang, “Zoom, Pruning, and Partial FFT for GPS Signal Tracking,” Proc. of ION-NTM’01, Long Beach, CA, Jan. 2001. C. Yang, J. Chaffee, J. Abel, and J. Vasquez, “Extended Replica Folding for Direct Acquisition of GPS P-Code

and Its Performance Analysis,” Proc. of ION-GPS’00, Salt Lake City, UT, Sept. 2000. C. Yang, M. Miller, and T. Nguyen, “Symmetric PhaseOnly Matched Filter (SPOMF) for Frequency-Domain Software GPS Receivers,” ION Journal: Navigation, Vol. 54, No. 1, Spring 2007. C. Yang, A. Soloviev, M. Veth, and E. Blasch, “Fast Acquisition with Simultaneous Frequency Search,” Submitted to IEEE Trans. n Wireless Communications, January 2016. C. Yang, J. Vasquez, and J. Chaffee, “Fast Direct P(Y)Code Acquisition Using XFAST,” Proc. of IONGPS’99, Nashville, TN, Sept. 1999. H.B. Zhao, W.Q. Feng, X.D. Xing, C. Sun, and X.M. Guan, “A Novel PN-Code Acquisition Method Based on Local Frequency Folding for Beidou System,” Proc. of IONITM, Monterey, CA, Jan. 2016.