Design of Peak-finding Algorithm on Acquisition of Weak GPS Signals

2006 IEEE Conference on Systems, Man, and Cybernetics October 8-11, 2006, Taipei, Taiwan

Design of Peak-finding Algorithm on Acquisition of Weak GPS Signals W. L. Mao, A. B. Chen, Y. F. Tseng, F. R. Chang, H. W. Tsao, W. S. Huang 

Abstract—GPS is the system combined CDMA technique and trilateration to obtain precision position information. The main goal of this paper is design of high-sensitivity receiver on acquisition of weak GPS signals. A new peak-finding algorithm is developed to search the C/A code phase in the FFT-based correlation domain. This method can estimate the peak location accurately and provides a faster performance in the software-based signal acquisition. The integration time deciding algorithm is also utilized to enhance the capability of waveform search under indoor environments. Simulation results demonstrate that our proposed scheme can acquire the GPS signal efficiently under the power level of -155dBm in frequency domain processing.

below those found outside. The high sensitivity GPS [1, 2] which make use of real-time convolution processor instead of an early-late correlator is proposed. The convolution processor contains over 2000 correlators per satellite, and become the most computation consuming operation. The DSP-based approach is a store-and-process method that performs the convolution in frequency domain. The code averaging correlation [3] use the FFT-based correlator for implementation the acquisition process in outdoor environments. They are converted to frequency domain using 1024-point complex FFT and multiplied by the conjugate of the FFT of the averaged-local-code.

I. INTRODUCTION

For indoor GPS operation, the extra millisecond of data can be integrated and then yielded the SNR gains that approach

G

lobal positioning system (GPS) that provides accurate positioning and timing information has become a commonly used navigation instrument for aircraft precision approaches, missile systems, automated vehicle guidance, and other civil applications. Each GPS satellite simultaneously transmits on two L-band frequencies denoted by L1 and L2, which are 1575.42 and 1227.60 MHz, respectively. The carrier of L1 signal consists of an in-phase and a quadrature-phase component. The in-phase component is biphase modulated by a 50-bps data stream and a pseudorandom code where chipping rate is 1.023 MHz. It is the basis for the vast majority of civil applications and will be the object of most of our attention in this paper. We call it as the civilian signal even through the military also uses this signal. The quadrature-phase component is also biphase modulated by the same 50-bps data stream but with a different psedudorandom code called the P-code, which has a 10.23 MHz chipping rate and a one-week period. The conventional GPS used outdoor can meet all the signal and measurement requirements for commercial applications. The biggest challenge of indoor GPS receiver is that the indoor signal is at power levels 30dB (one thousand times) M. L. Mao is with the Graduate Institute of Electrical Engineering and Department of Electrical Engineering, Mingchi University of Technology, Taipei county, Taiwan (e-mail: [email protected] ) A. B. Chen is with MediaTek Inc, Hsinchu City, Taiwan (email: [email protected] ) Y. F. Tseng is with Department of Electrical Engineering, University of California, Los Angeles, USA. (e-mail: [email protected] ) F. R. Chang and H. W. Tsao are with the Graduate Institute of Electrical Engineering and Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan. (e-mail: [email protected], [email protected] ) W. H. Huang is with Evermore Inc, Hsinchu City, Taiwan (e-mail: [email protected] ).

1-4244-0100-3/06/$20.00 ©2006 IEEE

N for each extra N millisecond. It takes more time for acquisition in indoor GPS receiver. The most important contribution of this paper is the weak signal can be acquired even the signal power is -155dBmW. The circular convolution in frequency domain is performed and average correlation method is then implemented. A peak-finding algorithm is developed to detect the C/A code delayed phase accurately in the autocorrelation domain. The integration time deciding algorithm is also presented to determine the suitable dwell time in each search bin. Simulation results show that our proposed peak-finding algorithm can be utilized in indoor applications even through the signal level is of -155 dBm. The remainder of this paper is organized as follows. Section 2 describes the GPS received signal and acquisition process. In Section 3, the acquisition in frequency domain using FFT scheme is introduced. The proper FFT size for averaging correlation is proposed. The peak-find and integration time deciding algorithms are developed for indoor GPS receiver. Simulation results are demonstrated in Section 5. Some conclusions are stated in the last section.

II. GPS SIGNAL AND ACQUISITION PROCESS The transmitted spread spectrum signal is represented as

S (t ) [ D (t ) CA(t )] cos(2f L1t )

(1)

where D(t ) is the binary data with duration T ( T= 20ms ).

CA(t ) represents the binary Gold Code with chip duration Tc

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( RC 1 / TC 1.023MHz ). f L1 is the L1 carrier frequency (1575.42MHz). The integer PG T / Tc 43dB is the processing gain of the GPS system. A GPS receiver must conduct a two-dimensional search in order to find each satellite signal, where the dimensions are C/A code delay and carrier frequency. Fig. 1 shows the block diagram of the serial search. It must be conducted across the full delay range of the C/A code for each frequency searched. A generic method for conducting the search is that the received waveform is multiplied by delayed replicas of C/A code, translated by various frequencies, and then passed through a baseband correlator containing a low-pass filter which has a relatively small bandwidth (perhaps 100-1000 Hz)[10]. The output energy of the detection filter serves as a signal detection statistic and will be significant only if both selected code delay and frequency translation match that of the signal. When the energy exceeds a predetermined threshold , a tentative decision is made that a signal is being received, subject to later confirmation. The value chosen for the threshold  is a compromise between the conflicting goals of maximizing the probability PD of detecting the signal when it is actually present at a given Doppler and code delay and minimizing the probability PFA of false alarm when it is not. Fig. 1 shows the block diagram of serial acquisition process.

Sequence in 0.5 chip increments through full span of 1023 chips

Code generator

Sequence frequency in 500 Hz increments through span of 10 KHz

Carrier NCO

90 0

Received signal r(t)

T

dt 

Digital IF 4.092MHz

I

Detection threshold  Declare signal if  and no signal if 

0

I 2 Q 2



0.5 chip increments. Each code phase search increment is a code bin. The combination of one code and one Doppler bin is a cell. Fig. 2 illustrates the two-dimensional search process. The center of the frequency interval is located at f c f d , where f c is the L1 carrier frequency and f d is the estimated carrier Doppler shift. A typical range for frequency search interval is f c 5 KHz. The frequency search is conducted in N discrete frequency steps that cover the entire search interval. For coherent processing used in many GPS receivers, the frequency bin width is approximately the reciprocal of the search dwell time. Typical values of frequency bin ( f ) are 250-1000 Hz. Assuming a 5 KHz frequency search range, the number N of frequency steps to cover the entire search interval would typically be 10-40. By the above discussion, if f =500 Hz, acquisition in time domain needs 2046 × 20 = 40920 bins to search. If signal is strong (above -120dBmW), the dwell time (integration time) is 1 ms, the total search time is needed 40.92 second. The acquisition process has a very large space to search and it takes a lot of time. So, acquisition process is the most time consumption in GPS receiver.

III. ACQUISITION IN FREQUENCY DOMAIN Acquisition in time domain needs more signal bins to search because the process has two-dimension search pattern. If the data are transformed to frequency domain, the search pattern can be reduced from two dimensions to one dimension. A multiple numbers of period data are transformed on frequency domain once. Fig. 3 shows this search pattern. The only one dimension searched is Doppler shift in frequency domain. The total searching steps are about 20 bins (from -5 KHz to +5 KHz). Besides FFT algorithm in DSP chip is easily to be implemented.

T

dt 

Q

0

T 0.5chip

Fig. 1 Signal search method

f  250 ~ 1000 Hz

6 4

T 0.5chip

2

f  250 ~ 1000 Hz

6

Search Direction Start of 1 Search

4 2

3 Start of 1 Search

Search Direction

5

3

7

5

Doppler search sequence

7 Doppler search sequence

2046 code positions

Fig. 2 Two-dimensional C/A code search pattern The following example assumes that a C/A code search is being performed and that all 2046 C/A code phase positions are being examined. The code phase is typically searched in

2046 code positions

Fig. 3 One-dimensional C/A code search pattern A. FFT Search Algorithm The conventional time-domain acquisition can compute autocorrelation function in a simplest way, but it takes a lot of number of operations. The convolution process in time domain equals to multiplication in frequency domain. If x[n]

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is the input signal, then

size can be reduced to 4K.

L

x[n]CA[(n m)L] x[n] CA[n]

R[m] 

n 1





(2)

IV. PROPOSED PEAK FINDING ALGORITHM

F 1 F ( x[n]) F (CA[n]) 

CA[n] is C/A code generator generates code,  means convolution, F is Fourier Transform and F-1 means Inverse Fourier Transform. Acquisition in frequency domain can reduce a large of number of operations and then increase the speed of computation. For example, in time domain, one period has 163668 points (16 × 1023), there are 163668 × 163668 multiplications and 163668 additions need be done. In frequency domain, FFT with 16K size will be used first. 16 1024 2 log 2  16 1024229376 2 complex multiplications (3.1) 2 16 1024 log 2  16 1024458752 complex adders (3.2) The block diagram of GPS frequency-domain acquisition is shown in Fig. 4.

Digital IF X[n]

I cos[wn]

R[m] FFT

IFFT Q

Intergration

I 2 Q 2

FFT

CA [n]

sin[wn]

Fig. 4 Block diagram of GPS acquisition in frequency domain Data Collection System

Decimate 16K to 4K

FFT

Decimate C/A code

FFT

NCO IFFT Peak Search and NCO Controller

Fig. 5 Block diagram of averaging correlation algorithm B. FFT Size If the size of the FFT is not a power of two, a mixed-radix algorithm can be used to manipulate the non–power–of–two FFTs. The sampling rate is set as 16.367667 MHz in our simulation, and almost 16368 points are calculated in one period. The 16K FFT and IFFT must be used to accomplish the transformation. It is difficult to implement the 16K FFT and become the circuit burden. If the incoming sampled GPS signal is down sampled from 16×1023 to 4×1023, the FFT size can be reduced to 4K. So the correctness and efficiency between 16K FFT and 4K FFT must be considered. In Fig. 5, the averaging correlation algorithm [3] is utilized and the FFT

Once the autocorrelation function is computed, the code delay phase needs to be determined from the time of peak value occurrence. Especially if signal power is weak, it is hard to get an optimal method to decide the correct peak time. For the strong signal, the peak value is always evident. The distance between maximum number (peak value) and second maximum number is not very far. Increasing the integration time is a simple way to become a higher peak value. It also increases the search time and then degrades the receiver efficiency. The determination of time of peak vale and integration time are essential to be considered in weak signal acquisition. Here, the Monte-Carlo method and statistical properties are combined to decide the suitable integration time. A. Peak finding algorithm In order to reduce the computation burden on receiver and obtain a better code phase effectively, a peak finding algorithm is proposed: Step1: Compute the autocorrelation function using the FFT method. Step 2: Find the maximum value, second maximum value and mean in correlation domain. Step 3: Normalize the autocorrelation function (i.e., maximum value =1). Step 4: If (maximum value－mean) > VTH1 , AND (maximum value－ second maximum value) > VTH2, then the time with peak value is code delay. Th eme a n i n gof“ pe a kv a l u e ”i st h ema x i mu mv a l u eof all. In order to avoid miscarriage, it is necessary to detect the difference between maximum value and second maximum value simultaneously. If the difference between maximum value and second maximum value is large enough, the probability of false alarm will be decreased. One vital index to evaluate the signal strength is SNR. The SNR in the autocorrelation function can be obtained as: max  d  n mean d  n SNR (dB) 10 log  (4)  s . t . d  d  n    where d  n is the data sample in autocorrelation function.

And combine the third step of the algorithm, it can be rewritten as (maximum value-mean) = SNR × (standard deviation) > VTH1 (5) Using the rule of thumb, a good performance signal has SNR more than 10dB and standard deviation is about 0.03 (after normalization). So SNR × (standard deviation) > 0.3, and VTH1= 0.3 (6) Then, the maximum value is assumed as two times bigger than second maximum value. The relationship can be represented as:

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SNR max_

value  dB SNR 2 nd _ max_ value  dB 3 dB

(6.1)

max  d  n mean d  n  log   s.t.d  d  n  

(6.2) 2nd max  d  n mean d  n  log  0.3 s.t.d  d  n   The equation becomes max d  n mean d  n max d  n 2nd max d  n  (6.4) 2 From the Eq. (6), max d  n mean d  n 0.3

max  d  n 2 nd max  d  n 0.15

(a)

(6.5)

The variable VTH2 is set as 0.15. These two parameters are decided and utilized in the acquisition process. B. Integration time deciding algorithm GPS receiver does not know the signal power level in the indoor situation. It is necessary to provide an efficient method to decide the integration time instead of increasing the integration time straight forward. The integration time deciding algorithm is follows: Step 1: The starting of integration time is 1ms. Step 2: The input signal is processed by peak finding algorithm, and then its code delay is found. Step 3: If the code delay cannot be found obviously, the integration time is increased to the next longer duration. The integration time step is as: 1ms →10ms →50ms →100ms →200 ms →500ms →1000ms . Using the algorithm above, the receiver can reduce the probability of false alarm and acquire an unknown power signal efficiently.

V. SIMULATION RESULTS The simulation results of the peak-finding method are conducted to verify the acquisition performances. The received signal is bandpass filtered, amplified and down-converted to IF and then digitized. The IF is fixed at 4.092 MHz, and a sampling frequency of 16.367667 MHz is selected. The averaging correlation algorithm is verified and then chose the most feasible number of FFT size. In the first experiments, the signal power is set as -120dBmW, noise power is set as -109 dBmW, integration time is 1 ms, and zero Doppler frequency shift. The size of FFT is varied as 1024(1K), 4096(4K) and 16368(16K) points. The output signal-to-noise ratio (SNR) and processing gain are compared for different FFT sizes. Fig. 6 shows the outputs of autocorrelation function with different FFT sizes using frequency domain method.

(b)

(c) Fig. 6 Autocorrelation function for different FFT sizes, (a) 1K FFT size, (b) 4K FFT size, and (c) 16K FFT size under the input SNR = -11dB. In the second experiment, the real data are recorded from the receiver with one-bit resolution. The signal power level is -125dBmW, noise power is -109 dBmW, integration time is 10 ms, Doppler frequency is 200 Hz, and the Clock offset is 44585 Hz. Fig. 7 shows the autocorrelation function with different FFT size under the input SNR of -16dB.

(a)

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approximates to 100%. Table 2 summarizes the integration time and the corresponding probability of peak location for indoor environments. Table 2 Integration time vs. detect probability of peak location

(b)

(c) Fig. 7 Autocorrelation function for different FFT sizes, (a) 1K FFT, (b) 4K FFT, and (c) 4K FFT under the input SNR=-15dB Table 1 shows the comparison between the 1K, 4K, and 16K FFT sizes under different input SNR conditions. The choice of 4K FFT size can have the better performance between processing gain and computation speed. The 16K FFT has the highest processing gain, but it takes more than 4 times multiplication operations. (Accurately speaking, it takes 16 1024 log 2  16 1024 4.67 times) Since the 4 1024 log 2  4 1024

Signal power (dBmW) Integration time (ms) Detect Probability (%) Signal power (dBmW) Integration time (ms) Detect Probability (%) Signal power (dBmW) Integration time (ms) Detect Probability (%)

-120

-125

-130

1

1

1

10

1

10

100

100

100

99.3

99.3

100

-140

-135

-145

10

50

100

100

200

500

3.6

97.3

100

0.3

98.9

100

-150

-155

200

500

1000

500

1000

1500

0.0

92.7

98.9

0.2

92.1

97.8

Based on the detective rate above, the adequate integration times are chosen and plotted on Fig 8. The proper dwell time are considered as the probability is near to 100%. Fig. 8 shows the very useful information to determine the integration time under variety of indoor circumstances.

processing gain of 1K FFT is not high enough, the probability of false rate can be increased Table 1 The output SNR and processing gain for different FFT lengths FFT size 1K 4K 16K output 18.47 21.73 26.53 SNR(dB) Experiment 1 processing 29.47 32.73 37.53 gain output 14.16 19.63 23.08 SNR Experiment 2 processing 30.16 35.63 39.08 gain

Fig. 8 Suitable integration time vs. different input SNR

VI. CONCLUSION The peak-finding algorithm is verified with difference input signal power. Each received data is simulated with 1000 runs, and noise power is -109 dBmW. The input SNR is varied from -120dBm to -155dBm. For the same signal power level, the dwell time is increased until the detect probability

This paper presents a new peak-finding algorithm used in GPS C/A code acquisition under indoor environments. The average correlation method for signal search was simulated in frequency domain. Three types of power-of-two-based FFTs,

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i.e., 1024-points, 4096-points, and 16368-points FFTs are implemented to evaluate the performances of output SNR and processing gain. The 4K-points FFT size can have the better performances between processing gain and computation speed. By including a longer integration time in each bin, the sensitivity can be increased to allow the indoor operation. The suitable dwell times for power level ranging from -120 dBm to -155 dBm have been simulated and summarized to the high-sensitivity acquisition process. ACKNOWLEDGMENT This research was supported by the Evermore INC., Hsinchu City, Taiwan, R. O. C. REFERENCES [1] [2]

[3]

[4] [5] [6] [7]

[8] [9]

[10] [11] [12]

[13]

[14]

[15]

F. van Diggelen and C. Abr a ha m:“ I ndo o rGPSTe c hno l o g y ” ,CTI A Wireless-Agenda, Dallas, May 2001. F. v a n Di g g e l e n:“ I ndo o rGPS t he o r y& i mpl e me nt a t i o n” :I EEE Position, Location & Navigation Symposium, vol.31, no.1, pp.240-247, 2002. J. Starzyk and Z. Zhu, “ Averaging correlation for C/A code acquisition and tracking in frequency domain” , MWSCS, Fairborn, Ohio, August, 2001. E. D.Ka pl a n:“ Unde r s t a ndi ng GPS:pr i nc i pl e sa nd Appl i c a t i o n” (Artech House, London, 1996.) P . Mi s r aa nd P . Eng e ,“ Gl o ba l po s i t i o ni ng s y s t e m: s i g na l s , me a s ur e me nt sa ndpe r f o r ma nc e s ” ,Ga ng a -Jamuna Press. J .B.Y.Ts ui :“ Funda me nt a l so fg l o ba lpo s i t i o ni ngs y s t em receivers, a s o f t wa r ea ppr o a c h”( J o hnWi l e y&So ns ,Ca na da ,2000. ) M. S. Br a a s c h, a ndA. J . Va nDi e r e ndo nc k, ” GPSr e c e i v e ra r c hi t e c t ur e a ndme a s ur e me nt ” ,Pr o c .o ft heI EEE. ,v o l . 87,no . 1,pp.48-64, Jan. 1999. Ralph Taylor and James Sennott, “ Na v i g a t i o nSy s t e ma ndMe t ho d” , United States Patent 4445118, NASA, Field May 22, 1981. M.S.Gr e wa l ,L.R.We i l la ndA.P .Andr e ws ,“ Gl o ba lpo s i t i o ni ng s y s t e ms ,i ne r t i a lna v i g a t i o na ndi nt e g r a t i o n” ,J o hnWi e l y&So nsI nc . , 2001. Ha r r yL.Va nTr e e s ,“ De t e c t i o n,e s t i mation, and modulation theory, Pa r t1” , Wi l e yI nt e rSc i e nc e ,2001. Mo ha me d Dj e bbo ur ia nd Dj a me lDj e bbo ur i :“ Fa s tGPS Sa t e l l i t e Si g na l Ac qui s i t i o n” ,El e c t r o ni c sLe t t e r s . c o m,no . 1/ 10/ 2003 A.J .Va n Di e r e ndo nc k,P .Fe nt o n,a nd T.Fo r d,“ The o r ya nd performance o fna r r o wc o r r e l a t o rs pa c i ng i n a GPS r e c e i v e r , ” Navigation: J. Inst. Navigation, vol. 39, no. 3, pp. 265–283, Fall 1992. A. J . Va nDi e r e ndo nc k, “ GPSr e c e i v e r s , ”i nGl o ba l Po s i t i o ni ngSy s t e m: Theory and Application, vol. I, B. W. Parkinson and J. J. Spilker, Jr., Eds. Washington, DC: American Institute of Aeronautics and Astronautics, 1996, ch. 8, pp. 329–407. P .Wa r d,“ Sa t e l l i t es i g na la c qui s i t i o na ndt r a c ki ng , ”i nUnde r s t a ndi ng GPS: Principles and Applications, E. Kaplan, Ed. Boston: Artech House, 1996, ch. 5, pp. 119–208. B.W.Pa r ki ns o n,a ndJ r .J .J .Spi l ke r :‘ Gl o ba lpo s i t i o ni ngs y s t e m: t he o r ya nda ppl i c a t i o ns ’( AAAI ,USA,1996. )

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