Fast M-Sequence Transform and Secondary Code ...

©2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other1works must be obtained from the IEEE.

Fast M-Sequence Transform and Secondary Code Constraints for Composite GNSS Signal Acquisition Daniele Borio1 , Member, IEEE

Abstract In this paper, the problem of coherently extending the integration time for the acquisition of new Global Navigation Satellite System (GNSS) signals is addressed. Unlike the acquisition of legacy GNSS signals, the presence of secondary codes allows the polarity of the transmitted signal to change each primary code period. These polarity changes have to be recovered and symbol combinations have to be tested before extending the coherent integration time. The hierarchical structure imposed by secondary codes and the presence of data/pilot channels are exploited to improve the acquisition process. A new algorithm, based on the fast m-sequence and Walsh-Hadamard transforms, is developed and used for efficiently testing all the possible symbol combinations. Secondary code constraints are included to further reduce the computational complexity of enumerating all symbol combinations. The proposed algorithms are analyzed in terms of Receiver Operating Characteristics (ROC) and an approximate expression for the probability of false alarm is derived exploiting results from Extreme Value Theory.

Index Terms Acquisition, Bit Estimation, Coherent Combining, Extremal Index, Global Navigation Satellite System, GNSS, m-Sequence Transform, QPSK, Receiver Operating Characteristic, ROC, Secondary Code, Walsh-Hadamard Transform.

1) Department of Geomatics Engineering, University of Calgary. 2500 University Dr NW Calgary, Alberta T2N 1N4 Canada. Email: [email protected]

ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

2

I. I NTRODUCTION The first stage of a Global Navigation Satellite System (GNSS) receiver is the acquisition block that aims at determining which satellites are in view and providing first rough estimates of the Doppler frequency and code delay of the received signals. This process usually consists of correlating the input samples with locally generated replicas of the signals transmitted by the different satellites. The correlated outputs are then processed using non-linear operators that allow the computation of the decision variable [1]–[3]. With the advent of new GNSS, new modulations [4], [5] have been introduced to provide more precise ranging information, simplify the bit synchronization process and increase the signal resilience against multipath and radio frequency (RF) interference. New GNSS signals are usually characterized by two components, namely the data and pilot channel, and by the presence of secondary codes that break the periodicity of the transmitted sequence, providing increased correlation properties and speeding up the bit synchronization process. The data/pilot structure of new GNSS signals has stimulated the development of combining strategies able to effectively integrate these two components and account for the presence of secondary codes. Non-coherent [6], semi-coherent [7] and coherent [8], [9] channel combining are just a few examples of the strategies developed for the joint acquisition of data and pilot channels. In [10], different combining strategies have been analyzed and expressions for the false alarm and detection probabilities derived when the coherent integration time is limited to one primary code period. In [11], the analysis is extended to different non-coherent techniques that allow the extension of the integration time over several code periods. However, [11] does not consider the case of coherent combining. Coherent combining is the optimal processing strategy, that provides the highest acquisition sensitivity [8] at the expenses of an increased computational load. Coherent combining requires the estimations of the relative signs between data and pilot channels and among subsequent periods of the incoming signals. This, in turn, requires testing all the possible symbol combinations that can occur between the two channels and for different code periods. In the following, the term “symbol” is used to denote an irreducible interval over which the polarity of the received signals is constant. Thus, a symbol models the combined effect of secondary codes and navigation message. The main focus of this paper is the development and analysis of efficient coherent combining


3

techniques for the acquisition of composite GNSS signals. The test of the different symbol combination is achieved by using specific combining sequences. This allows the use of efficient FFT-based algorithms for the parallel evaluation of the symbol combinations. It is noted, that the use of FFT-based techniques for signal acquisition is common in the GNSS context [12]–[14] for the parallel evaluation of all code delays and/or Doppler frequencies. This paper extends the use of FFT-based algorithms for the enumeration of the symbol combinations. Secondary codes impose additional constraints that effectively reduce the number of possible symbol combinations. In [15], the symbol combinations are evaluated by computing the evolutionary tree of the secondary code. At each step, the size of the tree doubles and different symbol combinations are tested by the different tree branches. [16] exploits the constraints imposed by the secondary code for progressively removing the most unlikely symbol combinations. None of those approaches exploit the efficiency of FFT-based algorithm. Moreover, [15] and [16] do not provide any indication about the process for fixing the decision threshold. All of the above aspects are considered in the paper and four cases are analyzed: •

exhaustive search: when no constraints are applied all the symbol combinations can be tested by correlating the input signal with a special sequence derived from maximum-length sequences (m-sequences) [17], [18];

•

single channel acquisition, data channel: the data secondary code and the navigation message are used to generate an extended code that allows one to account for the constraints imposed by the structure of the data channel;

•

single channel acquisition, pilot channel: the absence of a navigation message further limits the number of symbol combinations and this fact is exploited for the design of an FFT-based acquisition method;

•

dual channel acquisition: the data/pilot structure is accounted for in the design of an efficient FFT-based search method.

The proposed acquisition techniques are general and can be applied to the different modulations developed for the joint transmission of data and pilot components. More specifically, the proposed algorithms operate on the data/pilot correlator outputs, after primary code despreading, and thus do not depend on the modulation adopted for the transmission of the two components. For instance, the proposed methods directly apply to time-multiplexed signals, such as the GPS


4

L2C modulation, and their use is not limited to phase-quadrature transmissions. The exhaustive symbol search can also be applied to legacy GPS signals, for instance for extending the coherent integration time beyond the 20 ms bit duration of the GPS Coarse Acquisition (C/A) signal.

This paper provides a more detailed and complete analysis of the preliminary results presented in the conference paper [19]. In [19], combining algorithms were designed exploiting the property of cyclic codes. In this paper, a different approach is adopted. More specifically, it is shown that maximum length m-sequences allow the test of all symbol combinations in parallel, allowing the use of the fast m-sequence transform [18] for their evaluation. The equivalence between fast m-sequence and Walsh-Hadamard (WH) transforms [18] is exploited to further speed up the search process. This results is new and is one of the main contributions of the paper. The developed algorithms are analyzed in terms of Receiver Operating Characteristics (ROC) and an approximate expression for the probability of false alarm is derived using results from Extreme Value Theory [20]. This provides a simple criterion for fixing the decision threshold. Analytical results are supported by Monte Carlo simulations and the algorithms are demonstrated by means of real data from the experimental Galileo satellite, GIOVE-B. The remainder of this paper is organized as follows: Section II introduces the signal model and reviews the acquisition process. In Section III, a fast algorithm for the exhaustive search of the symbol combinations is derived whereas Section IV deals with the implementation of the secondary code constraints. The false alarm probability and the decision threshold is derived for the different techniques in Section V. In Section VI, the different integration techniques are further analyzed by means of Monte Carlo simulations and compared in terms of ROC. Some conclusions are drawn in Section VII. II. S IGNAL AND S YSTEM M ODEL The signal at the input of a GNSS receiver, in a one-path additive Gaussian noise environment, can be written as rRF (t) =

L p X

Ci yi (t) + ηRF (t)

(1)

i=1

that is the sum of L useful signals, emitted by L different satellites and with power Ci , and a noise term ηRF (t). ηRF (t) is a stationary Additive White Gaussian Noise (AWGN) with power ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

5

spectral density (psd) N0 /2. When considering composite GNSS signals with data and pilot components emitted with a 90 degree phase difference, yi (t) can be modeled as [4], [5]: yi (t) = eD,i (t − τ0,i ) cos (2π (fRF + f0,i ) t + φ0,i ) (2) + eP,i (t − τ0,i ) sin (2π (fRF + f0,i ) t + φ0,i ) where •

eD,i (t) and eP,i (t) are the data and pilot components;

•

τ0,i , f0,i and φ0,i are the delay, the Doppler frequency and phase introduced by the transmission channel on the ith signal;

•

fRF is the carrier frequency, i.e. 1176.45 MHz for Galileo E5a and GPS L5 and 1207.14 MHz for Galileo E5b.

In general the data and pilot components, eD,i (t) and eP,i (t), are given by the product of several terms eD,i (t) = di (t)sb,i (t)sD,i (t)cD,i (t) (3) eP,i (t) = sb,i (t)sP,i (t)cP,i (t) where di (t) is the navigation message, sbi (t) is the signal obtained by periodically repeating the subcarrier, sD,i (t) and sP,i (t) are the secondary codes or synchronization sequences for the data and pilot channels and cD,i (t) and cP,i (t) are the primary spreading sequences. The

di ( t )

sD , i ( t )

sb ,i ( t ) cD ,i ( t )

data

sb ,i ( t ) cP ,i ( t )

sP , i ( t )

navigation message secondary code subcarrier modulated primary code secondary code subcarrier modulated primary code

pilot

time

Fig. 1.

Schematic representation of a GNSS signal adopting a data/pilot structure. The data and pilot components are given

by the product of several terms including primary and secondary spreading sequences, the signal subcarrier and, in the case of the data channel, the navigation message.

structure of yi (t) is shown in Fig. 1 where the data and pilot components are transmitted on ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

6

two different channels. It is noted that the primary purpose of a secondary code is to speed up bit synchronization and, for this reason, the data secondary code, sD,i (t), and the navigation message, di (t), are time aligned and the duration of a secondary code period corresponds to a bit interval. The GNSS signal, yi (t), is filtered, downconverted, sampled at the frequency fs =

1 Ts

and

digitalized by the receiver front-end. These operations transform signal (1) into the sequence L p X rIF [n] = rIF (nTs ) = Ci e˜D,i (nTs − τ0,i ) cos (2π(fIF + f0,i )nTs + φ0,i ) i=1

+

L p X

(4)

Ci e˜P,i (nTs − τ0,i ) sin (2π(fIF + f0,i )nTs + φ0,i ) + ηIF (nTs )

i=1

where fIF is the receiver intermediate frequency, ˜· denotes the effect of front-end filtering and 2 . ηIF (nTs ) is a noise term supposed to be Gaussian, zero mean and with variance σIF

The quasi-orthogonality of the primary codes allows the separate analysis of the different GNSS signals. In this way, the following signal model can be adopted: √ rIF [n] =rIF (nTs ) = C e˜D (nTs − τ0 ) cos (2πF0 n + φ0 ) √ + C e˜P (nTs − τ0 ) sin (2πF0 n + φ0 ) + ηIF (nTs )

(5)

where the index i has been dropped and the quantity F0 = (fIF + f0 ) Ts introduced for ease of notation. A. Correlator outputs The input signal (5) is correlated with a delayed and modulated local code replica and a complex random variable, RX (FD , τ ), is produced. The index X denotes either the data or pilot channel and can assume the values D or P . In Fig. 2, the acquisition process is better described: the acquisition block tests different carrier frequencies, FD = (fIF + fd )Ts , that account for both intermediate and Doppler frequency, namely fIF and fd . When the signal is present, FD ≈ F0 and τ ≈ τ0 , the H1 hypothesis of correct signal alignment is verified and [6], [9]: RX (FD , τ ) =

√

C sin (πN ∆F ) dX R(∆τ ) exp(j∆φX ) + ηX 2 πN ∆F

(6)

where: ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

7

In-phase branch

1 N

cos (2π FD n )

∑

( ⋅)

I X (FD ,τ )

n=0

RX (FD ,τ )

τ

Frequency generator

rIF [n]

N −1

code generator

j

90°

− sin ( 2π FD n )

1 N

N −1

∑ ( ⋅) n=0

QX (FD ,τ )

Quadrature branch

Fig. 2. Coherent integration stage of a general acquisition system: the input signal is correlated with a delayed and modulated local code replica, producing the complex variable RX (FD , τ ).

•

R(·) is the cross-correlation between the local and the filtered incoming code;

•

∆F = F0 − FD is the difference between the frequency of the local carrier and of the incoming signal;

•

∆τ =

τ0 −τ Ts

is the difference between the local code delay and the delay of the incoming

code, normalized by the sampling interval, Ts ; •

∆φ is the difference between the phases of received and local carriers;

•

dX is a value in the set {−1, 1} that represents the effect of the navigation message and of the secondary code;

•

N is the number of samples used for the evaluation of the correlation between the incoming and the local signal. Tc = N Ts defines the coherent integration time and is assumed to be equal to the duration of a symbol, dX . Thus, the correlation (6) is evaluated over a time interval with constant dX ;

•

ηX is a complex Gaussian random variable obtained by processing the noise term in (5). The real and imaginary parts of ηX are independent and with equal variance σn2 .


8

The variance σn2 depends on the psd of the input noise, on the coherent integration time, on the type of filtering and on the sampling and decimation strategy adopted by the receiver front-end. A technique that can be used to practically estimate the variance σn2 is to correlate the input signal with a local code that is not broadcast by any satellite. When the signal is absent or not correctly aligned, the complex correlation, RX (FD , τ ), is made of noise only RX (FD , τ ) = ηX .

(7)

and the H0 hypothesis is verified. The basic task of any acquisition algorithm is to decide between the signal presence or absence. This decision is made by comparing a decision variable with a fixed threshold. K realizations of RX (FD , τ ) are obtained from subsequent portions of the incoming signal and from the two channels and stored in the vector R (FD , τ ) = [RX,1 (FD , τ ) , RX,2 (FD , τ ) , · · · , RX,K (FD , τ )]T .

(8)

In (8) the index X is not specified and the complex correlations in R (FD , τ ) can be evaluated from the data and pilot channel or from both. When both data and pilot channels are employed, the phase of the correlators from the pilot channel has to be corrected for accounting for the 90 degrees phase difference between the two signal components. In the following, proper phase alignment is assumed. An index has also been added to distinguish the different realizations of RX (FD , τ ). The complex correlations RX,i (FD , τ ) are evaluated over a symbol period, usually corresponding to a single primary code, and the correlation polarity, dX , can be different for each RX,i (FD , τ ). When using coherent combining [8], the integration time is extended by testing all the symbol combinations. In this way, new random variables are generated:    S1 (FD , τ ) RX,1 (FD , τ )     S (F , τ )   R (F , τ )  2 D   X,2 D S (FD , τ ) =  M ·  = |{z} . ..    .. .   Nc ×K  SNc (FD , τ ) RX,K (FD , τ )

      

(9)


9

where 

M1



  M 2  M= .  ..  MNc

     

is a combining matrix and Nc is the number of symbol combinations considered. Each row of M defines a symbol combination. For instance Mi = [1 − 1 − 1] leads to the random variable Si (FD , τ ) = RX,1 (FD , τ ) − RX,2 (FD , τ ) − RX,3 (FD , τ ) . When all the sign combinations are evaluated, the decision variable S (FD , τ ) is determined as [8]: S (FD , τ ) = max |Si (FD , τ )|2 . i

(10)

Eq. (10) is the high Signal-to-Noise Ratio (SNR) approximation to the optimal detection statistic determined in [8] and the signal presence is established by testing the condition S (FD , τ ) > β

(11)

where β is the decision threshold. β is usually obtained by fixing and inverting Pf a (β) = P (S (FD , τ ) > β|H0 )

(12)

that is the probability of false alarm [21], i.e., the probability of incorrectly declaring the signal presence. The false alarm and detection probability Pdet (β) = P (S (FD , τ ) > β|H1 )

(13)

characterize the acquisition performance and will be extensively analyzed in Section V and VI.


10

III. E XHAUSTIVE S EARCH In the exhaustive search all the possible symbol combinations between data and pilot channels and among different signal periods, are tested. It is noted that, when K correlator output are employed, 2K−1 valid combinations are possible. This is due to the fact that two combinations differing only by the absolute sign are equivalent. In this section a method for evaluating all the possible sign combinations by means of the FFT algorithm is detailed. The same algorithm can be implemented using the fast m-sequence and WH transforms [18]. In order to better illustrate the principle an example for K = 4 is at first presented. When K = 4, the input vector R (FD , τ ) has 4 elements and 8 symbol combinations are possible. These symbol combinations have to be chosen from the two sets in Table I. By opportunely TABLE I S ETS OF POSSIBLE SYMBOL COMBINATIONS . T HE TWO SETS DIFFER BY THE SIGN OF THEIR COMPONENTS .

−

+ 1

1

1

1

-1

-1

-1

-1

1

1

1

-1

-1

-1

-1

1

1

1

-1

-1

-1

-1

1

1

1

-1

-1

1

-1

1

1

-1

1

1

-1

1

-1

-1

1

-1

1

-1

1

-1

-1

1

-1

1

1

-1

1

1

-1

1

-1

-1

1

-1

-1

-1

-1

1

1

1

choosing the combining vectors Mi from Table I, it is possible to construct a string that contains all the possible combinations of 4 symbols. This process is illustrated in Table II where the string {1, 1, 1, 1, −1, −1, 1, −1} has been obtained by concatenating the symbol combinations such that two consecutive combinations have K − 1 = 3 common components. The scalar product between each symbol combination in Table I and the vector R (FD , τ ) can be rewritten in terms of the scalar product of the sequence {1, 1, 1, 1, −1, −1, 1, −1} and R (FD , τ ). Let cb = [1, 1, 1, 1, −1, −1, 1, −1]T

(14)


11 TABLE II S TRING OF MINIMUM LENGTH CONTAINING ALL THE POSSIBLE SYMBOL COMBINATIONS .

1

1

1

1

1

1

1

-1

1

1

-1

-1

1

-1

-1

1

-1

-1

1

-1

-1

1

-1

1

-1

1 1

1

1

1

1

1

1

1

-1 1

-1

-1

1

-1

be the vector containing the concatenated symbol combinations and ˜ (FD , τ ) = RT (FD , τ ) , 0, ..., 0 T R | {z }

(15)

2K−1 ×1

the vector with the signal components zero padded in order to have the same length as cb . Then the scalar product with the first symbol combination M1 = [1, 1, 1, 1] is given by ˜ (FD , τ ) . M1T · R (FD , τ ) = cTb · R The scalar product with the second symbol combination M2 = [1, 1, 1, −1] is given by ˜ (FD , τ ) , 1 M2T · R (FD , τ ) = cTb · cshift R ˜ (FD , τ ) by i positions. In general the where the operator cshift(·, i) circularly shifts the vector R scalar product with the ith symbol combination is given by T T ˜ Mi · R (FD , τ ) = cb · cshift R (FD , τ ) , i

(16)

Eq. (16) states that the components of the vector S (FD , τ ) are the samples of the circular ˜ (FD , τ ) that can be rapidly evaluated by means of FFT/IFFT pairs. correlation between cb and R ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

12

Combining sequence FFT

IFFT

S(FD ,τ ) R (FD ,τ )

(⋅)*

|⋅|2

FFT

max(⋅)

Correlation data

Frontend

Doppler frequency Code delay Correlation pilot

Fig. 3.

Zero padding

S (FD ,τ ) Decision variable

FFT-based symbol combination search: all symbol combinations are tested at once by using FFT/IFFT and circular

correlation.

In Fig. 3 the principle of this algorithm is depicted: for each delay τ and Doppler frequency FD , the vector R (FD , τ ) is evaluated and circularly correlated with the string of the concatenated symbol combinations by means of FFT/IFFT. The output of the circular correlation corresponds to the vector S (FD , τ ). The maximum of the correlation square modulus is the decision variable S (FD , τ ). This methodology can be used for each value of K by employing the appropriate symbol combination string. In Appendix A, it is shown that combining sequences of the type (14) can be obtained for an arbitrary K by mapping maximal length m-sequences from the binary values to {0, 1} to the real values {−1, 1}. In this way, the symbol combinations (9) can be evaluated by using the fast m-sequence transform [18]. The equivalence between fast m-sequence and WH transforms [18] can be exploited to further speed up the search process. In this way, vector (9) can be evaluated applying a single WH transform on sequence (15). The WH transform ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

13

can be modified to exploit the fact that the input vector is a zero padded sequence leading to Algorithm 1. It is noted that the proposed algorithm is multiplication free and does not require the evaluation and storing of the combining sequence. Input: The correlation vector R (FD , τ ) = [RX,1 (FD , τ ) , ..., RX,K (FD , τ )]T Output: The symbol combination vector S (FD , τ ) = [S1 (FD , τ ) , ..., SNc (FD , τ )] Initialization: •

i = 2(K−1) ;

•

Initialize the output vector with the correlation value:

size of a processing block (butterfly)

Sh (FD , τ ) = RX,1 (FD , τ ) for h ∈ {1, 2, ..., i}; •

k = 1;

•

Npb = 1;

index on the input data number of processing blocks per iteration

while i > 1 do i = i/2; k = k + 1; Npb = 2 · Npb ; for j ∈ {0, 2, ..., Npb − 2} do Sj·i+h (FD , τ ) = Sj·i+h (FD , τ ) + RX,k (FD , τ ) for h ∈ {1, 2, ..., i}; Sj·i+i+h (FD , τ ) = Sj·i+i+h (FD , τ ) − RX,k (FD , τ ) for h ∈ {1, 2, ..., i}; end end Algorithm 1: Modified fast WH transform for the evaluation of the different symbol combinations.

IV. S ECONDARY C ODE C ONSTRAINTS In the previous section, the decision statistic, S (FD , τ ), is obtained as the maximum of all possible symbol combinations, Si (FD , τ ). However, in new composite GNSS signals, both data and pilot channels are modulated by a secondary code. This implies that only a reduced number of symbol combinations can occur. Secondary codes limit the number of possible symbol combinations reducing the computational load required for the evaluation of the decision statistic. Different cases are considered and appropriate FFT-based algorithms are proposed in order to ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

14

efficiently test the symbol combinations allowed by secondary codes. In the following, only the case where the number of integrations, K, is lower than the secondary code duration is considered. When K equals the secondary code duration, primary and secondary code can be considered as a unique pseudo-random sequence and acquisition can be performed using the whole tiered code. In this case, testing the different symbol combinations is not required. To further extend the integration time, K should be set to an integer multiple of the secondary code duration and only the pilot channel should be used. Also in this case, the search for the different symbol combinations is not required. For these reasons, only the intermediate case where the coherent integration time is lower than the secondary code duration is considered. A. Single Channel Acquisition When considering the acquisition on a single channel, two cases are possible: •

acquisition on the data channel: data transitions can occur and the detection algorithm has to take into account the additional combinations allowed by the possible bit change at the boundary of the secondary code;

•

acquisition on the pilot channel: data transitions do not occur and no additional symbol combinations have to be accounted for.

When the data channel is considered, the number of combinations is bounded by Nc ≤ min 2K−1 , Nd + K − 1

(17)

where Nd is the length of the data secondary code. When K is low with respect to Nd , the secondary code allows all the possible symbol combinations and Nc is upper bounded by 2K−1 . In this case, the secondary code does not limit the number of combinations and the exhaustive search should be adopted. When K increases, not all the symbol combinations are allowed and Nc tends to be equal to Nd + K − 1. In the absence of bit transitions at maximum Nd combinations would be allowed; however, K − 1 additional combinations have to be considered for the bit transition. It is noted that Nd + K − 1 is only an upper bound, which does not account for possible repetitions of the same substring into the secondary code. However, due to the pseudo-random nature of secondary code, this bound becomes more and more accurate as K increases.


15

When the pilot channel is considered, the number of combinations is bounded by Nc ≤ min 2K−1 , Np

(18)

where Np is the length of secondary code. B. Data secondary code partial correlation When only a single channel is acquired, it is possible to jointly test all the possible bit combinations allowed by the secondary code and code delays by using a modified time-domain parallel acquisition algorithms [12]. A first solution is represented by constructing an extended secondary code that contains all the possible symbol combinations and circularly correlating it with a zero padded version of the input data block. The principle of this algorithm is illustrated in Fig. 4: the extended secondary code is formed by concatenating the three opportunely modulated portions of the secondary code. Only the first Nd +K −1 elements of the circular correlation are of interest and correspond to the Nd + K − 1 possible symbol combinations imposed by the secondary code. This acquisition scheme is similar to the one illustrated for the exhaustive search, where the combining sequence has been replaced by the extended secondary code. A second strategy consists in testing separately the cases in which a bit transition occurs or not. In Fig. 5 the extended secondary codes and the blocks of zero padded data are reported. At first all the Nd possible combinations without bit transition are tested. Then the K − 1 additional sequences, produced by a bit change, are analyzed by employing a second extended code. The maximum of the two circular correlations is taken as decision variable. C. Pilot Channel FFT-based algorithm The search for symbol combinations for the pilot channel is simplified by the absence of bit transitions. In this case a circular convolution with the pilot secondary code with a zero padded version of R (FD , τ ) allows the test of all possible symbol combinations. This algorithm corresponds to the first stage of the technique described in Fig. 5. D. Data/Pilot joint Acquisition When both data and pilot channels are employed, the input data block of K elements, can assume one of the different positions reported in Fig. 6. Because of the navigation message, the ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

16

Extended secondary code

S (FD ,τ )

FFT IFFT

R (FD ,τ )

(⋅)*

|⋅|2

FFT

max(⋅)

Single channel correlation

Front end

Doppler frequency Code delay

Zero padding

S (FD ,τ )

Reversed sign

Extended local code:

Decision variable

+

+

-

K -1

Nd

K -1

Last K – 1 elements of the secondary code

secondary code

First K – 1 elements of the secondary code

Zero-padded data: K blocks

0

Input samples

Fig. 4. FFT-based algorithm for symbol combination search: all symbol combinations are tested at once; only the first Nd +K −1 elements of the circular correlation are of interest and correspond to the Nd + K − 1 possible symbol combinations imposed by the data secondary code.

data secondary code can assume any sign value. Let H =

Np Nd

be the ratio between the pilot and

data secondary code lengths and assume that H is an integer number. When K is lower or equal than Nd it is possible to treat separately each segment in which the pilot code is divided by the data code. For each segment of the pilot secondary code, there are two possible cases depending if the input signal block crosses or not the data secondary code boundary. In the first case there are 2 (Nd − K + 1) possible combinations since the input data block can assume Nd − K + 1 different delays without crossing the boundary of the data secondary code. The factor 2 is due to the fact that the sign of the data secondary code can be either positive or negative. When


17

Case I + Nd

Secondary code

K

0

Zero padded input data

Circular correlation Test of Nd possible combinations

Case II +

-

K -1

K -1

K

0

Local replica: last K-1 secondary code elements followed by the first K – 1 secondary code elements with reversed sign Zero padded input data

Circular correlation: only the first K – 1 correlation samples are of interest. Test of K - 1 combinations not considered in the first case.

Fig. 5. FFT-based algorithm for symbol combination search: the cases with and without bit transition are dealt separately. The maximum of the correlations in the two cases is taken as decision variable.

the input data block crosses the data secondary code boundary there are 4(K − 1) possible combinations: the factor 4 is due to the different signs that the two consecutive portions of data secondary code can assume with respect to the pilot channel. K − 1 is the number of delays that make the input signal block cross the secondary code boundary. In this way, when considering both data and pilot channel and K ≤ Nd there are 2H(Nd + K − 1)

(19)

possible substrings into the combined data and pilot sequences. The number of possible substrings represents an upper bound for the number of possible symbol combinations that can occur when considering data and pilot combining. In particular the following condition holds: Nc ≤ min 22K−1 , 2H(Nd + K − 1) .

(20)

As K increases, the probability of finding two equivalent substrings tends to zero and the number of symbol combinations equals 2H(Hd + K − 1). In [8] the case of the GPS L5 signal was considered, and it was proven that, for Nd = 10, Np = 20 and K = 10, then the optimal detector


18

has to test Nc = 76 combinations. Eq. (20) can be considered an extension of this result, in fact: Nc = 2H(Nd + K − 1) = 2

20 (10 + 10 − 1) = 4 · 19 = 76. 10

The combined data and pilot acquisition can be performed by evaluating the extended secondary Nd

Nd

Nd

Np K -1

Nd

K -1 K -1

Nd

K -1 K -1

Fig. 6.

Nd

K -1

Data/pilot coherent combining: the extended data and pilot secondary code correlations are evaluated separately and

the two relative signs are tested for all possible data/pilot alignments.

code correlations separately and testing the two relative symbol combinations for all possible data/pilot alignments. The principle of the algorithm is illustrated in Fig. 6 where the data and pilot secondary correlations are evaluated separately and the two relative symbol combinations are tested for all possible data/pilot alignments. V. FALSE A LARM P ROBABILITY AND T HRESHOLD S ETTING In this section, an approximate formula for the false alarm probability for the detector (11) is provided. Under the H0 hypothesis, the correlator outputs consist of noise only and the random N

c variables {|Si (FD , τ ) |2 }i=0 are identically distributed.

It is noted that the decision variable, S (FD , τ ), is given by the general form (10), i.e. it is the N

c maximum of the set of random variables {|Si (FD , τ ) |2 }i=1 . Although the complex correlations,

RX,i (FD , τ ), are independent, the symbol combinations Si (FD , τ ) are not, since the correlation matrix CS = E S(FD , τ )SH (FD , τ ) = ME R (FD , τ ) RH (FD , τ ) MH = 2σn2 MMH

(21)


19

is not, in general, diagonal. Correlation (21) prevents the evaluation of an exact expression for the false alarm probability. The correlation among the elements of S(FD , τ ) implies that the random variables |Si (FD , τ )|2 are not independent. Moreover, CS is not, in general, Toeplitz Nc implying that |Si (FD , τ )|2 i=1 does not form a stationary random field. This prevents the use of classical results for the envelope of Gaussian random fields [22], [23]. A different approach, making use of the extremal index [20], is adopted here. If the |Si (FD , τ )|2 were independent, the false alarm probability (12) could be easily evaluated as [20] Pf a (β) = P max |Si (FD , τ )|2 > β = 1 − P max |Si (FD , τ )|2 < β i

i

=1−

Nc Y

Nc P |Si (FD , τ )|2 < β = 1 − P |Si (FD , τ )|2 < β

(22)

i=1

Nc = 1 − 1 − P |Si (FD , τ )|2 > β . Since |Si (FD , τ )|2 are not independent, a different approach is usually taken [20]. More specifically, it is noted that in the presence of correlation, the random variables |Si (FD , τ )|2 tend to cluster and the number of variables effectively impacting the maximum selection in (11) is lower than Nc . In this way, the following model is adopted [20]: θNc Pf a (β) ≈ 1 − 1 − P |Si (FD , τ )|2 > β

(23)

where θ < 1 is the extremal index [20] and Nef f = θNc is the number of random variables effectively impacting the maximum selection. Since Si (FD , τ ) are complex Gaussian random variables with independent real and imaginary components, |Si (FD , τ )|2 are χ2 distributed with 2 degrees of freedom [24]. Under H0 , Si (FD , τ ) are zero mean and β 2 P |Si (FD , τ )| > β|H0 = exp − . 2Kσn2

(24)

In this way, (23) becomes Pf a (β) ≈ 1 − 1 − exp −

β 2Kσn2

θNc .

(25)

Eq. (25) can be further simplified when the case of β 0 is considered. This condition is usually true, since a large β implies a low false alarm probability. When this assumption holds, (25) becomes

β Pf a (β) ≈ θNc exp − 2Kσn2

.

(26)


20

The determination of Nef f and thus of θ is, in general, a complex problem and some results are available for stationary random fields [20], [22], [23]. As already highlighted. the set |Si (FD , τ )|2 does not define a stationary field and the results from [22] do not apply. For this reason a Monte Carlo based approach has been adopted. In particular an estimate of the false alarm probability, Pˆf a (β), has been obtained by means of Monte Carlo simulations and Nef f obtained by solving

2

min Pˆf a (β) − Pf a (β) Nef f

10−2 ≤Pˆf a (β)≤10−4

ˆ

min Pf a (β) − Nef f exp − Nef f

2

β

. 2Kσn2 10−2 ≤Pˆf a (β)≤10−4

(27)

In this way, Nef f is the value that makes the theoretical model (26) fit Monte Carlo simulations the best. The range [10−2 − 10−4 ] has been chosen for the determination of Nef f because it represents typical values for the required false alarm probability. Eq. (26) can be easily inverted, providing a simple way for setting the decision threshold β: Pf a . (28) β = −2Kσn2 log Nef f In the following the results relative to the extremal index for the different combining strategies are reported and the validity of (26) discussed. A. Exhaustive search combining In this section, the value of Nef f is evaluated for the exhaustive search as a function of the number of integrations, K. More specifically, Nef f has been evaluated for K = 1, 2, .., 10 and the results are reported in Table III. In this case, Nef f has been rounded to the closest integer. It is noted that for K = 1, 2, the random variables |Si (FD , τ )|2 are independent [10], (25) is not an approximation and the values of Nef f are exact. From Table III, it emerges that Nef f grows exponentially as a function of K. More specifically, by linearly interpolating the natural logarithm of Nef f it is possible to find the following relationship: log Nef f = 0.6K − 0.42

(29)

and the false alarm probability (25) can be further rewritten as β . (30) Pf a (β) = exp {0.6K − 0.42} exp − 2Kσn2 In Fig. 7, results from Monte Carlo simulations are shown. The probability of false alarm obtained by means of Monte Carlo simulations is compared with the analytical approximation (30), showing its validity over the range [10−2 ; 10−4 ]. ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

21 TABLE III Nef f

10

FOR THE EXHAUSTIVE SEARCH COMBINING STRATEGY.

K

Nef f

2K−1

K

Nef f

2K−1

1

1

1

6

25

32

2

2

2

7

49

64

3

4

4

8

84

128

4

8

8

9

150

256

5

14

16

10

265

512

-2

Monte Carlo Simulations Analytical Approximation

K=7

K=2

K=8

K=3

K=9

K=4

K = 10

K=5 False Alarm Probability

K=6

Fig. 7.

10

-3

10

-4

0

50

100

150 Decision Threshold

200

250

300

Probability of false alarm as a function of the decision threshold, β, and the number of integrations, K. The post-

correlation variance, σn2 , has been normalized to 1. Monte Carlo simulations are compared with the analytical approximation (30), proving its validity.


22

B. Secondary Code Constraints: Data Channel In this section and in V-C and V-D, results of the fitting procedure (27) are briefly discussed when the constraints imposed by data and pilot secondary codes are implemented. Although specific focus is devoted to the E5a and E5b secondary codes, similar results can be obtained for other secondary codes. In the E5a case, the data and pilot secondary codes are 20 and 100 chip long, respectively. The E5b data signal is modulated by a shorter secondary code of length equal to 4 chips. The E5b pilot channel is 100 chip long [4]. Nef f for the Galileo E5a data channel is reported as a function of the number of integrations in Table IV. It is noted that Nef f is always less or equal than the number of symbol combinations allowed by the secondary code. For this reason, when the data channel is considered, Nef f is bounded by Nef f ≤ Nc ≤ min 2K−1 , Nd + K − 1 .

(31)

This condition is respected by the results reported in Table IV. In Fig. 8, Monte Carlo simulations TABLE IV N UMBER OF RANDOM VARIABLES EFFECTIVELY IMPACTING THE MAXIMUM SELECTION , Nef f , NUMBER OF INTEGRATIONS WHEN THE CONSTRAINTS OF

AS A FUNCTION OF THE

E5 A DATA SECONDARY ARE ACCOUNTED FOR .

K

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Nef f

2

4

8

11

13

18

20

24

26

29

28

28

26

31

28

32

35

36

33

are compared with model (26) when the constraints imposed by the data secondary code are used to reduce the number of symbol combinations. A good agreement is obtained between simulated and analytical results. C. Secondary Code Constraints: Pilot Channel The E5a and E5b pilot channels are characterized by 50 different secondary codes of length 100 chips. In Fig. 9, the results for Nef f are reported when these secondary codes are used. Although the different codes leads to different Nef f , a common trend can be observed. Nef f progressively increases as K, the number of integrations, increases. Moreover, Nef f saturates on the upper bound Nef f ≤ Nc ≤ min(2K−1 , Np ).

(32)


False Alarm Probability

23

10

-2

10

-3

E5a data secondary code Monte Carlo Simulations Analytical Approximation

K=2 K=4 K=6 K=8 K = 10 K = 12 K = 14 K = 16

K = 18 K = 20

10

Fig. 8.

-4

0

50

100

150

200 250 300 Decision Threshold

350

400

450

500

Probability of false alarm as a function of the decision threshold, β, and the number of integrations, K, when the

constraints imposed by the E5a data secondary code are used to reduce the number of symbol combinations. The post-correlation variance, σn2 , has been normalized to 1.

On both the E5a and E5b channel, the approximation Nef f ≈ Np = 100 holds for K > 25. D. Data and Pilot Combined Acquisition A similar analysis has been performed when both data and pilot channel constraints are used. The E5b channel has not been considered because of the reduced length of the data channel, Nd = 4. Nef f is reported in Fig. 10 as a function of K and also in this case a common trend ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

24 E5a Secondary Pilot Codes 100

Neff

80

a)

60 40 Mean over the 50 Secondary Codes Upper Bound Evelope for the 50 Secondary Codes

20 0

5

10

15

20 K

25

30

35

40

E5b Secondary Pilot Codes 100

Neff

80 60

b)

40 Mean over the 50 Secondary Codes Upper Bound Evelope for the 50 Secondary Codes

20 0

Fig. 9.

5

10

15

20 K

25

30

35

40

Number of random variables effectively impacting the maximum selection, Nef f , as a function of the number of

coherent integrations, K. The decision variable is obtained considering the pilot channel alone. a) Galileo E5a pilot secondary codes. b) Galileo E5b pilot secondary codes.

among the 50 pilot codes can be observed. Nef f progressively increases respecting the condition Nef f ≤ Nc ≤ min 22K−1 , 2H(Nd + K − 1) .

(33)

VI. S IMULATION AND R EAL DATA A NALYSIS In this section, Monte Carlo simulations are used for assessing the performance of the considered acquisition algorithms. Real data are also employed for demonstrating the feasibility of the different acquisition techniques.


25

E5a secondary code

350

300

Neff

250

200

150

100

50

0

Fig. 10.

Mean over the 50 Secondary Codes Upper Bound Evelope for the 50 Secondary Codes 2

4

6

8

10

K

12

14

16

18

20

Number of random variables effectively impacting the maximum selection, Nef f , as a function of the number of

coherent integrations, K. The decision variable is obtained coherently combining the data and pilot channel. Galileo E5a signal.

A. Monte Carlo Simulations Monte Carlo simulations are used for estimating the Receiver Operating Characteristics (ROC) [21] that are the plot of the detection probability as a function of the false alarm rate. ROC provides a statistical characterization of the acquisition performance allowing comparative analysis of the different algorithms [21]. A GNSS signal characterized by the parameters in Table V has been simulated and used for testing the different coherent combining techniques. The simulation parameters are the same as those employed in [10] and [11] for the analysis of single period and non-coherent acquisition algorithms. The coherent integration time, Tc , is equal to 1 ms and corresponds to the primary ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

26 TABLE V S IMULATION PARAMETERS .

Parameter

Value

Sampling frequency, fs

40.92 MHz

BIF = fs /2

20.46 MHz

Intermediate frequency, fi,E5 = fs /4

10.23 MHz

Code length N

10230 chip

Pre-detection integration time

1 ms

Samples/chip

4

Length of the data secondary code, Nd

20 chips

Length of the pilot secondary code, Np

100 chips

code duration of the Galileo E5a and E5b signals. Thus, the total integration time is equal to K ms. False alarm and detection probabilities have been obtained using error counting techniques and 5 · 105 trials have been used for each probability value. Detection probabilities are evaluated under the hypothesis of perfect code and carrier alignment, i.e., τ and FD match the code delay and Doppler frequency of the input signal. Moreover, only the single cell probabilities [25], (12) and (13), are considered. The ROCs for the data and pilot channel combining and the exhaustive search are evaluated considering the correlator outputs from a single channel. In Fig. 11, the case of K = 5 integrations is considered. It is noted that combining on data and pilot channel and exhaustive search have the same performance. This is due to the fact that, for K = 5, secondary codes do not provide any constraints on the possible symbol combinations. Thus, the three algorithms are equivalent. Joint data/pilot combining provides increased detection capabilities since useful power is recovered from both channels. The ROC obtained for non-coherent combining on a single channel is also reported as a comparison term. Coherent combining always outperforms its non-coherent counterpart although, for low C/N0 , the performance of the two classes of algorithms converges. This is in agreement with the results obtained in [8], [11] and reflects the fact that for low C/N0 the symbol estimation process becomes unreliable. In Fig. 12, the case of K = 10 is shown. Secondary codes reduce the number of symbol combinations to be tested and the exhaustive search has slightly lower detection performance. The case of K = 15 is considered in Fig. 13. The ROC for the exhaustive search is not shown since, for K = 15, Nc = 214 = 16384 bit combinations should be considered. ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

27

0

10

-1

10

-2

Detection Probability

10

K=5

C/N0 = 35 dB-Hz

C/N0 = 30 dB-Hz

C/N0 = 25 dB-Hz Data channel Pilot channel Exhaustive search Data&Pilot Non-coherent Combining

-3

10 -4 10

Fig. 11.

10

-3


10

-2

Receiver Operating Characteristic (ROC) for the different acquisition algorithms and for different C/N0 . Number of

integrations K = 5. Non-coherent combining is reported as comparison term.

This makes the exhaustive search unfeasible. As K increases the differences between coherent and non-coherent combining becomes more significant showing the advantage of first type of processing. From Figs. 12 and 13, it emerges that signal detection on the data channel has slightly better performance than on the pilot component. This can be explained by the use a shorter secondary code on the data channel. The data secondary code is five times shorter than the one on the pilot channel, resulting in a lower number of symbol combinations to be tested and thus in better acquisition performance.


28

10

K = 10

0

C/N0 = 35 dB-Hz

-1


10

C/N0 = 30 dB-Hz

10

-2

C/N0 = 25 dB-Hz

10

Fig. 12.

Data channel Pilot channel Exhaustive search Data&Pilot Non-coherent Combining

-3

10

-3


10

-2



B. Real Data Testing In this section the proposed algorithms are demonstrated by using real data. The sample results reported in the following are intended to show the feasibility of the acquisition schemes considered in the paper and do not provide any statistical characterization of the different techniques. The acquisition algorithms have been validated using live data from the second Galileo In-Orbit Validation Element, GIOVE-B. GIOVE-B is currently transmitting signals modulated according to [26], slightly differing from the final structure that will be adopted by the Galileo system


29

10

K = 15

0

C/N0 = 35 dB-Hz

-1


10

C/N0 = 30 dB-Hz

10

-2

C/N0 = 25 dB-Hz Data channel Pilot channel Data&Pilot Non-coherent Combining

-3

10 -4 10

Fig. 13.

10

-3


10

-2



[4]. The GIOVE-B signal has been collected by a NI PXI-5661 signal analyzer [27] using real sampling with sampling frequency fs = 50 MHz. The E5a modulation parameters and the NI PXI-5661 characteristics have been summarized in Table VI. The GIOVE-B E5a signal has been successfully acquired using the different techniques described in the paper and a C/N0 = 38 dB-Hz has been estimated. The normalized ambiguity functions for the different acquisition schemes are shown in Fig. 14 where K = 6 integrations have been used. The non-coherent combining strategy for the data and pilot channel has also been reported as comparison term. The parameters used for the acquisition process, such as the delay and Doppler bin size, are reported in Table VI. As expected, coherent combining provides better noise rejection and the signal peak ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

30 TABLE VI F RONT- END CHARACTERISTICS AND E5 A MODULATION PARAMETERS

Parameter

Value

Sampling frequency

50 MHz

Front-end intermediate frequency

8.45 MHz

Front-end bandwidth

22 MHz

Sampling type

real

E5a center frequency

1176.45 MHz

E5a chip rate

10.23 Mcps

Primary code length

10230 chips

Data secondary code duration

20 ms / 20 chips

Pilot secondary code duration

100 ms / 100 chips

Estimated C/N0

38 dB-Hz

Fig. 14. Normalized ambiguity functions for the different acquisition techniques. GIOVE-B E5a signal with K = 6 integrations.

clearly emerges from the noise floor. When the integration time is coherently increased the signal peak is narrowed along the Doppler dimension and a smaller frequency step is required for the


31

acquisition search. When both data and pilot channels are coherently combined, the noise level is further reduced at the expense of an increased computational load. The different acquisition strategies have been implemented in MatlabTM and the ambiguity functions in Fig. 14 computed using a 2.66 GHz Intel Centrino Core 2 Duo processor. The time requirements for the evaluation of the different ambiguity functions are reported in Table VII. The first row in Table VII indicates the time required for the evaluation of the K = 6 ambiguity functions before combining. A constant time of about 4.23 s is required for the computation of the ambiguity functions on a single channel. The second row indicates the time required for combining the correlator outputs and extend the integration time. When the number of correlators increase from 6 to 12, i.e., when moving from the single to the dual channel case, the computational load increases substantially, showing that the number of combinations to be tested is the main limiting factor for this type of acquisition algorithms. TABLE VII T IME REQUIREMENTS FOR THE COMPUTATION OF THE DIFFERENT AMBIGUITY FUNCTIONS IN F IG . 14

Doppler removal

Non-coherent

Exhaustive Search

Exhaustive Search

Exhaustive Search

Combining

Data Channel

Pilot Channel

Data&Pilot

8.46 s

4.26 s

4.23 s

8.42 s

1.17 s

4.04 s

4.03 s

234.6 s

and Correlation Combining time

VII. C ONCLUSIONS In this paper four strategies for coherently extending the integration time over several primary code periods for the acquisition of composite GNSS signals have been proposed and analyzed. The integration time is coherently extended by testing all possible symbol combinations. The properties of maximal length sequences have been used for the design of a fast algorithm that searches for all possible symbol combinations. The constraints imposed by secondary codes have also been exploited to reduce the number of admissible symbol combinations. Each method has been characterized from a statistical point of view and approximated expressions for the false alarm probabilities have been found exploiting results from extreme value theory. From the developed analysis, it emerges that secondary code constrains can significantly reduce the ACCEPTED FOR PUBLICATION ON IEEE T RANS . ON A EROSPACE AND E LECTRONIC S YSTEM , N OVEMBER 2009

32

computational load associated to the search for the different symbol combinations. As expected, coherent processing outperforms its non-coherent counterpart at the expense of an increased computational load. A PPENDIX A P ROPERTY OF BINARY M - SEQUENCES In this appendix, it is shown that maximal length binary m-sequences can be used for generating the combining sequence (14) for arbitrary K. More specifically, it is shown that a binary m-sequence, m, ¯ of length 2K−1 − 1 contains 2K−1 − 1 subsequences of length K such that if s¯ is a subsequence of m ¯ then the complementary of s¯, s¯c , is not a subsequence of m. ¯ In this context, two binary sequences are complementary if they sum to the constant sequence of all 1s. This implies that the elements of s¯c are obtained by negating the digits of s¯. The sequences of all 0s and 1s and length K, are not subsequences of m. ¯ The following properties are used for the proof; the m-sequence, denoted by m, ¯ is assumed of length 2K−1 − 1. 1) m ¯ contains all sequences of length K − 1, apart from the sequence of K − 1 consecutive 0s [17]. Each substring of length K − 1 is unique in m; ¯ 2) the longest sequence of all 1s in m ¯ has length K − 1. This property follows directly from the previous one. If there were a subsequence of all 1s and length K, there would be two identical subsequences of length K − 1, thus negating Property 1; 3) the sequences obtained by cyclically shifting m, ¯ are m-sequences that form, with the sequence of all zeros and length 2K−1 − 1, a vector space. The operations of addition and multiplication by a constant are defined with respect to the binary Galois field, GF(2) [17]. The vector space generated by m, ¯ is denoted by M. The proof is made by contradiction and the condition that s¯ and s¯c are both substrings of m ¯ is assumed to be true. By cyclically shifting m ¯ of L positions, the sequence m ¯ L is obtained. L is chosen such that s¯c in m ¯ L is aligned with s¯ in m. ¯ The sequence m ¯H = m ¯ +m ¯L

(34)

is a cyclically shifted version of m, ¯ because of Property 3. Moreover, m ¯ H contains the substring o¯ = s¯c + s¯,

(35)


33

that is a sequence of K consecutive 1s. This is due to the fact that s¯c and s¯ are complementary sequences of length K that have been aligned for the construction of m ¯ H . This also proves that m ¯ H is not the zero sequence and thus, because of Property 3, must be a cyclically shifted version of m. ¯ This, in turn, implies that m ¯ has a substring of length K of all 1s, in contrast with Property 2. This leads to a contradiction that implies that s¯c and s¯ cannot be substring of m ¯ at the same time. This implies that all the binary combinations with the exception of the constant string of all 1s are included present in m. ¯ The constant subsequence of K 1s can be included in the m-sequence, by extending by one element the longest subsequence of 1s already contained in m. ¯ For example, let be m ¯ = [1, 1, 1, 0, 0, 1, 0]

(36)

an m-sequence of length 24−1 − 1 = 7, then m∗ = [1, 1, 1, 1, 0, 0, 1, 0]

(37)

is the extended m-sequence containing 2K−1 = 8 subsequences of length K = 4, such that, if s¯ is in m∗ , s¯c is not. The combining sequence (14) is obtained by transforming the binary string m∗ using the following correspondence GF(2)

R

0

⇔ +1

1

⇔ −1.

(38)

R EFERENCES [1] E. D. Kaplan and C. J. Hegarty, Eds., Understanding GPS: Principles and Applications, 2nd ed.

Norwood, MA, USA:

Artech House Publishers, 2005. [2] J. B.-Y. Tsui, Fundamentals of Global Positioning System Receivers: A Software Approach.

Wiley-Interscience, May

2000. [3] P. Misra and P. Enge, Global Positioning System: Signals, Measurements and Performance. Ganga-Jamuna Pr, Dec. 2001. [4] “Galileo open service signal in space interface control document,” European Space Agency / Galileo Joint Undertaking, Draft GAL OS SIS ICD/D.0, May 2006. [5] “Interface specification NAVSTAR GPS space segment / navigation L5 user interfaces,” ARINC Incorporated, Tech. Rep. IS-GPS-705 (2005), Sept. 2005. [6] F. Bastide, O. Julien, C. Macabiau, and B. Roturier, “Analysis of L5/E5 acquisition, tracking and data demodulation thresholds,” in Proc. of ION GPS/GNSS, Portland, OR, Sept. 2002, pp. 2196 – 2207.


34

[7] C. Yang, C. Hegarty, and M. Tran, “Acquisition of the GPS L5 signal using coherent combining of I5 and Q5,” in Proc. of ION GNSS, 17th International Technical Meeting, Long Beach, CA, Sept. 2004, pp. 2184 – 2195. [8] C. J. Hegarty, “Optimal and near-optimal detector for acquisition of the GPS L5 signal,” in Proc. of ION NTM, National Technical Meeting, Monterey, CA, Jan. 2006, pp. 717 – 725. [9] C. Hegarty, M. Tran, and A. J. Van Dierendonck, “Acquisition algorithms for the GPS L5 signal,” in Proc. of ION/GNSS, Portland, OR, Sept. 2003, pp. 165 – 177. [10] D. Borio, C. O’Driscoll, and G. Lachapelle, “Coherent, non-coherent and differentially coherent combining techniques for the acquisition of new composite GNSS signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 3, pp. 1227–1240, July 2009. [11] ——, “Composite GNSS Signal Acquisition over Multiple Code Periods,” IEEE Trans. Aerosp. Electron. Syst., accepted for publication, Sept. 2008. [12] D. J. R. V. Nee and A. J. R. M. Coenen, “New fast GPS code-acquisition technique using FFT,” Electronics Letters, vol. 27, no. 2, pp. 158 – 160, Jan. 1991. [13] C. Yang, “FFT acquisition of periodic, aperiodic, puncture, and overlaid code sequences in GPS,” in Proc. ION GPS, Salt Lake City, UT, Sept. 2001, pp. 137 – 147. [14] D. Akopian, “Fast FFT based GPS satellite acquisition methods,” IEE Proc. Radar Sonar Navig., vol. 152, no. 4, pp. 277 – 286, Aug. 2005. [15] G. E. Corazza, C. Palestini, R. Pedone, and M. Villanti, “Galileo primary code acquisition based on multi-hypothesis secondary code ambiguity elimination,” in Proc. of ION/GNSS 20st International Technical Meeting of the Satellite Division, Fort Worth, TX, Sept. 2007, pp. 2459–2465. [16] N. Shivaramaiah, A. Dempster, and C. Rizos, “Exploiting the secondary codes to improve signal acquisition performance in galileo receivers,” in Proc. of ION/GNSS 21st International Technical Meeting of the Satellite Division, Savannah, GA, Sept. 2008, pp. 1497–1505. [17] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge University Press, July 2005. [18] M. Cohn and A. Lempel, “On fast m-sequence transforms,” IEEE Trans. Inform. Theory, vol. 23, no. 1, pp. 135–137, Jan. 1977. [19] D. Borio, “FFT sign search with secondary code constraints for GNSS signal acquisition,” in Proc. of IEEE 68th Vehicular Technology Conference (VTC 2008-Fall), Calgary, AB, Canada, Sept. 2008, pp. 1–5. [20] R.-D. Reiss and M. Thomas, Statistical Analysis of Extreme Values: with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd ed.

Basel, Switzerland: Birkhauser, July 2007.

[21] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume II: Detection Theory, 1st ed. Prentice Hall PTR, Mar. 1998. [22] R. Hill, R. Tough, and K. Ward, “False alarm curve for envelope of Gaussian random field,” IEE Electronics Letters, vol. 37, no. 4, pp. 258 – 259, Feb. 2001. [23] ——, “Distribution of the global maximum of a Gaussian random field and performance of matched filter detectors,” IEE Proc. Vision, Image and Signal Processing, vol. 147, pp. 297 – 303, Aug. 2000. [24] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed.

McGraw Hill Higher

Education, Jan. 2002.


35

[25] D. Borio, L. Camoriano, and L. Lo Presti, “Impact of GPS acquisition strategy on decision probabilities,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 3, pp. 996–1011, July 2008. [26] Galileo Project Office, “GIOVE-A + B navigation signal-in-space interface control document,” European Space Agency, First issue ESA-DTEB-NG-ICD/02837, Aug. 2008. [27] 2.7

GHz

RF

Vector

Signal

Analyzer

with

Digital

Downconversion,

National

Instruments,

http://www.ni.com/pdf/products/us/cat vectorsignalanalyzer.pdf, 2006.
