The sequel of this paper is organized as follows. In Section 2, the signal model for acquisition and tracking is described. The new TD correlation technique is ...
REDUCED-COMPLEXITY TIME-DOMAIN CORRELATION FOR ACQUISITION AND TRACKING OF BOC-MODULATED SIGNALS∗ Abdelmonaem Lakhzouri, Elena Simona Lohan, and Markku Renfors Institute of Communications Engineering, Tampere University of Technology P.O. Box 553, FIN-33101, Finland, Emails: {abdelmonaem.lakhzouri,elena-simona.lohan, markku.renfors}@tut.fi
Abstract Recent proposals for the modernized GPS and Galileo receivers suggested the use of a new modulation type, namely the Binary-Offset-Carrier (BOC) modulation, in order to get a better spectral efficiency and reduced interference compared to the traditional BPSK modulation. The acquisition and tracking of the code phase delay and Doppler shift is typically based on the correlation between the received signal and a reference code. The correlation can be performed in time-domain (TD) or in frequency-domain (FD). In this paper, we present a novel reduced-complexity TD correlation method. We compare its performance with two other TD correlation methods and with the traditional FD correlation method. Our new method takes into account the properties of the BOC waveform and re-uses the previously computed correlation values in such a manner to ensure the lowest number of additions and multiplications. The comparison will be carried out both in acquisition and tracking modes if long pseudorandom codes are employed, such as the 10230-chip length sequences in Galileo.
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I NTRODUCTION
CDMA receivers, such as Galileo and GPS receivers, spend a significant time in baseband processing for acquisition and tracking [1], [2]. This is due to the large number of required operations (e.g., additions and multiplications) to perform the correlation between the incoming signal and the replica code at the receiver with different candidate delays and Doppler shifts. There are two main methods to perform this correlation: either in Time Domain (TD) [3], or in Frequency Domain (FD) [2], [4]. In TD methods, a correlation value is computed for each code phase, usually in half-chip increments, until the full length of the pseudorandom (PRN) code is covered. The FD correlators employ FFT in order to search all the possible code phases in only one step. In both approaches, the procedure is repeated for all the possible Doppler shifts. The time-frequency search space depends on the receiver processing mode: in the acquisition mode (i.e., at cold start), the full PRN code length should be searched (i.e, 10230 chips for Galileo and modernized GPS receivers [5], [6]), while in the assisted-acquisition and in the tracking modes, only few tens to few hundreds of chips are usually enough. Recent proposals for Galileo and modernized GPS receivers suggested the use of a new modulation type, namely the Binary Offset Modulation (BOC) [6], [7] which triggers new challenges in the acquisition process, since the correlation waveforms are changed in the presence of BOC modulation. The features and properties of BOC modulated code sequences are still not well-understood in the context of fast acquisition algorithms. The goal of this paper is to analyze the TD and FD correlators for acquisition and tracking of BOC modulated PRN codes and to propose a new reduced-complexity TD correlation method which takes into account the properties of BOC signals. The sequel of this paper is organized as follows. In Section 2, the signal model for acquisition and tracking is described. The new TD correlation technique is presented in Section 3. Then, in Section 4, generic TD and FD correlation techniques are presented. Simulation results are shown in Section 5, and Section 6 gives the conclusions. ∗ This work was carried out in the project ”Advanced Techniques for Mobile Positioning” funded by the National Technology Agency of Finland (TEKES).
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2
T IME - DOMAIN CORRELATOR STRUCTURES
The acquisition and tracking modules are typically based on the correlation between the received signal and the replica code [1], [3]. The direct-form TD correlation can be expressed as y(τ ) =
1 Nc SF Ns NBOC
NX NBOC c SF NsX k=1
sref (l, k)rτ (l, k),
(1)
l=1
where y(τ ) is the correlation output with time lag τ , after coherent integration, N c is the number of coherent integration symbols, SF is the spreading factor or the number of chips per one coherent integration symbol, N BOC is the BOC modulation order (i.e., for BOC(n, m) modulation, NBOC , 2n/m [7]), Ns is the oversampling factor (i.e., the number of sub-samples per BOC sample), rτ (·) is the sequence of sub-samples of the received signal, starting with a time lag τ . Above, sref (·) is the reference PRN code, modulated via BOC and upsampled (i.e., having N s NBOC sub-samples per chip). The BOC waveforms, described in [7], and the real and imaginary parts of the PRN complex code are sequences of 1 and −1. Therefore, to perform the traditional time-domain correlation of eq. (1), only additions and sign inversions are required, but no multiplications.
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P ROPOSED REDUCED - COMPLEXITY TD APPROACH
The sum in eq. (1) can be computed iteratively, by re-using certain terms and by taking into account the sign alternations in the BOC waveform, in order to reduce the complexity. We define the term C τ0 (l) for l = 1, . . . , Ns NBOC as: Cτ0 (l) =
1 N c SF
NX c SF
cref (k)rτ0 (l, k).
(2)
k=1
where τ0 is the first time offset in sub-samples (e.g., τ0 = 0 if no assisted information about the time delay exists) and cref (·) is the PRN complex sequence at chip level (i.e., two real sequences of +1 and −1). Then, we can apply the following recursion: for τ = τ1 , . . . , τNmax (i.e., for each candidate delay) for l = 1, . . . , Ns NBOC (i.e., for each code shift within one chip) if l ≤ Ns NBOC − Nτ Cτ (l) = Cτ −Nτ (l + Nτ ) (i.e., re-use previously estimated correlations) else i.e., l > Ns NBOC − Nτ NX c SF Cτ (l) = cref (k)rτ (l, k) (i.e., compute k=1
end,
end.
a new correlation value)
where τNmax is the maximum time offset (in sub-samples). The block diagram of this method for N τ = 1 is shown in Fig. 1 (the generalization for Nτ > 1 is straightforward). Once the correlation values Cτ (l) are computed for every sub-sample within a chip l = 1, . . . , Ns NBOC and for every candidate code phase τ = τ1 , . . . , τNmax , we can build the overall correlation function y(τ ) according to the BOC waveform NsX NBOC 1 y(τ ) = w(l)Cτ (l). (3) Ns NBOC l=1
Here w(l) is a weight coefficient due to BOC modulation (it is either +1 or −1 [7]). To compute the number of multiplications and additions needed to perform the correlation, we notice that only additions and sign inversions are required, since the reference code and the weight coefficient w(l) are either 1 or −1. Moreover, only N τ cross-correlation terms are needed to be computed at each stage, because Ns NBOC − Nτ correlation terms have already been computed in the 2
previous stages. Therefore the number of additions per frequency bin and per coherent integration block for the proposed (Method 1) N (M 1) is given by: Dmax N (M 1) = Nτ (Nc SF − 1) + Ns NBOC − 1 × Ns NBOC − 1 + Ns NBOC (Nc SF − 1) Nτ (4) +Ns NBOC − 1. where Nτ is the time step in sub-samples, for example, if we scan the delay axis in increments (∆t) bin chips, then Nτ in sub-samples is given by Nτ = (∆t)bin Ns NBOC . (5) Given τ0 and τNmax , the maximum delay spread in sub-samples Dmax is defined as Dmax = (τNmax − τ0 )/(Ns NBOC ).
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(6)
G ENERIC CORRELATION TECHNIQUES
The proposed reduced-complexity TD approach is compared to 3 methods: • Method 2: Direct TD correlation: Using directly eq. (1) to generate the correlation output. The total number of additions per frequency bin is N (M 2) Dmax N (M 2) = Nc SF Ns NBOC − 1 × Ns NBOC , (7) Nτ • Method 3: Correlation at chip level: Here, we keep only one sub-sample per chip. In the absence of noise and delays, this will not affect the received signal, because all the sub-samples in one chip carry the same information (with a sign inversion at most). However, this downsampling below the Nyquist limit would tremendously affect the performance in the presence of noise and multipath delay. Therefore, this method can be kept only as a benchmark, in order to find out the lowest bound in the complexity of TD correlators. The output y chip (τ ) after coherent integration would become in this case y chip (τ ) = w(1)Cτ (1). (8) Here, the first sub-sample in each chip was considered, but the same holds if we used any of the sub-samples within one chip. The number of additions in this case is N (M 3)
= (Nc SF − 1) ×
Ns NBOC Dmax , Nτ
(9)
and, again, no multiplications are required, since the code chips are either +1 or −1. • Method 4: FD correlator structures: Here, the convolution in TD correlation is transformed to multiplication in FD. Therefore, for each code and signal symbol, we need 2 complex FFT blocks, 1 complex IFFT block and 1 complex multiplication. In order to test all the possible delays with a time lag Dmax , the length of one symbol (code or received signal) should be N = (SF + Dmax )NBOC Ns . (10) Typically, one complex FFT block of length N requires N log2 N complex multiplications and N/2log2 N complex additions. Then, the total number of complex multiplications and additions required for the FD correlation are: ( (M 4) Nc ⊗ = Nc (3N log2 N + N ) (11) (M 4) Nc ⊕ = 23 Nc N log2 N + (Nc − 1)N, Moreover, one complex multiplication is equal to 4 real multiplications and 2 real additions, and one complex addition is equal to two real additions. To compare the complexity of different correlation structures, we need to express the complexity of frequency-domain structure only in terms of real additions operations (the three other techniques have only additions). Therefore, we assume that the complexity of one real multiplication is α times higher than the complexity of one addition, where α is an integer higher than 1. Therefore, the number of equivalent additions for FFT-based correlation for N = (S F + Dmax )NBOC Ns is N (M 4) = 4αNc (3N log2 N + N ) + 9Nc N log2 N + 4N Nc − 2N, 3
(12)
Code DELAY
OFFSET
0
sample 0
x
sample 1
x
sample 2
x x
sample 3
SFNc
SFNc
SFNc
x w0
x
w1
+
x w2
x
x
sample Nchip
SFNc
....
w3
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rx
SFNc
x wNchip
x OFFSET
1
w0
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+
w1 w2
x x
sample Nchip
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LOOK FOR THE MAXIMUM
DELAY
wNchip -1
wNchip
DELAY
OFFSET
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w0
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+
wNchip-2
x x
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....... DELAY
x
NMAX
w0
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+
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x x x
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sample Nchip
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wNchip
Figure 1: Block diagram of the reduced-complexity time-domain correlation for BOC-modulated signal, N chip = Ns NBOC . 4
5
S IMULATION RESULTS
In this section, only the most significant simulation results are provided here. The simulation results showed that in all the cases the reduced complexity TD-correlation (Method 1) is less complex then the direct method (Method 2) and very close from the lowest bound complexity defined by Method 3. This conclusion is independent of the size of timefrequency uncertainty region (i.e., independent of being in acquisition or in tracking mode). The FD correlation method (Method 4) proved to be less complex than all the TD-correlation methods if we are in acquisition mode (i.e., the full code length has to be tested, see Fig. 2). However, if the tracking mode is active (i.e., only one part of the full code length has to be tested), Method 1 seems to have the best performance among all the tested techniques. The size of the uncertainty (the crossing point between FD and TD methods in Fig. 4), in this case, can be easily determined analytically and it is dependent of different parameters such as the factor α, the increments (∆t) bin , the oversampling factor Ns , and the BOC modulation order NBOC . N =4, N 12
10
s
=2, Max delay spread=10230 chips, time step=0.125 chips
BOC
11
Number of real additions
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Method 2, TD Direct method Method 1: TD reduced comp. (proposed) Method 3: TD Chip level (TD lower bound) Method 4: FD Correlation
8
10
7
10
0
20
40 60 Coherent integration time [ms]
80
100
Figure 2: Impact of the coherent integration length in cold start or acquisition mode: α = 2, N s = 4, NBOC = 2 Dmax = 10230 chips, (∆t)bin = 0.125 chips.
6
C ONCLUSIONS
In this paper, we have presented a novel time-domain correlation technique that can be used in acquisition and tracking for Galileo and modernized GPS signals. This technique takes into account the properties of BOC waveform adopted as modulation scheme for both systems (Galileo and modernized GPS). The main idea behind this correlation technique is to re-uses some of the previously computed correlation values to build the current one. simulation results proved that this method is less complex than the direct TD correlation technique and it is very close to the lowest complexity bound of TD correlation methods. Also, it has shown that this technique is less complex than the FD correlation in tracking and assisted acquisition modes.
References [1] Simon M. K., Omura J. K., Scholtz R. A., Levitt B. K., ”Spread Spectrum Communication Handbook”, McGrawHill, Inc., 1994. 5
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Ns=4, NBOC=2, Max delay spread=200 chips, time step=0.125 chips
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Number of real additions
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Method 2, TD Direct method Method 4: FD Correlation Method 1: TD Reduced comp. (proposed) Method 3: TD Chip level (TD lower bound)
7
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0
20
40 60 Coherent integration time [ms]
80
100
Figure 3: Impact of the coherent integration length in assisted acquisition and tracking mode: α = 2, N s = 4, NBOC = 2 Dmax = 200 chips, (∆t)bin = 0.125 chips.
12
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Ns=4, NBOC=2, Nc=50, time step=0.125 chips
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Number of real additions
10
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9
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Method 2, TD Direct method Method 1: TD Reduced comp. (proposed) Method 3: TD Chip level (TD lower bound) Method 4: FD Correlation
8
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1 multiplication = 8 additions
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6
10
0
2000
4000 6000 8000 Maximum delay spread [chips]
10000
Figure 4: Impact of the uncertainty region size: α = 8, Ns = 4, NBOC = 2 Nc = 50, (∆t)bin = 0.125 chips.
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[2] A. Alaqeeli, J. Starzyk, and F. van Graas, “Real-time acquisition and tracking for GPS receivers”, in Proc. of IEEE International Symposium on Circuits and Systems (ISCAS), vol. 4, pp. 500–503, May 2003. [3] E.D. Kaplan (ed.), Understanding GPS, Principles and applications, Artech House, Boston, 1996. [4] D.J.R. Van Nee and A.J.R.M. Coenen,“New Fast GPS code-acquisition technique using FFT”, Electronics Letters, vol. 27(2), pp. 158–160, Jan 1991. [5] G.W. Hein, J. Godet, J.L. Issler, J.C. Martin, T. Pratt, R. Lucas,”Status of Galileo Frequency and Signal Design”, in CD-ROM Proc. of ION GPS, 2002. [6] J.W. Betz, “Design and Performance of Code Tracking for the GPS M Code Signal”, MITRE Technical Papers, Sep 2000, Source: http://www.mitre.org/support/papers/tech papers99 00. [7] J.W. Betz, “The Offset Carrier Modulation for GPS modernization”, in Proc. of ION Tech. meeting, 1999, pp. 639– 648.
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