Robust Synchronization of Weak GPS Signals in ... - Semantic Scholar

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Robust Synchronization of Weak GPS Signals in Multipath Environments Using A Block Averaging Processing and Maximum Likelihood Estimation M. Sahmoudi and M. G. Amin

Abstract In this paper, we propose a robust approach for GPS synchronization in low signal-to-noise ratio (SNR) and multipath environments such as that encountered in indoor applications. Exploiting the replication property of the C/A code within each data bit, we develop a block averaging pre-processing (BAP) of coherent integration to improve the SNR by enhancing receiver robustness against noise and interferences. The proposed processing consist in compressing the data sequence by averaging the accumulated blocks to obtain a new short and strong data sequence. We introduce a new compensation technique to take account of the phase change and we present a theoretical analysis and several properties of this technique. The developed software GPS receiver involves both acquisition using an efficient and fast FFT-based approach, and tracking using the Maximum-Likelihood (ML) principle. Unlike to the existing acquisition methods, we process the newly generated signal so as the spectrum peak will be strong enough to be dominant, and we propose two novel satellites search algorithms. The first algorithm consists of searching the stronger spectrum peak over a carrier frequency of Doppler offset plus frequency compensation range while cross-correlating with the local code. In the second implementation, we exploit the BAP to suggest a very fast acquisition scheme. To formulate the problem of code tracking with multipath mitigation, we exploit the fact that during a period with no data bit edges the propagation delay causes only a circular shift to the spreading C/A code block. This allows the decomposition of the averaged data vector into a constant component plus an undesired signals component. An efficient temporal whitening transform is derived from the sample covariance matrix and applied to suppress strong colored interferences. The ML estimation of the multipath parameters becomes tractable, and then, a computationally efficient procedure for solving the complex ML optimization problem is considered. Indeed, the ML delay estimator is computed for each path using a finite differences -based maximization

M. Sahmoudi and M. G. Amin are with the Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA. This work is supported by ONR/NSWCCD under contract no. N65540-05-C-0028. March 27, 2007

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technique in a sequential scheme. The performance of the developed GPS receiver is evaluated through real data and simulation studies.

Index Terms Global positioning system (GPS), weak GPS signal, block averaging pre-processing (BAP), acquisition, tracking, interferences, multipath, maximum-likelihood estimation, robustness.

I. I NTRODUCTION Global Navigation Satellite Systems (GNSS), like the US GPS and the European Galileo, are based on the principle that one’s position can be determined from distances measured to objects with known positions [13]. From the propagation time measurements to at least four satellites the user’s coordinates in three-dimensional space can be determined, including an estimate of the clock offset between the user and system clocks [13], [23], [31]. Despite the ever increasing civilian applications, significant limitations of the current GPS arise from interference and, more importantly, multipath propagation as well as insufficient signal strength when operating indoor [12], [31], [43], [47]. The effect of interference is to reduce the signal-to-noise ratio (SNR) of the GPS signal such that the GPS receiver is unable to obtain measurements from the GPS satellite [23], [31], [43]. The direct-sequence spread-spectrum (DSSS) scheme, which underlines the GPS signal structure, provides a certain degree of protection against interference. However, when the jammer power is much stronger than the signal power, the spreading gain alone is insufficient to yield any meaningful information [31], [43]. Interference is typically mitigated prior to the correlation loops [3], [4], [39]. In these loops, conventional GPS algorithms process blocks of one millisecond of signal coherently and extend the integration time non-coherently [23], [31]. For weak signals, receiver sensitivity can be amplified if longer durations of signal are processed coherently [2], [10], [37], [43]. However, without external aiding, the existence of navigation bit transition limits the coherent integration period to 20 ms. In the case when the external assistance is not available, the navigation data can be estimated while the tracking process is on-going [52], [53]. The estimated binary data can be then used in a feedback scheme for the self-aiding to extend the coherent integration time of the tracking loop [52]. In this contribution, we deal with this problem during the acquisition stage for long coherent integration purpose. We consider a data segment of 20 ms length and we use a successive sign reversal of 1 ms signal by block for data bit transition detection and correction. We choose the data-bit edge free version of the data segment corresponding to the data version of best signal power (i.e. of best correlation results). This idea is adopted from [51], in which authors propose originally this sign March 27, 2007

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reversal technique before summing blocks of correlation in the despread correlator. Thus, the proposed acquisition algorithm can encompass the entire twenty blocks free from data bit transition. However, for the tracking mode we resolve this problem using a new strategy based on the phase/frequency relation, without data bit demodulation, as a simple alternative to some existing techniques [10], [26], [43], [53]. For indoor GPS applications, the data stream maybe provided from cellular networks [10], [12]. It is widely known that the dominant error source in GPS navigation systems is due to the multipath signal propagation [12], [13], [23]. Multipath errors refer to errors in the code tracking loop due to reception of the direct signal from the satellite and one or more reflections from the ground and objects in the vicinity of the receiver. The respective pseudorange errors are within the tens of meter level and, as such, should be mitigated for high precision positioning. Influence of multipath on the carrier phase is at the centimeter level and, therefore, is neglected in this paper [7], [31]. Many approaches for code tracking in presence of multipath have been proposed in recent years, including Narrow Correlator [44], Multipath Eliminating Technique (MET) [42], Multipath Estimating Delay Lock Loop (MEDLL) [45], Pulse Aperture Correlator (PAC) [28], and other techniques that are based on the maximum likelihood theory [38], [45], [48]. These techniques differ in their abilities to remove multipath errors, specially at low SNR and/or in the presence of interfering signals. Another well known challenge of existing approaches is the presence of closely-spaced multipath reflections. In addition, all existing methods are based on the Gaussian noise assumption and can’t cope with the cases when the GPS system receive other non-Gaussian undesired signals. In these cases, effective, high-resolution, and computationally efficient signal processing tools are required to estimate the parameters of the superimposed signals in GPS receivers. Since MEDLL like receivers are considered to be the best multipath mitigation method for GPS positioning, the aim of this work is to revise the ML-based approaches in order to develop a robust maximum-likelihood (RML) algorithm for positioning in multipath and weak signal environments. In this paper, we propose and combine two useful signal processing techniques to improve the GPS synchronization and to achieve robust acquisition and tracking against weak signal effects and undesired signals. We perform a block averaging pre-processing of long coherent integration (the so called BAP) to improve the GPS signal strength by a factor of used blocks number. This scheme enables the acquisition of very weak GPS signals, and we use the powerful ML approach as a second tool for signal tracking and multipath mitigation. For acquisition, we apply the Fast Fourier Transform (FFT)-based parallel code phase search algorithm to the block average of the properly accumulated data blocks. More precisely, we introduce first a new technique to take account of the phase change during the blocks accumulation by searching of the frequency compensation over a carrier frequency offset range, then we estimate March 27, 2007

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the code phase and Doppler frequency by cross-correlation in the frequency domain. We propose also another fast implementation that is suitable for moderate and high SNR signals acquisition. For tracking and multipath mitigation, it is assumed that the received signal has been converted to complex form at baseband, where 50-bps GPS data modulation and Doppler shift have been removed by a proper technique and a refined carrier tracking loop. The introduced block averaging pre-processing is exploited again to derive a block average model and used to reduce the interference contribution to a colored Gaussian noise effect during the ML delay estimation process. The root square inverse of the estimated covariance matrix acts as a whitening filter that effectively suppresses the strong colored residuals of the undesired signals. Then, the classical estimation problem of superimposed multipath signals in colored noise is efficiently solved by adopting a maximum likelihood approach. The asymptotic Gaussian distribution of the averaged interference and noise term serves to simplify the estimation algorithm and the development of an alternative parametrization of the GPS signal component, which facilitates the use of computationally efficient iterative techniques for the complex ML optimization problem. The contribution of this work is in the use of the block processing technique for both GPS signals acquisition and tracking in a novel block averaging -based scheme. For the GPS signals tracking task, a block average model -based maximum likelihood technique is investigated. This idea is inspired from [5], in which authors assume that the desired user, in the CDMA context, send a fixed preamble (i.e. a constant) during the acquisition. Here, we adopt the same strategy for the GPS synchronization problem by accumulating and averaging a large blocks number of data during which the C/A code sequence repeated periodically every block. In [5], a recursive approach based on finite differences was used to optimize the likelihood cost function. This technique is extended here to multipath conditions using a sequential procedure for each path parameters estimation. This paper is organized as follows. The next section describes the general GPS signal model. The proposed block averaging pre-processing and GPS signal acquisition schemes are described in Section III. In Section IV, we develop a block average model with Gaussian distribution. The pre-processing technique reduces the effect of undesired signals to a colored Gaussian noise, then we discuss a whitening mitigation of the residual colored noise. Section V addresses the maximum likelihood time delay estimation for signal tracking and multipath mitigation. Computer simulations and the conclusions are presented in Section VI and VII, respectively. II. GPS S IGNAL M ODEL In conventional GPS, the received signal in a multipath environment is an M -path model composed of the direct path signal and (M − 1) reflected rays, plus the interference term J(t), and the Gaussian March 27, 2007

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additive noise w(t). Each path wave is characterized by its amplitude Ak , phase shift from the direct signal φk (i.e. φ1 = 0), and time delay τk , which are all assumed to be constant over the observation period. In the complex form, the received signal is modeled by r(t) =

M X k=1

Ak x(t − τk )ejφk + J(t) + w(t),

(1)

where x(t) is the satellite DS-SS signal which consists of navigation data symbols, {d(i)}, multiplied by the spreading waveform c(t) of the pseudo-noise (PN) sequence {c(n)}, x(t) =

+∞ X

i=−∞

d(i)c(t − iT ).

(2)

The spreading waveform is a binary phase-shift keying (BPSK) signal, c(t) =

P −1 X n=0

c(n)g(t − nTc ),

(3)

obtained by convolving the PN impulse sequence of length P = T /Tc with a rectangular chip-shaping pulse g(t), where Tc is the pulse duration and T is the integration time of the DLL in the GPS receivers. The C/A code has a chip rate of 1/Tc = 1.023 MHz, so a chip is of period Tc = 977.5 ns (1023 × 103 chips/second). We recall that one period of the spreading C/A code consists of 1023 chips, hence it spans 1 millisecond, and there are twenty 1 ms code epochs (blocks) in each GPS BPSK data bit. The def

interference plus Gaussian noise term v(t) = J(t) + w(t) is assumed to be a zero-mean random signal. III. B LOCK AVERAGING P RE -P ROCESSING

FOR

W EAK GPS S IGNAL ACQUISITION

Recently, the application of block processing techniques to GNSS signals has received widespread interest due to their potential to improve the performance and capabilities of navigation receivers [15], [20], [32], [50]. A block processor operates on blocks of data rather than processing the data samples sequentially. The block size is usually chosen to be an integer multiple of the length of the spreading code. For GPS C/A code processing, the minimum block size is the length of the C/A code, which spans 1023 chips of duration 1 ms. The GPS receiver uses typically one spreading code; i.e. one block for the acquisition task. However, weak GPS signals require longer amount of data. Existing approaches, typically, process blocks of the cross-correlation output [15], [20], [32], [50]. In this paper, a different approach is proposed which calls for pre-processing (accumulating and averaging) blocks of the received signal before despreading. This requires the compensation of the phase change across several consecutive data blocks so that block averaging improves the SNR. The proposed pre-processing enables pre-correlation noise plus interference reduction and allows fast receiver startup. March 27, 2007

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A. Block averaging pre-processing of weak GPS signals Typically, consideration of multipath in a GNSS context emphasizes its effect on signal code and carrier tracking accuracies, since these receiver functions are more sensitive to multipath degradation than signal acquisition or data modulation [23]. Furthermore, when the direct and the reflected path are in-phase and then added up constructively, these constructive multipaths can help the signal acquisition in some even rare situations [23]. So, in the reminder of this section we ignore the multipath signals during the acquisition period. Thus, in the complex form, the input receiver signal from a given satellite can be expressed as, def

r(t) = Ad(t)c(t)ej[2π(fc +f )t+ϕ0 ] + J(t) + w(t) = s(t) + v(t)

(4)

where A is the GPS signal amplitude, fc = 1575.42 MHz is the L1 carrier frequency, f is the frequency offset due to Doppler which is typically within the range of f ∈ [−10 KHz, +10 KHz] [43], and φ0 is the phase offset. Based on the GPS front-end configuration, the received GPS signal is downconverted and sampled using a sampling frequency fs . In the above equation, s(t) denotes the signal portion of the received data. We assume fs to be an integer multiple of 1 KHz so that the number of data samples in each block is an integer. With knowledge of the GPS L1 carrier frequency, the received signal can be translated to baseband, so the sampled data model can be expressed as,

r(n) = Ad(nTs )c(nTs )ej(2πf nTs +ϕ0 ) + v(nTs ) = s(n) + v(n) ,

(5)

where Ts = 1/fs is the sample period. Pre-correlation noise and interference mitigation is a key task for GPS acquisition in many situations [23]. Although extending the coherent integration interval is a viable approach, it still limited by several technical challenges including data bit transitions. Additionally, with increased coherent integration, the frequency search bin size is reduced, leading to significant increase in search complexity [43]. Motivated by these constraints, next we propose a pre-processing method for both undesired signals mitigation, and data length compression. Consider N samples of r(n) for each 1 ms data block and accumulate signal samples over L ms, where L is an integer such that L ∈ {1, · · · , 20}. So, the total number of data samples considered is N L. Exploiting the replication of the C/A code, and since N Ts = 1 ms we can write c(nTs ) = c((n + iN )Ts ) for any integer i. Averaging the input data over L consecutive blocks to March 27, 2007

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obtain the averaged data block, rL (n)

def

=

L−1 1 X r(n + iN ) L

(6)

i=0

=

j[2πf nTs +ϕ0 ]

Ad(nTs )c(nTs )e

L−1 1 X HL (f ) + v(n + iN ) L i=0

=

HL (f )s(n) + vL (n)

(7)

P j2πf iN Ts represents the block average signal coefficient and v (n) = where HL (f ) = L1 L−1 L i=0 e 1 PL−1 i=0 v(n + iN ) is the block average of the undesired signal component. Using the sign correction L

technique [51], the data bit was considered constant over L data blocks in the above equation. We note that HL (f ) can be simplified as,   L−1 1 X j2πf iN Ts 1 sin(πf N Ts L) jπf N Ts (L−1) HL (f ) = e = e . L L sin(πf N Ts )

(8)

i=0

Figure 1 illustrates the proposed block-averaging signal pre-processing (BAP), which is equivalent to

Block 0

…… r(n)

Block 1

Block (L-1)

……

…… r(n+N)

…… r(n+(L-1)N)

1 L

Ȉ

……

rL(n)

One block

Fig. 1.

Block-averaging for long coherent integration over L data blocks of the received weak GPS signal.

applying a finite impulse response (FIR) filter with impulse response,  L−1  1 for 0 ≤ n ≤ L − 1, 1 X L N hL (n) = δ(n + iN ) =  0, otherwise. L

(9)

i=0

where δ(.) is a unit sample function. The BAP filter acts as a frequency selective bandpass filter. The

filter non-decimated version applies rectangular window, i.e. hL (n) = (1/L)wr (n), where wr (n) is the March 27, 2007

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rectangular window of length L [30]. The above function assumes peaks values at frequencies which are a multiple integer of 1 KHz. The zero-crossing bandwidth is 2/L KHz. Accordingly, it can be concluded that if the GPS signal with a carrier frequency within the range of [k ± 2/L] KHz, where k is an integer number, lies in the mainlobe of the filter and will benefit from the BAP step. Jammer signals outside that frequency band will encounter power reduction, whether they are narrowband or broadband. It is straightforward to show that the noise variance will be reduced by L, as a result of BAP. Figure 9, depicts the amplitude response function |HL (f )| for different block number L. It is clear that larger values of L lead to finer resolution, but, the sidelobes become more pronounced. This forms a compromise between signal enhancement and noise suppression. The use of different windows in place wr (n) may change the nature of this tradeoff, however, does not remove it. One way to resolve the above problem is to estimate the unknown frequency offset from the k KHz frequency and compensate for it prior to averaging by shifting the incoming spectrum to the BAP filter band response. To implement this idea, we define a new block-averaging with phase compensation while assuming constant Doppler variation over the integration time, which is a valid assumption for most civil applications [43]. That is, we multiply the baseband signal with a frequency component ej2πδf as,

r˜(n) = r(n)ej2πδf ˜

= Ad(nTs )c(nTs )ej[2πf nTs +ϕ˜0 ] + v(nTs )

(10)

where f˜ = f + δf and ϕ˜0 combines the initial phase ϕ0 and the constant phase difference between the local carrier and the incoming signal. It is important to note that the intent is to estimate δf such that f˜ is ˜ integer multiple of 1 KHz due to the fact that for any f˜ integer multiple of 1 KHz, ej2πf iN Ts = 1 for each

i ∈ {0, · · · , L − 1}. The main difference between the sequence r˜(n) and r(n) is only the phase, while

the C/A code information remains intact during the generation of this new sequence. This modification is ˜

necessary since the phase of r(n) and r(n + iN ) differ by ej2πf iN Ts . Thus, the phase change is corrected by the frequency compensation which makes the block averaging possible over multiple blocks. To determine an optimal value of δf , a simple search is proposed next during the acquisition process for the weak signal case, and a different scheme will be presented as well to design a very fast acquisition algorithm. The proposed block averaging scheme can be seen as an attractive coherent data compression technique for GPS signals pre-processing. March 27, 2007

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B. Acquisition of weak GPS signal using the block averaging processing 1) Acquisition scheme: We will apply the parallel code phase search acquisition algorithm to the frequency-compensated block averaged sequence r˜L (n). Compared to other acquisition methods, this algorithm cut down the search space to the different carrier frequencies [1]. The Fourier transform of the local code must only be performed once for each acquisition. To estimate δf , we consider a search over a range of 1000 Hz due to the periodicity of this value with period 1 KHz. Furthermore, if we find the optimal value within [0, 500] Hz, we can compute the other value within [500, 1000] Hz due to the symmetric about 500 Hz. Then, we have to consider only the range of [0, 500] Hz. During this search, we need a fine frequency resolution, e.g., for L = 20 ms a frequency step of 25 Hz is a reasonable choice. Then, we apply an acquisition method to estimate jointly f and δf once the 2-D spectrum peak is detected at (f + δf 0 ). Equivalently, the optimal frequency compensation δf 0 is estimated by considering the block of properly averaged data, r˜L (n) = HL (f + δf0 )s(n) + vL (n) ≈ s˜(n) + vL (n),

(11)

where s˜(n) is the signal portion of the new data sequence and vL (n) is the reduced noise component. During the 2-D spectrum peak search, we apply the parallel code phase search acquisition method [1], as described in Table I, to search the code phase within one code and the Doppler frequency f within a range of ±10 KHz. 2) Example of acquisition using real GPS satellites signals: Here we use a data set downloaded from the web site of the University of Colorado, USA [1]. These GPS signals was collected using a SIGE SE4110L front end GPS receiver. The parameters necessary for processing this data are as follows: Sampling Frequency of 38.192 MHz, Intermediate Frequency of 9.55 MHz (nominal), and the signed character is (8 bit) sample format. In Fig. 2, the typical time - and frequency -domain representations of the considered data are illustrated. It is clear that the signal is similar to a Gaussian noise and no discernable structure is visible. In the frequency-domain, we notice the presence of the bandpass filter of 6 MHz bandwidth applied before sampling. Additionally, we see that in the middle there is sinc function type behavior. This seems to be the main lobe of the 2.046 MHz of the sinc spectrum of the GPS signal. Figure 3 shows the acquisition results using the parallel code phase space search method. We observe that the four strong signals PRN 15, PRN 18, PRN 21 and PRN 22 were easily detected and acquired. However, as this method use only one block of data it is unable to detect and acquire the other weak signals. In Figure 4, we show that using the proposed acquisition scheme over twenty data blocks, the March 27, 2007

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' •

$

Parallel code phase search using the averaged data block

Start the data bit detection: Consider a 20 ms data segment and process data bit detection with successive bit sign reversal [51]. Other technique also maybe used for this purpose. – PART 1: B LOCK AVERAGING P REPROCESSING ∗ Step 1 - Fine multiplication by a local carrier within 500 Hz: The incoming signal is multiplied by a locally generated carrier signal within a range of

1 2NTs

Hz for BAP frequency compensation. We

use a frequency step of 25 Hz, then we generate 20 sequences in this step. ∗ Step 2 - Block averaging: For each frequency bin, accumulate and average the twenty frequency compensated blocks together. This is equivalent to multiply one ms of data by a rectangular window. – PART 2: PARALLEL CODE PHASE SEARCH USING THE AVERAGED DATA BLOCK ∗ Step 3 - Multiplication with a local carrier within ±10 KHz for Doppler frequency estimation: A frequency resolution of 500 Hz is a good choice compared to the typical value of 1 KHz. ∗ Step 4 - Correlation in frequency domain using FFT and IFFT. Detect the 2-D correlation peak, then note the code phase value and the Doppler plus the optimal frequency compensation value. Finally, estimate the Doppler frequency f by subtracting the optimal value δf 0 . •

End of data bit detection: Compare the twenty computed coherent sums. When a sufficiently strong peak is detected, then the acquisition is achieved and the receiver can begin tracking the estimated frequency and C/A

&

code phase as needed to estimate the user position. TABLE I

%

ACQUISITION A LGORITHM 1: BAP- BASED WEAK SIGNALS ACQUISITION

Time domain plot

Frequency domain plot

6 0 4 −5

Magnitude

Amplitude

2 0 −2 −4

Fig. 2.

−15 −20

−6 −8

−10

−25 0

0.005 0.01 0.015 Time (ms)

0

5 10 15 Frequency (MHz)

The considered real GPS signal representation in time and frequency domain.

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additional weak signals PRN 3, PRN 6, PRN 9 and PRN 26 can be acquired. For these signals, Figure 5 shows that the correlations peaks are not visible enough to be detected by the GPS receiver. As mentioned Acquisition results using L = 1 ms 10 Not acquired signals Acquired signals

9 8

Acquisition Metric

7 6 5 4 3 2 1 0

Fig. 3.

0

5

10

15 20 PRN number

25

30

Acquisition results using the parallel code phase search with integration over only one block: L = 1 ms.

Acquisition results 15 Not acquired signals Acquired signals

Acquisition Metric

12.5

10

7.5

5

2.5

0

Fig. 4.

0

5

10

15 20 PRN number

25

30

Acquisition results using the proposed method for coherent integration of twenty blocks: L= 20 ms.

before, Figure 6 illustrates the fact that after enhancing the weak signals power using a longer coherent March 27, 2007

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(a) Correlation output for the PRN number 3

(b) Correlation output for the PRN number 6

(c) Correlation output for the PRN number 9

(d) Correlation output for the PRN number 26

Fig. 5. Correlation output of the weak signals PRN 3, PRN 6, PRN 9 and PRN 26 using the parallel code search method over a time integration of 1 ms.

integration time of 20 ms, they were detected because their correlation peaks become significant to be detected, i.e., those satellites become visible to the GPS receiver. As predicted by theoretical analysis, comparing Figure 5 and Figure 6, we observe clearly the advantage of the BAP step in mitigating the zero-mean noise effect, that is, the secondary false peaks are greatly reduced by the BAP. 3) Computational load of the proposed acquisition scheme: In Table II, the computational complexity of each step in Matlab flops is roughly shown and compared with that of the code phase search acquisition algorithm using 20 ms of data. The major advantage of the proposed acquisition approach, in comparison to the existing long coherent integration based methods, is that the SNR was boosted using less FFT operations because it reduces the data sequence of L blocks size to a new data sequence of one block size. When the frequency information is not available, the compensation step (i.e. Part 1 in Table I) is performed using 25N additional multiplication in step 1. We have eliminated the number of blocks L March 27, 2007

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(a) Correlation output for the PRN number 3

(b) Correlation output for the PRN number 6

(c) Correlation output for the PRN number 9

(d) Correlation output for the PRN number 26

Fig. 6.

Correlation output of the weak signals PRN 3, PRN 6, PRN 9 and PRN 26 using the proposed acquisition algorithm

over a time integration of 20 ms.

because we can accumulate and average the blocks and multiply the result (i.e. N multiplications) by the component exp(j2πδf nTs ) instead of multiplying this latter with the data sequence point by point (i.e. LN multiplications). From Table II, we can see clearly the advantage of the proposed procedure. Interestingly, the required number of flops is independent of the number of blocks L. C. Fast BAP-based acquisition implementation Based on the above BAP data processing, we develop a fast and computationally efficient FFT-based signals acquisition approach. First, we square the phase-compensated sequence yielding, ˜

r˜L2 (n) = e2j[2πf nTs +ϕ˜0 ] + vL2 (nTs ) +

2 ˜ vL (nTs )d(nTs )c(nTs )ej[2πf nTs +ϕ˜0 ] . L

(12)

We use the property d(nTs )2 = c(nTs )2 = 1. Using a sufficient number of blocks, the FFT-based spectrum of r˜L2 (n) will exhibit a strong dominant peak at the frequency 2f˜. The goal of the above squaring is March 27, 2007

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Step

BAP-based procedure

Parallel Code Phase Search Acquisition

20N

Part 1: Phase compensation Part 2 : for each δf value,

41N

41LN

N log(N )

LN log(LN )

41N log(N )

41LN log(LN )

41N

41LN

IFFT

41N log(N )

41LN log(LN )

Total

20(102N + 83N log(N ))

82LN + 83LN log(LN )

Total for L = 20

2040N + 1660N log(N )

1640N + 1660N log(20N )

Step 3 Step 4: Cross-correlation FFT of the local code FFT of the data sequences Multiplication

≈ 8280N + 1660N log(N ) TABLE II C OMPUTATIONAL COMPLEXITY COMPARISON OF THE PROPOSED ACQUISITION METHOD AND THE EXISTING ONE .

to despread the C/A code to be able to estimate the carrier wave of the new sequence. However, the FFT-based spectrum measure may be not useful for some very noisy signals and other spectrum power measures should be used. Therefore, we suggest to use the averaged periodogram, a consistent estimator of the power spectral density (PSD) [24] and a block processing estimator, that is, for each data block we compute a periodogram as, 1 Pˆ (i) (f ) = N

N −1 2 X ri (n)exp(−j2πf n) ,

(13)

n=0

and then average all the periodograms together to yield the final PSD estimate as, L−1 1 X ˆ (i) Pˆav (f ) = P (f ). L

(14)

i=0

This procedure can not be applied for the original GPS signal because SNR is too low. After BAP, the signal becomes strong enough to detect the peak of the FFT-based spectrum of the squared sequence. Then, we can estimate (δf + f ) and deduce the Doppler frequency f , as explained in Table III. Finally, we correlate with the local C/A codes to determine each PRN number and code phase. Although we don’t know which PRN gives this peak, we can try all 32 possible PRNs, the main difference is that we only need to perform a one-dimensional search. Indeed, part 1 requires 20N flops, step 3 requires 20N log(N ), and step 4 requires 128N log(N ). The total of required flops is 20N + 148N log(N ), thus,

we reduced greatly the computational load. Figure 7 illustrates the effectiveness of the squaring step. March 27, 2007

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15

Decoupled estimation of Doppler and code phase using the averaged data block

$

Start the data bit detection: Consider a 20 ms data segment and process data bit detection [51]. – PART 1: B LOCK AVERAGING P REPROCESSING ∗ Step 1 - Fine multiplication by a local carrier within 500 Hz: The incoming signal is multiplied by a locally generated carrier signal within a range of

1 2NTs

Hz for BAP frequency

compensation. We use a frequency step of 20 Hz, then we generate 25 sequences in this step. ∗ Step 2 - Block averaging: For each frequency bin, accumulate and average the twenty frequency compensated blocks together. – PART 2: D ECOUPLED D OPPLER FREQUENCY AND CODE PHASE

ESTIMATION

∗ Step 3 - Square the block averaged data and apply the averaged periodogram to detect its spectrum peak. Estimate (f + δf ) once the peak is detected. Conclude the Doppler frequency by subtracting the corresponding optimal δf value. ∗ Step 4 - Correlation with local codes: Since the Doppler has been estimated from step 3, the remain search is only one-dimensional in the code phase line.

• End of data bit detection. &

TABLE III

%

ACQUISITION A LGORITHM 2: BAP- BASED FAST SIGNALS ACQUISITION ALGORITHM

6

6

x 10

x 10

8

FFT of the squareed block−averaged signal

FFT of the squared original signal

7 6 5 4 3 2 1

2000 4000 6000 8000 10000 Frequency in Hz

Fig. 7.

7 6 5 4 3 2 1

2000 4000 6000 8000 10000 Frequency in Hz

Comparison of the FFT of a squared GPS signal and the FFT of its squared block-averaged sequence.

Figure 8 illustrates the benefit of the BAP step. D. Effect of the block averaging pre-processing to undesired signals More interestingly, we can show that the block averaging operation reduce the noise and/or interference magnitude and power before the correlation processing. In order to consider the main jamming scenarios March 27, 2007

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x 10

5

The original PRN6 signal

x 10 Block−average of PRN6 signal 2.5

FFT−based correlation spectrum

2

2

1.5

1.5

1

1

0.5

0.5

2000 4000 6000 8000 10000 Frequency in Hz

2000 4000 6000 8000 10000 Frequency in Hz

Fig. 8. Comparison of the FFT-based spectrum of a GPS signal and its block-averaged sequence. Digital frequency resolution is 500 Hz.

of the GPS signal, three types of jammers have been considered in addition to the Gaussian noise. 1) Gaussian noise: If the GPS signal is corrupted only by a zero-mean Gaussian noise w(t), it is evident to see that the magnitude of wL goes to zero as L becomes large. We can also write the following equality regarding the effect to the noise variance, L 1 X 1 V ar(wL ) = 2 V ar(w) = V ar(w). L L

(15)

i=0

Thus, the BAP reduces the random noise power by a factor of L. The SNR improvement that can be achieved by accumulating and averaging multiple periods of the phase compensated C/A code is given in the following proposition. Proposition 1: In Gaussian noise environments, the proposed block averaging processing increases the GPS signal strength by a factor of L or equivalently increases the SNR by a factor of 10log10 (L) dB. For example, if we average a frequency-compensated sequence of an entire data bit period (i.e. 20 blocks), then we enhance the system SNR by 13 dB, which it is definitely an important gain for weak signal acquisition and processing. We note that it is not the only way to increase the GPS signal power using long coherent integration but the proposed block averaging concept has many advantages. First, unlike to the accumulating block processing or the standard coherent integration it reduces the data input to one block (see Fig. ??), which is very important for fast search of the C/A code beginning and Doppler shift during the acquisition process. Second, the resulting short vector (1 ms interval) permits the use of a block average model for the maximum-likelihood delay estimation during the tracking and multipath mitigation process, as we present in the next section. 2) Narrowband continuous wave (CW) jammer: It is one of the most frequently encountered jamming signals, representing RF spikes, and could appear at any time in the spectrum. We express the CW March 27, 2007

DRAFT

17

interferer as J(nTs ) = J cos(2πf0 nTs +θ), where J is the interference amplitude, f0 is the CW frequency offset relative to the GPS carrier, and θ is its random uniform phase. The block averaging effect on the CW interferer over L blocks is given by, L−1 1 X J [(n + iN )Ts ] = L

L−1

1 X J cos [2πf0 (n + iN )Ts + θ] L i=0   J(nT s) if f0 = 0 mod(1KHz) =  1 sin(πf0 L) J [nT + πf (L − 1)] ; otherwise s 0 L sin(πf0 )   J(nT s) if f0 = 0 mod(1KHz) =  |H (f )|J [nT + φ (f )] ; otherwise.

i=0

L

0

s

L

(16)

0

= J(nTs ) ∗ hL (n).

(17)

Block averaging weight function dor different L 1 L=20 L=10 L=5 L=2

0.9

Block averaging weight function

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1

Fig. 9.

−0.5

0 Frequency offset in KHz

0.5

1

Block-averaging gain function versus frequency offset for different blocks number L.

The following observations summarizes the BAP effect on a CW interference: 1) If the frequency offset f0 of the CW jammer is within the frequency set {k/L ; k ∈ Z∗ } KHz, then BAP eliminates the jammer entirely. 2) If the frequency offset f0 of the CW jammer corresponds to the first sidelobe of the gain March 27, 2007

DRAFT

18

coefficient, i.e. |HL (f0 )| = S , then block averaging reduces the CW jammer power approximately by 20log10(L/S) = 13 dB, and by more than 20 dB if f0 is within other sidelobes of |HL (f0 )|. 3) In presence of a CW jammer with frequency offset f0 = 0 mod(1 KHz), the BAP has no effect on it and keep it power intact. This is the worst case of a narrowband CW jammer. 3) Partial-band jammer: A partial-band jammer is an undesired signal that spreads its average power Jav evenly over some frequency range Wj , which is a subset of the GPS bandwidth Ws . We may express

jammer power as Jav = J0 Ws , where J0 is the value of the power density spectral of an equivalent wideband jamming signal across the GPS band. This partial-band interference may be characterized by its power spectral density given by Pj (f ) =

 

Jav Wj

=

J0 Ws Wj

|f | ≤ 12 Wj

if |f | > 12 Wj ,

0



if

where Ws ≫ Wj . Then, the mean square of the output of the BAP filter may be computed as, Z +∞ Z Jav +Wj /2 2 2 Ej = | HL (f ) | Pj (f )df = | HL (f ) |2 df W j −∞ −Wj /2 2 Z +Wj /2 Jav sin(πf L) df. = 2 Wj L −Wj /2 sin(πf ) The above integral can be computed in a Fourier series form [41], [46], yields, " # L−1 sin(kπWj ) Jav 1 2 X 2 Ej = 1+ (L − k) . L L πWj k

(18)

(19)

(20)

k=1

In appendix I, we simplify further the above equation into a closed form solution valid for large L as,   2Υ(L, Wj ) Jav 1 2 Ej = − (21) , L Wj LπWj πWj πWj j where Υ(L, Wj ) = [cos( πW 2 ) − cos((L − 1/2) 2 )]/2 sin( 2 ). Again, from equations (20) and (21)

we conclude that the block averaging is a useful processing to reduce the sub-band jammer power using a large number of blocks L. In the limit as Wj becomes zero, the interference becomes an impulse at the carrier. In this case, the interference is a pure frequency tone (CW). Then, the power spectral density is given as, Pcw (f ) = Jav δ(f0 ) and the corresponding power is given by, Jav sin(πf0 L) 2 2 2 Ecw = Jav | HL (f0 ) | = 2 . L sin(πf0 )

(22)

4) In-band Chirp jammer: It is a sinusoidal interference including Doppler shift effect, i.e., a non

stationary signal that contains the jammer frequency variation in time. The spectrum of such a jammer may spreads on the GPS frequency range. The Chirp jammer can be defined by the formula 2

J(t) = JejπW t March 27, 2007

/T

,

− T /2 ≤ t ≤ T /2.

(23) DRAFT

19

Since the phase of J(t) varies quadratically versus t, the frequency changes linearly versus t. It called also a swept CW jammer because it is a pulse whose time duration equals T seconds. Over the life of the Chirp pulse, the changing frequency sweeps from −W/2 to +W/2 Hz. Thus in the frequency domain, the spectrum of the Chirp is concentrated in the range |f | < W/2. The parameters that can be specified for this jammer are the Chirp power level according to the interference to signal ratio (ISR) in dB, the Chirp pulse’s length T , called also the sweep duration (units generally in microseconds) and the Chirp’s swept bandwidth W (units generally in MHz). It is interesting to note that we can approximate the Fourier transform of |J(t)| with a rectangle that extends from f = −W/2 and f = +W/2. Figure 10 below shows that a Chirp is bandlimited. If the encountered Chirp jammer has a bandwith of W = 2

Spectrum for an LFM waveform 300

250

Amplitude spectrum

200

150

100

50

0 −2

Fig. 10.

−1.5

−1

−0.5

0 Frequency − MHz

0.5

1

1.5

2

Typical spectrum of a Chirp jammer with sweep duration of T = 10 Microsecond, and swept bandwidth of Wj = Ws = 2 MHz.

MHz, then it is a broadband jammer in this case which will spread over the total frequency range of the GPS signal spread bandwidth Ws . This broadband interference may be characterized by its power spectral density given by   J if |f | ≤ 1 W 0 2 s Pj (f ) =  0 if |f | > 1 W . 2 s

(24)

In that case, the achieved gain using the block averaging preprocessing is given in the following proposition. March 27, 2007

DRAFT

20

Proposition 2: The power of a broadband jammer after the block averaging preprocessing becomes, Z +∞ Z +Ws /2 Ej2 = | HL (f ) |2 Pj (f )df = J0 | GL (f ) |2 df −∞

=

J0 L2

−Ws /2

Z

2 sin(πf L) df = J0 . sin(πf ) L

+1

−1

(25)

To prove this proposition, it is sufficient to know that the above integral is equal to L [41]. Thus, the preprocessing stage reduces also a broadband jammer by a factor of the used blocks number L.

E. Fine frequency estimation of weak GPS signal using 20 ms As long as the signal is found during the acquisition process an additional frequency refining could be necessary for tracking loops. Commonly known as the fine frequency estimation task [43], needed for the transition of the receiver function from the acquisition mode to the tracking mode. Next, we describe the commonly used method based on the phase/frequency relation within 1 ms [43], then we extend it to longer coherent integration. After despreading, the input signal becomes a continuous wave given by, y(n) = Ad(nTs )e−j2πf nTs +φ0 + w(n),

(26)

in which y(n) was sampled at 1 KHz rate (i.e. y(n) is the correlation point output over 1 ms ). If the highest frequency component in one block of data at time m is Ym (f ), the initial phase θm (f ) of the input can be found from the acquisition outputs as, θˆm (f ) = tan−1



ℑ(Ym (f )) ℜ(Ym (f ))



,

(27)

where ℑ(.) and ℜ(.) represent the imaginary and real parts of a complex number, respectively. Similarly, we assume that after a short time interval the frequency component Yn (f ) of one block of data is the strongest one. The frequency f is the same because the input frequency will not change that rapidly during a short time. Hence, the phase angle θˆn (f ) of the input signal at time n is obtained via equation (27) as, tan−1 (ℑ(Yn (f ))/ℜ(Yn (f ))). These two phase angles are used to find a fine frequency estimate as, fˆ = (θn − θm )/2π(n − m) [23], [43]. This method performs integration only over one code period, which is not sufficient in weak GPS signal environments. Here, we propose a blind frequency estimation method by extending the above technique to longer coherent integration that doesn’t need the data bits and bit edges information. The proposed strategy consist of subdividing each data bit interval into several parts and find two consecutive parts, which are free from data bit transitions. If we subdivide a 20 ms March 27, 2007

DRAFT

21

signal segment into four parts of 5 blocks, the post-correlation block averaged signal is expressed as, y˜i (n) =

k=5 X

Ad(nTs )e−j(2πf nTs +φ0 ) + w(n) ; i = 1, 2, 3, 4.

(28)

k=1

where n = 5(i − 1) + k and k is the length of the coherent integration that we choose 5 ms in this paper. p We compute the magnitudes and phases of each part as ℜ(˜ yi )2 + ℑ(˜ yi )2 and using equation (27), respectively. Then, we consider the phase difference corresponding to the minimum magnitude difference yi ) = 5/1000 s. Then, a time and we normalize it by 2π(n − m) = π/100 because n − m = length(˜

period of 10 ms without data bit transition is localized and used for fine frequency estimation.

IV. GPS S IGNAL P ROCESSING

FOR

ROBUST M ULTIPATH D ELAY E STIMATION

In this part, we ignore the Doppler and phase change and we assume that are being tracked and compensated by PLL and FLL loops [23]. Characterizations of the phase shifts φk effects on direct signal delay ML estimation show that the worst case for estimation accuracy is when the direct and reflected signals are in/out phase, i.e. φk = 0 or φk = 180 [23], [38]. So, we will only consider that are in phase with the direct path, and we restrict our attention to time-delay estimation. For simplicity, the data modulation issue is not tackled in this part, so as to demonstrate the potential of the proposed multipath mitigation method. Before applying the ML approach, we manipulate the observations to both apply the proposed block averaging preprocessing to the multipath GPS signal model and to apply a Gaussian model.

A. Problem formulation of the time delay estimation Using long consecutive data blocks for improving SNR, a software multipath mitigation approach will be undertaken in this contribution. Define α ¯ ∈ CM ×1 and τ¯ ∈ CM ×1 as vectors containing complex amplitudes and delays of all the M multipath signals, respectively. We define a row vector of the delayed multipath signals as, s(t) = [x(t − τ1 ), · · · , x(t − τM )], thus s ∈ C1×M . Then, the signal model given in Eq. (1) can be expressed as r(t) = s(t)¯ α + v(t). First, we form a discrete-time observation by sampling the output of a filter matched at a rate q/Tc , where q is an integer which represents the oversampling factor. Typically, q = 1 or q = 2 in conventional receivers, but a higher value can be considered for fine delay search. The received vector yi , is formed by stacking N samples, associated with the ith data block, from the sampled discrete-time signal into a sequence of cyclostationary random vectors, yi = [r(iN ) r(1 + iN ) · · · r(N − 1 + iN )]T ∈ CN March 27, 2007

(29) DRAFT

22

where the nth element is yi,n = r(n + iN ). Accordingly, a matrix form of the signal model can be expressed as yi = S(¯ τ )¯ α + vi , where the matrix S(¯ τ ) ∈ CN ×M is the basis-function contains N samples from each of the delayed M multipath signals. Again, we use the block averaging processing for both interference plus noise mitigation and data compression to reduce the computation complexity. The new block average data vector, L−1 1 X mL = yi = [rL (0), rL (1), · · · , rL (n), · · · , rL (N − 1)]T , L

(30)

i=0

where rL (n) is the n-th block-averaged sample point over L successive block. For simplicity of presentation, next we ignore the oversampling factor, i.e., q = 1. 1) Effect of the block averaging on multipath signals: In the GPS receiver, a closely spaced multipath signal may arrive with a fractional delay. To account for this fact, we define the integer νk ∈ {1, · · · , N } and the fraction γk ∈ [0, 1) such that the time delay τk = (νk + γk )Tc be searched within one code. Since the GPS C/A code is periodic of period N (1 ms), we have shown in [35] that the block average sequence of the observation vectors yi can be expressed as, mL =

L−1 1 X yi = S(¯ τ )¯ α + vL , L

(31)

i=0

def 1 L

where vL =

PL−1 i=0

vi . Intuitively, since each path signal (without carrier wave and data modulation)

consists of the periodic C/A code, after block averaging this path remains approximatively the same signal. Thus a block average of data can be decomposed into two terms; a multipath C/A code term and an additive component of undesired signals. In other words, the block averaging reduces the interference τ )¯ α, and noise components by a factor of L but has no effect on the multipath GPS signal term S(¯

assuming, of course, that the frequency and phase change being tracked using PLL and FLL units. 2) Time delay estimation of superimposed multipath signals: It is important to recall that the best way to mitigate the multipath effect is to estimate those reflected signals and then subtract their contributions before or during the correlation procedure [23], [45]. To de so, we consider the problem of estimating the unknown parameters τ¯ and α ¯ using the maximum likelihood approach and the signal model given by Eq. (31). Due to the repetition of the spreading C/A code, the GPS signal exhibits strong self-coherence [4], [39] which can be seen as a ”non zero-mean” property when averaging several blocks of data. This feature can be utilized in two effective ways. It allows suppression of a large class of zero-mean interferers. Further, it reinforce the Gaussianity property of the noise and interference which permits the derivation of the ML algorithm. Indeed, the block averaging processing,reduces the interference effect to a colored Gaussian noise one by the central limit theorem (CLT), i.e., vL converges in distribution to March 27, 2007

DRAFT

23

a complex Gaussian random vector (r.v.) as L → ∞. It is worthy to recall that the use of large number of blocks L is not a major problem during the tracking stage. Because vL may contain interference and colored noise, the assumption of white noise is not realistic. Then, we model vL as a Gaussian colored noise with covariance matrix KL ∈ CN ×N of rank K . Therefore, mL is also asymptotically complex Gaussian r.v. with the density function,

where λJ =

 fmL (mL ) = λL exp −[mL − S(¯ τ )¯ α]H K−1 τ )¯ α] , L [mL − S(¯

1 det(πKL ) ,

(32)

and (.)H denotes the conjugate transpose operator.

Figure 11 compares the multipath error envelopes for matched filter (MF) DLL (wide correlator spacing with 1 chip), narrow correlator spacing (0.1 chip) early-late DLLs, and the maximum-likelihood Multipath Estimating DLL (MEDLL). Here we assume a two path channel without additive noise. The curves above the abscissa correspond to the case when the reflected signal was at 0 deg in phase with respect to the direct signal, i.e. φ2 = 0, and the curves below the abscissa correspond to the out of phase case, i.e. φ2 = π , that are the two worst possible situations. The error lies within this envelop for all other phase

values. In this experiment, we assume a signal-to-multipath (SMR) ratio of A1 /A2 = 2, i.e. 6 dB. The signal bandwidth was chosen to 2.046 MHz corresponding to the bandwidth of the C/A-code signal. This figure shows that the MEDLL receiver has a good and the best multipath rejection than the conventional DLL and the narrow correlator. It eliminates any multipath biases for delays greater than 0.1 chip. This 100 1 chip spacing DLL 80

60 0.1 chip spacing narrow correlator

Tracking error [m]

40

20

0

−20 MEDLL −40

−60

−80

−100

0

0.5

1

1.5

Multipath delay [C/A chips]

Fig. 11.

Error envelopes for the conventional DLL (1 chip spacing), non-coherent narrow correlator (0.1 chip spacing) and the multipath

maximum likelihood estimator (MEDLL). The parameters used are SNR = -20 dB, and MSR = 6dB.

comparison study permit us to conclude that the ML based approach is the most robust and promising March 27, 2007

DRAFT

24

method for GPS positioning in a multipath environment. Therefore, we focus in this work on the ML principle for GPS delay estimation to derive a robust maximum-likelihood (RML) algorithm for weak GPS signal and interference environments. The use of ML estimation in presence of non white noise suggests the need for a robust whitening to decorrelate the data and improve the signal power.

B. Robust whitening pre-processing The whitening processing is motivated by the fact that in GPS, the desired signals are usually 20 to 30 dB bellow the noise floor. As such, the total received signal power is dominated by the interference signals. Then, the covariance matrix KL of the block average mL can be approximated as K L ≈ K vL = K J + K w .

(33)

The vector vL is contained in at most a K -dimensional subspace of CN with K < N . The eigendecomposition of the covariance matrix is KL = VDVH , where D is a diagonal matrix of the eigenvalues (λn ). Furthermore, λn =

  

1 2 L (dn + σ ), 1 2 Lσ ,

if

n≤K

(34)

otherwise ,

where dn is the variance of the interference along the nth eigenvector and σ 2 is the Gaussian noise variance. Then, the K largest eigenvalues of KL correspond to the eigenvectors forming the subspace of interference signals spanned by VI ∈ CN ×K . Thus, V can be partitioned as V = [VI VN ], where the −1/2

columns of VN span the noise subspace. The matrix W = D−1/2 V, denoted commonly as W ≡ KL

,

serves as a whitening filter for mL . We can see from equation (34) that as the power of the interference term increases, the diagonal elements of D−1/2 corresponding to VI approach zero. In the limit, the null space of W converges to the interference subspace, i.e., the stronger the interference, the more it is suppressed by the whitening processing. The matrix KL can be estimated using the sample covariance ˆ L as, matrix R L−1 1 X H ˆ ˆ KL = yi yiH − mL mH L = RL − mL mL . L i=0

Compared to existing interference excision techniques [22], [3], [39], our procedure suppress the jammer without distortion nor energy reduction of the desired GPS signal. Next, an efficient algorithm will be presented to resolve the optimization problem for ML multipath parameters estimation. March 27, 2007

DRAFT

25

V. S EQUENTIAL ML

FOR

M ULTIPATH D ELAY E STIMATION

A. Single-path estimation With a single-path channel (i.e. τ¯ ≡ τ and α ¯ ≡ α), the ML estimation of τ and α is based on the value of mL that maximize fmL |τ,α (mL |τ, α) with respect to τ and α. The solution of this standard signal parameter estimation in colored Gaussian noise is [34], τˆ = arg max τ

α ˆ=

2 ˆ −1 m |s(τ )H K L ˜ L| , ˆ −1 s(τ ) s(τ )H K

(35)

L

ˆ −1 m s(ˆ τ )H K L ˜L , −1 H ˆ s(ˆ τ ) K s(ˆ τ)

(36)

L

ˆ −1/2 mL is the new block average data sequence after the whitening processing. We note ˜L =K where m L

that the conventional matched filter (MF) estimator is equivalent to assuming KL to be the identity matrix, i.e. assuming that the signal is received in a white Gaussian noise environment. Once the delay estimate τˆ is known, α ˆ can be immediately computed. Below, we propose a low-complexity implementation of

the above equations (35) and (36). 1) Time delay estimate computation: Ideally, we would like to differentiate the objective function (35) with respect to τ . However, the delay lies within an uncertainty region τ ∈ [0, τ˜) and s(τ ) is only piecewise continuous on this interval. Furthermore, as illustrated in Fig. 14, this objective function generally has numerous local maxima. To counter these problems similarly as presented in [5], we divide the uncertainty region into N cells of width Tc and consider a single cell Cν = [νTc , (ν + 1)Tc ). for τ ∈ Cν such that τ /Tc = ν + γ , the signal vector can be expressed as [5], s(τ ) = (1 − γ)s(ν) + γs(ν + 1),

(37)

∂s(τ ) = s(ν + 1) − s(ν) = a constant. ∂τ

(38)

and

Thus, within a given cell, we can differentiate the likelihood function and solve for the local maxima in closed form. We first compute the following 3N sufficient statistics: ˆ −1 s(ν), z1 (ν) = s(ν)H K L ˆ −1 s(ν + 1), z2 (ν) = s(ν)H K L

(39)

ˆ −1 ˆH z3 (ν) = m i KL s(ν). March 27, 2007

DRAFT

26

Within the cell Cν we define a

def

=

(z1 (ν) − z1 (ν + 1))ℜ{z3 (ν)¯ z3 (ν + 1)}

+(|z3 (ν + 1)|2 − |z3 (ν)|2 )ℜ{z2 (ν)} b

def

=

−z1 (ν)|z3 (ν + 1)|2 + z1 (ν + 1)|z3 (ν)|2 , z1 (ν)|z3 (ν + 1)|2 − z1 (ν + 1)|z3 (ν)|2

(40)

−2z1 (ν)ℜ{z3 (ν)¯ z3 (ν + 1)} +2ℜ{z2 (ν)}|z3 (ν)|2 , c

def

=

z1 (ν)ℜ{z3 (ν)¯ z3 (ν + 1)} − ℜ{z2 (ν)}|z3 (ν)|2 .

where ℜ{z} denotes the real part of z and z¯ is the complex conjugate. The likelihood function has stationary point at τ = (ν + γ˜ )Tc , where [5]   a˜ γ 2 + b˜ γ+c=0  γ˜ ∈ [0, 1).

(41)

If Cν contains a local maximum, that local maximum must occur at either γ = 0 or at one of the stationary points. Since the global maximum for the entire uncertainty region is by definition a local maximum of one of the cells, we have identified at most 3N candidates for τˆ. We compute the likelihood function at each of these points and select the one with the greatest value. The main computation burden lies in computing the sufficient statistics given in (39). However, we implemented a low complexity computation using recursive closed form calculations of equations (39). The key idea is using closed form expressions for the rank-one modification of a matrix inverse based on the sample correlation matrix instead the sample covariance matrix, and exploiting the low complexity Cholesky factorization of the correlation matrix, which has a recursive expression, H ˆ L = L − 1R ˆ L−1 + 1 yL yL R . L L

2) Covariance estimation: Since for GPS applications, fast parameter estimation is desired, the sample covariance estimator has some limitations. Indeed, if the number of observation vectors is less than the code length N , then the sample covariance matrix does not have a full rank. Thus, the ML cannot be computed until at least N blocks have been received. However, if K ≪ N , it should be possible to obtain a good estimate of the interference subspace from fewer than N vectors. Since only K vectors are needed to span VI , the covariance matrix must have the form K = BBH + σ 2 I, where B ∈ CN ×K . Thus, we can estimate a matrix of this form using the least square error cost function, ˆ σ ˆ − BBH − σ 2 Ik2 (B, ˆ 2 ) = arg min2 kK B,σ

March 27, 2007

(42) DRAFT

27

Using the fact that the two-norm of a symmetric matrix is maxn |λn |, this problem has a simple solution σ ˆ2 = ˆ B

1 2 (λK+1

+ λN ) 1/2

(43)

= [v1 , · · · , vK ]DI

ˆ and DI = diag(λ1 − σ where the vi ’s are the eigenvectors of K ˆ 2 , · · · , λK − σ ˆ 2 ). Thus, ˆ =B ˆB ˆH + σ K ˆ2I

is equivalent to replacing the N −K smallest sample covariance eigenvalues by σ ˆ 2 . In using this covariance estimator, the interference subspace remains unchanged, so we have not compromised interference ˆ has full rank and white noise subspace. rejections. Yet, K

B. Multipath estimation ˆ L is Gaussian with 1) ML for multipath parameters estimation: For a given pair (¯ τ, α ¯ ), the vector m ˆ L . Thus the standard MK estimator of (¯ mean s(¯ τ )¯ α and covariance K τ, α ¯ ) is given by [21] ˜H ˜ L} τˆ ¯ = arg max{m τ )m L Φ(¯

(44)

ˆ −1 s(¯ ˆ −1 m ˆ ˜ L, α ¯ = [sH (¯ τ )K τ )]−1 sH (¯ τ )K

(45)

τ¯

and

where Φ(¯ τ ) is a square matrix defined by ˆ −1 s(¯ ˆ −1 s(¯ ˆ −1 . Φ(¯ τ) = K τ )[sH (¯ τ )K τ )]−1 sH (¯ τ )K

Since there is no closed form solution to (44), the delays estimation requires a multidimensional search over the set spanned by τ¯ = [τ1 , · · · , τM ]T . This makes the computational load prohibitively high even with few propagation paths. Next, we propose a sequential procedure to reduce this multidimensional problem to a succession of M one-dimensional optimization problems. Then, a simple grid search can be performed or one can apply the above recursive update method given by equations (39)-(41). 2) Sequential estimation of multipath parameters: To avoid the multi-dimensional search in (44) we introduce an iterative algorithm in which complex amplitude and delay of each path are estimated sequentially. The estimation procedure consists of M steps repeated several cycles as follow: •

First cycle: parameter estimation. At each step we look for the dominant path parameters. – Step 1: We estimate the first path parameters (ˆ α1 , τˆ1 ) ˜ L: – Step 2: We subtract the contribution of this path from the observation m (2)

˜L =m ˜L−α m ˆ 1 s(ˆ τ1 ) March 27, 2007

DRAFT

28

(2)

˜ L instead of m ˜ L. Then we iterate the previous procedure over m

– Step m (1 ≤ m ≤ M ): Define (m) ˜L m

m−1 X

˜L− =m

α ˆ k s(ˆ τk ),

k=1

and (m) 2 ˆ −1 m |s(τ )H K L ˜L | τˆm = arg max , ˆ −1 s(τ ) τ s(τ )H K

(46)

L

αˆm =

(m) ˆ −1 m s(ˆ τm )H K L ˜L . ˆ −1 s(ˆ s(ˆ τm )H K τm )

(47)

L

Step M: In the last step, we compute (ˆ αM , τˆM ) to conclude the first cycle. •

(1st)

(1st) M )m=1

αm , τˆm Second cycle: refinement. We denote (ˆ

to refer these estimates to the first

cycle. In this 2nd cycle, we will refine the estimation procedure as follow. The main difference with the first cycle is that at each m-th step, all the paths with indices other than m are subtracted from ˜ L as m (m)

˜L m

˜L− =m (2nd)

and we compute the refined estimates (ˆ αm •

X

(1st)

α ˆ k s(ˆ τk

),

k6=m

(2nd) M )m=1

, τˆm

using equations (46) and (47).

Last cycle of refinement. Further cycles can be contemplated until reaching a convergence criterion. From simulations, we remark that two cycles are sufficient to achieve convergence.

The proposed procedure remove the multipath contribution in the direct path delay estimation by subtracting the reflected signals sequentially one by one. It is important to note that the existing MLbased approaches (e.g. MEDLL and ML2) mitigate the multipath by subtracting the multipath correlation contribution from the over all correlation function. Then, another advantage of our ML multipath mitigation method is the fact that it doesn’t need a large number of correlators to estimate those correlation functions and their derivatives.

VI. P ERFORMANCE A NALYSIS

AND

S IMULATION R ESULTS

First we give some practical tricks for the implementation issue of the proposed RML scheme. In the majority of the multipath scenarios, the amplitudes of secondary path signals are smaller than that of the direct path. To exploit this fact, we observe through extensive simulations that the performance can be significantly improved by introducing a constraint on the power of the received reflected signal to March 27, 2007

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29

enhance the capability of the receiver to detect the presence of a reflected signal with influence on the positioning algorithm. This constraint is formulated as, 20 log(

A1 ) < 10 dB. Ak

(48)

This means that when the direct to reflected signal power ratio (48) is greater than 10 dB, the influence of the reflected signal can be neglected. Since we have no information about the secondary path signals, we limit the search interval by introducing another constraint formulated as, τ1 ≤ τk ≤ τ1 + 1.5Tc .

(49)

Indeed, it is commonly known that a multipath signal has a significant effect when arrives with a relative delay less than 1.5Tc . The interference-to-signal ratio (ISR) is defined as ISR = 10 log 10 (Pi /Ps ), where Pi is the interference power and Ps is the signal power. To evaluate the performance of the proposed

iterative delay estimation algorithm, a number of simulations have been performed according to some realistic scenarios [23], [43]. Statistical performance has been obtained through 200 Monte Carlo runs.

A. Performance in additive white Gaussian noise The performance of both the proposed RML and MEDLL estimators as a function of the SNR is illustrated in Figure 12 and Figure 13. According to [6], we assumed an integration time of 600 ms (i.e. L = 600), which corresponds to the duration of 30 GPS data bit. This long correlation time yields to a

SNR higher than 30 dB after correlation, such that the covariance matrix and the delay estimation can be computed with a sufficient accuracy. We can see from Figure 12 that for the multipath free case both methods deliver a bias free estimation. This figure shows the variance of the estimated tracking error as a function of the SNR at the input of the GPS navigation receiver. The considered SNR values are between -30 dB and -10 dB, typical values at the input of GPS receivers [23]. We recall that the SNR becomes positive after despreading (i.e. correlation). It can be seen that the variances for both methods are nearly identical except for very low SNR values when the RML becomes superior. This results show the advantage of the proposed block average preprocessing in the maximum likelihood time of arrival estimation for weak GPS signals. Next in figure 13, one additional reflected path scenario is considered, with a short relative delay of ∆τ = 0.25Tc and SMR of 6 dB. Once again, both RML and MEDLL tend to have the same performance

in moderate SNR. However, as expected when the SNR becomes very small, the RML emerges as the March 27, 2007

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30

superior solution due to the used BAP which cleans the averaged data before the maximum likelihood estimation task. 0.04 RML MEDLL 0.035

2

Tracking error variances [m ]

0.03

0.025

0.02

0.015

0.01

0.005

0 −30

Fig. 12.

−28

−26

−24

−22

−20 SNR [dB]

−18

−16

−14

−12

−10

Tracking error variances versus SNR in multipath free and additive Gaussian noise.

0.06 RML MEDLL

2

Tracking error variances [m ]

0.05

0.04

0.03

0.02

0.01

0 −30

Fig. 13.

−28

−26

−24

−22

−20 SNR [dB]

−18

−16

−14

−12

−10

Tracking error variances versus SNR in a typical multipath environment.

Thus, we can conclude from these experiments that exploiting long time coherent integration before the correlation processing is more advantageous than exploiting it during the correlation stage. March 27, 2007

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31

B. Robustness to interference and non-Gaussian noise effects. 1) Robustness to partial-band CW jammer: The following proposition justifies and explains the robustness of the used Gaussian model for the RML estimation in presence of a CW interference. Proposition 3: The RML correlator transforms a continuous wave (i.e. sinusoidal) interference effect to a colored Gaussian noise effect. Proof: For clarity of presentation, we derive the proof of this result in appendix II. Basically the ML receiver compute the delay estimation through the early and late correlators before searching the correlation peak which corresponds to the ML delay estimate. Therefore, using the central limit theorem we proved in appendix II that the early and late integrators convert the CW interference effect to a colored Gaussian noise effect. To illustrate this result through simulation, we present an example that demonstrates the receiver interference suppression capability using the block averaging processing and the RML scheme. The interference assumes a single tone continuous wave (CW) [39], J(t) = J cos(2πf0 t + θ),

(50)

where J is the CW jammer amplitude, f0 is the jammer frequency offset from the frequency fL1 of the GPS signal and θ is a random phase uniformly distributed over the interval [0, 2π). Signal strength losses are more important in the case of jammers centered in the main lobe of the GPS signal spectrum, i.e. with frequency close to fL1 . However, any CW interference cause considerable problems to the GPS receiver [27]. Here we consider a worst case scenario when f0 = 0 mod(1 KHz), so we set f0 = 5 KHz. As illustrated in Fig. 14 (the top), direct application of the log-likelihood function produces local maxima. Although it peaks at the correct delay τ = 0.4Tc , there are numerous false peaks caused by the uneven weighting of the noise and interference subspace. Fig. 14 (the bottom), shows that the proposed preprocessing mitigates this problem and significantly reduces the false peaks. Now, we compare the time-delay estimation performance of the proposed robust ML (RML) scheme with the ML2 algorithm that has been developed recently in [38], as a refined version of MEDLL [45]. We consider a direct path and one multipath delayed by 0.25Tc such that the signal-to-multipath ratio (SMR) is 6 dB (i.e. A1 /A2 = 2). In order to describe the behavior and evaluate the performance of both the ML2 and the RML algorithms, we apply the root mean square error (RMSE) performance measure. The averaging is performed coherently over 100 code periods (i.e. L = 100 ms), which is a considered reasonable observation interval [38]. Figure 15 depicts the result of the estimate τˆ1 in the case of one multipath (i.e. M = 1). Since it was not derived with a jammer in mind, the ML2 fails to estimate the time-delay if the CW interference arrives with ISR more than 15 dB, whereas the proposed March 27, 2007

DRAFT

32 Before Robust Whitening 35

Log−Likelihood

30 25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5 Delay [Chips]

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

After Robust Whitening

35

Log−Likelihood

30 25 20 15 10 5 0

Fig. 14.

0

0.1

0.2

0.3

0.4

0.5 0.6 Delay [Chips]

Log-likelihood function calculated with covariance matrix estimated from 100 observation vectors. The GPS signal strength is

according to SNR = -20 dB. The interference signal is a CWJ with JSR = 30 dB and f0 = 5 KHz.

RML continues to provide good results up to ISR = 30 dB. The RMSE of the RML method for τˆ1 is approximately between 3 and 20 meters, as the range of one chip for the GPS C/A code is 293 meters. 1 ML2 0.9

RML

0.8

0.7

RMSE

0.6

0.5

0.4

0.3

0.2

0.1

0

Fig. 15.

0

10

20

30 ISR [dB]

40

50

60

RMSE of the time-delay of the direct path versus the interference to signal ration ISR. Used parameters: M = 1,

L = 100 ms, SNR = -20 dB and SMR = 6 dB.

2) Robustness to broadband Chirp jammer: Figure 16 shows the performance results of the proposed RML and the well known MEDLL receiver in presence of Gaussian noise, the typical reflected path March 27, 2007

DRAFT

33

(SMR = 6 dB and ∆τ = 0.25Tc ), and a broadband Chirp jammer with bandwidth of W = 2 MHz. As expected, the conventional maximum-likelihood scheme given by the MEDLL algorithm has no 1 MEDLL (ML2) RML

0.9

0.8

0.7

RMSE

0.6

0.5

0.4

0.3

0.2

0.1

0

0

10

20

30 ISR

40

50

60

Fig. 16. RMSE of the time-delay of the direct path versus the Chirp interference to signal ration ISR. Used parameters: M = 1, L = 500 ms, SNR = -20 dB and SMR = 6 dB.

capability to combat strong and broadband interference while the block-averaging and prewhitening based maximum likelihood approach given by the RML algorithm is only negligibly affected by even a 30 dB Chirp jammer. To evaluate the effect of the blocks number for each considered algorithm, we plot the RMSE index performance measure versus L in Figure 17. We can see that the performance of the derived RML method can be significantly improved using a large number of blocks in contrast of the MEDLL which still suffering from the presence of this Chirp jammer with power level of 30 dB. 3) Robustness to deviation from the Gaussianity assumption: Here, we compare the proposed RML, the narrow DLL and MEDLL in term of their robustness to non-Gaussian noise effect. For this purpose, we consider a two paths GPS signal model corrupted by an impulsive non Gaussian noise. The generalized Gaussian (GG) distribution has a density proportional to exp(−|x|p ), p > 0. A p less than 2 gives a distribution suitable as an impulsive noise model. By inferring p a wide class of probability distributions can be characterized including uniform, Gaussian, Laplacian and other sub- and super- Gaussian densities. In Figure 18, we plot the RMSE versus the parameter p that measures the deviation from the Gaussian model. We compare these methods in case of an additional reflected path delayed by ∆τ = 0.25Tc and SMR of 6 dB. As can be observed from figure 18, the narrow DLL and MEDLL fail to estimate correctly March 27, 2007

DRAFT

34 1 RML MEDLL (ML2)

0.9

0.8

0.7

RMSE

0.6

0.5

0.4

0.3

0.2

0.1

0 100

Fig. 17.

200

300

400

500 600 L: blocks number

700

800

900

1000

RMSE of the time-delay of the direct path versus the integration time blocks number L. Used parameters: M = 1,

ISR = 30 dB, SNR = -20 dB and SMR = 6 dB.

0.9 RML Narrow DLL MEDLL (ML2)

0.8

0.7

RMSE

0.6

0.5

0.4

0.3

0.2

0.1

0

Fig. 18.

0

0.2

0.4

0.6 0.8 1 1.2 1.4 The charecteristic exponent p of the impulsive noise

1.6

1.8

2

RMSE of the time-delay of the direct path versus the exponent characteristic p of the generalized Gaussian noise.

Used parameters: M = 1, L = 100 ms, SNR = -20 dB, SMR = 6 dB and ∆τ = 0.25Tc .

the time delay as the noise parameter p deviate from 2, i.e. as the noise becomes more impulsive and less Gaussian. This results make sense because narrow DLL and MEDLL are based on the Gaussian noise assumption that is not verified in this example. In general, this illustrate the robustness of the RML compared to the existing methods with respect to the environment modelization errors. We conclude that the performance of the conventional correlator and MEDLL (and ML2 of course), can be significantly deteriorated in presence of interference and/or non-Gaussian noise because it does not account for the non-Gaussianity environment. March 27, 2007

DRAFT

35

C. Multipath error envelopes comparison In this experiment, we compare the multipath-resistant for time of arrival estimation of the proposed RML scheme with the conventional narrow correlator (DLL) and the MEDLL algorithm using its refined version ML2 [38]. To do so, we consider the commonly used experiment for multipath error evaluation as described for Figure 11. For convenience, we recall that a direct path signal and one additional multipath are superimposed on the GPS receiver such that the signal-to-multipath ratio (SMR) is 6 dB. For the MEDLL we chose a fine grid of ∆τ = 0.1023Tc for the correlations computation that allows to separate short-delayed multipath signals. The equivalent parameter used in the ML2 implementation was set ε = 0.1Tc as the correlator spacing. Equivalently MEDLL computes ten correlation values per chip. For RML we show performances for q = 2 and q = 5. The results are given in Figure 19. The 20 Narrow DLL Narrow DLL 15

ML2 ML2

10

RML, q=5

Tracking error [m]

RML, q=5 RML, q=2

5

RML, q=2 0

−5

−10

−15

−20

0

0.5

1

1.5

Multipath delay [C/A chips]

Fig. 19.

Error envelopes for the narrow DLL (0.1 chip spacing) and the multipath maximum likelihood estimators ML2 and RML. The

parameters used are SNR = -20 dB, and MSR = 6dB.

error envelope of the proposed RML exhibits two regions of residual errors. On one hand, for a wide range of delay differences between 0.3Tc and 1.5Tc , both MEDLL and RML perform well and present a bias free delay estimation. The other region is between 0.1Tc and 0.3Tc . For this interval of short delayed reflected paths, it’s seem that the ML2 perform relatively as the superior algorithm. However, it is shown that multipath effects may be reduced using higher sampling rates. Thus, we remark that if it is possible to increase sufficiently the oversampling factor used in the proposed RML this algorithm could also reduce the effect of closely-spaced multipath. We believe that it is too expensive to use a factor of q = 10 for the RML implementation although it is a realistic rate for the modern systems [33]. We note

that this is a real trade-off problem between robustness and optimality issues of the maximum-likelihood estimator implementation. Indeed, RML uses more multiplication operations than a simplified ML2 (by March 27, 2007

DRAFT

36

assuming the amplitudes are constant) but it is more robust to non-Gaussian noise and interference as shown by simulations. Overall, the performance of the RML is similar to the performance of the MEDLL in a multipath but jemmer-free environment. For both methods, errors for small relative delays can be reduced if more number of samples per chip is used.

VII. C ONCLUSION A new block averaging processing to enhance the GPS signal power relative to the noise and zeromean interferences is introduced. The proposed block average model relies on the repetitive structure of the C/A code which remains intact when averaging the code over its replicas. Based on this enhanced block signal, we introduced two efficient and fast FFT -based acquisition algorithms. To mitigate the multipath effect on the time of arrival estimation, a robust maximum likelihood approach was proposed using the developed block-average model. It is based on applying a sample mean model to reduce all zero-mean interferences effect to a colored Gaussian noise effect. A whitening processing was then used to mitigate the high colored noise associated with the interference signals. Another advantage of using a long block averaging lead to Gaussian random variables which, in turn, allow a ML solution. Thus, we improved all GPS synchronization steps to provide a complete and robust software -based GPS receiver. We conclude that the weak GPS signal acquisition and tracking can be improved substantially with the aiding information provided by cellular networks, and/or exploiting the particular GPS temporal signal structure over a long block averaging coherent integration. The simulations results presented in this paper show that the proposed GPS synchronization approach provides good robustness against both interference and multipath which is highly attractive when operating in hostile and urban environments.

A PPENDIX I P ROOF

OF PROPOSITION

III-D.3

Proof: In this appendix, we compute the output power of a sub-band jammer after BAPCI, that was expressed through the following integral, Z Jav +Wj /2 1 sin(πf L) 2 2 Ej = df Wj −Wj /2 L2 sin(πf )

(51)

First of all, we may express the function under the integral using Fourier series as [41], L−1 1 sin(Lω/2) 2 1 2 X = + 2 (L − k) cos(kω) L2 sin(ω/2) L L

(52)

k=1

March 27, 2007

DRAFT

37

Using this formula into (51), we obtain ( ) Z +Wj /2 L−1 X J W 2 av j + 2 Ej2 = (L − k) cos(k2πf )df Wj L L −Wj /2 k=1

=

=

L−1 X

Jav 2Jav sin(kπWj ) + (L − k) 2 L Wj L kπ k=1 " # L−1 sin(kπWj ) Jav 1 2 X 1+ (L − k) . L L πWj k

(53)

k=1

Now, we would like to simplify further the above equality by computing the included sum. First, we rewrite the above sum and we recall two sum sines formulas for 0 < x < 2π . L−1 X k=1

L−1 X

sin(kx) =

k=1

L−1 X k=1

sin(kx) = k

Z

0

L−1

L−1

k=1

k=1

X sin(πkWj ) X sin(kπWj ) (L − k) =L − sin(πkWj ) k k

x

cos( x2 ) − cos(L − 12 )x 2 sin( x2 ) +∞

sin(L − 12 )x x X sin(kx) π−x dx − ≈ = when L is large enough. x 2 sin( 2 ) 2 k 2 k=1

We include these equalities and then we may write, L−1 X k=1

(L − k)

sin(kπWj ) k

= L

L−1 X k=1

πW πW sin(πkWj ) cos( 2 j ) − cos((L − 1/2) 2 j ) − k 2 sin( πWj ) 2

(1 − Wj ) = Lπ − Υ(L, Wj ) as L → ∞ . 2

(54)

πWj πWj j where Υ(L, Wj ) = [cos( πW 2 ) − cos((L − 1/2) 2 )]/2 sin( 2 ). Thus, we obtain a simple closed form

expression of a sub-band jammer BAPCI output power for large L as,   Jav 1 2Υ(L, Wj ) 2 Ej = − . L Wj LπWj

(55)

A PPENDIX II P ROOF

OF PROPOSITION

3: ROBUSTNESS

OF THE

ML

CORRELATOR TO A

CW

JEMMER

Here, we evaluate the CW interference term at the output of the maximum-likelihood correlator. Indeed, we proof that during the ML time-delay estimation procedure, the interference term is converted to a colored Gaussian noise at the output of the integrators. Proof: For simplicity, the received signal is modeled as the GPS signal of interest plus a continuous wave interference, as r(t) = s(t) + J(t) = As x(t − τ ) + AJ ej(2πfJ t+ϕJ ) , where fJ and ϕJ are the March 27, 2007

DRAFT

38

frequency and phase offsets of the CW interference relative to the GPS signal. It is convenient to recall that x(t) =

N −1 X n=0

d(n)c(t − nT ),

where d(n) is the data symbol and c(t) =

PL−1 l=0

t ∈ [o, N T ]

(56)

cl g(t − lTc ), is the spreading waveform, that is, the

result of modulating the C/A code cl ∈ {−1, +1} via the pulse shape g(t). During the cross-correlation process with the C/A code replica the ML operate as a correlator and deliver the early/late correlation outputs Rrc (n) = Rsc (n) + RJc (n), where Rsc (n) is the cross-correlation between the received GPS signal and the local code, and RJc (n) is given by Z (n+1)T 1 ± RJc (n) = Ai ej(2πfJ t+ϕJ ) c(t − τ − nT ± δTc )dt T As nT

(57)

where T is the integration time of the DLL. It can be expressed as Z (n+1)T L−1 X AJ ± RJc (n) = ej(2πfJ t+ϕJ ) cl g(t − τ − nT − lTc ± δTc )dt T As nT l=0 √ Z L−1 (n+1)T ISR X jϕJ = cl e ej2πfJ t g(t − τ − nT − lTc ± δTc )dt T nT l=0 √ Z L−1 ISR X jϕJ cl e g(t − τn,l )ej2πfJ t dt ; τn,l = τ + nT + lTc ± δTc = T l=0 √ L−1 ISR X jϕJ = cl e F [g(t − τn,l )] (−fJ ), T l=0

def

where F [g(t)] (f ) = G(f ) denotes the Fourier Transform of a function g applied in a frequency f . Then, we get ± RJc (n)

=

=

√ √

L−1 ISR X j(2πfJ τn,l +ϕJ ) cl e G(−fJ ) T l=0

L−1

ISR G(−fJ ) j[ϕJ +2πfJ (τ +nT ±Tc δ)] X j2πfJ lTc e cl e L Tc l=0

=

√

j[ϕJ +2πfJ (τ +nT ±Tc δ)]

ISRsinc(fJ Tc )e

L−1 1 X j2πfJ lTc cl e , L

(58)

l=0

where sinc(.) denotes the sinc function that is the spectrumof the rectangular pulse shape g(t). By the P j2πfJ lTc converges for large L to a Gaussian random variable (RV) central limit theorem, 1/L L−1 l=0 cl e

because cl are independent identically distributed (iid) RVs. We observe also from the above equation + − that the early RJc (n) and late RJc (n) outputs are fully correlated. Thus, RJc (n) is a complex Gaussian March 27, 2007

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39

RV, and then the CW interference is converted to a colored Gaussian noise for large blocks number L.

R EFERENCES [1] Denis M. Akos, ”A Software Radio Approach to Global Navigation Satellite System Receiver Design,” PhD Dissertation, Ohio University, August 1997. [2] D. M. Akos et al., ”Low Power GNSS Signal Detection and Processing,” in Proceedings of the ION conference GPS’2000, Sept. 2000. [3] M. G. Amin, Liang Zhao and A. R. Lindsey, ”Subspace array processing for the suppression of FM jamming in GPS receivers,” in IEEE Transactions on Aerospace and Electronic Systems, 40(1), pp: 80 - 92, Jan. 2004. [4] M. G. Amin and W. Sun, ”A Novel Interference Suppression Scheme for Global Navigation Satellite Systems Using Antenna Array,” in IEEE Journal on Selected Areas in Communications, Vol. 23 No. 5, 2005. [5] S. E. Bensley and B. Aazhang, ”Maximum-Likelihood Synchronization of a Single User for CDMA Communication Systems,” in IEEE Trans. On Comm., 46(3), March 1998. [6] R. Bischoff, and al., ”Multipath-Resistant Time of Arrival Estimation for Satellite Positioning,” Elsevier Int. J. Electronics and Communications, Vol. 58, pp: 3-12, 2004. [7] M. S. Braasch and A. J. Van Dierendonck, ”GPS receiver architectures and measurements,” in Procedings of IEEE, 87(1), pp: 48-64, Jan. 1999. [8] M. S. Braasch, ”Performance comparison of multipath mitigating receiver architectures,” in Proc. of IEEE Aerospace Conference, Vol. 3, March 2001. [9] Rod Bryant, ”Assisted GPS (Using Cellular Telephone Networks for GPS Anywhere),” in GPS WORLD, May 1, 2005. [10] I. H. Choi et al., ”A Novel Weak Signal Acquisition Scheme for Assisted GPS,” in Proc. of ION GPS’02, Sept. 2002. [11] A. Dempster, N. Laird, and D. Rubin, ”Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society, Series B, 39(1):138. 1977. [12] G. Dedes and A. G. Dempster, ”Indoor GPS Positioning: Challenges and Opportunities,” in Proc. of the IEEE VTC’05 Conf., USA, 2005. [13] A. El-Rabbany, Introduction to GPS : the Global Positioning System, Boston, MA : Artech House, 2002. [14] E. Ertin, U. Mitra, and S. Siwamogsatham, ”Maximum-likelihood-based multipath channel estimation for code-division multiple-access systems,” IEEE Trans. on Communications, Vol. 49, Issue 2, Feb 2001. [15] G. Feng and F. Van Graas, ”GPS Receiver Block Processing,” in proceedings of the ION conference GPS’99, Nashville, TN USA, Septemebr 1999. [16] Feder, M. and Weinstein, E. ”Parameter Estimation of Superimposed Signals Using the EM Algorithm,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 36, April 1988. [17] J. A. Fessler and A. O. Hero, ”Space-Alternating Generalized Expectation-Maximization Algorithm,” IEEE Trans. Signal Proc., 42(10), Oct. 1994. [18] B. H. Fleury et al, ”Channel Parameter Estimation in Mobile Radio Environments Using the SAGE Algorithm,” IEEE JSAC for Wireless Comm. Series, 17(3), March 1999. [19] P. D. Groves, ”GPS Signal to Noise Measurement in Weak Signal and High Interference Environments,” in Proceedings of the ION conference GNSS2005, Long Beach, CA USA, 2005. March 27, 2007

DRAFT

40

[20] S. Gunawardena, F. Van Gaas and A. Soloviev, ”Real Time Block Processing Engine for Software GNSS Receivers,” in Proceedings of the ION conference NTM 2004, San Diego USA, January 2004. [21] Carl W. Helstrom, Elements of Signal Detection and Estimation, Prentice Hall, 1995. [22] R. Iltis and L. Milstein, ”Performance analysis of narrowband interference rejection techniques in DS spread-spectrum systems,” IEEE Transactions on Communications, vol. 32, no. 11, pp. 1169-1177, November 1984. [23] E. D. Kaplan (Ed.), Understanding GPS: Principles and Applications, Artech House Publisher, Second Edition 2006. [24] Steven Kay, Intuitive Probability and Random Processes Using MATLAB, Springer, 2006. [25] J. M. Kelly and M. S. Braasch, ”Validation of theoretical GPS multipath bias characteristics,” in Proccedings of IEEE Aerospace Conference 2001, Vol. 3, March 2001. [26] M. Kokkonen, S. Pietila, ”A New Bit Synchronization Method for a GPS Receiver,” in Proceedings of the IEEE conference PLANS’2002, April 2002. [27] R. Jr. Landry, P. Boutin and A. Constantinescu, ”New anti-jamming technique for GPS and Galileo receivers using adaptive FADP filter,” in Elsevier Journal of Digital Signal Processing, Vol. 16, Issue 3, Pages 255-274, May 2006. [28] Jason Jones and Patrick C. Fenton, ”The Theory and Performance of NovAtel Inc.’s Vision Correlator,” NovAtel Inc. http://www.novatel.ca. [29] D. Moelker, E. van der Pol, and Y. Bar-Ness, ”Multiple antennas for advanced GNSS multipath mitigation and multipath direction finding,” in Proceedings of the ION conference GPS’97, 1997. [30] A. V. Oppenheim, A. S. Willsky With S. H. Nawab, Signals and Systems, 2nd Edition, NJ: Prentice-Hall, 1997. [31] B. W. Parkinson and J. J. Spilker (Eds.), Global Positioning System: Theory and Applications, Vol. 1 and Vol. 2, Progress in Astronautics and Aeronautics, 1996. [32] Mark L. Psiaki, ”Block Acquisition of Weak GPS Signals in a Software Receiver,” in Proceeding of the ION Conference GPS 2001, Salt Lake City, UT USA, September 2001. [33] H. Saarnisaari and V. Tapio, ”A Simple Multipath Delay Estimator Based on Alternating Projection Algorithm,” in Proceedings of the IEEE/ION conference PLANS’2006, San Diego, April 2006. [34] Mohamed Sahmoudi, Robust Separation and Estimation of non-Gaussian and/or non-Stationary Sources, Ph.D. dissertation, Univ. Paris-Sud (Paris XI), Orsay, France, December 2004. [35] M. Sahmoudi and M. G. Amin, ”Improved Maximum Likelihood Time Delay Estimation for GPS Positioning in Multipath, Interference and Low SNR Environments,” in Proceeding of the ION/IEEE Conference PLANS’2006, San Diego, April 2006. [36] M. Sahmoudi and M. G. Amin, ”A Novel Maximum-Likelihood Synchronization Scheme for GPS Positioning in Multipath, Interference and Weak Signal Environments,” in Proceedings of the IEEE conference VTC’2006 Fall, Montr´eal, Canada, September 2006. [37] S. Soliman et al., ”gpsOneT M : a hybrid position location system”, in Proc. of IEEE Sixth ISSSTA, Vol. 1, Sept. 2000. [38] J. Soubielle, I. Fijalkow, P. Duvaut and A. Bibaut, ”GPS Positioning in a Multipath Environment,” IEEE Trans. on Signal Processing, Vol 50, No. 1, 2002. [39] W. Sun and M. Amin, ”A Self-Coherence Anti-Jamming GPS Receiver,” IEEE Trans. on Signal Processing, Vol. 53, No. 10, October 2005. [40] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice Hall, NJ. 1992. [41] Georgi P. Tolstov, Fourier Series, (Translated from the Russian by R. A. Silverman), Prentice-Hall, Inc. New Jersy 1962. [42] B. R. Townsend and P. Fenton, ”A Practical Approach to the Reduction of Pseudorange Multipath Errors in a L1 GPS Receiver,” in Proceedings of ION GPS-94, Salt Lake City, USA, 1994. March 27, 2007

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[43] J. Bao-Yen Tsui, Fundamentals of GPS Receivers: A software approach, Second Edition, Wiley, 2002. [44] A. J. Van Dierendonck et al. ”Theory and Performance of Narrow Correlator Spacing in a GPS Receiver,” Journal of the Institute of Navigation, vol. 39, no. 3, Fall 1992. [45] R. D. J. Van Nee et al, ”The Multipath Estimating Delay Lock Loop: Approaching Theoretical Accuracy Limits,” in Proc. IEEE PLANS’1994, April 1994. [46] Philip L. Walker, The theory of Fourier series and integrals, Chichester; New York : Wiley, 1986. [47] P. W. Ward, ”GPS receiver interference monitoring, mitigation and analysis techniques,” Journal of the ION, Vol. 41, No. 4, Winter 1995. [48] Lawrence R. Weill, ”Multipath Mitigation Using Modernized GPS Signals: How Good Can it Get ?,” in Proceedings of ION GPS’2002, Portland, OR USA, 2002. [49] Lawrence R. Weill, ”Multipath mitigation. How good can it get with new signals ?,” GPS World, June 2003. [50] L. Winternitz, M. Moreau, G. J. Boegner and S. Sirotzky, ”Navigator GPS Receiver for Fast Acquisition and Weak Signal Space Applications,” in Proceeding of the ION congerence GNSS 2004, Long Beach, CA USA, September 2004. [51] Chun Yang and Shaowei Han, ”Block-Accumulating Coherent Integration over Extended Interval (BAPCIX) for Weak GPS Signal Acquisition,” Proceedings of the ION conference GNSS’2006, Fort Worth TX, USA, September 2006. [52] Qin Zhengdi, ”Self-Aiding in GPS Signals Tracking,” in Proceeding of the ION conference GPS’2002, Portland OR, USA, September 2002. [53] Nesreen I. Ziedan, GNSS Receivers for Weak Signals, Artech House, August 2006.

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