Multirate Signal Processing: a Solution for Wide-band ... - CiteSeerX

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Multirate Signal Processing: a Solution for Wide-band GNSS Signal Recovery Daniele Borio, Salvatore Fazio, G´erard Lachapelle PLAN Group, Department of Geomatics Engineering, University of Calgary

BIOGRAPHIES

ABSTRACT

Daniele Borio received the M.S. degree in Communication Engineering from Politecnico di Torino, Italy, the M.S. degree in Electronics Engineering from ENSERG/INPG de Grenoble, France, in 2004, and the doctoral degree in electrical engineering from Politecnico di Torino in April 2008. He has been a senior research associate at the PLAN group of the University of Calgary, Canada, since January 2008. His research interests include the fields of digital and wireless communication, location, and navigation.

In this paper, a new approach based on multirate signal processing is proposed for the recovery of new wide-band GNSS signals. The use of two or more narrow-band front-ends is suggested and different portions of the spectrum of the wide-band signal are recovered separately. The wide-band signal is then reconstructed from its narrow-band components. Algorithms for compensating phase offsets and amplitude imbalances have been developed and used to equalize possible differences among the narrow-band components. Specific focus is given to the case where two front-ends are employed and live GNSS data from the Galileo In-Orbit Validation Elements, GIOVE-A and GIOVE-B, are used for demonstrating the effectiveness of the developed multirate algorithms.

Salvatore Fazio is a M.S. student in Communication Engineering at the Politecnico di Torino, Italy. He received his B.Sc in Electrical Engineering, majoring in Telecommunications from the Politecnico di Milano in 2005. Since July 2008, he has been a visiting student at the PLAN Group of the Department of Geomatics Engineering at the University of Calgary where he is completing his master thesis on multirate signal processing for GNSS applications.

Professor G´erard Lachapelle holds a CRC/iCORE Chair in Wireless Location in the Department of Geomatics Engineering, the University of Calgary, where he has been a professor since 1988. He heads the Position, Location And Navigation Group and supervises over 25 research engineers and graduate students. He has been involved in a multitude of Global Navigation Satellite Systems (GNSS) R&D projects since 1980, ranging from RTK positioning to indoor location and GNSS signal processing enhancements. He has received numerous awards for his accomplishments.

KEYWORDS Global Navigation Satellite Systems, GNSS, Frontend, Multirate Signal Processing, Power Symmetric Filter, Wideband Signals 1

INTRODUCTION

New Global Navigation Satellite System (GNSS) signals are wide-band in nature [1]. This is a fundamental choice made to satisfy the growing demand of location based services, improving the performance of navigation signals and fully exploiting the technologies currently available. Wide-band signals have narrow correlation peaks that allow more precise range measurements and provide an increased resilience to multipath and radio frequency (RF) interference. The new European navigation system, Galileo, will provide three wide-band signals: the AltBOC in the E5 frequencies, the cosBOC(15, 2.5) on the Public Regulated Service (PRS) channel in the E1 band and the E6 modulation [1, 2]. These signals

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have bandwidths 10 to 20 times larger than the legacy GPS Coarse Acquisition (C/A) signal, requiring new and faster front-ends able to accommodate the increased signal rate without significantly distorting the received signal. The problem of recovering and digitizing wide-band signals becomes even more critical when a Software Defined Radio (SDR) approach is used. In SDR, a front-end is used to provide a high data rate, digital representation of the incoming signal directly from the antenna [3]. The digital samples are then processed using a general purpose platform, such as a desktop PC, allowing testing and analysis of the recovered signal without any additional hardware design. The drawback of SDR is that all the hardware complexity is moved into the front-end. The design of a front-end able to accommodate the large bandwidth of new Galileo signals is still a challenge and alternative solutions need to be considered. One potential solution entails splitting the problem complexity over several front-ends. In this way, each front-end recovers only a portion of the spectrum of interest and the final signal is obtained by combining the different data streams. In this paper, the use of several narrow-band front-ends for the recovery of new GNSS signal is investigated. More specifically, the spectrum of a GNSS signal is divided in two or more components, and each part is recovered by a different front-end. Multirate techniques [4] are then used for the reconstruction of the original wide-band signal. This type of approach is, to the authors’ knowledge, new, and represents one of the main contributions of this paper. The development of this approach has required the selection of appropriate reconstruction filters and the design of estimation techniques for recovering the relative phase between signal components. When two or more front-ends are used for recovering the different portions of the GNSS spectrum, different Phase Lock Loops (PLL) are required for the signal down-conversion. This leads to constant phase offsets even if the same clock is used for driving the PLLs. These phase offsets need to be compensated for, requiring the design of two new phase estimation algorithms. The relative phase information is then used for the wide-band signal reconstruction. In the simplest configuration, two front-ends are used for collecting the upper and lower portions of the spectrum of a wide-band GNSS signal. The two components are filtered, up-sampled and modulated in order to compensate for the constant random phase shift introduced by the front-ends and possible frequency mismatches. Once processed, the upper

and lower components are summed together, leading to the reconstructed wide-band signal. Power Symmetric Finite Impulse Response (PS-FIR) filters [4] have been chosen for shaping the spectra of the upper and lower band components. These filters are required for removing common components in the input signals, avoiding mutual interference and allowing the correct matching of adjacent portions of the input spectra. The proposed techniques have been tested using live GNSS data, recovered from the Galileo experimental satellites, GIOVE-A and GIOVE-B, showing the validity of the developed theory and the effectiveness of the multirate approach.

The remainder of this paper is organized as follows: in Section 2 the basic principles of Multirate signal processing are introduced with particular emphasis on their applicability to the GNSS context. Section 3 provides a general overview of the technique developed for the reconstruction of wide-band GNSS signals with a description of the selected reconstruction filters. In Section 4, the two different approaches developed for estimating phase differences among the different signal components are detailed. The experimental setup adopted for testing the developed algorithms with live GNSS data is detailed and the validity of the multirate approach demonstrated on wide-band signals broadcast by the GIOVE satellites.

2

THE MULTIRATE APPROACH

Multirate refers to all those signal processing techniques that involve changes in the sampling rate of the digital signals being processed. Multirate techniques have been used extensively in different research areas, such as audio and image processing, signal compression and communications. Changes in sampling rate allow computationally efficient algorithms and simplify operations that would be extremely complex if implemented using a constant sampling frequency [4]. Multirate techniques are usually based on the use of filter banks for:

• analysis: the input signal is split into sub-band components with lower sampling rate; • synthesis: several sub-band components are merged to produce a higher rate output signal.

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1207.14 MHz

yup[n] Front-end 1 E5b

Wide-band signal

Amplitude/ phase/frequency recovery

y(t)

Multirate Engine

y[n]

Front-end 2 E5a

ylo[n] 1176.45 MHz

Figure 1: Multirate approach for the recovery of wide-band GNSS signals. Two different front-ends recover different portions of the signal spectrum. The relative phase between the two spectral components is estimated and used for the reconstruction of the wideband signal.

the output of the different front-ends do not in general respect this requirement and additional processing is required for compensating these misalignments. The last condition is achieved using analysis filters that remove spectral components that would alias in contiguous bands. All three requirements are addressed in the following sections and specific algorithms are developed for compensating for possible amplitude, phase and frequency mismatches. In this paper, specific focus is given to the case where two front-ends are used. However, the algorithms presented are general and can be extended to use several front-ends. 3

In this paper, the latter application is considered and adapted to the case of wide-band GNSS signals. More specifically, an analog wide-band signal, y(t), is split into two or more sub-bands that are recovered by different front-ends. The sub-band components are then processed and fed into a multirate engine that aims at reconstructing a digital version, y[n], of y(t). An example of such processing is provided in Fig. 1, where the specific case of the Galileo AltBOC modulation is considered. Two front-ends are used for recovering the E5a and E5b components of the AltBOC signal. In order to produce a digital signal, y[n], that closely approximate the analog wide-band signal, y(t), the components obtained at the output of the different front-ends have to mimic the sub-band components that would have been obtained by using an analysis filter bank applied on a wide-band digital signal. For this reason, three requirements must be satisfied by the components:

SIGNAL RECONSTRUCTION

In this section, the procedure adopted for the reconstruction of wide-band GNSS signals is detailed. It is noted that the two side-band components cannot be directly recombined, since they have different centre frequencies and phases; they have to be up-sampled to the sampling frequency of the wide-band signal to be reconstructed. The following operations have to be performed:

• pre-filtering for reducing out-of-band noises; • amplitude equalization; • up-sampling for doubling the sampling frequency; • frequency and phase compensation: the two side-band components have to be aligned in phase and share a common centre frequency.

• they must be correctly equalized; • they must be aligned in phase and frequency; • they must not to interfere on adjacent bands. The first condition implies that side-band components should have similar power levels in contiguous bands, i.e., the front-ends have to scale the input signal in the same way. If multi-bit front-ends are used, Automatic Gain Controls (AGCs) can scale the side-band components differently , providing signals with different magnitudes. If this effect is not compensated, the reconstruction of the wide-band signal cannot be performed without significant distortions. The second condition requires the alignment of side-band components in both phase and frequency. The signals at

These operations are better detailed in Fig. 2 where two side-band components are at first filtered for reducing out-of-band noise. After pre-filtering, the signal at the output of the second front-end is scaled by the gain β to avoid amplitude distortions. In this case, since the two frontends have similar frequency response and the input signal, before de-spreading, is dominated by the noise component, the two side-band components should be characterized by the same power. Thus, β is determined as σ ˆ1 (1) β= σ ˆ2 where σ ˆ 21 and σ ˆ 22 are estimates of the power of the sub-band components. This operation corresponds to sub-band amplitude equalization.

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exp{ j 2πΔf1nTs }

Digital signals with sampling frequency fs Front-end 1

Up-sampling 2

ℜe{} ⋅

B1 ( z ) Hilbert Transform

a)

j Analytic signal

Front-end 2

2

β

B2 ( z )

Hilbert Transform

Pre-filtering

ℜe{} ⋅

b)

j Analytic signal

Δφˆ

Phase estimation

Phase rotation

exp { j 2πΔf 2 nTs }

Figure 2: Operations required for the correct signal reconstruction. The components from the two front-ends are at first filtered and up-sampled. Phase and frequency compensation are then performed for the correct alignment of the two side-band signals. Phase and frequency compensation is achieved in different stages. At first the analytic signal of the input components is evaluated. The use of analytic signals simplifies frequency shift and phase rotation of the side-band components. The two side-band components are then up-sampled and the sampling rate is multiplied by a factor two. This doubles the range of digital frequencies, allowing the correct alignment of the two signal components. After up-sampling the

New frequencies available after up-sampling

Frequency alignment

ℑm

ℑm

Side-band components Phase alignment

f

f

ℜe

fs 2 a)

f s' = fs 2

ℜe

f s' = fs 2

side-band components can be aligned in frequency and the phase difference can be compensated. The principle of frequency and phase alignment is illustrated in Fig. 3: the two signal components are shifted in frequency and aligned with respect to a common centre frequency. The phase difference between the two signals is obtained from the phase estimation block that implements one of the two techniques described in Section 4. At this point only the real part of two side-band components is retained. In this way, the analytic complex signals are transformed back to real signals. It is noted that this operation can potentially cause aliasing. However, the real operation makes the side-band components fold into portions of the spectrum that will be removed by the analysis/synthesis filters, producing an alias-free reconstruction.

3.1

Wide-band signal reconstruction

b)

Figure 3: Phase and frequency alignment of the sideband components. The side-band components are shifted in frequency and aligned with respect to a common centre frequency. Phase differences are also compensated. a) Before alignment. b) After alignment.

After the operations described in the previous section, the wide-band GNSS signal can be finally reconstructed. The processing required for the wideband signal reconstruction is detailed in Fig. 4. The two side-band signals enter two analysis filters that shape the boundaries of the spectra of the two components. In this way it will be possible to recombine the

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filters are related by the following conditions: a)

H0(z)

F0(z)

H1 (z) = −z−N H0∗ (−z) α

b)

H1(z)

Real signals with sampling frequency 2fs

Analysis filters

F1(z)

Reconstructed wideband GNSS signal

Synthesis filters

Figure 4: Wide-band signal reconstruction: the two processed side-band components are recombined using an analysis/synthesis filter bank.

two side-components avoiding significant distortions at the spectra boundaries. The signals at the output of the analysis filters approximate the high and low components that would have been obtained by splitting a wide-band signal into sub-bands. Synthesis filters are then used for reconstructing the wide-band signal. In Fig. 4, the reconstructed wide-band signal is further multiplied by a constant gain α. This is because the obtained signal will be saved in a new binary file with a limited number of bits. The gain α is used for limiting the quantization noise in the reconstructed file, allowing the use of the full quantizer dynamic range. When more than two front-ends are used, the amplitude, phase and frequency mismatches among the different signal components have to be compensated using the algorithms described above and in Section 4. Moreover, M-channel filter banks have to be employed for the signal reconstruction.

3.2

F0 (z) = z−N H0∗ (z)

To disk

Power Symmetric FIR Filters

The filter bank reported in Fig. 4 plays an essential role for the reconstruction of wide-band GNSS signals. More specifically, it is required for removing common components in the side-band signals and for properly shaping their spectra. For these reasons, the four filters, H0 (z), H1 (z), F0 (z) and F1 (z), have to be carefully chosen in order to produce an alias-free reconstructed signal and avoid phase and amplitude distortions. Necessary conditions ensuring these properties can be found in [4] where several classes of filters fulfilling those requirements are also described. In this paper, Power Symmetric FIR filters have been adopted. These filters have been chosen for their stability and for the design simplicity [4, 5]. The four

F1 (z) = z

−N

(2)

H1∗ (z)

where N is the filter order and has to be odd. (·)∗ denotes complex conjugate. Conditions (2) ensure that the filter bank is alias-free. Moreover, if H0 (z) satisfies the power symmetry condition: H0∗ (z)H0 (z) + H0∗ (−z)H0 (−z) = 1

(3)

the filter bank provides perfect reconstruction (PR) of the input signal, i.e., the output of the filter bank is a scaled and delayed version of the original signal. H0 (z) is a low-pass filter and has been designed using the procedure described in [5, 4]. 4

PHASE RECOVERY

The correct reconstruction of the wideband signal can be obtained only if the input components are synchronously sampled and their phase aligned. The first condition can be achieved by driving the front-ends with a common clock whereas phase alignment can be achieved by estimating and compensating possible phase differences. Two different methods for phase recovery are considered and detailed in the following subsections. 4.1

The Prony-Like Method (PLM)

The first method represents a generalization of Prony’s method for frequency estimation [6]. According to Prony’s method, the frequency of a complex sinusoid can be estimated by evaluating the projection of the signal over a delayed version of itself [6]. More specifically, let be y[n] a complex exponential with unknown initial phase, φ, and frequency fc : y[n] = A exp { j2π fc nT s + jφ} ,

(4)

where T s is the sampling interval and A is the signal amplitude. Then, according to Prony’s method, the unknown frequency, fc , can be determined as 1  ∠ y[n]y∗ [n − 1] 2T s 1  = ∠ exp { j2π fc nT s + jφ} 2T s · exp {− j2π fc (n − 1)T s − jφ} .

fc =

(5)

If the input signal is corrupted by a zero mean white noise,

European Navigation Conference 2009 - Naples, Italy - May 3-6, 2009

y[n] = A exp { j2π fc nT s + jφ} + η[n],

(6)

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the frequency estimate produced by (5) will be noisy, but can be improved by averaging as follows   N    1 X  1   ∗ ∠ fˆc = y[n]y [n − 1] .      2T s N − 1 n=1

(7)

A similar approach can be adopted for estimating the phase difference between two correlated signals when the difference between their central frequencies is known. Prony’s method can be at first generalized for determining the phase difference between two complex sinusoids characterized by a known frequency offset and affected by uncorrelated noise sequences: E

h

η1 [n]η∗2 [n]

i

= 0.

(8)

Let y1 [n] and y2 [n] be the two complex sequences y1 [n] =A1 exp j2π fc,1 nT s + jφ1 + η1 [n]  y2 [n] =A2 exp j2π fc,2 nT s + jφ2 + η2 [n] 



(9)

where

• A1 and A2 are the amplitudes of the useful signal components; • fc,1 and fc,2 are signal frequencies whose difference, ∆ f = fc,1 − fc,2 , is known; • φ1 and φ2 are two unknown phases; • η1 [n] and η2 [n] are two uncorrelated noise sequences. Similarly to (7), the phase difference, ∆φ = φ1 − φ2 , can be evaluated as   N−1     1 X   ∗ y [n] , ∆φˆ = ∠  y [n] exp j2π∆ f nT } {−   1 s 2   N n=0 (10) that is the phase of the projection of y2 [n] on y1 [n] after that the frequency difference, ∆ f , has been compensated for. Eq. (10) can be also used when the useful components of y2 [n] on y1 [n] are two modulated signals with a non-zero correlation:  y1 [n] =A1 [n] exp j2π fc,1 nT s + jφ1 + η1 [n]  y2 [n] =A2 [n] exp j2π fc,2 nT s + jφ2 + η2 [n]

(11)

with h i E A1 [n]A∗2 [n] = R1,2 , 0.

(12)

In this case, the projection of y2 [n] on y1 [n], after frequency compensation, becomes N−1 1 X y1 [n] exp {− j2π∆ f nT s } y∗2 [n] N n=0

=

N−1 n o 1 X A1 [n]A∗2 [n] exp j∆φˆ N n=0

N 1 X + η1 [n] exp {− j2π∆ f nT s } η∗2 [n] N n=0

+

(13)

N−1  1 X A1 [n] exp j2π fc,2 nT s + jφ1 η∗2 [n] N n=0

N−1  1 X + η1 [n]A2 [n]∗ exp − j2π fc,1 nT s − jφ2 . N n=0

For N → ∞ and by exploiting the law of large numbers, it is possible to substitute the average operators in (13) by the statistical mean, leading to N−1 1 X lim y1 [n] exp {− j2π∆ f nT s } y∗2 [n] N→∞ N n=0 n o h i = E A1 [n]A∗2 [n] exp j∆φˆ h i + E η1 [n]η∗2 [n] exp {− j2π∆ f nT s } h i  + E A1 [n]η∗2 [n] exp j2π fc,2 nT s + jφ1 h i  + E η1 [n]A∗2 [n] exp − j2π fc,1 nT s − jφ2 n o = R1,2 exp j∆φˆ

(14)

where the uncorrelation between η1 [n], η2 [n], A1 [n] and A2 [n] and the fact that η1 [n], η2 [n] are zero mean, have been exploited. Eq. (14) proves the validity of (10) for estimating the phase difference between two correlated modulated signals. Similar results hold for real signal, when the complex exponentials in 11 replaces by real sinusoids. This algorithm has been named the PronyLike Method (PLM) since it represents a generalization of Prony’s frequency estimation technique. The principle of the algorithms is better illustrated in 5, where the phase difference between y1 [n] and y2 [n] is obtained from the projection of one signal over the other, after compensating for the frequency difference. The PLM requires that two correlated signals are present in the two components collected by the different front-ends. Moreover, the difference between their centre frequencies has to be known. This can be achieved by making the two signal components share a common portion of the spectrum, containing a training signal. For example, for the reconstruction of the cosBOC(15, 2.5) of the GIOVE E1a signal, the lower and upper band components were made

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{

exp − j2π ( fc,1 + fd ) nTs

y1[n]

1 N

y2[n]

N −1

∑ (⋅) i =0

∠

}

c1[n − τ ]

y1[n]

(⋅ )

*

{

exp j2π ( fc,1 − fc,2 ) nTs

}

N −1

1 N

∑ (⋅)

1 N

N −1

i =0

∠

Integration y2[n]

∑ (⋅) i =0

(⋅ )

*

Integration

Figure 5: Prony-Like Method: the phase difference between two signals is estimated as the phase of the projection of y1 [n] over y2 [n] after compensating for known frequency differences.

overlapping around the L1 centre frequency (1575.42 MHz). Consequentely, GPS L1 Coarse/Acquisition (C/A) signals were present in both data sets and used for phase recovery. The advantage of the PLM is that it does not require any a-priori knowledge about the correlated signals in the common portion of the spectrum. Moreover, it requires a modest computational load, allowing on-line phase estimation. It is noted that the PLM behaves similarly to differentially coherent detection techniques for directsequence code acquisition [7, 8]. In differentially coherent detection [8], the input signal is delayed and multiplied by itself in order to remove the effect of the frequency modulation. In this case, the two signal are multiplied to remove the effect of common frequency modulation, allowing phase estimation Similarly to the case of differentially coherent detection, the drawback of the PLM is the noise amplification caused by the multiplication of the noise components in the two correlated signals. This effect can however be compensated by using long integrations.

4.2

Correlation Based Method (CBM)

The second method, called the Correlation Based Method (CBM), exploits the properties of the wideband GNSS signal to be reconstructed. More specifically, GNSS signals usually adopt a direct-sequence spread spectrum (DSSS) modulation and it is usually possible to acquire and track the signal components captured in both lower and upper bands independently. Thus, the phase difference can be estimated after despreading the side-band components. This requires precise knowledge of the signal frequency, since a residual frequency offset would lead to a timevarying phase difference. The general scheme for the CBM is shown in Fig. 6. The phase difference is esti-

{

exp − j2π ( fc,2 + fd ) nTs

}

c2 [ n − τ ]

Figure 6: Phase difference estimation by using the Correlation Based Method. The two signals are first despreaded and the phase is estimated from the correlation. mated as ∆φˆ = n o = ∠ Rc,1 ( fd , τ)R∗c,2 ( fd , τ)  N−1    1 X = ∠ y1 [n]c1 [n − τ] exp −2π( fc,1 + fd )nT s  N n=0  N−1 ∗   1 X      ·  y2 [k]c2 [k − τ] exp −2π( fc,2 + fd )kT s     N k=0 (15) where Rc,1 ( fd , τ) and Rc,2 ( fd , τ) are the crosscorrelations of the incoming signals with two locally generated replicas. τ and fd are the code delay and Doppler frequency of the GNSS signal. c1 [n] and c2 [n] are the local codes used for despreading the incoming signals and they do not need to be the same. For example, the two side components of the AltBOC signal can be interpreted as two QPSK modulations with different spreading codes, from the E5a and E5b modulation [2]. In the cosBOC(15, 2.5) case c1 [n] and c2 [n] are the same. The Doppler frequency, fd , and the code delay, τ, have to be the same for the despreading of both y1 [n] and y2 [n]. For this purpose, one of the two components was used two drive a PLL the frequency estimate of which was, in turn, employed for despreading both lower and upper-band signals. Despreading allows significant noise reduction in the input signals leading to a less noisy phase estimate with respect to the PLM, when the same integration time is employed and the same signal power is available. The drawback of the CBM is that it requires a precise knowledge of the code delay and Doppler frequency, implying a high computational load. The CBM is also affected by processing biases that arise when different types of processing are adopted for the evaluation of the cross-correlations. For ex-

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ample, when side-lobe processing [9] is used for despreading the lower and upper components, a phase bias occurs. This is due to the fact that BOC side lobes are not, in general, symmetric with respect to their centre frequency. Because of this asymmetry, the two side components experience different phase rotations. The resulting phase bias depends on both the sub-carrier spectrum and front-end characteristics and is the same for different datasets when the same type of processing is used. It can therefore be estimated and compensated using training signals. Because of these drawbacks, i.e. the computational complexity and the processing biases, the PLM should be preferred.

cosBOC(15, 2.5) Lower lobe

GPS C/A

a)

Phase estimation results

Both algorithms were tested on several data sets. In this section, sample results for the phase difference estimation are shown. More specifically, two cases have been considered:

• the Galileo cosBOC(15, 2.5) signal in the L1 band;

f

1575.42 MHz Front-end 1

Front-end 2

Training sequence E5b

E5a

b)

f

1191.795 MHz Front-end 1

4.3

cosBOC(15, 2.5) Upper lobe

BOC(1,1) (Giove-A)

Front-end 2

Figure 7: Spectrum configuration for the recovery of wide-band GNSS signals. a) CosBOC(15, 2.5): the two front-ends recover a common part of the spectrum around the L1 centre frequency, the totality of the GPS C/A signals are used as training sequence for phase recovery. b) AltBOC: a continuous wave is added to the input signals using an RF combiner in the E5 centre frequency. This pilot tone is used for the estimation of the phase difference.

97

• the Galileo AltBOC in the E5 frequencies.

The adopted configuration are shown in Fig. 7. In the first case, the totality of the GPS L1 C/A signals are used as training sequence in a common portion of the spectra collected by the two front-ends. In the AltBOC case a Continuous Wave (CW) is injected at the 1192 MHz frequency and used as training signal. Results for the cosBOC(15, 2.5) case are shown in Fig. 8. In this case, the two side-band components both contain the L1 C/A and Galileo E1b and E1c signals. A PLL was used to track the E1c signal and the same type of processing was used for evaluating the correlator outputs on both components. Thus, no processing bias was found and both the PLM and the CBM provide the same results. In Fig. 8, the PLM provide smoother phase estimates when the same total integration time is adopted. This is due to the much higher power that the PLM is recovering, that is all L1 signals transmitted by both the GPS and GIOVE satellites. For the CBM, only the E1c signal was used and the coherent integration time was limited to 8 ms. The total integration time was further increased by using a moving average filter on the phase estimations. For the evaluation of the phase difference in the AltBOC case, only the PLM method was used.

Phase Difference [deg]

96

95

94

93

92

91 Correlation Based Method Prony-Like Method

90 0

10

20

30

40

50 Time [s]

60

70

80

90

100

Figure 8: Phase difference estimation for the reconstruction of the cosBOC(15, 2.5) signal. The same total integration time of 0.4 s has been used for both the CBM and PLM. 5

EXPERIMENTAL SETUP AND REAL DATA ANALYSIS

This section describes the experimental setup and the methodologies adopted for testing the developed multirate algorithms. At first, the case of the BOC(1, 1) signal is detailed. Although the BOC(1,1) signal cannot be considered wide-band, it is characterized by

European Navigation Conference 2009 - Naples, Italy - May 3-6, 2009

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Roof antenna

BOC signal LNA Full signal

Front-end 1

Reference signal

Front-end 2

Multi-rate processing

Front-end 3 Side-band components

Figure 9: Experimental setup adopted for the reconstruction and analysis of the BOC(1,1) signal. Three front-ends have been employed for the recovery of the full-band and side-band components. b) Normalized Correlation Function

Normalized Correlation Function

a) 1 0.8 0.6 0.4 0.2 0 3.355

3.356

3.357 3.358 delay [ms]

3.359

1 0.8 0.6 0.4 0.2 0 3.355

c) Power/frequency [dB/Hz]

-60 -70 -80 2 4 Frequency [MHz]

3.356

3.357 3.358 delay [ms]

3.359

d)

-50

0

5.1

National Instruments PXI-5661

Splitter

Power/frequency [dB/Hz]

two lobes that can be easily split into side components. This allows the recovery of both full-band and side-band components that, in turn, allows comparative analysis between the original and reconstructed signals. The BOC(1,1) signal is also used for testing the impact of phase estimation errors. The reconstruction of the cosBOC(15, 2.5) is then detailed and the experimental setup adopted for the AltBOC modulation is briefly described. The results presented in this section have been obtained using live data from by the Galileo In-Orbit Validation Elements, GIOVE-A and GIOVE-B [10]. The signals have been collected by a National Instruments (NI) system composed of three NI PXI-5661 front-ends [11]. This system allows the use of the three front-ends in synchronous mode, i.e., the three front-ends can be driven by the same clock leading to time-synchronized sampling. NI PXI-5661 frontends allow the selection of several sampling frequencies with a maximum rate of 50 MHz real sampling. The local frequency, for the signal down-conversion, can be selected in the range [1 MHz − 2.7 GHz] [11]. Since the three front-ends use different PLLs for the signal down-conversion, phase differences are observed among the recovered signals. This motivates the development of the techniques described in Section 4. The side and full-band components have been processed using a customized version of the University of Calgary’s software receiver, GSNRxTM [12].

6

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2 4 Frequency [MHz]

6

BOC(1, 1) processing

The BOC(1, 1) signal has been recovered according to the experimental setup described in Fig. 9 which shows an open-sky signal split among the three frontends of the NI system. The first front-end has been used for recovering the full BOC(1, 1) signal. This signal was used as reference for assessing the quality of the reconstructed signal. Front-ends 2 and 3 have been used for the recovery of side-band components. The side-band components were then combined using the techniques described in Sections 4 and 3. Some sample results for the reconstructed signal are reported in Fig.s 10 and 11. In Fig. 10 depicts both the correlation and Power Spectral Density (psd) of the reconstructed signal. Fig. 10 also shows the role played by the process of phase estimation and compensation. In Figs. 10 a) and c), the signal is reconstructed without recovering the phase difference between the side components. In this case, significant distortions can be observed on both psd and correlation function. More specifically, phase errors make

Figure 10: Impact of the phase estimation on the signal reconstruction. a) Correlation function of the reconstructed signal in the presence of phase estimation errors. b) Correlation function of the reconstructed signal with correct phase recovery. c) Psd of the reconstructed signal in the presence of phase estimation errors. d) Psd of the reconstructed signal with correct phase recovery. the two portions of the reconstructed spectrum add destructively, producing the notch observed in Fig. 10 c). When the phase difference is correctly recovered, the spectrum components are added in phase and no significant distortion can be observed at the junction point. This is shown in Figs. 10 b) and d) where the full BOC(1, 1) signal is correctly reconstructed. Original and reconstructed signals have been further compared in terms of Carrier-to-Noise density ratio (C/N0 ). The GSNRxTM software has been used for acquiring and tracking the side-band and the full

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45 Full-band signal Reconstructed signal Lower-band component Upper-band component

44 43

C/N0 [dB-Hz]

42 41 40 39

Table 2: Front-end parameters for the cosBOC(15, 2.5) reconstruction. Parameter Value Sampling frequency 50 MHz Front-end bandwidth 22 MHz Centre frequency #1 5.42 MHz Centre frequency #2 20.42 MHz Sampling Type Real

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Figure 11: C/N0 of the different signals involved in the BOC(1,1) reconstruction and analysis. Table 1: Average C/N0 of the different signals involved in the BOC(1,1) reconstruction and analysis. Signal C/N0 Full-band 41.5 dB-Hz Reconstructed 41.1 dB-Hz Lower-band 39.6 dB-Hz Upper-band 38.4 dB-Hz

BOC(1, 1) signals and the Narrow-Wideband Power Ratio (NWPR) method [13] has been used for estimating the C/N0 of each component. The evolution of the C/N0 as a function of time is shown in Fig. 11. It is noted that the full-band signal collected from the first front-end and the one obtained by combining the two side components are characterized by almost identical C/N0 values and only a difference of about 0.4 dB is observed. The side-band components are characterized by a C/N0 of 39.4 and 38.4 dB-Hz. This difference can be explained by the fact that it was not possible to split the BOC(1, 1) side lobes in a completely symmetric way. The front-end used for recovering the lower lobe was also collecting part of the upper component, justifying the increased signal power measured by the C/N0 estimator. The average C/N0 values are summarized in Table 1: a 3 dB gain is observed in the reconstructed signal with respect to the weaker of the side components. 5.2

cosBOC(15, 2.5) reconstruction

In this section, results for the cosBOC(15, 2.5) reconstruction are provided. In this case only two frontends have been used and the parameters adopted for

the data collection are reported in Table 2. The centre frequencies reported in Table 2 refer to the values to which the L1 centre frequency (1575.42 MHz) has been mapped by the respective front-ends. It is noted that each single front-end does not allow the recovery of the full cosBOC(15, 2.5) signal that has a reference bandwidth of 32.736 MHz. Only the use of two frontends and the multirate techniques described above allow the full recovery of this wide-band signal. In Fig. 12, the psd of the reconstructed cosBOC(15, 2.5) signal is shown. The reconstructed signal is characterized by a sampling frequency f s = 100 MHz and effective bandwidth of more than 40 MHz. It is noted that the bandwidth of the reconstructed signal is less than twice the single front-end bandwidth. This is due to the common portion of spectrum recovered by the two front-ends to allow phase recovery and a better reconstruction of the signal components at the intersecting boundaries of the spectra. The L1 centre frequency has been mapped to the 25 MHz frequency. No significant distortion can be observed in the reconstructed spectrum and the two side components are properly merged at the new centre frequency. It is noted that the spectrum in Fig. 12 has a maximum at the centre frequency and that the signal is progressively attenuated as the spectrum boundaries are reached. This effect is likely due to the employed antenna that was not designed for the reception of such a wide signal. The antenna provides a good gain in correspondence of the L1 centre frequency and progressively attenuates the signal components away from it. The reconstructed signal was processed using the GSNRxTM software and the results are shown in Fig. 13. The GSNRxTM software was able to process both the full-band signal and the side components. Single-side processing [9] was adopted for independently tracking the cosBOC(15, 2.5) side components. It is noted that the Doppler estimates of the three signals considered in Fig. 13 differ slightly. This is due to the different centre frequencies of the three signals that impact the Doppler effect. In Fig. 13, different estimated C/N0 are also shown. As for the BOC(1, 1) case, the average C/N0 of the recon-

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cosBoc(15, 2.5)

Welch PSD Estimate 1

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Normalized Autocorrelation Function [unitless]

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E1a lower lobe centre frequency 1550.42

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Figure 12: Psd of the reconstructed cosBOC(15, 2.5) signal.

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

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Figure 14: Autocorrlation of the cosBOC(15, 2.5) signal: theoretical and estimated from the reconstructed signal.

Doppler Estimate [Hz]

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Figure 13: Doppler frequency and C/N0 estimates obtained using the GSNRxTM software on the reconstructed cosBOC(15, 2.5) signal.

structed signal is about 3 dB higher than the one of the weakest side-band component. Differences between lower and upper components are present and can be due to several factors such as different front-end filtering and antenna gain. Further analysis is required for better understanding these differences. The quality of the reconstructed signal is further analyzed in Fig. 14 where the autocorrelation function of the cosBOC(15, 2.5) is shown. More specifically, the cosBOC(15, 2.5) autocorrelation function has been estimated from the reconstructed signal and compared with the corresponding theoretical model. A good agreement is found between theoretical and empirical results, further supporting the effectiveness of the developed algorithms.

Figure 15: Experimental setup adopted for the reconstruction of the AltBOC signal.

5.3

The AltBOC signal

As highlighted in Section 4, the AltBOC signal reconstruction required the use of a pilot tone for the estimation of the phase difference between the two side components. For this reason the experimental setup shown in Fig. 15 has been adopted. In this case an RF signal combiner has been used for injecting a continuous wave at the 1192 MHz frequency. The reconstructed signal was processed using the GSNRxTM software that was able to successfully process the E5a and E5b components independently. However, due to the complexity of the AltBOC modulation, it was not possible to completely assess the quality of the reconstructed signal. For this reason, a methodology for determining the quality of the recovered AltBOC signal is under development and the analysis of the AltBOC reconstruction is left for fu-

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ture work. 6

CONCLUSIONS

In this paper, multirate techniques have been developed for the reconstruction of wide-band GNSS signals from their side-band components. The developed algorithms have been tested using real data, leading to the successful reconstruction of wide-band Galileo signals. This has proven the feasibility of recovering wide-band signals using several narrow-band frontends, showing the potential of multirate signal processing techniques in the treatment of GNSS signals. Future work includes the extension of the obtained results to the case of several front-ends and a characterization of the reconstructed AltBOC signal. ACKNOWLEDGMENT The authors would like to thank Tom Williams for reviewing the document.

dissertation, National University of Ireland, Cork, Jan. 2007. [8] C.-D. Chung, “Differentially coherent detection technique for direct-sequence code acquisition in a rayleigh fading mobile channel,” IEEE Trans. Commun., vol. 43, no. 234, pp. 1116– 1126, Mar. 1995. [9] P. Fishman and J. W. Betz, “Predicting performance of direct acquisition for the M-code signal,” in Proc. of the International Technical Meeting of the Institute of Navigation (ION/NTM), Anaheim, CA, USA, Jan. 2000, p. 574582. [10] Galileo Project Office, “GIOVE-A + B navigation signal-in-space interface control document,” European Space Agency, First issue ESA-DTEB-NG-ICD/02837, Aug. 2008. [11] 2.7 GHz RF Vector Signal Analyzer with Digital Downconversion, National Instruments, http://www.ni.com/pdf/products/us/cat vectorsignalanalyzer.pdf, 2006.

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[12] M. G. Petovello, C. O’Driscoll, G. Lachapelle, D. Borio, and H. Murtaza, “Architecture and Benefits of an Advanced GNSS Software Receiver,” in Proc. of International Symposium on GPS/GNSS, Tokyo, Japan, Nov. 2008, p. 11. [13] A. J. Van Dierendonck, “Gps receivers,” in Global Positioning System: Theory and Applications, B. W. Parkinson, J. J. Spilker, P. Axelrad, and P. Enge, Eds., vol. 1. Amer. Inst. for Aeron. and Astron., 1996.

[3] J. Mitola, “The software radio architecture,” IEEE Commun. Mag., vol. 33, no. 5, pp. 26–38, May 1995. [4] P. P. Vaidyanathan, Multirate Systems And Filter Banks. Prentice Hall PTR, Oct. 1992. [5] F. Mintzer, “Filters for distortion-free two-band multirate filter banks,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 33, no. 3, pp. 626 – 630, June 1985. [6] B. Porat, Digital Processing of Random Signals: Theory and Methods. Dover Publications, Feb. 2008. [7] C. O’Driscoll, “Performance analysis of the parallel acquisition of weak GPS signals,” Ph.D. European Navigation Conference 2009 - Naples, Italy - May 3-6, 2009