URSI GASS 2020, Rome, Italy, 29 August - 5 September 2020

Raw GNSS Data Compression using Compressive Sensing for Reflectometry Applications

Bushra Ansari*, Vishal Kaushik, Sanat K. BiswasDepartment of Electronics and Communication Engineering, IIIT Delhi, New Delhi, India

Abstract

Global Navigation Satellite System (GNSS) is a passiveand in-expensive technique for remote sensing applications.GNSS reflectometry (GNSS-R) aims at analyzing the multi-path signal reflected from the surface surrounding the re-ceiving antenna. One of the challenge is the size of the rawGNSS-R I/Q data. The data size is very large even for ashort duration of time. To address this issue an algorithmis proposed using the Compressive Sensing (CS) theory, tocompress and reduce the size of data that is to be trans-mitted for further processing. The data is first compressedusing the CS theory and then reconstructed using convexl1 minimization algorithm. In addition, the performanceof different sparsifying and measurement matrices that areused in the compression are compared and reconstructionof the original signal is performed. The algorithm proposedis verified for the raw Global Positioning System (GPS) andNavigation with Indian Constellation (NavIC) data set.

1 Introduction

GNSS-Reflectometry (GNSS-R) has gained attention dueto its unique and inexpensive characteristics. In GNSS-Rmethod, the GNSS signals are acquired in bi-static or multi-static radar configuration. The main objective of GNSS-Ris to receive GNSS multi-path signals and derive useful in-formation about the reflecting surface. The signal proper-ties (signal-to-noise ratio (SNR), signal power etc.) of themulti-path signal deviates from that of the directly receivedGNSS signal. The change in signal properties depends onthe surface properties of the ground and hence provides awide range of opportunities. Some of its applications in-clude soil moisture and vegetation height measurement [1],snow height retrieval [2], and altimetry estimation [3].

In the GNSS-R technique, the direct and the reflected sig-nal is received using a GNSS antenna. After that, raw I/QGNSS data is to be sent to the remote server through wire-less channel for further processing as shown in Fig.1. Thedata used in GNSS-R has a size of around 100 thousandsamples for a very short duration, which take long trans-mission time while sending it through the wireless channelfor GNSS-R analysis. Hence it is necessary to develop analgorithm to reduce the amount of data to be stored on boardand transmitted, for GNSS-R applications.

Figure 1. GNSS-R Geometry

The above-mentioned problem can be solved by compress-ing the GNSS data first and then reconstructing it at thereceiver end. The problem is segmented into three steps,the first step is the sparse representation of the GNSS datain a sparse domain. The second step is the formation of themeasurement vector of lower dimensions than the originalsignal. The last step is the reconstruction of the originalsignal from a reduced number of measurement. Accordingto previous work done in this area, [4] a two-stage deter-ministic CS algorithm has been proposed. The determin-istic CS matrix is not robust to noisy signal. A study isperformed in [5], which uses multiple measurement vec-tor as the reconstruction algorithm, this algorithm is com-plex as it uses multiple measurement CS theory. To solvethese problems a compression and reconstruction algorithmbased on compressive sensing is used in this paper which isapplicable to the noisy input signal as well. The proposedmethod has low algorithmic complexity than the previouslyproposed algorithms [4, 5]. Finally, the proposed algorithmis applied on both direct and multipath GPS data collectedby TechDemo satellite (TDS)-1 and simulated NavIC sig-nal collected using bladeRF SDR. Additionally, the Nor-malised Mean Square Error (NMSE) performance of dif-ferent sparsifying and measurement matrix is compared.The algorithm is further validated by generating the DelayDoppler Maps (DDMs) for both the original and the recon-structed signal.

The rest of the paper is organised as follows: In section II,the mathematical formulation for the compression and re-construction of the GNSS signal is explained. Section IIIprovides the details of the simulation setup. Results anddiscussions are included in section IV and section V con-

cludes the paper outlining the future work.

2 Mathematical Formulation

Compressive sensing is an emerging signal processing tech-nique. It is a sampling method which provides efficient sig-nal compression and reconstruction by using the fact thata sparse signal can be reconstructed from a lesser numberof samples. In the compression process, a certain numberof samples in the sparse domain are discarded, assumingthat the majority of samples are non-significant. Hence lessnumber of measurements are acquired and can be used toreconstruct the complete signal. The missing informationcan be reconstructed using the CS sparse recovery algo-rithms [6]. The CS theory can be applied to GNSS signalsas well, as it has a sparse representation in both DiscreteCosine Transform (DCT) and Discrete Wavelet Transform(DWT) domain, as shown in Fig. 2.

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Figure 2. Sparase Representation of signal in DCT andDWT Domain

Consider a signal x ∈ Rm, of size m× 1 that has a sparserepresentation in a transform domain. The signal can berepresented sparsely in a particular transform domain, bymultiplying the signal with the sparsifying matrix i.e. DCTor DWT matrix. The signal with a reduced number of sam-ples, constructed using the sparse signal, is called the mea-surement vector. The measurement vector y ∈ Rn of sizen×1 where n < m will be used for original signal recovery.The vector y ∈ Rn is a linear measurement of the signal xgiven by y = Ax, where A is the measurement or sensingmatrix of size n×m. This system is under-determined, sothe reconstruction of the signal x is an ill-posed problem.Therefore, an optimization algorithms can be used to ob-tain an optimal solution for this system. The sparse signalrecovery algorithms are based on the convex-optimizationtechnique. One of the method that is based on convex l1minimization which gives near optimal solution can be de-fined as:

min‖x‖1 sub ject to y = Ax (1)

This convex minimization approach is called as Basis Pur-suit (BP). However, in case of noisy measurements, the lin-ear measurement vector is represented as y=Ax+N, whereN is the noise and ‖N‖2 ≤ ε , where ε is sufficiently small

value. This optimization approach is called Basis PursuitDenoising (BPDN) and is defined as [6]:

min‖x‖1 sub ject to‖y−Ax‖2 ≤ ε (2)

The BPDN algorithm can be solved using SPGL-Toolboxwhich is a Matlab solver for large-scale sparse reconstruc-tion. The procedure followed in compression and recon-struction of GNSS I/Q data set is summarized in the Algo-rithm 1 given below:

Algorithm 1: CS based reconstuction algorithmINPUT: x (GNSS Signal)STEP1: Compression of GNSS signal i.e. sparserepresentation and generation of mesurement vector .

Dx = ψ

where D is the sparsifying marix and ψ is the sparserepresentation of GNSS signal.

y = Ax i.e. y = AD−1ψ

where y is the measurement vector and φ is themeasurement matrix.STEP2: Wireless transmission of measurement vector.STEP3: Obtain the sparse signal using convex l1minimization algorithm.

min‖Dx‖1 sub ject to‖y−Ax‖22 ≤ ε

min‖ψ‖1 sub ject to∥∥y−AD-1ψ

∥∥22 ≤ ε

STEP4: Reconstruction of original signal from thesparse vector.

x̂ = D-1ψ.

OUTPUT: x̂ (Reconstructed Signal)

The performance of the algorithm for GNSS signal com-pression and reconstruction is further verified by generat-ing the Delay Doppler Maps (DDMs) for both the originaland the reconstructed data set. DDM is defined as the am-plitude/power distribution of the reflected signal in a 2-Darray of delay offsets and doppler shifts around the specu-lar point.

3 Simulation Setup

The study presented in this paper is conducted on the real-time raw GPS and the NavIC data-set. The GPS IF datais from TDS-1 satellite which is available at FTP serverftp://ftp.merrbys.co.uk. The data file is first processed us-ing the softGNSS [7] algorithm to obtain the data vector.The data is collected for a duration of 2 minutes 20 second,with the sampling rate of 16.367 MHz and the intermedi-ate frequency of 4.188 MHz. For experimentation usingNavIC data-set, a simulation testbed is developed as shownin Fig.3. The NavIC data for Reflected (multipath) sce-nario is simulated using the GNSS Simulator SIMAC2. TheMultipath data is simulated for two different surface prop-erties (i.e. dielectric constant) using physics based model.

After that, the bladeRF SDR is used to receive the NavICdata sets. The NavIC data sets are collected for a durationof 10ms, with the sampling rate of 5 MHz. Finally, twodata sets are collected for two different surface properties(i.e.dielectric constant). The Dielectric constant for Dataset-1 is 5 and Data set-2 is 20. These data sets are furtherprocessed for reflectometry application in MATLAB envi-ronment.

Figure 3. Simulation setup for NavIC signal Processing

4 Results and Discussion

The TDS-1 I/Q data and simulated and sampled NavIC I/Qdata are compressed and then reconstructed using the algo-rithm mentioned in section-II. The normalised mean squareerror (NMSE) in signal reconstruction is calculated at dif-ferent sampling ratio to show the comparison between dif-ferent matrices. The NMSE is defined as:

NMSE =‖x̂− x‖2

2

‖x‖22

(3)

where x̂ is the recovered signal and x is the original signal.The performance of DCT and DWT sparsifying matrix iscompared in terms of NMSE at different sampling ratio forthe Toeplitz, Bernoulli and Gaussian measurement matrix.One of the case with toeplitz matrix is presented in Fig. 4.It is observed that the NMSE of the DCT sparsifying ma-trix is less than that of the DWT matrix, which states thatthe DCT matrix is performing better than the DWT matrix.Similarly, the performance of Toeplitz, Bernoulli and Gaus-sian measurement matrix is evaluated as shown in Fig. 5.

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Figure 4. Comparison between DCT and DWT sparsifyingmatrix for different sampling ratio.

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Figure 5. Comparison between Toeplitz, Bernoulli andGaussian measurement matrix for different sampling ratio.

The performance of Toeplitz measurement matrix is bet-ter as compared to Bernoulli and Gaussian matrix in termsof signal reconstruction. Now the GNSS data compressionand reconstruction is performed using the DCT sparsify-ing matrix and toeplitz measurement matrix. The GNSSdata is reconstructed from 40% of original data samples.Additionally, after performing the frequency-domain corre-lation of the reconstructed data with the locally generatedGPS/NavIC code (depending on the data used), the samesatellite PRN i.e. PRN 24 and code delay is found as de-tected for the original data. The frequency domain acquisi-tion for the original and reconstructed GPS data for satellitePRN 24 is presented in Fig.6 and Fig.7 respectively.

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Figure 6. Frequency domain acquisition for the OriginalGPS data.

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Figure 7. Frequency domain acquisition for the Recon-structed GPS data.

For reflectometry application, the DDMs of two NavIC datasets for different surface properties (i.e. dielectric constant5 and 20) are generated. Now according to bi-static radarequation [8], the coherent power levels for two differentsurfaces will vary depending on the surface properties. TheDDMs for two different NavIC data sets are presented inFig.8 and Fig.9. The correlation power levels of the DDMshows variation depending on the surface properties. More-over, the DDMs of the data which is reconstructed using theproposed algorithm also shows the variation in the corre-lation power. The experimental results of DDMs obtainedfrom the original and reconstructed data set are sumamrizedin the Table.1

Table 1. Change in correlation peak with different surfacedielectric constant

DielectricConstant

Correlation peak(Original Data set)

Correlation Peak(Reconstructed Data set)

5 2848 476.820 4176 849.9

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Figure 8. DDM for the multi-path NavIC signal for Dielec-tric Constant 5

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Figure 9. DDM for the multi-path NavIC signal for Dielec-tric Constant 20

The results presented in Fig.8 and Fig.9 implies that thesimilar variations in correlation power can be achieved withthe reconstructed GNSS signal as shown by the originalGNSS signal DDMs. Hence, only few significant GNSSdata samples can be transmitted and then reconstructed atreceiver’s end for further processing. It will reduce theamount of data to be stored on board and transmitted forfurther processing.

5 Conclusion

In this paper, Compressive Sensing based reconstruction al-gorithm is proposed that is reducing the data size for re-flectometry application. The BPDN convex optimisationalgorithm of compressive sensing theory is applied to theGNSS signal. From the experimental results, it has beenconcluded that for compression of the signal, DCT sparsi-fying matrix with Toeplitz as the measurement matrix hasbetter performance. The satellite acquisition is also per-formed for both the original and reconstructed data sets andthe same satellite PRN and code delay is detected for bothdata. Additionally, it is feasible to differentiate between twosurface properties (i.e. dielectric constant) from the DDMsgenerated from the reconstructed data. However, further re-search is required to validate and generalize this algorithmfor more variations in surface properties.

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