Aalborg Universitet. Compressed Sensing with Rank Deficient Dictionaries

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1 Downloae from vbn.aau.k on: januar 11, 2019 Aalborg Universitet Compresse Sensing with Rank Deficient Dictionaries ansen, Thomas Lungaar; Johansen, Daniel øjrup; Jørgensen, Peter Bjørn; Trillingsgaar, Kasper Fløe; Arilsen, Thomas; Fyhn, Karsten; Larsen, Torben Publishe in: Globecom. I E E E Conference an Exhibition DOI (link to publication from Publisher): /GLOCOM Publication ate: 2012 Document Version Accepte author manuscript, peer reviewe version Link to publication from Aalborg University Citation for publishe version (APA): ansen, T. L., Johansen, D.., Jørgensen, P. B., Trillingsgar, K. F., Arilsen, T., Fyhn, K., & Larsen, T. (2012). Compresse Sensing with Rank Deficient Dictionaries. Globecom. I E E E Conference an Exhibition, General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights.? Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research.? You may not further istribute the material or use it for any profit-making activity or commercial gain? You may freely istribute the URL ientifying the publication in the public portal? Take own policy If you believe that this ocument breaches copyright please contact us at vbn@aub.aau.k proviing etails, an we will remove access to the work immeiately an investigate your claim.

2 c 2012 IEEE. Personal use of this material is permitte. Permission from IEEE must be obtaine for all other uses, in any current or future meia, incluing reprinting/republishing this material for avertising or promotional purposes, creating new collective works, for resale or reistribution to servers or lists, or reuse of any copyrighte component of this work in other works. Publishe at IEEE Globecom Compresse Sensing with Rank Deficient Dictionaries T. L. ansen, D.. Johansen, P. B. Jørgensen, K. F. Trillingsgaar, T. Arilsen, K. Fyhn an T. Larsen Department of Electronic Systems, Aalborg University, Denmark Abstract In compresse sensing it is generally assume that the ictionary matrix constitutes a (possibly overcomplete) basis of the signal space. In this paper we consier ictionaries that o not span the signal space, i.e. rank eficient ictionaries. We show that in this case the signal-to-noise ratio (SNR) in the compresse samples can be increase by selecting the rows of the measurement matrix from the column space of the ictionary. As an example application of compresse sensing with a rank eficient ictionary, we present a case stuy of compresse sensing applie to the Coarse Acquisition (C/A) step in a GPS receiver. Simulations show that for this application the propose choice of measurement matrix yiels an increase in SNR performance of up to 5 10 B, compare to the conventional choice of a fully ranom measurement matrix. Furthermore, the compresse sensing base C/A step is compare to a conventional metho for GPS C/A. I. INTRODUCTION The framework of compresse sensing (CS) has receive much research interest in recent years, see e.g. 1] an 2]. Compresse sensing allows signals to be sample below the Nyquist rate (subsample) while still allowing for reconstruction. To o this, it is require that the signal has a sparse representation in some known ictionary Φ C N Q. Then we can write the noisy signal z C N 1 as z = Φx + w, (1) where w C N 1 is i.i.. zero-mean noise (typically Gaussian istribute) an x C Q 1 is a sparse vector, i.e. it has very few non-zero entries, x 0 Q. A measurement matrix, Θ C M N, which satisfies certain requirements, is chosen to obtain the sub-sample (compresse) measurements y C M 1 y = Θz. (2) As M N the number of samples is significantly reuce. From the sub-sample signal y, a sparse estimate of x can be foun by solving the optimization problem 2]: min x x 1 s.t. y Ax 2 < ε (3) where we have efine the compresse ictionary A = ΘΦ an ε is a small fixe constant. Solving (3) is known as reconstruction. Several iterative greey algorithms exits, which fins a sparse x, e.g. 3], 4]. In CS it is esire that the A = ΘΦ matrix satisfies the Restricte Isometry Property (RIP), as this gives certain guarantees for successful reconstruction. It has been shown that choosing the measurement matrix Θ as a ranom matrix such as Gaussian or Bernoulli matrices gives an A matrix which satisfies RIP with high probability. Such measurement matrices are therefore wiely eploye 5], 6]. The ictionary Φ must be chosen such that the signal of interest is sparse, when the columns of the ictionary are use as the basis of representation. Often a square non-singular matrix is use, e.g. a iscrete Fourier matrix. In this case the columns of Φ constitute a basis of C N 1. In other cases Φ is fat (Q > N) an constitutes an over-complete ictionary of C N 1. In both cases, the ictionary has full row-rank. In this paper, we explore the application of CS to cases where the ictionary is rank eficient, i.e. it oes not have full row rank an thus its columns oes not span C N 1. Specifically, it is foun that choosing the rows of the measurement matrix as a ranom combination of the columns in the ictionary, increases the Signal-to-Noise Ratio (SNR) in the compresse samples. It is shown by simulations that this in turn results in better reconstruction performance. We present one example of a CS application, where a rank eficient ictionary arises; the Coarse Acquisition (C/A) step in a receiver for the Global Positioning System (GPS). CS is here applie with the hope of achieving a reuction in computational complexity. GPS is base on Coe Division Multiple Access (CDMA) an in the C/A step the receive signal must be correlate with a very large number of locally generate Pseuo Ranom Noise (PRN) sequences. We take a software efine raio approach an assume that the GPS signal is sample above the Nyquist rate an the C/A is subsequently performe exclusively in software. The C/A step can then be pose as a sparse ecomposition problem. In 7] compresse sensing was use as a preprocessor to reuce the computational complexity of sparse ecompositions of auio an speech signals. This preprocessing consists of applying a measurement matrix igitally an then the sparse ecomposition is obtaine irectly from performing the reconstruction given by (3). With this in min, we apply CS as a preprocessor to GPS C/A. In this case a rank eficient ictionary matrix is obtaine, because the columns contain the above Nyquist rate sample signal. Another approach of applying CS to a GPS system is given in 8], where the measurement matrix is applie in analog harware. The paper is structure as follows. In Section II we present

3 a signal moel for the consiere ictionary matrix. From this the measurement matrix is erive in section III. Section IV escribes the application of compresse sensing to GPS C/A. Numerical simulations on the GPS system are presente in Section V followe by conclusions in Section VI. II. SIGNAL MODEL In the following we elaborate on the signal moel in (1) to clearly specify the consiere scenario, in which a special structure of the measurement matrix will enhance the SNR. The consiere ictionary Φ C N Q can be either square (N = Q) or fat (Q > N). We start by introucing the ictionary using its SVD Φ = UΣV, (4) where U C N N an V C Q Q are unitary matrices an Σ C N Q is a matrix with the singular values in escening orer on its iagonal. The column space of Φ is equal to the span of those columns in U, which correspon to nonzero singular values. We now consier ictionaries with K singular values that are approximately zero. ence the ictionary can be represente well by a low rank approximation. a the singular values been exactly zero, the ictionary woul be rank eficient. The following erivations are however also true for singular values that are approximately zero. Define the matrices U high C N (N K), U low C N K, Σ high C (N K) (N K) an Σ low C K K such that U = U high U low ], Σ = Σhigh Σ low 0 ], (5) where 0 is the zero matrix. The matrices U high an Σ high are therefore associate with the N K singular values that are significant, an U low an Σ low the K singular values that are approximately zero. From this we efine the two subspaces S high an S low by the bases U high an U low. It is note that these subspaces are orthogonal. The SVD can therefore be written as Φ = U high Σ high V + U low Σ low V. (6) Since singular values in Σ low are approximately zero, the ictionary Φ is well approximate by only the first term in (6). In the special case where Φ is rank eficient, the secon term isappears since Σ low is the zero matrix. It is now assume that the signal of interest is perfectly represente as Φx, an hence the noisy signal z can be written as in (1). The vector x C Q 1 is consiere as a zeromean ranom sparse vector, where the entries are uncorrelate an ientically istribute. As the vector x is sparse, each entry is istribute such that it has a value of zero with high probability. The specific istribution of the entries in x is not important, as we are only intereste in the covariance matrix of x given by C xx = σ 2 xi R Q Q, (7) where σ 2 x enotes the variance of each entry in x an I enotes the ientity matrix. It is note that although the variance is the same for all elements in x, the vector is still assume to be sparse. The noise vector w in (1) is assume to be i.i.. zero-mean with covariance matrix C ww = σ 2 wi R N N, (8) where σ 2 w is the variance of the noise samples. III. SELECTING TE MEASUREMENT MATRIX In the following it is shown that choosing the rows of the measurement matrix in the space spanne by the ictionary, yiels a larger SNR in the compresse samples, compare to ranom Gaussian or Bernoulli matrices. This result is similar to the ranom sample kernels erive in 8] for GPS C/A using analog compresse sensing. This result is ifferent from what is conventional in CS, where completely ranom matrices are typically use an the following thus presents a performance improving technique. Consier the i th compresse sample which is efine accoring to (1) an (2) as y i = Θ i Φx + Θ i w for i = 0,..., M 1, (9) where Θ i C 1 N enotes the i th row of the measurement matrix Θ. In the compresse sample y i, the average power of the signal of interest is E Θ i Φx 2] an the noise power is E Θ i w 2], thus we efine the SNR of the i th sample as SNR yi (Θ i ) = E Θ i Φx 2] E Θ i w 2 ] = Θ iφc xx Φ Θ i Θ i C ww Θ i = E ] Θ i Φxx Φ Θ i E ] Θ i ww Θ i = σ2 x Θ i ΦΦ Θ i σw 2 Θ i Θ i. (10) We assume that achieving a high SNR for each compresse sample yiels better reconstructions. This assumption is justifie by simulation in the case stuy of GPS C/A. Therefore we seek conitions for the row Θ i such that SNR yi is high. Aitionally, the rows of the measurement matrix Θ i must be linearly inepenent such that each compresse sample contains ifferent information. The fraction in (10) is known as a Rayleigh quotient 9]. It attains a maximum value equal to the largest eigenvalue λ max of ΦΦ when Θ i is the eigenvector corresponing to λ max. The Rayleigh quotient can similarly be minimize to the smallest eigenvalue λ min when Θ i is the eigenvector corresponing to λ min. In general, when Θ i is equal to an eigenvector of ΦΦ, the Rayleigh quotient is equal to the corresponing eigenvalue. Now, we write ΦΦ in terms of the SVD given by (4) as ΦΦ = UΣV VΣ U = UΛU, (11) where Λ = ΣΣ. This can be seen to be the Eigenvalue Decompostion (EVD) of ΦΦ, with eigenvectors as columns in U an eigenvalues on the iagonal of the matrix Λ. The column space of ΦΦ is equal to the span of those columns in U, which correspons to nonzero eigenvalues. As these are the same as the nonzero singular values of Φ (consier that

4 Λ = ΣΣ ), the column space of ΦΦ is thus equal to the column space of Φ. The EVD of ΦΦ can also be written in terms of U high an U low similarly to (6) as ΦΦ = U high Λ high U high + U low Λ low U low, (12) where Λ high = Σ high Σ high an Λ low = Σ low Σ low. Suppose now that Θ i is chosen as an arbitrary row vector in C 1 N. As the union of S low an S high spans C N any vector in C 1 N can uniquely be ecompose into ˆΘ i = g low U low + g high U high, (13) where g low C 1 K an g high C 1 (N K). Inserting ˆΘ i into (10) yiels SNR yi ( ˆΘ i ) = σ2 x σ 2 w = σ2 x σ 2 w ˆΘ i ΦΦ ˆΘ i ˆΘ i ˆΘ i g low Λ low glow + g highλ high ghigh g low glow + g highghigh. (14) We reach (14) by inserting (12) an (13) an by noting that U low U high an U high U low are zero matrices. We now show that Θ i = g high U high yiels a better SNR than using ˆΘ i as normally one 5]. By noting that all eigenvalues of Λ high are strictly larger than Λ low, it can now be shown that SNR yi ( ˆΘ i ) = σ2 g x low Λ low glow + g highλ high ghigh σw 2 g low glow + g highghigh is satisfie if < σ2 g x high Λ high ghigh σw 2 g high ghigh = SNR yi (Θ i ), (15) g low Λ low g low g low g low < g highλ high ghigh g high ghigh. (16) This inequality is satisfie since both sies of the inequality are Rayleigh quotients. ence the left han sie has maximum equal to the largest eigenvalue in Λ low an the right han sie has minimum equal to the smallest eigenvalue of Λ high. The inequality in (15) implies that using Θ i yiels a better SNR than ˆΘ i. Therefore Θ i shoul be chosen in the subspace S high. It follows that the measurement matrix can be represente as Θ = GU high, (17) where G C M (N K) is a matrix chosen such that A = ΘΦ satisfies RIP. It is now assume that choosing G as a ranom Gaussian matrix makes A satisfy RIP. This is justifie by the theorems from compresse sensing, which state that choosing Θ as a ranom Gaussian matrix makes A satisfy RIP with high probability. owever, as U high is often not available irectly, the following two approaches can be use for selecting a suitable measurement matrix that is on the form given by (17) without using the eigenvectors of ΦΦ explicitly. In the following approaches, the matrices G 1 C M Q an G 2 C M N are chosen as ranom Gaussian matrices. 1) It is note that the space spanne by S high is approximate by the column space of Φ. ence an appropriate choice of the measurement matrix is Θ = G 1 Φ (18) = G 1 VΣ }{{ low U } low + G 1 VΣ high U }{{} high. (19) G low G high This matrix is approximately on the form given by (17) since the entries G low are low compare to G high. This choice of measurement matrix is use in the case stuy in Section IV. 2) By choosing the measurement matrix as Θ = G 2 U high U high, (20) the compresse ictionary is given by A = ΘΦ = G 2 U high U high(u high Σ high + U low Σ low )V = G 2 U high Σ high V G 2 Φ, (21) where the last approximation follows since the ictionary is low rank approximate well by only the significant eigenvalues in Σ high. The matrix A now has a structure similar to what is normal in CS. It is note that U high U high can be forme by the matrix prouct U b U b where U b is any orthonormal basis of S high. The propose measurement matrix can be interprete as follows; when calculating the compresse samples y, the Nyquist samples z are orthogonally projecte onto the subspace S high whereafter the compresse measurements are generate by multiplication with a ranom fat matrix G 2. In the projection, all signal energy in the subspace S low is remove. Since most energy of the signal of interest is in the subspace S high only a small amount of signal energy is remove. On the other han, the noise energy is equally sprea onto all eigenvectors U, hence a large amount of noise energy is remove. IV. APPLICATION TO COARSE ACQUISITION STEP OF A GPS RECEIVER In this section we investigate the application of the CS framework to the C/A step of a software efine GPS receiver. This is an example of a system in which the ictionary becomes rank eficient, because the signal is oversample. See 10] for an implementation of such a receiver. The ultimate goal of applying CS in a GPS receiver, is to reuce the computational requirements of the C/A step. In this paper we are however not investigating the computational complexity in etail, as we are more concerne with whether it is possible to apply CS to this application an what are the effects of the choice of measurement matrix. A. GPS Signal Moel Each satellite transmits two signals esignate as L1 an L2 with carrier frequencies of Mz an Mz respectively. The L2 signal is use for military purposes an is therefore not consiere in the following.

5 For the L1 signal each satellite is assigne a unique Gol coe 11], which in GPS is known as C/A coes. These C/A coes exhibit low correlation between each other an between time-shifts of the same coe. Each satellite transmits a lowrate (50 z) ata signal, which is moulate by the high-rate (1.023 Mz) C/A coe. As the C/A coes are of length 1023, they are repeate every 1 ms. There are 32 ifferent C/A coes in use but at any given time only a subset of these are within the fiel of view of the GPS receiver 12]. Due to a high velocity ifference between the satellites an the receiver, a significant Doppler shift is introuce. The satellites an receiver are not synchronize in time, hence the phase of each transmitte C/A coe (coe phase) is unknown. The purpose of the C/A step is to ientify which satellites are within the fiel of view an to etermine the Doppler shift an coe phase of these. Navigation ata an the C/A coes are moulate onto the in-phase component of the L1 frequency, while navigation ata an the so-calle P-coe are moulate onto the quarature component of the same carrier frequency. The P-coe is a pseuo ranom noise sequence an its transmit power is 3 B lower than that of the C/A coe. For the C/A step of a GPS receiver only the C/A coe is of any interest an the P-coe is therefore consiere as co-channel noise. As the power of the P-coe is typically lower than the noise floor of the receiver, it is ignore. When neee, the P-coe can be extracte from the noise ue to the processing gain of CDMA. The navigation ata moulate onto the in-phase component is neither inclue in the moel. It is thus assume that no navigation ata shifts occur uring the C/A step. This is fair as the uration of one navigation ata bit is much longer than the time-span over which the C/A step is performe. In a real receiver there exist techniques to hanle the situation where a ata shift occur in the signal-uration where the C/A step is performe 10]. The ieally receive baseban signal from the s th satellite is thus moelle to solely consist of the C/A coe spreaing waveform, enote as g (s) (t). It is perioic with 1 ms an can be written as g (s) (t) = ĝ (s) (t kt chip N 1 ), (22) k= where ĝ (s) is the contribution from one perio, N 1 = is the length of the C/A coe an T chip = Mz is the uration of one chip. The contribution from one perio is given by ĝ (s) (t) = N 1 1 n=0 c (s) n] v(t nt chip ), (23) where c (s) n] { 1, 1} is the bipolar C/A coe of length N 1, an v(t) is the impulse response of the channel incluing transmit an receive filters. Assuming a signal linear receiver, its frequency response is given by V (f) = R t (f) R x (f) (f), (24) where the transmit filter R t (f) is moele as an ieal sharp cut-off low-pass filter with a one-sie banwith of at least Mz 12], an the channel frequency response (f) is constant since multipath effects are not consiere. The receive filter R x (f) is ominating an is moele as an ieal low-pass filter with a one-sie banwith equal to half the sampling frequency f s. It is also acting as anti-aliasing filter. Uner the effects of channel attenuation, Doppler shift, time elay, Aitive White Gaussian noise (AWGN) an after ownconversion to baseban, the summe signal from all satellites in complex baseban representation is given by S 1 z(t) = s=0 α (s) g (s)( ) ( t t (s) exp j ω (s) t + θ (s))] + w(t), (25) where α (s) is the channel attenuation from the s th satellite to the receiver, t (s) is the time elay or coe phase of the C/A coe for the s th satellite, ω (s) is the Doppler frequency for the s th satellite, θ (s) is the phase of the carrier at the receiver an w(t) is complex-value aitive white Gaussian noise. B. Coarse Acquisition as a Sparse Decomposition Problem The problem of C/A is to etermine the Doppler shift ω (s) an the coe phase t (s) in (25) for each satellite. The signal moel oes not have an obvious sparse structure that can be exploite for formulation of the sparse ecomposition problem since the parameters t (s) an ω (s) are consiere unknown, continuous-value constants. To ientify a sparse structure these values are quantize similarly to the proceure in 8]. Denote the number of ifferent Doppler shifts as D an the number of ifferent equally space coe phases as L. In worst case the Doppler shift can eviate up to ±10 kz, an it is in most cases sufficient to know the Doppler frequency in steps of 500 z 10]. In this case D = = 41. For a coe phase resolution of 1 chip, L = 1023 which is the length of the C/A coe. The signal moel is approximate as follows S 1 z(t) D 1 L 1 s=0 =0 l=0 g (s) (t lt c ) exp j (D ω t + θ (s))] α (s,,l) + w(t), (26) where D = { 20, 19,..., 19, 20} is the set efining possible Doppler shifts along with the step size ω = 2π500 ra s, T c = 1 ms L is the step size for the coe phase an α(s,,l) is the amplitue associate with the s th satellite, the th Doppler shift an the l th coe phase. Note that α (s,,l) only has one non-zero element for each s, because the amplitue of each satellite only pertains to one Doppler shift an one coe phase. This is what makes the signal have a sparse representation. Rewriting (26) as S 1 z(t) D 1 s=0 =0 l=0 L 1 φ (s,,l) (t) x (s,,l) + w(t), (27)

6 zn] (0,0,0) n] (S 1,D 1,L 1) n]. NX 1 ( ) n=0. X ( ) N 1 Fig. 1. Conceptual illustration of coarse acquisition. The receive signal zn] is correlate with the C/A coes uner all combinations of coe phases an Doppler-shifts. The number of correlators is SDL, where S is the number of C/A coes, D is the number of Doppler-shifts an L is the number of coe phases consiere. The largest correlation values signify the present satellites, their Doppler-shift an coe phase values. n=0 Pick largest λ i Magnitue of Eigenvalues of ΦΦ Eigenvalue Number, i Fig. 2. Magnitue of eigenvalues (sorte) of ΦΦ, for a ictionary Φ generate accoring to (33) with S = 32, D = 41, L = 1023, N = an sampling frequency f s = Mz. where φ (s,,l) (t) = g (s) (t lt c ) expjd ω (t lt c )], (28) x (s,,l) = α (s,,l) expjd ω lt c ] exp jθ (s)], (29) shows that the approximation of z(t) is sparse with respect to the basis functions φ (s,,l) (t), since α (s,,l) is only nonzero for a few triplets (s,, l). 1 The GPS C/A problem can be solve by fining the triplets for which α (s,,l) is nonzero. Since it is esire to solve the problem in the igital omain, we sample the signal z(t) with a sample frequency f s = 1 T s an thereby efine the following vectors: z = z(t 0 ),, z(t N 1 )] T C N 1 (30) φ (s,,l) = φ (s,,l) (t 0 ),, φ (s,,l) (t N 1 )] T C N 1 (31) w = w(t 0 ),, w(t N 1 )] T C N 1, (32) where N is the number of obtaine samples an t n = n T s. Denote the ictionary matrix Φ C N SDL that has columns given by φ (s,,l) for all combinations of s, an l Φ = φ (0,0,0), φ (0,0,1),, φ (S 1,D 1,L 1)]. (33) The ictionary Φ is fat since SDL N. The sparse signal moel can now be expresse in matrix form: z Φx + w, (34) with sparse x C SDL 1 given by x = x (0,0,0), x (0,0,1),, x (S 1,D 1,L 1)] T. (35) The naive way to fin x is by correlating the receive signal with the SDL combinations of C/A coes, Doppler frequencies an coe phases as epicte in Fig. 1. This metho is computationally heavy ue to the large number of correlators (SDL = ). The computations can be performe more efficiently using a Fast Fourier Transform (FFT) base approach known as Parallel Coe Phase Search (PCPS) 10]. 1 Implementation etail: The inclusion of jd ω lt c into (28) allows us to write a ictionary as a concatenation of partial circulant matrices (we efine a circulant matrix as in 13]). A circulant matrix is iagonalizable by the Fourier matrix, which facilitates faster matrix-vector multiplication. Solving the problem in (3) iteratively can thus be implemente more efficiently. V. SIMULATIONS & RESULTS In orer to assess the performance of the propose measurement matrix an the compresse sensing base C/A scheme we carry out a numerical simulation 2. First we show that the ictionary is inee rank eficient. To o this, the signal is sample with a sample frequency 5 times the chipping rate, f s = Mz. We sample two perios of the C/A coe which correspons to N = samples. From this the ictionary is constructe an the eigenvalues of ΦΦ are calculate. The magnitues of these eigenvalues are shown in Fig. 2. It is clearly seen that only a few of the eigenvalues of ΦΦ are significant. From (11) follows that the singular values of Φ is the square root of the eigenvalues of ΦΦ. Therefore, only a few of these singular values are significant an Φ is thus rank eficient. To solve the optimization problem in (3) efficiently, we use the state-of-the-art reconstruction algorithm CoSaMP 4] elaborate with the concept of moel-base CS 14]. The moelbase approach is basically restricting the CoSaMP algorithm to not prouce more than one nonzero element of x for each C/A-coe. To reuce the computational requirements for the simulations all ranom matrices are implemente as ranom circulant matrices. That is, the first column is generate as ranom Gaussian an the remaining columns are foun as circulant shifts of this. In 15] simulations show that in many cases this oes not affect the reconstruction performance. Throughout all numerical experiments the performance of the system is evaluate through its ability to correctly ientify present satellites an their corresponing Doppler shift an coe phase. We efine the success rate performance metric as the ratio of correctly ientifie satellites to the total number of satellites present. A satellite is correctly ientifie if its estimate Doppler frequency is 0.6 ω = 300 z from the reference frequency an its estimate coe phase is within 0.6 T chip 587 ns from the reference coe phase. The factor of 0.6 is somewhat arbitrarily chosen, such that two ajacent search steps are consiere successful, when the true parameter falls in between these two. 2 To comply with the reproucible research paraigm, the (Python base) source coe use for the simulations can be foun online at

7 Success Rate -] Success Rate, M= C/N 0 B-z] PCPS, f s =5.115 Mz PCPS, f s = Mz PCPS, f s = Mz Θ =GΦ, f s =5.115 Mz Θ =GΦ, f s = Mz Θ =GΦ, f s = Mz Gaussian Θ, f s =5.115 Mz Gaussian Θ, f s = Mz Gaussian Θ, f s = Mz Fig. 3. Success rate versus C N 0 for ifferent Nyquist sample frequencies. For the compresse sensing base approach, 984 compresse samples are use. Parallel Coe Phase Search (PCPS) is utilizing all Nyquist samples from a 2 ms signal. Approximate 99 % confience intervals are shown. The use measure for channel noise is the carrier-to-noiseensity ratio C N 0 z], where C is the average power of the passban signal an N 0 is the noise spectral ensity. Typically C N 0 ranges from 35 to 55 Bz 16]. The signal from one satellite is generate in complex baseban with an average sample power P s. The noise is a zero-mean iscrete white process where the amplitues have a complex Gaussian istribution. It can be shown that the variance σw 2 of the complex noise is given by P s σ 2 W = C N 0 1 f s. (36) For each simulation the signals with equal power is generate for 8 ifferent satellites ranomly chosen from the set {0,..., 31}. When running the simulations the reconstruction algorithm is given the number of satellites present in the signal. Each ata point is average over 100 simulations, each with a new ranomly generate signal an measurement matrix Θ. To illustrate the effect of choosing a measurement matrix accoring to the result in Section III, two choices of the measurement matrix are use; Θ = GΦ an Θ equal to a ranom Gaussian matrix of imensions M N with inepenent entries. The result of this simulation is shown for ifferent sampling frequencies in Fig. 3. First of all it is note that the conventional metho of Parallel Coe Phase Search (PCPS) performs equally well for all simulate sampling frequencies. The same is the case for Θ = GΦ. owever when using Θ equal to a Gaussian matrix, the ability to ientify the correct support of the signal ecreases with the sampling frequency. The propose measurement matrix thus significantly improves performance an this is expecte to generalize for all cases where rank eficient ictionaries are use. It is note that the propose compresse sensing base approach to GPS C/A oes not perform as well as PCPS. The propose approach might however still fin applications in receivers which are known to operate in the high SNR region, if the computational requirements can be reuce. In this paper we have not explore the very important aspect of the computational requirements for the propose metho compare to conventional methos such as PCPS. VI. CONCLUSION In this paper we have consiere rank eficient ictionaries in the context of compresse sensing. We have shown that to increase the SNR in the compresse samples, the rows of the measurement matrix have to be chosen within the subspace spanne by the ictionary. Through a case stuy of compresse sensing applie to the C/A step in a GPS receiver, the propose measurement matrix is shown to increase performance compare to the usual choice of a ranom measurement matrix. This is expecte to generalize for all applications of compresse sensing with rank eficient ictionaries. The case stuy emonstrate that compresse sensing can be applie to the C/A step of a GPS receiver. It is subject of further research to investigate if a reuction of computational requirements can also be achieve. REFERENCES 1] E. Canes, J. Romberg, an T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, vol. 52, no. 2, pp , Feb ] D. 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