Real-Time Software Receiver for GPS Controlled Reception Pattern Antenna Array Processing

Transcription

1 Rea-Time Software Receiver for GPS Controed Reception Pattern Antenna Array Processing Yu-Hsuan Chen, Jyh-Ching Juang, Nationa Cheng Kung University, Taiwan David S De Lorenzo, Jiwon Seo, Sherman Lo, Per Enge, Stanford University, USA Dennis M Akos, University of Coorado Bouder, USA BIOGRAPHY Yu-Hsuan Chen is a Ph D candidate in eectrica engineering of Nationa Cheng Kung University, Taiwan He visited GPS Laboratory at Stanford University in He received his MS degree in eectrica engineering from Nationa Cheng Kung University, Taiwan in David S De Lorenzo is a research associate at the Stanford University Goba Positioning System (GPS) Laboratory He received his MS in Mechanica Engineering from University of Caifornia, Davis and his PhD in Aeronautics and Astronautics from Stanford University Jyh-Ching Juang received the Ph D degree in eectrica engineering from University of Southern Caifornia, Los Angees, in 987 He had been with Lockheed Aeronautica Systems Company before he went back to Taiwan in 99 Currenty, he is a Professor at the Department of Eectrica Engineering, Nationa Cheng Kung University, Tainan, Taiwan His research interests incude DSP-based contro appications, GNSS navigation design, sensor networks, and advanced signa processing Jiwon Seo is a Postdoctora Schoar at Goba Positioning System Laboratory and Space Environment and Sateite Systems Laboratory at Stanford University He received his BS degree in mechanica engineering (division of aerospace engineering) from KAIST (Korea Advanced Institute of Science and Technoogy) and received MS degrees in aeronautics/astronautics and eectrica engineering from Stanford and a PhD degree in aeronautics/astronautics from Stanford in His research interests incude ionospheric effects on GPS aviation; aternative positioning, navigation, and timing; and atmospheric remote sensing Dr Seo was a recipient of the Samsung Lee Kun Hee Graduate Feowship for five years Sherman Lo is a senior research engineer at the Stanford University Goba Positioning System (GPS) Laboratory He is the Associate Investigator for the Stanford University efforts on the Federa Aviation Administration (FAA) Aternate Position Navigation & Time (APNT) study Per Enge is a Professor of Aeronautics and Astronautics at Stanford University, where he is the Keiner-Perkins, Mayfied, Sequoia Capita Professor in the Schoo of Engineering He directs the Stanford University GPS Research Laboratory Dennis M Akos is an Associate Professor with the Aerospace Engineering Science Department at the University of Coorado at Bouder Dr Akos competed the PhD degree in Eectrica Engineering at Ohio University within the Avionics Engineering Center ABSTRACT This paper demonstrates a rea-time software receiver supporting GPS L C/A controed reception pattern antenna (CRPA) processing The software receiver is impemented on a widey-avaiabe recent generation muti-core processor and is capabe of processing array signas from either radio frequency (RF) front-end modues or coected datasets Most importanty, the receiver aocates dedicated sateite-tracking antenna patterns for each baseband receiver channe The architecture of such a software receiver needs to be carefuy deveoped so that it can process signas from each individua antenna, cacuate the appropriate beam formed composite signas (one for each sateite/receiver channe) and then process those signas Mutitasking and synchronization mechanisms were deveoped to support

2 the tracking of mutipe channes in rea time To achieve rea-time capabiity, parae operations are necessary to reduce computation compexity Bit-wise operations are expoited and impemented in the correator Additionay, Singe Instruction Mutipe Data (SIMD) instructions are used to efficienty cacuate the covariance matrix for the beam steering agorithm The architecture supports at east eighty tracking channes (ten channes from each of seven antennas pus ten composite beam formed signas) in rea time The CRPA software receiver was architected to operate without extensive set up and pre-caibration enhancing its suitabiity for commercia users The agorithm design was architected to aid ease of set up Whie conventiona antenna array system receivers are used with the geometry of antennas and cabe engths known in advance, the agorithm impemented aows for operation without such a priori knowedge Two beam steering techniques were tested First is deterministic beam steering An adaptive agorithm, Minimum Variance Distortion Response (MVDR) agorithm, was impemented to adaptivey maximize signa power The architecture determines of the carrier phase difference between signas from different antennas for a singe sateite in order to buid the steering vector An experiment was conducted to show the enhanced C/No and controed reception patterns through directing the CRPA toward the direction of sateite of interest For evauating interference rejection, a LI GPS simuator is used to buid an environment with CDMA and CW interferences The resut shows that the MVDR agorithm has reiabe performance than non beam steered and deterministic beam steering under the both type of interferences INTRODUCTION GPS provides hour a-weather position, navigation, and timing (PNT) services wordwide However, GPS and GNSS signas are reativey weak and thus vunerabe to deiberate and unintentiona interference An eectronicay-steered antenna array system provides an effective approach to mitigating interference by controing the reception pattern and steering beams/nus As a resut, so caed controed reception pattern antenna (CRPA) array have been depoyed by organizations such as the US miitary which seek high eves of anti-jam performance However, there is a tradeoff in increased cost and computationa compexity which to-date has not been acceptabe to commercia GNSS users The research conducted in this paper brings the directive gains and interference rejection benefits of eectronicay-steered antennas coser to commercia users by impementing CRPA processing on software receivers A software receiver was deveoped for study a civi use of GNSS beam steering It uses a 7-eement antenna array with one RF front-end per antenna to coect the desired signas The digitaized signas are then transferred to PC by a USB microcontroer board The signa processing, positioning, and beamforming are accompished by software [][] Conventionay, antenna array system receivers perform CRPA with the geometry of the antennas and cabe engths known in advance In the deveoped software receiver, the agorithm impemented aows for operation without such a priori knowedge As the carrier phase difference is reated to geometry of antenna as we as ine biases of the cabes, this can be used as weights for deterministic beamforming or constraints for adaptive beamforming The software receiver contains ten channes per antenna for assessing the carrier phase So for seven antennas without beamforming we have seventy independent channes This resuts in high computationa compexity In order to achieve rea-time capabiity, singe instruction mutipe data (SIMD) instructions [] and assemby programming is used to acceerate the operation Additionay, muti-threading programming is adopted to fuy expoit the muti-core resources of the processor An experiment was conducted to demonstrate that the carrier to noise ratio (C/No) is enhanced by array processing With an injected interference, the deveoped software receiver is aso tested with ow power CW and CDMA interferences Comparisons are made between a singe antenna, CRPA by deterministic beamforming and MVDR adaptive beamforming The paper is organized as foows First, the agorithms of the CRPA are introduced Then, the impementation of the CRPA in our deveoped software receiver is described The architecture of the software receiver, incuding hardware and software components, is expained in detai For verifying the rea-time capabiity, the software anayses, incuding thread activities and timing, are shown A caibration procedure by carrier phase precise positioning is impemented for cacuating the controed pattern resuts The experiments for enhancing C/No and interference rejection are described and resuts show the performance of the CRPA software receiver Finay, some concuding remarks are made ALGORITHMS OF THE CONTROLLED RECEPTION PATTERN ANTENNA The primary goa of a controer reception pattern antenna is to enhance the carrier-to-noise ratio of a seected sateite and reject interference [] The basic agorithm is deterministic beamforming which combines the signa of

3 antennas, mutipied by compex weights and then summed over a antennas as shown in equation () M s( t) = si ( t) wi = W S () i= where s i (t) is the signa from ith antenna and w i is the weight associated to the ith antenna Traditionay, the signa of one of the antennas is set as a reference and its weight is set to The other weights are set as phase shifts reative to the reference antenna represented in equation () wi = exp ( j( ϕ i + δli ) () where ϕ i is the phase difference based on antenna geometry and the direction of desired sateite δli is the phase difference resuting from the cabe ength difference and phase center inaccuracy ϕ i can be cacuated as r πpi rˆ ( φ, θ ) ϕi = () λ where p r is the baseine vector of the ith antenna and i r ˆ ( φ, θ ) is the unit vector to sateite as shown in the figure z θ T [ I R] W T Wn + = γ µ n + () where R is the covariance matrix and T is the steering vector Then, in order to keep the magnitude of weight of the reference signa as unity, the weight vector is normaized as shown in equation () W W w () n+ = n+ r, n+ IMPLEMENTATION OF A CRPA IN A SOFTWARE RECEIVER For deterministic beamforming and the adaptive MVDR scheme, obtaining the steering vector is the key to impementing a CRPA However, as mentioned in the previous section, parameters of the steering vector are obtained through caibration A software receiver has the fexibiity of impementing channes to process mutiantenna signas A set of channes with the same PRN assignment are configured to process the signa of an individua antenna, respectivey The measurement of the phase ocked oop is integrated carrier phase (ICP) which integrates carrier phase throughout tracking The ICP is often used to smooth code pseudorange for improving accuracy of positioning In our software receiver, ICP differences between different antennas are taken to buid the steering vector Without any caibration prior to execution, the software receiver uses ICP differences to perform CRPA instead of the azimuth/eevation to the sateites and baseine vectors of antennas Figure shows the bock diagram of the impementation of the CRPA in our software receiver Ant r rˆ y x p r i φ Ant i Figure The antenna geometry and direction of sateite showing cacuation of ϕ i Because p r i is a known prior and r ˆ ( φ, θ ) can be cacuated after positioning, ϕ i can be obtained However, γ i is needed to be re-caibrated whenever any part of antenna array hardware is changed Compared to deterministic beamforming, the adaptive scheme uses feedback to find the optima weights for the beam The MVDR agorithm [] uses weights of the deterministic beamforming as the steering vector to constraint the gain of the desired direction to unity whie steering the nus to reject any interference The nu steering is accompished by cacuating the appropriate weights iterativey The impementabe updating equations of MVDR is written as ϕ = ϕ ϕ M M M ϕ N = ϕ N ϕ j ϕ e j ϕ M e Figure Bock diagram of the impementation of CRPA in a software receiver Once a of the channes assigned to a given sateite are tracking, the ICPs are needed to re-initiaize according to in-phase and quadrature-phase of individua correator output It takes severa miiseconds to average the phase Then, the ICPs are continuousy integrated and the phase difference between the reference antenna and others are computed at runtime Any of the tracked sateite can be seected by the operator and then the ICP difference is utiized to cacuate the steering vector For deterministic

4 beamforming, the weight vector is equa to steering vector For the MVDR agorithm, it is obtained through an adaptive procedure as expressed in equations () and () every miisecond The beam formed signa is cacuated by mutipied weights with signas and then sum over a antennas comparative operations The used processor is the quadcore Inte Core i7 which runs eight threads at a time and supports SIMD instructions sets incuding MMX and SSE (,,,,, ) The hardware set-up of the CRPA software receiver is depicted in the figure 7 eement Antenna Array ARCHITECTURE OF SOFTWARE RECEIVER The deveoped CRPA rea-time software receiver runs on a PC patform and uses USB for data input It currenty operates on GPS L C/A signa There are a tota of 8 channes of which ten are aocated to process rea data for each antenna as we as ten tracking channes to process compex data for the beam formed composite signa ICP measurements from each channe assigned to track the same sateite are coected to buid the steering vector Ony one sateite is seected for beam steering (signa beam) The weight updating rate of MVDR agorithm is KHz Moreover, positioning is dedicated to a beam formed composite signa There is a GUI to show ICP differences, weights and C/No of a channes to iustrate the beamforming performance as we as the positioning resut A HARDWARE The 7-eement antenna eements are arranged in circe with one waveength between the each on a circuar pane of auminum In the RF front-end, the L signa is downconvertered to a MHz intermediate frequency and samped at 7 MHz The front-end outputs -bit rea IF data The cocks of the front-ends must be perfecty synchronized for array signa processing A function generator set as sinusoida wave output at 7 MHz It is divided to two branches by a -to- spitter One is used the drive the cock input of USB microcontroer The other one is further divided by a -to-8 spitter to eight branches which are used as reference cock for each front-end The USB microcontroer serves as a bridge from RF front-end to PC Its -bit parae digita interface I/O is connected to seven -bit RF front-ends outputs The resuting data rate is MByte/s cose to the imit of USB data rate of MByte/s It is necessary to have an efficient strategy to reach such a high data rate The hardware incuding antenna array, RF front-end and USB microcontroer board was deveoped in [] Since the deveoped software receiver impements as many as 8 channes, it required a muti-core processor with mutithread support Further, the processor needs to support singe instruction mutipe data (SIMD) instructions to account for the high computationa compexity The impementation of SIMD instructions for Inte is caed as MMX/SSE/SSE [7] The 8-bit register of SSE can be divided to severa items in terms of bytes, words, or doube words and perform parae mathematica and to 8 Spitter RF Front end Box to Spitter USB Controer Board Function Generator PC with Inte Core i7 CPU Figure Diagram of hardware set-up of CRPA software receiver B SOFTWARE The software is deveoped with Visua Studio under - bit version of Microsoft Windows XP Most of source code is programmed using C++ The functions with high computationa compexity are programmed by inine assemby such as correation operation and covariance matrix cacuation The components used are isted in the Tabe Tabe Components ist of software architecture Component Description USB driver Provided by chip manufacturer USB C++ ibrary Software correator Hand-coded inine assemby using SSE instruction set based on bitwise parae agorithm [8] Tracking Use GPL-GPS [9] open source Positioning codeand modify interface to MVDR adaptive beamforming -Covariance Cacuation -Weight Update Weight-and-Sum and quantization of composite signa System Program software correator [] Hand-coded inine assemby using SSE instruction set for cacuating covariance Hand-coded C++ code for weight update Hand-coded inine assemby using SSE instruction set Arrange threads to achieve reatime capabiity The IF data transfer uses the C++ ibrary provided by chip manufacturer for communicating with USB driver The procedure of the IF data transfer is depicted in the figure

5 The width of data transfer for one sampe is -bits for the entire IF data stream from a antennas The data is further separated into a circuar queue of individua antennas The size of the entry of the queue is C/A code period (one msec) New IF data is stored into the rear index of the queue and the processing of the data is started from front index of queue acquisition/tracking are performed within each receiver thread Next, the weight-and-sum operations for beam forming are separated into 8 threads and run simutaneousy The combined data is further quantized to compex -bit outputs, and then processed within a composite antenna receiver thread In the composite receiver thread, not ony is software correation, acquisition and tracking performed, but the position soution is aso cacuated In parae, the covariance matrix is cacuated by averaging over one msec within N threads where N depends on the number of eements in the covariance matrix In addition, the MVDR weight update is performed in another thread The whoe procedure must be finished within one msec to achieve rea-time capabiity CODING EXAMPLES TO ENABLE REAL-TIME Figure Procedure of IF data transfer from the USB interface to the circuar queues To fuy expoit the resources of Inte Core i7 CPU, mutipe threads are created and arranged in a way such that at most 8 threads need be executed Figure shows the fowchart of the threads } } } The bit-wise parae agorithm [8] represents an incoming signa as a bit of a variabe or register, and then performs a parae ogic operation instead of mutipication In the software receiver, the agorithm is impemented using the SIMD instructions 8 sampes can be processed at a time using the 8-bit XMM registers After the bit-wise operation, the resuts are stored within the bits of the register and an accumuation (counting the number of bit set to one) is required Conventionay, this operation is performed by addressing a ook-up tabe in the memory The memory size for -bit tabe is KB [8] However, with a CPU with SSE can use POPCNT instruction [] which can count the number of bits set to one for a -bit register in a -bit operating system An exampe of counting the number of correation resuts being six using SSE instructions shows in the figure Figure Fowchart of threads every msec The main thread contros a the working threads At first, seven receiver threads process the IF data obtained from the individua queue of antennas Software correation and Figure Exampe of counting the number of correation resuts being using SSE instructions

6 As mentioned in the previous section, the weight-and-sum operation is to combine the IF data by mutipying the compex weights and summing over a antennas This operation can be performed in parae using SSE instructions Figure 7 shows impementation of the weight-and-sum operation using SSE instruction At first, the rea inputs and compex weights are oaded into the XMM registers in terms of bytes Then, the mutipy packed signed integers and store ow (PMULLW) instruction is performed to mutipy rea inputs with compex weights in parae Finay, a parae addition is performed three times to obtain the summations of rea and imagery components Figure 7 Exampe of the weight-and-sum operation using SSE instructions CALIBRATION OF THE ANTENNA ARRAY BY CARRIER PHASE PRECISE POSITIONING Athough the software receiver can act as a CRPA in reatime without any prior knowedge, the caibration of the antenna array in the post-processing mode wi aid in examining the CRPA performance of receiver The gain pattern of the composite antenna is critica to show the performance of CRPA The gain is cacuated using reative positions of antennas and weights as shown in equation () M r πp = i rˆ ( φ, θ ) GP( φ, θ ) wi exp j + γ i () i= λ where baseine vector p r i and cabe ength differences γ i are obtained through caibration Figure 8 shows the used caibration procedure of the antenna array In the CRPA software receiver, the ICP measurements are coected from the channes of the individua antennas for one minute The azimuth/eevation of sateites are aso obtained using the beam formed composite signa The singe difference ICP between signas of antennas and reference antenna j is represented as []: k k k k ϕ ij = λ rij + δlij + Nij + εij (7) where rij is differentia range toward the k sateite k between ith and jth antenna, Nij is the integer associated k k to ϕ ij, ε ij is the phase error The doube difference ICP between sateites and reference sateite is represented as: k k k k ϕ ij = λ rij + Nij + εij (8) The cabe ength difference term is subtracted in the doube difference Based on the distance of the antenna position cose to one waveength, equation (8) can be written as: k k k ϕ = λ rˆ rˆ p + N + ε (9) where ij k ( ( ) k rˆ is the unit vector to sateite k, ij ij ij p ij is baseine vector between ith and jth antenna By combining a the doube difference measurements of the pair ijth antennas, the observations equation is represented as: ( ) ϕ ij rˆ rˆ Nij εij ϕij rˆ ( rˆ ) = + Nij + εij λ p ij I K M M M M () K K K K ϕij rˆ ( rˆ ) Nij εij Γ = λ Gp ij + N + Ε From the positioning resuts of composite channes, the azimuth and eevation of sateites are used to manipuate matrix G Before soving the reative antenna positions, the integer vector N needs to be resoved The LAMBDA method, specified in reference [], is used By substituting the soutions of N and p ij into the equation (7), the cabe ength differences are cacuated after fitering the noise term CRPA Software Receiver Antenna# Channe #~#N Tracking Antenna# Channe #~#N Tracking Antenna#7 Channe #~#N Tracking Composite Channe #~#N Positiong ICP Measurements ICP Measurements ICP Measurements Sateite s Azimuth Eevation Antenna Array Precise Positioning Singe Difference Measurements Doube Difference Measurements Resoving Integer Ambiguity by LAMBDA Method Matrix Manipuation Of G Matrix Soving Reative Antenna Positions Soving Cabe Length Differences Figure 8 Bock diagram of the caibration procedure of antenna array

7 ANALYSIS OF THE THREAD ACTIVITIES AND TIMING PERFORMANCE In order to anayze the activities of the threads of the CRPA rea-time software receiver, the Inte Thread Profier coector program is utiized This is a thread anayzer appication to understand the threading patterns in muti-threaded software This coector program is executed with our software receiver and outputs a timeine diagram containing execution fow of a the threads as shown in the figure 9 The resuting execution fow exacty foows the designed fow shown in figure Tabe Execution time of threads Thread Name Mean Execution Time (µsec) Process IF Data of 79 Antenna # ~ #7 Weight and Sum 7 Composite Receiver 8 and MVDR weight update Tota 9998 EXPERIMENT FOR ENHANCING THE C/NO An experiment is conducted in open fied (ow mutipath environment) to examine the performance of CRPA Figure shows the hardware set-up of the software receiver Figure 9 Execution fow of the threads of the CRPA reatime software receiver coected from the Inte Thread Profier coector program The execution time of the threads is measured by counting the cock cyces of CPU The threads needed to execute in one msec are divided into three parts and execution times is measured trias The resuts are represented as a box pot in the figure The mean execution times of each part are isted in the tabe The tota mean execution time is ess than one msec and shows that the CRPA software receiver can achieve reatime capabiity Time (µsec) 8 Execution Time of Threads for one msec Data Figure Hardware set-up of the software receiver To determine the beam formed antenna gain pattern for each sateite in view, the software receiver runs in postprocessing mode In each run, the receiver performs CRPA by MVDR adaptive beamforming toward one of sateites Figure shows the fitered C/No of a sateites in view The CRPA begins to perform from th second There is more than db gain through the CRPA for a sateites tested Antenna 7 Weight_n_Sum Composite_n_MVDR Tota Figure Box pot of the execution time iustrating the rea time capabiity of the CRPA

8 Fitered C/No PRN PRN9 PRN C/No (db-hz) PRN PRN PRN PRN PRN PRN 9 PRN PRN Time (sec) Figure The fitered C/No of the software receiver with and w/o CRPA beam steering The ICP measurements and azimuth/eevation of sateites are recorded for seconds Then, the caibration of antenna array using carrier phase precise positioning is performed and the resut is shown in the figure It is cose to the physica antenna array geometry Moreover, with reative antenna positions and weights from CRPA, the gain patterns of composite antenna after CRPA toward one sateite are cacuated by equation () Figure shows the resuting gain patterns toward the seected sateite The corresponding sky pot is shown in the center of the figure There is a high gain in the direction of sateite which is the basis as to why the C/No is enhanced by the CRPA processing N (λ) Ant# Reative Position of Antennas Ant# Ant# Ant# 8 PRN PRN Figure The gain patterns of the composite antennas by the CRPA toward the specified sateite in a sky pot format EXPERIMENT FOR INTERFERENCE REJECTION In order to examine the interference rejection performance of the CRPA software receiver, a singe channe GPS simuator provides injected, via spitters, interference which is combines with received signa from four eements of the antenna array The hardware set-up is depicted in figure The GPS simuator generates two types of interference One is CDMA interference obtained by setting a PRN number which is currenty unaocated The other is CW interference by turning off C/A code spreading Figure shows the power spectra density of IF signa for three cases: w/o interference, with CDMA interference, and with CW interference The spectrum of the signa with CDMA interference has a higher power obe within MHz bandwidth than no interference case The spectrum of signa with CW interference has a peak in the center frequency of IF 9 Skypot PRN 8 PRN PRN Ant#7 Ant# Ant# E (λ) Figure Reative position of the antennas from the carrier phase precise positioning Figure Diagram of the hardware set-up of the CRPA software receiver with the interference using GPS simuator

9 Power Spectrum (db) Power Spectra Density of IF Signa w/o Interference w/ CDMA Interference w/ CW Interference - Frequency (Hz) 7 8 x Figure Power spectra density of IF signa with and without interference The CRPA software receiver processes these signas in the post-processing mode and ony combines the four signas antennas which are subject to the interference Figure 7 shows fitered C/No of the PRN in the presence of CDMA and CW interferences In both cases, the interference starts seconds into the run Without CRPA, the software receiver wi ose ock on the PRN signa when interference is present The software receiver performs impements the CRPA at th second and contributes over db gain on C/No When interference is not present, the performance of MVDR is cose to deterministic beamforming However, after interference is present, MVDR has gain about db higher than deterministic beamforming It shoud be noted that these are reay simpe tests of nu steering and meant to verify that the agorithms performing correcty PRN Fitered C/No with CDMA PRN Fitered C/No with CW w/o CRPA w/o CRPA C/No (db-hz) C/No (db-hz) CONCLUSIONS We impemented a rea-time CRPA software receiver for GPS L C/A in a PC with a modern processor to demonstrate the feasibiity of CRPA technoogy for civi appications The deveoped agorithms impement a compete CRPA without any a prior The resut of experiments shows its performance of enhancing C/No as we as interference rejection The software receiver can be extended to a GPS L by repacing RF hardware components with minima changes to the software architecture The Federa Aviation Administration Aternate Position Navigation and Time study is interested in the use of the CRPA with the L signa for robust time [] Other future work incudes everaging a GPU, which is we known for its parae structure, to impement a-inview beamforming impementation and process higher resoution data The current bit resoution does not provide significant interference rejection as high power interference can saturate the anaog to digita converter Hence processing higher resoution data is needed for robust interference rejection We are aso interested in reducing the number of channe and impementing CRPA software receiver in a singe-core processor for mobie device are panned as future activities This may be usefu for aowing CRPA technoogy to fiter into ower cost civi appications ACKNOWLEDGMENTS The authors gratefuy acknowedge the invauabe hardware provision of Staffan Backén for antenna array The authors aso acknowedge the eectronic assistance of Doug Archdeacon for setting up PC and RF devices This work has been supported by Federa Aviation Administration and Nationa Science Counci, Taiwan under contract NSC98-97-I-- REFERENCES Time (sec) Time (sec) Figure 7 Fitered C/No of the software receiver performing CRPA by MVDR and deterministic beamforming in the cases with CDMA and CW interference [] DS De Lorenzo, Navigation Accuracy and Interference Rejection for GPS Adaptive Antenna Arrays, PhD thesis, Stanford University, 7 [] SP Appebaum, Adaptive Arrays, IEEE Transactions on Antennas and Propagation, vo, no, pp 8-98, 97 [] K Borre, DM Akos, N Bertesen, P Rinder, and SH Jensen, A Software-defined GPS and Gaieo Receiver: A Singe-Frequency Approach, Birkhäuser Boston, 7 [] DM Akos, A Software Radio Approach to Goba

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