Implementation of 2D MIMO Radar in MATLAB by Gowshigan Selvarasa A capstone project report submitted in partial fulfilment of the requirements for the degree of Bachelor of Science in Engineering Telecommunication Engineering Examination Committee: Dr. Attaphongse Taparugssanagorn (Chairperson) Dr. Mongkol Ekpanyapong Dr. Teerapat Sanguankotchakorn Nationality: Sri Lankan Asian Institute of Technology School of Engineering and Technology Thailand May 2015
Transcript
Implementation of 2D MIMO Radar in MATLAB
by
Gowshigan Selvarasa
A capstone project report submitted in partial fulfilment of the requirements for thedegree of Bachelor of Science in Engineering
Telecommunication Engineering
Examination Committee: Dr. Attaphongse Taparugssanagorn (Chairperson)Dr. Mongkol EkpanyapongDr. Teerapat Sanguankotchakorn
Nationality: Sri Lankan
Asian Institute of TechnologySchool of Engineering and Technology
ThailandMay 2015
Acknowledgements
I wish to express my sincere thanks to my adviser Dr. Attaphongse Taparugssanagorn whoprovided the insight and expertise that greatly supported the project. His ardent support inteaching me difficult theories and guidance through email in sharptime should be greatlyappreciated.
I would like to also express my sincere thanks to my project members Dr. Teerapat San-guankotchakorn and Dr. Mongkol Ekpanyapong , who gave me advice on the mistakesfound in my project and constructive feed backs to complete my project successfully.
My sincere appreciation is extended toward the friends who do Phd and masters in samedepartment who helped during when I troubled with MATLAB programming. At last mythanking goes to all who supported and motivated me directly and in directly.
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Abstract
Radar is a device used to locate a target using the parameters , velocity , distance fromthe radar, Angle of Departure (AOD), Angle of Arrivals (AOA), Radar- Cross- Coefficients(RCS), etc. The conventional radars suffer from less parameter identiablity, less resolu-tion,non adaptive techniques,etc. MIMO concept is a growing topic in wireless communi-cation. It is described in research studies that ”MIMO means superiority”. If MIMO hasgreat applications in wireless communications , why it is not in radar communication. Thisproject, the algorithms used to estimate the target parameters , velocity , distance, AOA,AODare implemented for both one dimensional array processing and two dimensional array pro-cessing, In the last chapter obtained results are explained considering various scenarios suchas SNR vs Angle error.
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Table of Contents
Chapter Title Page
Title Page iAcknowledgements iiAbstract iiiTable of Contents ivList of Figures vList of Tables vi
1 Introduction 1
1.1 Background ( History of Radar) 11.2 Problem Statement 81.3 Objectives 91.4 Limitations and Scope 9
2 Literature Review 11
2.1 Information 11
3 MIMO Radar: Overview 13
3.1 MIMO Radar: Signal Model 133.2 MIMO Radar Ambiguity Function 143.3 Parameter Identiablity 153.4 DOA and DOD Method 15
4 Methodology 22
5 Results and Conclusion 23
5.1 Results 235.2 Conclusion 31
References 32
iv
List of Figures
Figure Title Page
1.1 Hybrid phased array where steering is done mechanically 81.2 Picture of a phased shifters 83.1 Rectangular co-ordinate system of bi static radar 143.2 Classical beam forming 155.1 CAPON spectrum of one target 235.2 MUSIC spectrum of one target 235.3 Root-MUSIC estimated target angle 245.4 Estimated AOA and AOD using Root-MUSIC 245.5 Estimated AOA and AOD using CAPON method for six targets 255.6 Estimated AOA and AOD using Root-MUSIC method for six targets 255.7 CAPON spectrum of target no of six 265.8 MUSIC spectrum of target no of six 265.9 CAPON spectrum of target no of six with high SNR 275.10 MUSIC spectrum of target no of six with high SNR 275.11 Estimated angles using both algorithm 285.12 Comparison between estimated angles and true angles 285.13 2D Root-MUSIC Algorithm 305.14 Comparison of estimated angles using 2D Root- MUSIC 30
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List of Tables
Table Title Page
5.1 This table contains SNR values associated error in estimated AOAand AOD 29
5.2 This table contains SNR values associated error in estimated az-imuth Angle and elevation Angle 31
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Chapter 1
Introduction
1.1 Background ( History of Radar)
The ” RADAR” word stands for radio detection and ranging. It is an electromagnetic device used fordetection of target’s the speed, range, altitude, and some other parameters like AOD, AOA of eithermoving or stationary object. Since its’ innovation to now, it has many application in various fieldssuch as Air-Traffic Control, Air-Defence System, Marine RADARs , Aircraft anti-collision systems,ocean altimetry and radar for geological observations.
The discovery of electro-magnetic light theory by James clerk Maxwell, electromagnetic waves byHeinrich Rudolf Hertz, invention of super hydronic receiver by French engineer Lucien levy, inven-tion of Magnetron by Albert Wallace Hull, contributed to the theoretical concepts of Radar. Afterthat American physicist Albert H. Taylor and Leo C. Young located a wooden ship for the first timeand then ships are equipped with radars. Then the important components of radar an amplifier and anoscillator tube was invented by George Metcalf and William C. Hahn.
Sir Robert Watson-Watt considered as inventor of RADAR, and the patent was granted in 1935. MITRadiation Laboratory contributes to the significant development RADAR
1. Developed and fielded advanced radar systems for war use
2. Exploited British 10cm cavity magnetron invention
3. Grew to almost 4000 persons(9 received the Nobel prize)
4. Designed almost half of the radars deployed in world war II
5. Created over 100 different radar systems
Gradually there were several improvements in radar technology. Once the concept of array antennaswere introduced. Phased-array RADAR development began around in 1958 in MIT Lincoln Labora-tory
1.1.1 Radar Fundamentals
This device basically emit a electromagnetic wave which reflect on target and return back to the Radar.Based on the structure of antennas and receivers radar can be categorized into two. They are Mono-static radars and bi static radars. Mono-static radars’ transmitters and receivers are located in samelocation . In some cases same device is used for both transmitting and receiving. Bi static radars aretype of radar where the antennas and receivers planted in different location , distance is much higherthe the wave length. Kay (2013)The signal transmitted by the radar is called probing signals. Different types of probing signal are fordifferent applications.Main categorizes are continuous waveform radars and pulse radars.
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1.1.2 Radar Equation
Equations can be used to explain and evaluate the radar performance.The received signal of the radar can be given below. Skolnik (1970)
Pr =Ps
4πr2Gσ (Equation 1.1)
wherePs : reflected power from the target
G : Gain of the antenna
σ : radar cross section
r: distance from the radar
1.1.3 Doppler Shift
Relative between radar and target causes two phenomenon in received signal , one is dialation in thereceived signal and other one is frequency shift Dialation which cause distortion in the reconstructedimages and this can be compensated effectively by the narrow band and wide band process.
Doppler shift is useful to distinguish the moving objects and resting objects based on the observedfrequency. We usually observe phenomenon when sound emitting object move towards or outwardsus. when the movement when it moves towards us, we see the sound is approaching us and pitch isincreasing and when the sound goes away from us, pitch is decreasing.
This section illustrated the Doppler shift in Bi-static radar. Even the tiny motion of either radar ortarget cause change in Doppler frequency. it implies that it is necessary to analyse how the Dopplershift affect the received signal.
The Doppler shift is basically the time rate or total displacement change by the wavelength normalizedwave length λ
In this, transmitter is static point and target is moving.
fd =1
λ
[d (RT +RT )
dt
](Equation 1.2)
where the symbols V ,δ, β respectively represent the velocity ,aspect angle angle between the trans-mitter and receiver with vertex.
The projection of the target velocity vector in to line of sight (LOS) of receiver to target is in the formof
dRRdt
= ν cos
(δ +
β
2
)(Equation 1.3)
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whereθT look angle of transmitter
θR look angle of receiver
Similarly the project transmitter to -target LOS of target’s velocity vector into transmitter
dRtdt
= V cos
(δ − β
2
)(Equation 1.4)
The equations Equation 1.2 and Equation 1.3implies
fd =2V
λcos (δ) ∗ cos
(β
2
)(Equation 1.5)
V : velocity
β2 : bi static bisector
consider the special case where R = RT = RR Equation 1.5 can be modified
fd =2V
λ= 2
dRλdt
(Equation 1.6)
This is a well known expression of Doppler shift which is used communication theories. it is obviousthat when β is reduced to, 0 above equation converges to mono-static radar equation. when β
2 is equalto 90, doppler shift approaches to zero
1.1.4 Delay in Received Signal
Delay between transmitted and received signal can be written as
y(t) = x(t− T0) (Equation 1.7)
x(t) y(t) τ are respectively transmitted signal , received signal and the delay. Delay of the nonmoving target can be expressed as t0 = 2d0
c , in which c is velocity of light , d0 is the distance . It isassumed that both radar and targets are in rest.
According to the above scenario shown in the picture, the time delay can be derived as below
d1 = c4t (Equation 1.8)
d1 = d0 + v4t (Equation 1.9)
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From Equation 1.8 and Equation 1.9 we can get
4 =d0
c− v(Equation 1.10)
d0 is the distance at the beginning of the time.In the similar way the following equation can be derived
T0 = 2d0
c− v(Equation 1.11)
1.1.5 Derivation of Doppler stretch factor
Consider the scenario below
t0 is the time signal transmitted from the radar, At a time t the distance Target travelled is
dt = d0 + v(t− t0) (Equation 1.12)
The distance target travelled at t+4t is equal to
dt+t0 = dt + v4t (Equation 1.13)
dt+4t = c4t (Equation 1.14)
By combining Equation 1.13 Equation 1.14 , it implies
c4t = d0 + v(t+4t− t0) (Equation 1.15)
from that
4 =d0 + v(t− t0)
c− v=
dtc− v
(Equation 1.16)
Lets assume that sampling period of the signal is Ts. The time interval between the transmitted andreceived signal of first sample is t
′0 − t0. Similarly for the second sample T
′s
t′0 = t0 + 24t0 (Equation 1.17)
t1 = t0 + Ts (Equation 1.18)
t′0 = t1 + 24t1 = t0 + Ts + 2
dt1c− v
(Equation 1.19)
t′1 − t
′0 = t1 − t0 + 2(4t1 −4t0) = Ts +
2(dt1 − dt0c− v
(Equation 1.20)
(Equation 1.21)
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The distance dt1 − t0 can be equated to vTs
T′s = Ts + 2
vTsc− v
= Tsc+ v
c− v(Equation 1.22)
frequency of the received signal is
f′
=c− vc+ v
f (Equation 1.23)
The Doppler shift is the different between f1 and f . It is apparent that transmitted signal y(t) , whicharrives at time t1 = 24t1 can be shown
y(t1 + 24t1) = x(t1) (Equation 1.24)
y(t1 +2d0 + 2vt1 − t0
c− v) = x(t1) (Equation 1.25)
this can be equated to (Equation 1.26)
y(t1c+ v
c− v+
2(d0 − vt0)
c− v= x(t1) (Equation 1.27)
t′1 = (t1
c+ v
c− v+
2(d0 − vt0c− v
(Equation 1.28)
y(t′1) = x(t
′1
c− vc+ v
− 2d′0
c− v) (Equation 1.29)
The Doppler stretch factor can be given
α =c− vc+ v
(Equation 1.30)
1.1.6 Doppler stretch factor for approaching target
Same concept can be applied to approaching target aswell. In this case , the relationship between tand dt can be expressed.
dt = d0 − v(t− t0) (Equation 1.31)
when time is (t+4t) , the target is at
dt+4t = dt − v4t = d0 − vt+ vt0 − v4t
4t =d0 − v(t− t0
c+ v=
dtc+ v
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Lets derive Doppler stretch factor
T′s = t
′1 − t
′0 = t0 + Ts +
2dt1c+ v
− t0 −2d0
c+ v(Equation 1.32)
T′s = t
′1 − t
′0 = t0 + Ts +
2dt1c+ v
− t0 −2d0
c+ v(Equation 1.33)
T′s = Ts +
2dt1 − dt0c− v
= Tsc− vc+ v
(Equation 1.34)
y(t1c− vc+ v
+2d0 + 2vt0c+ v
) = x(t1)After substitution t′1 = (t1
c− vc+ v
+2d0 + 2vt0c+ v
(Equation 1.35)
it impliesy(t′1) = x(t
′1
c+ v
c− v− 2d
′0
c− v) (Equation 1.36)
doppler strech can be expressed (Equation 1.37)
α = (c+ v
c− v) (Equation 1.38)
It is apparent from Doppler stretch factor that, when the target approaching spectrum of the signalwill be broader and when resides the spectrum will be widened. Now received signal can be shown
y(t) = x(αt− T0) (Equation 1.39)
1.1.7 Ambiguity Function (AF)
The ambiguity function is analytical tool which is used in radar communication to evaluate radar’sperformance against objects that interrupt waves such as clutter , noise, etc. AF was derived inWoodward (1953).Basically this is a surface plot between the doppler shift range and ambiguityfunction. The plot helps to
• detection of the waveform
• parametric measurement accuracy
• range
• doppler shift
• clutter rejection
The general form if AF as implied in Levanon and Mozeson (2004)
| χ (τ, ν) |=|∫ +∞
−∞υ (t) υ? (t+ τ) exp (2πt) dt | (Equation 1.40)
1.1.8 Properties of Ambiguity Function
Only the main properties of the AF is listed
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• Maximum at (0, 0) ⇒| χ (τ, ν) |≤| χ (0, 0) |= 1 implies that AF can be higher on only inorigin. In other way it means that when the reflect signal is matched to the original signal, AFvalue get higher
• Constant volume: the volume of the AF function under normalized is equal to unity and it isirrelevant of waveform. Prop 1 and Prop 2 together says that if we try to squeeze the AF , itcannot exceed the total volume 1 and it has to happen at any point
• Symmetry around the origin , implies that quadrant of AF may give solution but we need toconsider only the positive value
1.1.9 Target Models
Modelling targets are important to evaluate the performance of the radar. There are two types oftargets which we consider. They are fluctuating targets and non fluctuating targets. The RCS ofthe fluctuating targets varies with the time. so for every scan and every pulse different RCS will beechoed. There is a requirement for tabulation of basic functions for fluctuating targets. First studywas made by Swerling during 1950s. Swerling has proposed five types targets models
General target model
p (σ) =m
Γ (m)σav
(mσ
σav
)m−1
emσav (Equation 1.41)
• SwerlingI : In this model Chi-Squared PDF is with two degrees freedom (m=1). All scattersare independent and equal areas. RCS is constant for the first scan but varies with scan to scan.so different scattered signal will be result in receiver with time
• SweringII : Random RCS values to each pulses
• SwerlingIII:Chi-Squared PDF is with degrees of freedom (m=2) similar to Swerling 1, RCS isconstant for a single scan
• SwerlingIV: same as SwerlingIII, different RCS for different pulse
• SwerlingV Infinite degree of freedom( m=inf) and constant RCS
1.1.10 Phased-Array RADAR
Phased-array consists several transmitting array elements , each with phased shifter. By means ofshifting the phased in each elements beams are formed, and it is also useful to steer beam in desireddirection
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Figure 1.1: Hybrid phased array where steering is done mechanically
This technique is attractive as steering is done electrically and get rid of the disadvantage of mechan-ical steering and was called as digital beam form technique.
Figure 1.2: Picture of a phased shifters
Though the phased-array radar performed well than the conventional radars, it couldn’t last for longas the modern communication requires superior capabilities and and this concept wasn’t adaptive tothe requirements. Mean while Multi static radars were in use and MIMO concept was boomed intowireless communication. This leads to the entry of MIMO radars
1.2 Problem Statement
The previous conventional RADAR systems suffers from
• less parameter identiablity ,
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• in efficient transmit beam patterns
• unwanted side lobes
• less angle resolution
• less Doppler resolution.
Current research shows MIMO concept has enhanced the modern wireless communication.if MIMOshows it’s superiority in wireless communication thy why it is not it in radar communication. Therefore aim of this project is to simulate a 2D MIMO radar in matlab
1.3 Objectives
Overall objective,
1. Implementing 1D MIMO radar in MATLAB
• Estimating the target parameters velocity, Distance, AOA, AOD
• Implementing Root-MUSIC algorithm in MATLAB
• Implementation of CAPON algorithm in MATLAB
2. Implementing 2D MIMO radar in MATLAB and find 3D location of Target
• Estimating the target parameters velocity, Distance, AOA, AOD
• Extending Root-MUSIC algorithm to support 2D arrays ( Planar arrays)
3. Demonstrating the Capabilities of MIMO radar using the simulation results
1.4 Limitations and Scope
This study focus in learning signal processing techniques in MIMO Radar and learning the algorithmsused estimate parameters
1. No hard ware will be used and MATLAB will be used to simulate the result
2. AWGN Communication channel assumed
3. Mutual couplings of the antenna is not considered
4. Free space propagation is considered
5. Ideal antennas with omni direction is considered
6. Radial Movement of Targets is considered
7. The only noise associated with the simulation is AWGN
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8. Channel fading is ignored
9. Parameters related radiators , systems noise is not considered
10. Point targets is considered
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Chapter 2
Literature Review
2.1 Information
MIMO- which is a short term of Multiple Input and Multiple Output, is a concept applied in mod-ern wireless communication system. MIMO, was introduced by Fisheler in Fishler et al. (2004)as new concept in radar signal processing. It is tabulated in the study that there are major differ-ences between the conventional phase radars and MIMO radars. MIMO radars are exaggerated forits’ higher performance for high SNR. It is also noted that phased-array radars has slightly betterperformance in low SNR.In the proposed concept , the transmit elements emits orthogonal and in-dependent waveform where as in traditional phased radars the emitted wave forms phase co related.The Cramer-Rao-Bound ( CRF) is applied to prove it’s dramatically improvements inter of DF.
MIMO radars superior abilities are illustrated in J. Li and Stoica (2007). A comparison between thetraditional radar and MIMO radar has been done using the factors parameter identifiablity, robust andadaptive beamforming , probing signal design. MIMO radars has reduced side-lobes in beam formingand it’s resolution using the CAPON and GIRT methods and increase in the detection performance ofmultiple targets.
Further more J. Li et al. (2007) , proved that the parameter identiablity - the ability to identify themaximum targets over the phased-arrays using CRB technique. Moreover the superiority of MIMOover its’ phased counter parts due to the wave form diversity.
The basic works on detection performance of radar to different type of targets was done in (Swerling,1960) bu Peter Swerling. Four types of radar targets were introduced . Type 1, TypeII , TypeIII,TypeIV, TypeV. The more explanation about these model are in the introduction section David.M hasextended swerling target models Chi-Square target models in (Drumheller, 1994). A theoreticalevaluation is carried out in (Du et al., 2008) with the finite number of small scatters considered andformula has derived for the probability of detection performance of the MIMO radar. That was namedas non time consuming simulation.
(J. Li & Stoica, 2007)states that MIMO radar with co-located antennas implies significant superi-ority over it’s phased-array system , such as improved parameter identiablity , direct applicability ofadaptive techniques for parameter estimation as well as superior flexibility of transmit beam patterndesigns. Similarly (Haimovich, Blum et al. 2008) has found widely separated antennas can be usedto obtain a diversity gain for target detection and for estimation of various parameter such as angleof arrival and Doppler. Both study suggest that different configuration of antenna enhance differentfeatures of radar. However this study is focused on patched antenna types thus propagates in twodimensional.
As a new concept and combination of old phased-array radars and MIMO radars phased MIMO wasintroduced in (Hassanien & Vorobyov, 2010). Basic idea is without losing the advantages offered byconventional phased radar systems and combining the waveform diversity of MIMO a propagating acoherent signal. In this concept, every arrays are partitioned into sub arrays. It is proved that the newtechniques has several advantages over the conventional phased-arrays and pure MIMO radars such
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as reduced side lobe levels in beam pattern , Higher robustness, adaptive beam forming , excellentadaptive interference. Moreover in (Hassanien and Vorobyov 2010),it is found that MIMO radartrade-off the phased-array in
1. Angle resolution , detecting higher number of targets, improving parameter identiablity andextending array aperture
2. Using beam forming techniques at both the transmitting and receiving end efficient design ofoverall beam pattern of the virtual array
3. Resolution and robustness against beam-shape loss
4. Improved robustness strong interference
Similar to the above concept but with small different, in (Browning et al., 2009),was introduced withname hybrid phased-MIMO. same as the above , arrays are portioned into sub arrays and cub arrayconfiguration can be changed But it is concluded that there are more works needed to be done as thenumber of open problems raised.
But in Hassanien and Vorobyov (2010), the one of problem rise with hybrid-phasedMIMO is solved, a selection matrix is introduced with steering vector to choose subsets thus the configuration can bechanged
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Chapter 3
MIMO Radar: Overview
Based on the structure of antenna and receiver, MIMO radars are categorized into two.They are mono-static radar and bi-static radar. Mono static radars are called co- located radars which means thattransmitter and receiver located in close distance. Theories and Equation discussed in the followingsection are associated with with co-located MIMO radars. Details of the symbols is given in frontsection
3.1 MIMO Radar: Signal Model
consider a MIMO radar with Mt transmit elements and Mr receive elements θ is a generic param-eter , generally it is azimuth angle, in case of two dimensional array processing another additionalparameter φ is introduced
the general model transmitted signal is given by
Mt∑m=1
e−2πfτ(θ)xm (n) = a∗ (θ)x (n) (Equation 3.1)
τ , f ,θ represents the time taken reach the target, some times referred as one way delay, career fre-quency and azimuth angle
x(n) can be expandedx[n] = [x(1)x(2)x(3)x(4)x(5)x(6)x(7)...........x(n)]
The steering vector a(θ) is given bya(θ) = [ei2πτ1θei2πτ2θei2πτ3θ.......ei2πτmθ]
b(θ) = [ei2πτ1θei2πτ2θei2πτ3θ.......ei2πτmθ]steering vector of receiving array
in which τ is delay between the target and receiver m is the number of receiving elements
The received signal can be represented under assumption targets are point targets this final receivedsignal can be given
y(n) =K∑k=1
βkbc(θk)a
∗(θk)x(n) + ε(n), n = 1, ..., N. (Equation 3.2)
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in the above equation K is number of targets that scatter the transmitter signal towards the receiverand βk are the complex amplitude related RCS of the targets θK are location parameters and ε(n) iscombination of noise and all other interference including clutter.
3.2 MIMO Radar Ambiguity Function
All MIMO radar theories were generalized from the traditional radars. Ambiguity function inherentthe properties of MIMO radar system. AF of a mono-static radar is given by
Θmimo = |X(τ, wD)| = |∫ + inf
− infs(t)S(t− τ)ejwdt| (Equation 3.3)
where τ and wD are respectively time and frequency shift. It is proved in (You et al., 2007) usingthe right angle co-ordinate system as shown in picture, AF of bi-static radar can be written
Figure 3.1: Rectangular co-ordinate system of bi static radar
Since MIMO radar waveforms are mutually orthogonal and each of the received signals are matched,the AF of MIMO radar can be written.
dm.j= complex coefficient for (m, j) the transmit and receive Channel τm,j(p) two-way delay fmj(θ)is Doppler shift Zj(t) is white gaussian noise observed of jth receive antenna.
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3.3 Parameter Identiablity
The term Parameter identiablity is introduced in (J. Li & Stoica, 2009) which means’s the maximumnumber of target can be detected. Waveform diversity in MIMO systems contributes to achieve thehigh parameter identiablity . It is proved that Kmax of a MIMO radar is given by
Kmax ∈ [2(Mt +Mr)− 5
3,2MtMr
3] (Equation 3.7)
Except the case for Mt = 1 for a phased-array radar
Kmax 62Mr − 3
3(Equation 3.8)
3.4 DOA and DOD Method
DOA is Direction of Arrival some times referred as Angle of arrival and DOD is degree of Departureand sometimes referred as AOD the angle of departure. In the following section the popular twomethods are explained and Root-Music method is extended based on (Hassanien et al., 2013)
3.4.1 Delay and SUM Method
Figure 3.2: Classical beam forming
The simplest and classical beam-former methods are delay-and -sum method. The figure ?? depictsa structure of classical narrow-band beam former. The out put y(K) can be expressed as a linearweighted sum of the sensor elements. That is, y(k) = wHu(k)
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3.4.2 MUSIC and Root-MUSIC
MUSIC - is a algorithm used to distinguish the multiple signals. The general problem faced receivingelements are signal coming from arbitrary location has arbitrary directional characteristics (polariza-tion,gain ,or phase) and undergoes noise interference and thus an arbitrary covariance matrix. Thisalgorithm can be used to estimate
• number of signals
• Direction of A
• strength and cross co -relation among the direction wave forms
• polarization
• strength of noise /interference
Multiple signal classification is shortened as MUSIC. It is a super method with high resolution , whichis used to estimate the number of arrived signal and thus it can be used estimate DOA. The fundamen-tal ideal exploit the Eigen value of a input covariance matrix.Eigen decomposition will yield Eigenvectors from sample covariance matrix or Singular Value Decomposition (SVD) of the data matrix.The research study shows one requires K > 2N so that SNR within 3dB to be optimum. This resultcannot be directly applied to estimation of DOA
Lets consider that number of arrays arrive in the plane in D. It can be considered input of m-elementarray. This array includes the noise vector and incident signal.That is
u(t) =
D−1∑l=0
a(Φl)sl(t) + n(t) (Equation 3.9)
and
u(t) = [a(Φ0)a(Φ1)...a(ΦD−1)[
s0(t)s1(t)sD−1(t)
+ n(t) = As(t) + n(t) (Equation 3.10)
In the above equations ST (t) = [s0(t)s1(t)...sD−1(t)] is the vector of reflected signal in planen(t) = [n0(t)n1(t)...nD−1(t)] is the vector caused noise , anda(φ) is the steering vector of jthsignalcorresponding to the Direction of Arrival.. To make it simple, we shall skip the time factors from U ,s,n from this point onwards.
Generally , to visualize the received vector u and steering vector a(φ) can be considered as Mdimensional space vector. the received vector is linearly combined with steering vector array ands0, s2, sD−1 is the coefficient of the combination. Considering the above signal model, a covariancematrix Ruu can be developed as shown below.
In the above equation signal correlation matrix of E[ssH ] is denoted as Rssλ0λ1......λM−1 is assumed as the eigen values of Ruu and
[|Ruu − λiI] = 0] (Equation 3.13)
we can further expand this as
[|ARssAH + σ2nI − λiI| = |ARssAH − (λiI − σ2
n)I| = 0] (Equation 3.14)
Eigen values , vi can be derived from ARssAH are
[vi = λi − σ2n
Matrix A is constructed with linearly independent steering vectors. The incident signal are not highlycorrelated as the correlation matrix Rss is a non-singular matrix. Since Rss is a non- singular matrixand matrix A has full column rank. It implies that if number of received signal D less than arrayelementsM , the rank ofM ∗M dimensional matrixARssAH will beD. It can be shown thatM−Dof the eigen value Vi of matrix ARssAH are zero. This implies that eigen value M − D of autocorelation matrix Ruu will be equal to noise variance σn2 . After sorting eigen values of Ruu fromlargest to small such that largest λ0 and smallest λm−1
λD........., λM−1 = σ2n (Equation 3.15)
The finite sample data can be used to estimate the auto corelation matrix Ruu. Noise power whichcontributes to the generation of eigen values won’t be identical. once the multiplicity factor K deter-mined , number of signal can be obtained from the relation m = D + k so it can be given.
[hatD = M −K] (Equation 3.16)
qi is the eigen vector related to the particular eigen value λi
[Ruu − λiI = 0] (Equation 3.17)
The eigen vector related with M −D smallest eigen value
[(Ruu − σ2nI)qi = ARssA
Hqi + σ2nI − [sigma2
nI = 0] (Equation 3.18)
[ARssAHqi = 0] (Equation 3.19)
Since matrix A has full rank Ruuand is non-singular matrix.It implies that
[AHqi = 0] (Equation 3.20)
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[
aH(Φ0)qiaH(Φ1)qi
.
.aH(Φn−1)qi
= [
00..0
(Equation 3.21)
The above signal model implies that small eigen values associated withM −D is orthogonal steeringvector D, which are used to construct A
By observing the steering vector that are orthogonal to the eigen vector associated with the auto corelation matrix of received signal, that almost equal to σ2
n the real steering vectors of received signalcan be found.
It is obvious now that the eigen vectors yielded by covariance matrix Ruu is constructed from twoorthogonal sub space. The both sub spaces are signal subspace ( eigen sub space) and noise sub space( non -principle eigen subspace). Signal sub space consists the steering vector of the DOA. Hence thissub space is orthogonal to the noise vector. So the DOA can be estimated by searching all orthogonalsteering vectors.To make the searching simple, lets form noise sub space
[Vn = [qDqD+1 .........qDM−1] (Equation 3.23)
Because of the fact that , steering vector of signal components and eigen vector of noise sub space areorthogonal to each other aHVnV H
n a(Φ) for all angle of arrival φ . Therefore DOA of received signalscan be determined from finding peak values of received MUSIC spectrum.
[PMUSIC(Φ) =1
aHVnV Hn a(Φ)
(Equation 3.24)
Or, alternatively,
PMUSIC(Φ) =aHa(Φ)
aHVnV Hn a(Φ)
(Equation 3.25)
The denominator will be decrease by the fact that steering vector aφ and noise space Vn are orthogo-nal. This cause the peaks in the MUSIC spectrum.
3.4.3 Root MUSIC
There are many different approaches has been found to modify MUSIC algorithm to increase theresolution of the angle estimation and decrease the computation complexity. The one accepted bymany and used by many researchers is called Root-MUSIC which was developed by Barbell, whichis based on the idea exploring the polynomial roots of linear spaced arrays. Signal space eigen vectorsis used to define a rational spectrum function with improved resolution capability
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3.4.4 Extending Root-MUSIC to support two dimensional propagation
Extending Root-MUSIC algorithm to 2D needs a transmit array interpolation method that enableswhich helps to jointly estimation of elevation and azimuth parametersAssume that location of transmit antennas are Pm , [xm, ym]T , m = 1, 2....M and location ofantennas are measured in wave lengths. The steering array of (M ∗ 1) dimensional steering vectorcan be defined as
using Equation 3.30 and Equation 3.31 following two equations can be derived
Yu(τ) = [y1(τ), ..., yMd(τ)]T =≈
L∑l=1
βl(τ)u(θ, φ)bT (θl, φl) + Zv(τ) (Equation 3.35)
and
Yv(τ) = [yMd+1(τ), ..., yMd+Nd(τ)]T =≈L∑l=1
βl(τ)u(θ, φ)bT (θl, φl) + Zu(τ) (Equation 3.36)
In the above two equations size of Yu and Yv areNd∗N andNd∗N andZv(τ) = [ZMd+1(τ), .......ZMd+Nd(τ)Using Yu(τ) a covariance matrix Md ∗Nd can be formed as
R̂u =
Ts∑τ=1
Yu(τ)Y Hu (τ) (Equation 3.37)
and Ts number of radar pulses. A straight forward Root-MUSIC algorithm can be made estimate isςl = sin(θl)cos(φl)
and another covariance matrix can be formed
R̂υ =
Ts∑τ=1
Yυ(τ)Y Hυ (τ) (Equation 3.38)
The Root-MUSIC algorithm can be applied to above equation as well. The estimate is υl = sin(θl)sin(φl).There for estimate of angles are
ˆθi = sin−1(|Xi|φl = ](Xi)
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3.4.5 CAPON
The idea behind the delay-and sum method is, it searches for the arrived strongest signal and it can beconsidered as the best estimate of the arrival direction. The different from other method to CAPONmethod is, it tries to solve the problem related with classical beam forming (delay and sum method).The output power from CAPON spatial spectrum is given below
Pcapon(Φ) =1
aH(Φ)R−1uua(Φ)
(Equation 3.39)
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Chapter 4
Methodology
since this is research type project , The progress which used to finish the task is explained in the belowpicture
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Chapter 5
Results and Conclusion
5.1 Results
The simulation has been done for different SNR values and different number of transmit and receiveelements,
Figure 5.1: CAPON spectrum of one target
From the image 5.1 we observe that a peak is created to angles which are equal or closely equal totrue angles of AOD and AOA. This figure is the CAPON spectrum vs angles
Figure 5.2: MUSIC spectrum of one target
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This figure 5.2 is MUSIC spectrum vs angles
Figure 5.3: Root-MUSIC estimated target angle
In the figure 5.3, it is shown that estimated AOD and AOD using CAPON method in 2D Plot.
Figure 5.4: Estimated AOA and AOD using Root-MUSIC
In the figure 5.4, it is shown that the esimated AOD and AOA using Root-MUSIC algorithm.CAPONalgorithm and root MUSIC algorithm performances are same for one target. Let’s consider multiplenumber targets cases, consider no of targets six,
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Figure 5.5: Estimated AOA and AOD using CAPON method for six targets
The figure 5.5 shows estimated angles using CAPON method. we could able to observe that resolutionof some angles are not enough to locate them in 2D plot.
Figure 5.6: Estimated AOA and AOD using Root-MUSIC method for six targets
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Figure 5.7: CAPON spectrum of target no of six
Figure 5.8: MUSIC spectrum of target no of six
But we still see the spectrum resolution not enough to distinguish targets, so let’s increase the SNRvalue to 40dB where as it was 10dB in previous case.
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Figure 5.9: CAPON spectrum of target no of six with high SNR
Figure 5.10: MUSIC spectrum of target no of six with high SNR
Once SNR value has been increased, resolution of the angles clear, accuracy of angle estimation alsohigher
27
(a) Estimated elevation and azimuth angles withCAPON
(b) Estimated elevation and azimuth angles withRoot-MUSIC
Figure 5.11: Estimated angles using both algorithm
Figure 5.12: Comparison between estimated angles and true angles
The above diagram clearly depicts the estimated angle’s accuracy has been increased. In the previouscase less than six targets were clearly identified. Comparing 5.5 and 5.6 , we can come to conclusionthat root MUSIC has more resolution than CAPON and it is said root MUSIC is fastest algorithm.
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Table 5.1: This table contains SNR values associated error in estimated AOA and AOD
SNR RMSQ-AOD RMSQ-AOA
0 54.3460 20.1010
10 12.2421 14.0289
20 2.1184 2.6337
30 0.1235 0.0295
40 0.0399 0.0271
50 0.0017 0.0044
60 0.0024 0.0012
70 0.8165 0
100 0 0
The 5.1 shows root-mean squre error in estimated angles and related SNR values. It is clear that whenthe snr values increase resolution and accuracy increases.Now consider the scenario where two dimensional array processing is applied. As mentioned inliterature review it is using the 2D array processing we can three dimensional location.
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Figure 5.13: 2D Root-MUSIC Algorithm
Figure 5.14: Comparison of estimated angles using 2D Root- MUSIC
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Table 5.2: This table contains SNR values associated error in estimated azimuth Angle and
elevation Angle
SNR RMSQ-Elvation Angle RMSQ-Azimuth Angle
0 0.4660 0.8756
10 0.5372 0.6345
20 0.4232 0.5221
30 0.3297 0.4321
40 0.0375 0.0281
50 0.0018 0.0008
60 0.0005 0
70 0 0
100 0 0
Root MUSIC algorithms is performed and Root mean square error is calculated. If we know elevationangle, azimuth angles and direct distance it is easy to estimate three dimensional coordinated fromconverting from polar coordinates from rectangular coordinates Velocity and distance estimation canestimated by calculating Doppler shift, and since we can get the time taken to signal two way propa-gation, we can calculate the velocity and distance. The codes used in this case is not integrated withinthe main codes but this idea can be used to estimate the velocity and distance
5.2 Conclusion
This study just focus on implementing available AOA, AOD estimating algorithms and implementingnewly proposed 2D Root MUSIC algorithm. From results shown above we can evaluate performanceof MIMO varying with SNR and changing the no of transmit antennas and receiver antennas and alsochanging the no of integrated pulses. By using two dimensional transmit beam forming, Azimuthangle and Elevation angle can be estimated. By calculating Doppler shift, distance and velocity canbe calculated with the information available about time taken to a pulse its two way propagation.Implementing MIMO radar in MATLAB is not like implementing radar in practical case. To avoidthe complexity clutter models, Rayleigh fading, and some other important facts which is considered inreal environment has been ignored which is a passive approach. As future plans, integration hardwareparts with SDN (software defined radio) is suggested as SDN is low cost when considering aboutimplementation cost and it is growing research topic. Furthermore the concept MIMO can extendedto massive MIMO as it is the current hot research in wireless communication.
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