DESIGN OF AN AIRBORNE MULTI-INPUT
MULTI-OUTPUT RADAR EMULATOR TESTBED FOR
GROUND MOVING TARGET IDENTIFICATION
APPLICATIONS
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of
Science in the Graduate School of the Ohio State University
By
Evgeny Yankevich, BS
Graduate Program in Electrical and Computer Science
The Ohio State University
2012
Master’s Examination Committee:
Prof. Emre Ertin, Advisor
Prof. Lee Potter
c Copyright by
Evgeny Yankevich
2012
ABSTRACT
Multi input multi output (MIMO) radar is a radar system with multiple receive
and transmit antennas, that can transmit independent waveforms on each transmit
elements. Although many traditional multi-antenna radar concepts such as phasedarray, receive beamforming, synthetic aperture radar (SAR), polarimetry, and interferometry can be seen as special cases of MIMO radar, the distinct advantage of a
multi-antenna radar system with independent transmit waveforms is the increased
number of degrees of freedom leading to improved resolution and performance in
detection and parameter estimation tasks.
A promising application of MIMO radar is the identification of slowly moving
targets using airborne MIMO radar platforms. The advantage of using MIMO in this
configuration is its ability to synthesize a larger virtual array with relatively fewer
antennas. This allows higher spatial resolution and better separation of returns from
ground clutter and targets. The space-time adaptive processing (STAP) methods
originally developed for Single-input, Multiple-output (SIMO) radar are applicable
to MIMO radar systems after proper pre-processing of the received signals. The
performance of STAP algorithm critically hinges on the structure of the clutter covariance matrix; therefore, MIMO STAP methods will benefit greatly from theoretical
and empirical study of the clutter statistics.
The contribution of this work can be summarized in three parts. First, we present
ii
a design of a rooftop MIMO radar testbed that emulates a MIMO GMTI system
mounted on airborne platform. Second, we give results of a simulation study of
ground clutter for the testbed rooftop geometry, highlighting potential issues with the
relatively close range. Third, we extend previous results on clutter covariance matrix
rank for MIMO systems with orthogonal waveforms to the case of MIMO systems
employing nonorthogonal waveforms. Relationship between rank of covariance matrix
of orthogonal and nonorthogonal waveforms was established.
iii
To my family: my wife Uliana and sons Tamir and Tal;
and my parents who supported and motivated me so much during this time.
iv
ACKNOWLEDGMENTS
I would like express my gratitude to my advisor Prof. Emre Ertin for his supportive
position and help during entire program period. He open the world of radars for me
and taught the proper way and methodology of conducting research. I appreciate his
way of intuitive and formal explanations of very complicated issues in radar signal
processing and system design.
I would like to thank Siddharth Baskar for his help in implementation of the MIMO
GMTI emulator design.
v
VITA
1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Born in Tashkent, Uzbekistan
1991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BS in Electrical Engineering from Moscow
Institute of Electronic Technologies (MIET)
1993-2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Research Engineer at Qualcomm Israel,
Infineon Technologies, Zoran Microelectronics.
2010-2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Assistant and Fellow
in the Department of Electrical and Computer Engineering at The Ohio State University
FIELDS OF STUDY
Major Field: Electrical and Computer Engineering
Specialization: Signal Processing and Communication
vi
TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
CHAPTER
1
2
3
4
PAGE
MIMO Radar Concept . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Notations and Glossary . . . . . . . . . . . . . . . . . . . . . . . .
3
Review of GMTI and STAP Processing . . . . . . . . . . . . . . . . . .
8
2.1 Doppler Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Processing for Ground Moving Target Indication . . . . . . . . . .
9
12
MIMO Model for GMTI . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.1 Orthogonal Waveforms Case . . . . .
3.1.1 Virtual Array Concept . . . .
3.1.2 Uniform Linear Array . . . . .
3.2 Nonorthogonal Waveforms Case . . .
3.3 Rank of Clutter Covariance Matrix . .
3.3.1 Orthogonal waveforms case . .
3.3.2 Nonorthogonal waveforms case
.
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19
20
22
25
27
28
31
MIMO GMTI: TestBed Description . . . . . . . . . . . . . . . . . . . .
33
4.1
4.2
4.3
4.4
34
36
39
41
42
Reliable Detection Range . .
Antenna Array Requirements
Transmitter . . . . . . . . . .
Receiver . . . . . . . . . . .
4.4.1 Noise Figure . . . . .
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vii
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4.4.2
4.4.3
4.4.4
4.4.5
4.4.6
Linearity . . . . . . . . . . . . . . .
VGA gain and Tx / Rx isolation .
RF switches . . . . . . . . . . . . .
BPF bandwidth and aliasing . . . .
Local oscillator synchronization and
clock . . . . . . . . . . . . . . . . .
4.5 Micro Controller Board . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
the digitizer reference
. . . . . . . . . . . . .
. . . . . . . . . . . . .
43
44
45
45
5
MIMO GMTI: Matlab Simulation . . . . . . . . . . . . . . . . . . . . .
49
6
Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
CHAPTER
47
47
PAGE
A
Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
B
Receiver Link Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
C
Amplifier Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
C.1 1-dB Compression Point . . . . . . . . . . . . . . . . . . . . . . . .
C.2 IP3 Intermodulation Point . . . . . . . . . . . . . . . . . . . . . .
61
61
viii
LIST OF FIGURES
FIGURE
PAGE
2.1
Doppler for moving platform . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Data cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.1
Side-looking radar . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Virtual antenna rray due to spatial convolution . . . . . . . . . . . .
21
3.3
Equivalent SIMO array with one Tx and NM Rx antennas . . . . . .
22
3.4
Relative speed of the radar platform and a target . . . . . . . . . . .
23
3.5
Plane wave striking antenna array . . . . . . . . . . . . . . . . . . .
24
4.1
Block Diagram of the Emulator . . . . . . . . . . . . . . . . . . . . .
34
4.2
Tx and Rx antenna arrays . . . . . . . . . . . . . . . . . . . . . . . .
35
4.3
SNR calcution. Ouptut power 30 dBm (1W), signal bandwifth 125
MHz; Integration time 1, 64 pulses . . . . . . . . . . . . . . . . . . .
37
4.4
Transmitter Block Diagram. One channel. . . . . . . . . . . . . . . .
40
4.5
Receiver Block Diagram. One channel. . . . . . . . . . . . . . . . . .
41
4.6
Noise figure with low isolation . . . . . . . . . . . . . . . . . . . . . .
42
4.7
Noise figure with high isolation . . . . . . . . . . . . . . . . . . . . .
43
4.8
Saturation Levels
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.9
BPF as antialising filter . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.10
BPF as antialising filter . . . . . . . . . . . . . . . . . . . . . . . . .
46
ix
4.11
Micro Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.1
Sorted eigenvalues of the estimated clutter covariance matrix for simulation Scenario I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Sorted eigenvalues of the estimated clutter covariance matrix for simulation Scenario II . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Receiver link budget . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.2
B.1
x
CHAPTER 1
MIMO RADAR CONCEPT
Multi-input Multi-output (MIMO) radar is an active research area. First envisioned
for enhancing the performance of digital communication communication systems, this
idea has been adopted and modified to different radar applications. In general, any
system that employs multiple antennas can be considered as MIMO radar [3], [4]. In
the literature, MIMO radars are distinguished based on the geometry of the receive
and transmit centers. There are two main categories. MIMO radars with widely
separated Tx and Rx arrays provide statistically independent measurements of the
illuminated scene and are categorized as a statistical MIMO radars. If antennas are
relatively close to each other, so that for each scatterer in the illuminated scene the
angle of arrival is approximately the same for all phase centers, then the system
is referred to as a coherent MIMO radar. Synthetic aperture radar (SAR) is also
considered as a special case of MIMO system [8] since it processes coherently the
data collected over the points on the flight trajectory. Another example of MIMO
radar is fully polarimetric radar, since formally there is no difference between spatially
separated antennas and polarimetrically separated ones. A phased array antenna
system is a special case of coherent MIMO radar where a single receive antenna is
used to process signals from multiple Tx antennas transmitting the same waveform
coherently after applying appropriate phase shifts to direct the beam.
The main advantage of the coherent MIMO radar is its ability to synthesize a large
1
virtual array with fewer antenna elements for improved spatial processing. In this
work, we will focus on the coherent MIMO radar case with collocated elements.
Although a collocated MIMO radar resembles the phased array (PA), there is a
fundamental difference between these two approaches. PA coherently transmits the
same waveform from all elements, steering the beam by applying different phases to
the phase centers. MIMO radar transmits independent waveforms from all antennas
omni-directionally while each receiver antenna receives a superposition of all transmitted signals. Beam forming is done after the receive by applying weights to the
received signal. Typically, a MIMO radar system employs orthogonal waveforms on
its transmit antennas. Orthogonality of the transmit waveforms enables the receivers
to separate the channel response from each transmit element, equivalent to synthesizing a much larger virtual array. This is accomplished at each receiver antenna by
feeding the input signal into a filter bank with M matched filters. Orthogonality
of the waveforms makes the channels between the receiver and the transmitters distinguishable at the output of the matched filters. Each of the N receivers receives
M signals providing a signal space with N M degrees of freedom [2]. M transmit
antennas and N receive antennas can provide a virtual aperture which is equivalent
to SIMO configuration with M N elements. It can be shown that although phased
array radar has higher power concentrated instantaneously in a certain range cell,
the average power collected by MIMO radar will be greater or equal to phased array
over the coherent processing interval.
One of the promising applications of MIMO radar is identification of slowly moving targets using airborne MIMO radar platforms. The advantage of using MIMO
configuration is its ability to synthesize a larger virtual array with relatively fewer
antennas. This allows higher spatial resolution and better separation of returns from
ground clutter and targets. The space-time adaptive processing (STAP) methods
2
originally developed for Single-input, Multiple-output (SIMO) radar are applicable
to MIMO radar systems after proper pre-processing of the received signals. The
performance of the STAP algorithm critically hinges on the structure of the clutter
covariance matrix; therefore, MIMO STAP methods will benefit greatly from theoretical and empirical analysis of the clutter statistics.
The contribution of this work can be summarized in three parts. First, we present
a design of a rooftop MIMO radar testbed that emulates a MIMO GMTI system
mounted on airborne platform. Second, we give results of a simulation study of
ground clutter for the specific rooftop geometry we adopt for the testbed, highlighting potential issues with the relatively close in range. Third, we provide extension
of the results reported in [2] and [10] regarding the rank of the covariance matrix for
the case of nonorthogonal waveforms.
The rest of the thesis is as organized as follows: in chapter 2 we review the GMTI
problem and the standard STAP solution for target detection in ground clutter. In
chapter 3 we layout the MIMO radar model for the GMTI scenario and review clutter
covariance matrix derivation for orthogonal waveforms; also, we extend the analysis
to the case of nonorthogonal waveforms. In chapter 4 we discuss the design of the
MIMO GMTI testbed in detail. In chapter 5 we present simulation results for the
rooftop geometry. We conclude in chapter 6 with some remarks and suggestions for
the further work.
1.1
Notations and Glossary
This short section represents adopted notations and abbreviations that will be used
A) will be used for matrices and bold lower case
in the other sections. Bold capital (A
x) stands for notation of vector.
(x
A ∗ : conjugate of A .
3
A H : Hermitian of A , the sequence of conjugate and transpose operations on A .
A T : transposed matrix A .
a bb = [a0 b0
a1 b 1
...
aN −1 bN −1 ]: Hadamard product: element by element prod-
uct of two vectors.
a ⊗ b : Kronecker product operation: for a of length N and b of length M




b
b
 0 
 0 




 b1 
 b1 




a ⊗ b = (aabT )T = a0 
 , . . . , aN −1  . 
 .. 
 . 
 . 
 . 




bM −1
bM −1
kaak2 =
√
√
aH a = < a, a > is the second norm of vector a
4
(1.1.1)
AOA
angle of arrival
ADC
analog to digital converter
AWG
arbitrary waveform generator
BW
actual bandwidth of the transmitted signal
c
light speed in vacuum
BPF
band pass filter
CPI
coherent pulse interval
D
effective aperture length
DAC
digital to analog converter
DCPA
displaced phase center aperture
∆T x
distance between Tx antennas
∆Rx
distance between Rx antennas
FT
Fourier transform
FFT
Fast Fourier transform
FIR
finite impulse response
φn (t)
the waveform transmitted from nth Tx antenna
fD
Doppler frequency shift, Hz
Fc
carrier frequency
Fs
sampling frequency
Ga
Antenna gain, dBi
GMTI
ground moving target indicator
IF
intermediate frequency
5
L
number of pulses in CPI
LFM
linear frequency modulation
LO
local oscillator frequency
LSB
least significant bit
LPF
low pass filter
λ
wavelength of the carrier frequency, m
M
number of transmitter antennas
MIMO
multi input multi output
MTI
moving target indicator
MDV
minimum detectable velocity
N
number of receiver antennas
Nc
number of scatterers in the illuminated clutters
NF
noise figure of receiver front-end
PA
phased array
PRI
pulse repetition interval
PRF
pulse repetition frequency
PSWF
prolate spheroidal wave functions
R
radar maximum range
RF
radio frequency
RCS
radar cross section
Rx
receiver
6
SAR
synthetic aperture radar
SCR
signal to clutter ratio
SCV
sub-clutter visibility
SFDR
spurious free dynamic range
SIMO
single input multi output
SIR
signal to interferer ratio
SNR
signal to noise ratio
σ
radar cross section, m2
STAP
space-time adaptive process
T
pulse duration
Tx
transmitter
VGA
variable gain amplifier
Vt
target velocity vector
Vp
radar platform velocity vector
ULA
uniform linear array
x T,m
vector coordinates of mth Tx antenna
x R,n
vector coordinates of nth Rx antenna
7
CHAPTER 2
REVIEW OF GMTI AND STAP PROCESSING
One of the important applications of modern radar systems is surveillance of moving
objects. In this mode, the radar system processes waveform returns to detect presence
of moving targets. For a ground based air surveillance radar, the problem can be
formulated as detecting reflections from isolated targets in noise. But in the case
of an airborne radar platform attempting to locate moving targets in the ground,
the detection problem is more complicated due to the presence of reflection from
stationary reflectors in the ground or clutter. Slow moving targets cannot be isolated
based on their Doppler frequency since returns from the ground overlap in Doppler.
Clutter returns depend on many factors . For example, properties of land clutter have
dependence on radar parameters like frequency, polarization and incident angle and
also characteristics of the surface itself like topology, vegetation, moisture, season of
the year.
Typically, combined return from ground clutter is much stronger than the signal reflected from the desired targets. Since the Clutter power also increases with
transmitted power, increasing transmitted power does not improve the signal to clutter ratio (SCR). SCR can be improved through processing of the received echoes to
amplify the target signal while suppressing the clutter returns.
8
2.1
Doppler Phenomena
A single radar pulse can be used to detect the distance to the target, since the
measured phase shift is related to the pulse traveling time. If a sequence of coherent
pulses are used, then a sequence of phase measurements is available in each coherent
processing interval, which reveals the rate at which the distance the target changes.
Phase differences of successive pulses can be interpreted as the frequency shift caused
by the motion. If we denote the distance between a narrow band radar with a carrier
frequency Fc (with corresponding wavelength λ) and stationary target as R, then the
phase of the reflected signal is given by
φ=
2π 2R
λ
. If the target recedes from the radar radial with radial speed Vt then the phase
increments between successive pulses causes a frequency shift of
fD = −
1 dφ
1 d 2π 2R 2 dR
2Vt
2Vt Fc
=−
=−
=−
=−
2π dt
2π dt
λ
λ dt
λ
c
(2.1.1)
Fourier transform of the received returns across the pulses in the CPI can reveal
the radial speed of an isolated targets. To avoid aliasing the pulse repetition frequency
(PRF) has to be at least twice higher than Doppler frequency shift in the scene. PRF
is also referred as the slow time sampling frequency.
For the case of an airborne radar interrogating moving targets on the ground, the
frequency domain of the returns is the superposition of two spectral elements: moving
target and clutter Doppler spectra. Ground patches experience relative range speeds
with respect to the moving airborne platform, as a result both clutter and target
returns with exhibit frequency shifts as determined by (2.1.2) .
fD =
2Vt Fc
cos (θ)
c
(2.1.2)
9
Figure 2.1 depicts a moving radar system mounted on an airborne. All objects in
the scene this case experience a relative velocity with respect to the radar coordinate
system. Specifically, for a ground clutter patch at an azimuth angle θ the relative
range speed is Va sin(θ). As a result scatter returns in the illuminated area that defined
by beam-width angle Θ3dB will exhibit different Doppler shifts. As an example,
consider three clutter points P1 , P2 , P3 chosen on the isorange contour in the spot
defined by squinting angle θ as it is shown on the figure. The radial speed of P1 and
P3 will differ from the P2 .We can calculate the spreading of the Doppler frequencies
around the center frequency as the same range bin as 2.1.3,[1]. Due to this Doppler
spread of clutter responses, they can overlap with the target returns in the Doppler
domain for slow moving targets, complicating their detection.
2Va Θ3dB Θ3dB sin θ −
− sin θ +
λ
2
2
Θ3dB 4Va
2Va Θ3dB
sin
cos (θ)
=
cos θ ≈
λ
2
λ
SpreadfD =
(2.1.3)
GMTI Data Cube
After transmission of a pulse modulated with a chosen waveform the radar switches
to the listening mode. In this mode radar collects and analyses echoes returned from
the illuminated area. The receiver samples the output of its RF front-end at the
fast sampling frequency after baseband conversion. To avoid aliasing the sampling
frequency equal to F s ≈ 2BW where BW is actual bandwidth of the transmitted
1
signal. For unmodulated signal BW ≈ , where T is duration of the transmitted
T
pulse, or BW ≈ the occupied frequency bandwidth if the system uses compressed
pulse like in the case with linear frequency modulation (LFM) or any other type of
phase/amplitude modulation. After match filtering with the transmitted waveform,
the radar collects samples from the output of the match filter in time interval that
10
corresponds to minimum and maximum range in the radar beam. If radar has K
range bins, it acquires K samples for each pulse. These range samples form a column
vector of K × 1 size.
To estimate speed of a target traditional mono-static radar transmits sequence of
L pulses during coherent pulse interval (CPI). System collects K samples after each
pulse forming K × L matrix of complex-valued samples. If system has Q receivers
then each of them collects Q × K × L samples per CPI. Stacking these matrices on
the top of each other this data can be represented in a form of the data cube as
it is shown on the Figure 2.2. Each row of horizontal slice corresponds to the same
1
seconds. Taking FFT of this vector we gain
range bin illuminated with delay of
P RF
frequency characterization of the illuminated range bin. It FFT of the signal exceeds
the threshold then it indicates not only about presence of a target, but also allows
calculate radial velocity of the target too since frequency bin and PRF are known.
Figure 2.1: Doppler for moving platform
11
In the MIMO case when the system has M transmitter antennas and N receiver
antennas, as it will be shown later, the match filter outputs provide Q = M N measurements and the dimensions of the data cube are K × L × M N .
2.2
Processing for Ground Moving Target Indication
In order to increase the signal to clutter ratio and distinguish between clutter and slow
moving targets, GMTI data cube has to be processed to suppress clutter response
while maintaining gain on the moving targets. There are several methods proposed
in the literature to accomplish this task. Spectrum of the ground reflections from a
ground patch and the spectrum of the return of a moving object at that ground patch,
differ by their Doppler frequency shift. The frequency shift is defined by equation
(2.1.2). Since clutter and moving target have different radial speeds the separation
can be achieved on basis of the different Doppler frequency shift between clutter and
target after spatial processing to isolate returns in azimuth angle domain.
Figure 2.2: Data cube
12
STAP
In the following we review STAP algorithm [9]. Space-time adaptive processing
(STAP) takes advantage of joint processing in Doppler and Angle domain to separate clutter returns from returns from moving targets.
Consider a plane wave A exp(j2πFc t) of the carrier frequency Fc impinging on
antenna array the signal received by q th antenna (or antenna element) can be represented as:
yq (t) = Ae
x sin θ +φ
j2πFc t− q∆
0
c
(2.2.1)
The a single sample taken at the time moment t0 from q th antenna:
x sin θ +φ
j2πFc t0 − q∆
0
c
y[q] ≡ yq (t0 ) = Ae
−j2πq∆x
sin θ
λ
= Âe
(2.2.2)
for ∀ q = 0,..., Q-1
Figure 2.3: Clutter (from: http : //sclab.kaist.ac.kr/ bwjung/page/research.html)
13
Here  absorbs common factor  = Aej2πFc t0 +φ0 . The snapshot y from entire array
acquired taken at the time moment t0 can be written as a vector:
T
y = y[0] y[1] . . . y[Q − 1]
−j2π(Q−1)∆x
−j2π1∆x
sin θ
sin θ T
λ
λ
= Â 1 e
... e
T
= Â 1 e(−jKθ ) . . . e(−j(Q−1)Kθ )
(2.2.3)
= Â a s (θ)
2π∆x
sin θ represents the spatial frequency and a s (θ) is a spatial steering
λ
vector of antenna array. Applying weight function to the snapshot y it is possible to
Where Kθ ≡
perform beam-forming in required direction. For the special case when the beam is
directed to the angle θ0 , the filter coefficients can be written in the following way:
h = [h0
h1
= [w0
. . . hN −1 ]T
w1 ejKθ0
...
wN −1 ej(N −1)Kθ0 ]T
(2.2.4)
=
[w0 a0s (θ0 )
w1 a1s (θ0 )
...
−1)
wN −1 a(N
(θ0 )]T
s
= w a∗s (θ0 )
Then the output from this spatial filter that corresponds to the time t0 and steering
direction defined by θ0 is:
w a ∗s (θ0 )]T Âaas (θ)
z(θ0 ) = h T y = [w
Q−1
= Â
X
wq aqs (θ0 )∗ aqs (θ)
q=0
(2.2.5)
Q−1
= Â
X
jqKθ0 −jqKθ
wq e
e
q=0
Q−1
= Â
X
wq e−jq(Kθ −qθ0 )
q=0
A vertical slice of the data cube shown on the Figure 2.2 corresponds to signal
composed of the returns from a sequence of pulses received during one CPI from the
14
same range bin. Each pulse represents temporal sample delayed on the ej2πlfD , where
l = 0, . . . , L−1 is the pulse number. In addition to the spatial steering vector a s (θ) we
can define a temporal steering vector a t (fD ) = [1 ej2πfD
ej2π2fD
...
ej2π(L−1)fD ].
Then the samples of the matrix that corresponds to the range bin k0 can be written
as:
(0)
(1)
Y [k0 , l, q] = [at (fD )aas (θ) at (fD )aas (θ)
...
(L−1)
at
(fD )aas (θ)]
Vectorized matrix Y [k0 , l, q] takes the following form:


(0)
a (fD )aas (θ)
 t

 (1)

 a (fD )aas (θ) 
 t

Y [k0 , l, q] = 
 = a t (fD ) ⊗ a s (θ)
.


..




(L−1)
at
(fD )aas (θ)
(2.2.6)
(2.2.7)
In order to maximize SNR, or in more general case signal to interferer ratio (SIR),
we will have to find a filter that is optimal for this specific combination of Doppler
frequency shift fD and angle of arrival θ: hopt (fD , θ) that correspond to the received
signal s = Y [k0 , l, q]. As it was shown in Appendix A, equation A.0.12, coefficients
of the optimal Doppler-angle filter can be found:
∗
h opt = Σ −1
w s
(2.2.8)
where Σ w denotes covariance matrix of combined interfering signal.
Clutter response is combined returns from number of scatterers that are in the
isorange sector illuminated through the current CPI. The clutter response is modeled
as a sum of individual responses of the scatterers. Let clutter consists of Z scatterers
with intensity defined by eq. (2.2.9) which differ because of RCS variation.
ρz =
Pt Tp G2a λ2 σz L
(4π)3 R4
(2.2.9)
15
Received signal coming from single scatterer takes the following form in term of
temporal a ct (fDz ) and spatial a cs (θz ) steering vectors:
C z = ρza ct (fDz ) ⊗ a cs (θz )
(2.2.10)
Then the total signal coming from the clutter:
C=
Z−1
X
ρza ct (fDz ) ⊗ a cs (θz )
(2.2.11)
z=0
Covariance matrix of the interferer consisting of the clutter only is given by:
∗
T
C C ]=E
Σ c = E[C
Z X
Z
hX
C ∗i C Tj
i
i=1 j=1
=
Z X
Z
X
h
i
∗ T
E CiCj
(2.2.12)
i=1 j=1
=
Z
X
∗
∗
ρi [aact (fDi )∗a ct (fDi )T ] ⊗ [aacs (θi )∗a cs (θi )T ]
i=1
This is a square block matrix of the size defined by the number of transmitters M:
M × M . Each element is also square matrix of the size is equal to the number of
receivers N, N × N .
The performance of STAP filtering is determined by the covariance structure of the
clutter. Specifically on the number of eigenvectors of the and their associated power
determines the SCR at the output of the STAP filter.
In the next chapter we will extend STAP algorithm to MIMO arrays and review
the derivation of the clutter covariance matrix for the MIMO configuration for the
case of orthogonal waveforms and extend it to the case of nonorthogonal waveforms.
16
CHAPTER 3
MIMO MODEL FOR GMTI
In this section, we extend STAP processing to MIMO radar systems that employs
multiple receive and transmit antennas with independent waveforms on the different
transmit antennas. Specifically, we consider a radar system that have N receiver
antennas and M transmitter antennas on an airborne platform with a side-looking
antenna array To simplify the analysis of the system, we make the following assumptions. We assume that the aircraft moves linearly along the x- axis x with a ground
speed of Va as given in Figure 3.1. We assume that the distances between the antennas are much smaller than the distance between the antenna and the target so
that the staring angle to any given reflector can be taken the same for all Tx and Rx
antennas and the reflected electromagnetic wave can be considered as a plane wave.
We assume the illuminated scene consists of a single target and Nc clutters. The
following parameters further define the problem :
V a will denote vector the airborne linear ground speed and Va its absolute value;
V t is a vector of target ground speed and Vt its absolute value;
x T,m and x R,n are R3 coordinates of mth Tx and nth Rx antennas respectively;
u ∈ R3 is the unit vector pointing to the target, u ck is the unit vector pointing to the
k’th clutter cell ;
ρ is the target RCS and ρck is denotes RCS of the k th clutter cell;
T is a pulse length and φm (τ ) is mth baseband waveform;
17
Figure 3.1: Side-looking radar
18
Assuming plane wave propagation of the receive waveform and narrowband radar, the
received demodulated signal for the l’th pulse at the nth Rx antenna can be written
as a sum of M of waveforms reflected from the target, and Nc clutters, combined
with additive white Gaussian noise:
yn (lT + τ ) =
+
M
−1
X
ρ φr (τ ) exp(j
r=0
N
−1
c −1 M
X
X
2π T
V a lT + V t lT + x R,n + x T,m ))
u (V
λ
ρck φr (τ ) exp(j
k=0 r=0
2π c T
V a lT + x R,n + x T,m ))
u (V
λ k
(3.0.1)
+ w(lT + τ )
3.1
Orthogonal Waveforms Case
MIMO radar systems employing orthogonal waveforms on transmit, synthesize a separate channels between each transmit and receive antenna, which can be recovered
on receive by match filtering with each transmit waveform. This implies that each
receiver is be able to distinguish the channel to each transmit antenna from the received mix of M transmitted waveforms. This is accomplished by passing the mix of
waveforms coming from all transmitters to a bank of matched filters corresponding
to the set of all transmit waveforms. Signals transmitted from two different transmit
antennas during pulse period T sec are orthogonal. The inner product of these two
signals is given by:
Z
< φn (t), φm (t) >=
T
φn (t) φ∗m (t)dt = δnm
0
19
(3.1.1)
3.1.1
Virtual Array Concept
Due to orthogonality of the waveforms output of mth matched filter of nth Rx antenna
will be non-zero for the mth waveform only and substituting eq.(3.1.1) into eq.(3.0.1):
s(n, m, l) =
M
−1 Z T
X
2π
V a lT + V t lT + x R,n + x T,m ))dτ
=
ρ φr (τ ) φ∗m (τ ) exp(j u T (V
λ
0
r=0
N
−1 Z T
c −1 M
X
X
2π
V a lT + x R,n + x T,m ))dτ
+
ρck φr (τ ) φ∗m exp(j u ck T (V
λ
0
k=0 r=0
(3.1.2)
+ w(n, m, l)
2π T
V a lT + V t lT + x R,n + x T,m ))
u (V
λ
N
c −1
X
2π
V a lT + x R,n + x T,m )) + w(n, m, l)
+
ρck exp(j u ck T (V
λ
k=0
= ρ exp(j
Plainly, the output of a bank of M matched filters is M waveforms received
separately. A different interpretation of this results is that a MIMO radar system
with N + M antennas can provide the same channel information as a SIMO radar
system array with one transmit antenna and N M receive antennas array as shown
in Figure 3.2 and 3.3 [2]. Locations of phase centers of the virtual SIMO array are
xT,m + x R,n } , where n=0,...,N-1; and m=0,...,M-1 The phase centers of
defined by {x
the virtual SIMO array are given by spatial convolution of phase centers of transmit
and receive antennas. If locations of the receive antennas are defined as
x) =
gR (x
N
−1
X
x − x R,n )
δ(x
(3.1.3)
n=0
and the location of the transmit antennas are defined as
x) =
gT (x
M
−1
X
x − x T,m )
δ(x
(3.1.4)
m=0
20
Then the positions of the virtual SIMO array antennas can be expressed as convolution of these two arrays:
x) =
gV (x
N
−1 M
−1
X
X
x − (x
xT,m + x R,n ))
δ(x
(3.1.5)
n=0 m=0
Angle resolution of a side-looking GMTI airborne mounted radar depends on effective
antenna aperture size [5]. MIMO radar systems effectively synthesize a larger virtual
array with higher spatial resolution. This leads to better separation of the clutter
and target returns in Doppler-Angle domain, reducing the the minimum detectable
velocity (MDV) [5].
Figure 3.2: Virtual antenna rray due to spatial convolution
21
3.1.2
Uniform Linear Array
In the following we study the case when both transmit and receive antenna arrays are
uniform linear arrays (ULA) in detail. We assume both transmit and receive antennas
are parallel to the direction of the aircraft movement. The spacing between receive
antennas will be denoted as ∆Rx and spacing between transmit antennas as ∆T x
correspondingly and ∆T x = N ∆Rx . In this case mutual coordinates of the Tx/Rx
arrays elements form virtual uniform linear array stretched along the x-axis i.e.
xT,m + x R,n } = m∆T x + n∆Rx , where n=0,...,N-1; and m=0,...,M-1
{x
(3.1.6)
and the received demodulated signal defined in (3.0.1) becomes
yn (lT + τ ) =
+
M
−1
X
ρ φVr t lT + τ +
r=0
N
−1
c −1 M
X
X
2R
+ r∆T x sin Θ + n∆Rx sin Θ
c
ρck φVr k lT + τ +
k=0 r=0
(3.1.7)
2R
+ r∆T x sin Θ + n∆Rx sin Θ
c
+w(lT + τ )
Figure 3.3: Equivalent SIMO array with one Tx and NM Rx antennas
22
φVmi represents the mth waveform shifted to the corresponding Doppler shift and is
defined as:
φVmt
= φm e
φVmk = φm e
j2πFc t
c
Vt sin ψt +Va sin Θt
j2πFc t
Va
c
sin Θt
Figure 3.4 illustrates the effective speed for Doppler shift calculation. Under listed
Figure 3.4: Relative speed of the radar platform and a target
bellow assumptions the equation (3.1.7) can be approximated and brought to the form
more suitable for the further derivations. The first assumption is that all transmitted
waveforms are narrow band signals allocated around carrier frequency Fc . Then the
Doppler frequency shift can be approximated by the shift received by the Fc . The
second assumption we are doing says that target and scatterers composing the clutter
are point targets. This means actually two things: the target spatial shift during pulse
23
duration is negligibly small T Vt ≈ 0 or T Vk ≈ 0 and the second one is that there is
no inner clutter motion during pulse interval. As before we assume reflectors are in
the far field as a result the received wave is assumed to be plane view inducing phase
delays across the elements of the receive antennas, as shown on the Figure 3.5
Figure 3.5: Plane wave striking antenna array
M −1
X
j2π
2R
yn (lT + τ +
)≈
ρφr (τ )e λ
c
r=0
+
N
−1
c −1 M
X
X
ρck
[2Va lT +r∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
φr (τ )e
k=0 r=0
2R +w lT + τ +
c
24
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
dτ
dτ
(3.1.8)
Then using the orthogonality of the waveforms the output of mth matched filer of nth
antenna during lth pulse can be represented in the following form:
s(l, n, m) =
Z T
M
−1
X
j2π
ρt
φr (τ )φ∗m (τ )e λ
+
0
r=0
N
−1
M
−1
c
X
X
ρck
[2Va lT +r∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
T
Z
φr (τ )φ∗m (τ )e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
dτ
dτ
(3.1.9)
0
k=0 r=0
+w(l, n, m)
= ρt e
+
j2π
λ
N
c −1
X
ρck
[2Va lT +m∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
e
j2π
λ
[2Va lT +m∆T x +n∆Rx ] sin Θk
+ w(l, n, m)
k=0
3.2
Nonorthogonal Waveforms Case
In many situations transmitted waveforms are not strictly orthogonal; instead, there is
some correlation between waveforms. Equation (3.2.1) describes correlation between
two transmitter waveforms m and n.


Z T
 δnm n = m
∗
φn (t) φm (t)dt =
< φn (t), φm (t) > =

0
 αnm n =
6 m
(3.2.1)
Since each receiver demodulates the received signal and sends it to the bank matched
filters corresponding to the full set of transmitted waveforms, we would like to consider
mutual correlation of all possible waveforms. Let φ = [φ1
(3.2.2) represents the cross-correlation matrix of the set.


1
α12 . . . α1M




Z T
 α21

1
.
.
.
α
2M


A=
φ (t) φ H (t)dt =  .

.
.


..
..
0
 ..



αM 1 αM 2 . . . 1
= α1 α2
...
αM
25
φ2
. . . φM ]T then eq.
(3.2.2)
Then for the nonorthogonal case, the output of mth matched filer of nth antenna
during lth pulse can be written as:
s(l, n, m) =
Z T
M
−1
X
j2π
ρt
φr (τ )φ∗m (τ )e λ
[2Va lT +r∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
dτ
0
+
r=0
N
−1
c −1 M
X
X
ρck
Z
T
φr (τ )φ∗m (τ )e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
dτ
0
k=0 r=0
(3.2.3)
+w(l, n, m)
=
M
−1
X
ρt αr,m
r=0
−1
N
c −1 M
X
X
ρck
+
k=0 r=0
e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
αr,m e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
+ w(l, n, m)
The first, second and third terms represent contribution of targets, clutter and noise
correspondingly. Since we are interested in the analysis of the clutter covariance
matrix let’s consider the term that represents contribution of clutter separately. The
clutter term in equation (3.2.3) is a sum of returns coming from the Nc clutter points
as a response to M transmitted waveforms collected at the output of mth matched
filter of the nth receiver during lth pulse:
sc (l, n, m) =
N
−1
c −1 M
X
X
ρck
αr,m e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
(3.2.4)
k=0 r=0
The output of nth antenna will correspondingly compose M × 1 vector combined from
outputs of the individual matched filters defined by eq. (3.2.4):
s c (l, n) = [sc (l, n, 0) sc (l, n, 1)
...
sc (l, n, (M − 1))]T
(3.2.5)
Set of ”clutter” samples that is collected by N Rx antennas during lth pulse period
is M × N matrix:
S c (l) = [ssc (l, 0) s c (l, 1)
...
s c (l, N − 1)]
26
(3.2.6)
It will be more convenient to represent these samples that correspond to the single
pulse as a N M × 1 vector

s c (l, 0)


 s c (l, 1)

S c (l) = 
..

.


s c (l, N − 1)









(3.2.7)
Then all samples received during single coherence interval can be represented as
N M × L matrix.
S c (0) S c (1)
S c = [S
...
S c (L − 1)]
(3.2.8)
In order to maximize the SIR, as it was shown earlier, we need to design the
∗
optimal filter h opt = Σ −1
w s . Where Σ w is the covariance matrix of the interfering
signal. To find Σ w matrix S c (l) should be first vectorized. The resulting vector Ŝ c
has size N M L × 1

S c (0)


 S c (1)

Ŝ c = 
..

.


S c (L − 1)









(3.2.9)
Then the clutter covariance matrix:
Σ w = E[Ŝ cŜ H
c ]
3.3
(3.2.10)
Rank of Clutter Covariance Matrix
Clutter mitigation problem relates to the problem of finding inverse clutter matrix
as was shown in the STAP section (2.2). As one can see, the size of the covariance
matrix is really big. This makes the computation of inverse matrices of covariance
27
unpractical especially in real time applications. This requires other more efficient
methods of estimation clutter covariance matrix.
The main problem that arises in clutter mitigation is getting reliable estimation of
b w . This can be done collecting enough statistics from
the clutter covariance matrix Σ
Nav
X
bw = 1
ss T . This method
samples, taken from the neighboring range bins Σ
Nav n=1
assumes there are no other targets in these range bins. For the case of regular STAP
with a SIMO array the rank of the clutter matrix is given by the Brennan’s rule [2]
2vT
.
as RST AP ≈ N + β(L − 1), where N is the number of receiver antennas, β =
∆Rx
An extension of the Brennan rule for the MIMO case was developed in [10]. In
the following we review their derivation and extend it to the case of non-orthogonal
waveforms.
3.3.1
Orthogonal waveforms case
As given in eq. (3.1.9), for the case of orthogonal waveforms, the output of mth
matched filer of nth antenna during lth pulse as
s(l, n, m) = ρt e
+
N
c −1
X
j2π
λ
ρck
[2Va lT +m∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
e
j2π
λ
[2Va lT +m∆T x +n∆Rx ] sin Θk
(3.3.1)
+ w(l, n, m)
k=0
following the notations from [10]
∆Rx sin Θt
λ
∆Rx sin Θk
fs,k =
λ
∆T x
γ=
∆Rx
2Va T
2Vt T
fD =
sin Θt +
sin ψt
λ
λ
the eq. (3.3.1) can be rewritten
fs =
s(l, n, m) = ρt e
j2πfD l j2πfs (n+γm)l
e
(3.3.2)
+
N
c −1
X
k=0
28
ρck ej2πfs,k (βl+n+γm) + w(l, n, m) (3.3.3)
For the further analysis the only the clutter part of the eq.(3.3.3) is relevant
sc (l, n, m) =
N
c −1
X
ρck ej2πfs,k (βl+n+γm) =
k=0
N
c −1
X
ρck c k
(3.3.4)
k=0
where, for convenience purposes, c k = ej2πfs,k (βl+n+γm) . To prevent appearing of
λ
grating lobes Rx antennas spacing was taken as ∆Rx = and this actually defines
2
the range of spatial frequency fs,k as ±0.5. Since sweeping ranges of parameters of
sc (l, n, m) are n=0,..,N-1; m=0,.., M-1; L=0,.., L-1, the size of s c is a N M L × 1.
As it was shown earlier, to maximize the SINR we have to find Σ w = E[sscs c T ].
Σw = E[sscsc H ] = E
c −1
h NX
ρcici
c −1
NX
i=0
=E
c −1 N
c −1
h NX
X
i=0
=
N
c −1
X
ρci (ρcj )∗
ρcj cj
H i
j=0
c i (ccj )
H
i
=
N
c −1 N
c −1
X
X
j=0
i=0
h
i
E ρci (ρcj )∗ c i (ccj )H
(3.3.5)
j=0
σi2c i (cci )H
i=0
Here two assumptions were applied. The first assumption is about statistical independence of the scatterers composing clutter and the second relationship was about
h
i
the variance of reflection of the i-th scatterer E ρci (ρci )∗ = σi2 . As we can see from
eq. (3.3.5), rank of the clutter covariance matrix is determined by the rank of matrix
c (cc)H
Σw ) = Rank
Rank (Σ
c −1
NX
σi2c i (cci )H = Rank [cc0 c 1
...
c Nc −1 ]
(3.3.6)
i=0
Examining the eq.(3.3.3) we can conclude that the vector c i = ej2πfs,i (βl+n+γm) can
be considered as a set of non-uniformly distributed samples of the function ej2πfs,i x
taken at the moments defined by (βl + n + γm). This function is band and time
limited since fs,k ∈ {±0.5} and x ∈ {0, [β(L − 1) + (N − 1) + γ(M − 1)]}
It is known that time and band limited function can be approximated by a finite set
of prolate spheroidal wave functions (PSWF). It was shown [2] that it is required
29
2WX+1 prolate spheroidal wave functions in order to represent function with time
duration from 0 to X and energy concentrated in the frequency range of [-0.5W,
0.5W]. In other words
ci ≈
R
c −1
X
ηtu t
(3.3.7)
t=0
Where the number of orthogonal PSWF composing the set is Rc = 2W X + 1 =
2 · 0.5 · [β(L − 1) + (N − 1) + γ(M − 1)] + 1 = β(L − 1) + N + γ(M − 1). Since any of
c i can be represented as a linear combination of PSWF, this set forms a basis of the
vector space defined by the covariance matrix Σw . Therefor,
Σ
Rank (Σ w ) = Rank [cc0 c1
...
c −1
RX
cNc −1 ] = Rank
u
ηt t
t=0
u0 u 1
= Rank [u
...
u Rc −1 ]
(3.3.8)
It was proven [2] that rank of clutter covariance matrix for the case when γ and β
are integer is determined by eq.(3.3.9)
Σw ) 6 min (N + γ(M − 1) + β(L − 1), Nc , N M L)
Rank(Σ
(3.3.9)
and since the number of scatterers forming clutter → ∞ it will be acceptable to
Σw ) ≈ N + γ(M − 1) + β(L − 1)
consider rank of the matrix as rank(Σ
The relationship with Brannen’s rule for SIMO configuration RST AP ≈ N + β(L − 1)
and MIMO case can be done though the virtual antenna array formed by Tx and Rx
antennas. The effective size of the virtual array is given by Nv = [N + γ(M − 1)].
Substitution of Nv into the Brannen’s expression gives RST AP ≈ Nv + β(L − 1) =
N + γ(M − 1) + β(L − 1) which is exactly the value that is given by the eq.(3.3.9)
for integer γ and β
30
3.3.2
Nonorthogonal waveforms case
The result of the previous section can be extended to the nonorthogonal case in
straightforward way. For the nonorthogonal case, the output of mth matched filer of
nth antenna during lth pulse can be written as given in the eq.(3.2.3)
s(l, n, m) =
+
M
−1
X
ρt αr,m e
r=0
N
−1
c −1 M
X
X
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θt +2Vt lT sin ψt
(3.3.10)
ρck αr,m e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
+ w(l, n, m)
k=0 r=0
The size of s c is a N M L × 1, since n ∈ {0, N − 1}, m ∈ {0, M − 1} and l ∈ {0, L − 1}.
Using notations (3.3.2), the clutter term in the above equation can be represented in
a form:
sc (l, n, m) =
=
N
−1
c −1 M
X
X
k=0 r=0
−1
N
c −1 M
X
X
ρck αr,m e
j2π
λ
[2Va lT +r∆T x +n∆Rx ] sin Θk
(3.3.11)
ρck αr,m ej2πfs,k (βl+n+γr)
k=0 r=0
In vector notation, we have
sc =
N
c −1
X
A)cck ,
ρck D(A
(3.3.12)
k=0
A) with block elements that are
where the N M K × N M L block-diagonal matrix D(A
waveform cross-correlation M × M matrices from eq. (3.2.2) is given by:


A 0 ... 0




 0 A ... 0 


A) = 
D(A

.
.
.
 . .

..
 . .



0 0 ... A
31
(3.3.13)
The the covariance matrix Σ w = E[sscs c H ] will be defined by eq.(3.3.14). Here we
used statistical independence of reflectivity of different clutters.
H
Σ w = E[sscs c ] =E
c −1
h NX
=
k=0
N
c −1
X
A)cck
ρck D(A
c −1
NX
A)ccr
ρcr D(A
H i
r=0
(3.3.14)
A)cck D(A
A)cck
σk2 D(A
H
k=0
Then the rank of covariance matrix Σ w :
Σw ) = Rank
Rank(Σ
c −1
NX
σk2
A)cck D(A
A)cck
D(A
H (3.3.15)
k=0
A)[cc0 c 1
= Rank D(A
...
c Nc −1 ]
Rank of Σw is actually defined by the product of two matrices. We can find it using
two basic matrix properties namely:
XY ) 6 min(Rank(X
X ), Rank(Y
Y ))
1. Rank(XY
X ) = k, Rank(XY
XY ) = Rank(Y
Y)
2. for the matrix X of size n × k with Rank(X
We can conclude that if the waveform cross-correlation matrix A is full rank then
A) is also full rank and therefore Rank(Σ
Σw ) will be given by eq.(3.3.9):
D(A
Σw ) 6 min (N + γ(M − 1) + β(L − 1), Nc , N M L)
Rank(Σ
(3.3.16)
Otherwise it will be:
Σw ) 6 min N + γ(M − 1) + β(L − 1), Nc , N M L, N LRank(A
A) (3.3.17)
Rank(Σ
32
CHAPTER 4
MIMO GMTI: TESTBED DESCRIPTION
In this chapter, we review the design of a MIMO radar testbed that can emulate
measurements from a 4×4 MIMO radar system on a moving airborne platform. The
system is composed of four independent transmit and receive channels. The block
diagram for the testbed is given in Figure 4.1. A four channel Arbitrary Waveform
Generator (AWG) is used to synthesize four independent waveforms an intermediate
frequency. The transmit RF frontend modulates the signal to X-band and feeds it to
the transmit antenna array. Electronically selected four receive antennas receive the
returns from the scene and converts it into baseband for sampling by a a four channel digitizer. The operation of the AWG, the RF switches and the digital-to-analog
converter is synchronized by a custom micro controller board. Each receiver signal
chain is connected to a bank of 16 switchable antennas. 64 Tx and 64 Rx antennas
are printed on a circuit board in two rows parallel to each other. The system can
emulate linear airborne motion along its long axis by sequentially selecting 4 consecutive Rx antennas out of the available 64 receive antennas. For each pulse four
independent waveforms are transmitted from four transmitters and received by four
antennas that are electronically selected. At subsequent pulses, waveforms transmitted from the fixed set of transmitters are received by a different set of four receiving
antennas shifted spatially kλ/2 relative to the previous set, where k ≥ 1 is an integer, corresponding to normalized platform speed of β = k. The emulated airborne
33
speed is given by the spacing between elements of the synthesized receivers Arrays in
subsequent pulses nd the pulse repetition frequency:
Vp =
k∆Rx PRF
2
(4.0.1)
The factor 2 in the denominator is due to the fact that the motion is only emulated
on the receive antennas, effectively reducing the platform speed by half. For a PRF
of 8 KHz, the emulated speed of the aircraft will be a multiple of 60 m/sec.
For transmitters any fixed set of four antennas among 64 available antennas is
possible. The most standard scheme of uniformly spaced linear array is represented
on the figure 4.2. In this scheme Tx antennas allocate sparsely with the step equal
to 2λ, the number of receiver antennas, but the Rx antennas are selected as a group
of 4 sequential antennas with λ/2 spacing between them.
4.1
Reliable Detection Range
This low power, low cost laboratory system is designed to provide reliable detections
at a range of 300 m at a transmit power level 1W. The system operates in the Xband with 10.5 GHz center frequency. Carrier frequency was chosen to make the
Figure 4.1: Block Diagram of the Emulator
34
physical dimensions of the array implementable in the laboratory conditions. Under
these initial requirements the reliable detection range was verified using the radar
range equation for compressed pulses.The following list represents the set of other
parameters that were used. A 3 dB noise figure was assumed for RF front end
and other losses due to system model inaccuracy and timing/quantization errors
are assumed to not exceed 4 dB. The bandwidth was chosen 125 MHz due since
affordable ADC solution with 16 bit of resolution operate at a maximum sped of 250
Msamples/sec.
Figure 4.2: Tx and Rx antenna arrays
35
Carrier Frequency
10.5 GHz
Tx power (Pt )
30 dBm
Antenna gain (Ga )
10 dBi
RF front end NF (F)
3 dB
Losses (Λ)
4 dB
Pulse duration
100 usec
Bandwidth (BW)
125 MHz
Number of pulses (L)
1, 64
Maximal Range (R)
300 m
RCS
1 m2
We assume reliable detections are possible at a given range if SNR > 15 dB, while
fixing output power, pulse duration, bandwidth and antenna gain for the system.
The radar range equation for L integrated compressed pulses has form represented
by equation (4.1.1). According to the this equation the expected SNR at 300 meter
for one pulse is 21.5 dB and for 39.5 dB for 64 pulses integration period. Figure 4.3
shows the SNR degradation as a function of range for L = 1 and L = 64 pulses.
SNR =
4.2
Pt Tp G2a λ2 σLΛ
(4π)3 R4 KB T F
(4.1.1)
Antenna Array Requirements
To achieve high SNR with relatively low output power we use 100 µs pulse in order
to coherently integrate returns from the reflectors in the scene. The reflected signal
from the scene appears at the receiver antenna on the order of 2R/c = 1µs. This
means that the transmitter and the receiver have to operate simultaneously. The
main problem of such a configuration is the severe requirement on the Tx/Rx power
isolation. Strong coupling of Tx power into the receiver antennas can cause saturation
36
Figure 4.3: SNR calcution. Ouptut power 30 dBm (1W), signal bandwifth 125 MHz;
Integration time 1, 64 pulses
37
of the entire Rx chain. This reduces the sensitivity of the system and add intermodulation products to the output signal.
We use a low-noise amplifier (Hittite HMC963LC4) with gain of 22 dB, noise
figure 2.5 dB and P1dB = 10 dBm. This results in an LNA input signal power that
given by :
PS in =
Pt Tp G2a λ2 σΛ
= −112.5dBm.
(4π)3 R4
(4.2.1)
Noise power at the LNA input is equal to:
PN in = KB T F BW = −174 + 10 ∗ log10(BW ) = −93dBm.
(4.2.2)
System has 10 log 10(T ∗ BW ) ≈ 41 dB of processing gain for a single pulse. The
processing gain is only applicable if the level of signal plus thermal noise is within the
ADC dynamic range. The effective number of bits of the ADC is lower than 16 bits
due to timing jitter and nonlinearities, Therefore we have chosen to cover two LSB(s)
by the mix of signal and noise. The widest available ADC with required sampling
rate and maximal acceptable price range was 16 bits ADC with 250 Msps. Maximum
input signal that corresponds to the full ADC range is maxADC = 2dBm ≈ 0.8Vpp .
The ADC dynamic range is given by DRADC = 6.02 ∗ 16 ≈ 96 dB. Therefore
processing gain is effective if (Noise > maxADC − 6.02 ∗ (16 − 2) ≈ 2 − 14 ∗ 6 = −82
dBm). This determines the required Rx chain gain to bring noise level to the ADC
operating range is given by GRxmin = −82 − (−93) = 11 dB. Here the noise power
level was considered as a reference since the contribution of the desired signal is
negligible (-112 vs -93 dBm).
The Tx power coupled to the Rx antennas could saturates ADC if the receiver
fronted has to much gain. The minimal Tx / Rx isolation can be found by finding
the largest coupling level for which the receiver operating with GRxmin can saturate
a component of the receiver chain and the ADC. To prevent ADC from staring we
38
require the maximum level of Tx signal at Rx input to be maxADC −GRxmin = 2−11 =
−9 dBm, which in turn determines the minimum Tx / Rx isolation as 30 -(-9) = 39
dB. Next we check the level of the coupled signal at the mixer input (Figure 4.5):
-9+22-1-6 = +6 dBm and find that it is 3 dB higher than the mixer’s compression
point. To insure linearity of operation we want signal level to be 3 dB below the
saturation point. Therefore non -saturation of the ADC and linear operation of the
mixer is guaranteed if the TX/RX isolation is at least 45 dB.
4.3
Transmitter
The block diagram of the transmitter is given in Figure 4.4. The transmitter waveforms built in the Matlab environment is loaded into the arbitrary waveform generator
(AWG). The transmitter can use any waveform with total passband bandwidth of 125
MHz. The system generates waveforms directly at intermediate frequency (IF) using
an AWG sampling frequency of 5.2 Gsps (A low pass filter (LPF) provides 40 dB
attenuation at 2.6 GHz). The output signal of AWG is the baseband signal digitally
modulated to the IF frequency of 1.8 GHz. This iF signal is fed to a low pass filter
of BW=2 GHz and then further up-converted to the carrier frequency of 10.5 GHz.
Next the signal is amplified to the required level and fed to the transmitting antenna
after passing through the band pass filter which suppresses sidebands and the LO
leakages.
Linearity of the transmitter is achieved by keeping each element in the transmit
chain far enough from their saturation (P1dB ) point. In order to minimize the quantization noise while avoiding the saturation of the mixer, the AWG IF signal is generated at maximum amplitude but then properly attenuated before the mixer. A bandpass filter with low insertion loss (0.6 dB) and a power amplifier with P1dB = 33dBm
were chosen to insure linearity at the final stage of the transmitter. Operating point
39
Figure 4.4: Transmitter Block Diagram. One channel.
40
of this A-B type power amplifier is 2.5 dB below its P1dB compression point. The
AWG quantization noise level is around -48 dBc in 8-bits mode and about -60 dBc
when all 10 bits are used. In order to use all 10 available bits of the AWG resolution,
an external source is use to trigger the AWG.
4.4
Receiver
The block diagram of for the receiver chain is shown in Figure 4.5. An RF switch
(SP16T) selects the active antenna active At each pulse. The RF switch directs
the signal from the LNA to a band pass filter which filters out the desired piece of
spectrum providing image rejection and band limiting of the received signal. Then
the received signal is demodulated to an intermediate frequency IF = 187.5 MHz and
sampled by the data acquisition unit at Fs = 250 Msps with 16 bits resolution.
Figure 4.5: Receiver Block Diagram. One channel.
41
4.4.1
Noise Figure
The noise figure(NF) of the receiver is mainly determined by the characteristics of
the first stage. We have chosen to use low noise amplifiers (LNA) immediately after
each antenna. The LNA (Hittite HMC963LC4) is used in the current design has 22
dB gain and 2.3 dB noise figure. The overall NF of Rx front end will not exceed 4 dB.
Figure 4.6 shows the NF degradation along the Rx chain. If the isolation between
Tx and Rx antennas can be reduced to a higher level of 60 dB then it is possible
to improve the NF by adding additional amplifier before BPF in the RF part of the
receiver. As an example, Hittite HMC903LP3E (NF = 1.8 dB, Gain 18 dB and P1dB
= +14 dBm) can be used immediately after antenna and before BPF ensures total
NF ≈ 2.2 dB only, as it shown on Figure 4.7.
Figure 4.6: Noise figure with low isolation
#1 LNA(HMC963), #2 Cable, #3 RF Switch,
#4 BPF, #5 Mixer, #6 IF Amp, #7 LPF
42
Figure 4.7: Noise figure with high isolation
#1 LNA(HMC903), #2 Cable, #3 RF Switch,
#4 LNA(HMC963) #5 BPF, # 6Mixer, #7 IF Amp, #8 LPF
4.4.2
Linearity
Saturation levels for each element of the receiver chain and the corresponding signal
level for an input power of -15 dBm is shown in Figure 4.8. We observe that the
system design guarantees linearity across all the elements. Figure B.1 represents
more detailed information about calculation of the main receiver parameters. The
first column in the right part of the table shows the gain propagation along the
elements of Rx chain. The second column gives NF. As it was already mentioned NF
can be improved by adding gain before mixer. IP3 and P1dB characteristics are given
in the fourth and fifth columns correspondingly. Signal level that saturates the chain,
from ”Cumulative Output Summary”, Psat-Gain = -11.4 dBm.
43
The third order inter-modulation products receive three times more gain than the
desired signal. Having total IP3 level, signal power and gain, we can calculate the
level of inter-modulation products at the output of Rx chain.
Figure 4.8: Saturation Levels and Signal Lever with Pin =-15 dBm
#1 LNA(HMC963), #2 Cable, #3 RF Switch,
#4 BPF, #5 Mixer, #6 IF Amp, #7 LPF
4.4.3
VGA gain and Tx / Rx isolation
The total attenuation in the receiver chain can be calculated by adding RF switch
loss ≈ 6 dB; BPF insertion loss ≈ 1 dB; Mixer conversion loss ≈ 7 dB; LPF insertion
loss ≈ 1 dB; and losses in connectors and cables ≈ 1dB. Overall this amounts to 16
dB attenuation across the chain. In order to bring input signal to the desired level
with LNA gain of 22 dB, the IF VGA has to provide additional 11-(22-16)=5 dB gain.
If the TX/RX isolation is better than 45 dB, then IF VGA gain can be increased
44
correspondingly. VGA gain is controlled by the filtered DAC signals coming from the
micro-controller board.
4.4.4
RF switches
Each of the four receive channels has an RF switch that enables selection of one active
element from the 16 available receive elements. Address of each sub channel has to be
set on the address bus sequentially by the micro-controller. To avoid ambiguity and
asynchronous switching of the banks caused by going through the high-Z state of a
regular buffer, a double latch buffer solution was chosen. We note that the electrical
delay experienced by different sub-channels of the RF switches should be measured
and compensated in the digital domain at the receiver as a calibration step.
4.4.5
BPF bandwidth and aliasing
A bandpass filter is placed between the LNA and the RF switch. The BPF plays
two main roles: Image rejection filtering and band-limiting the received signal before
sampling to preventing aliasing. Figure 4.9 demonstrate the problem with choice of IF
Fs
. In general, down-conversion onto the low IF with following
frequency as FIF =
4
low-pass filtering has certain advantage. The bandwidth of thermal noise will be
smaller than in the case shown on Figure 4.10 let’s call it as high IF configuration
3Fs
FIF =
. The main problem with the low IF configuration is that it can lead to
4
fluctuating DC levels due to cross leakage of the two sidebands.
The high IF configuration passes approximately twice more thermal noise, but
in this case BPF has enough ”room” to provide proper band limiting of the signal
before sampling. It is definitely more preferable scenario for us.
45
Figure 4.9: BPF as antialising filter
Figure 4.10: BPF as antialising filter
46
4.4.6
Local oscillator synchronization and the digitizer reference clock
Estimation of the Doppler frequency shift is accomplished by comparing phase differences between transmitted and received signals. This requires phase coherency
between the transmitter and receiver. In order to fulfill this requirement the transmit and receive LO sources, the control signals coming from micro controller and the
ADC unit have to be synchronized. This is accomplished supplying the same reference clock, a standard 10MHz reference source, to Tx and Rx LO synthesizers, ADC
unit and the micro controller. Inputs to the micro controller and ADC unit have to
be buffered to avoid reference source contamination.
4.5
Micro Controller Board
The micro controller module (MCM) performs several functions. First, it performs
simultaneous switching of the addresses for all receiver banks. Second, it controls the
gain of Rx VGA and attenuation of the variable attenuator in the Tx path. Third
it synchronizes the arbitrary waveform generators and the ADC acquisition board
providing four buffered output triggering signals. These buffers are able to supply a
50 Ohm load.
The MCM derives four independent control lines setting the receiver gain of each
channel. The voltage range of the of the VGA gain control line is 0-5V. This level is
controlled through writing proper values to the external DACs by micro-controller.
Noise and spurs on these lines will cause undesired modulation of the received signal.
In order to insure suppress the quantization noise on these lines the DAC outputs
have to be filtered. The recommended bandwidth of the LPF should not exceed
F sDAC
where F sDAC is the DAC sampling frequency.
10
AWG generates the transmitter waveforms at receiving triggering signal from the
47
MCM module. The same triggering signal is fed to the acquisition (ADC) board.
There are two ways t generate the triggering signal. The first is generation of triggering signal on interrupt coming from the programmable timer that sets up the pulse
repetition frequency. The second case is receiving external interrupt from an external
synchronizing generator.
Buffers on RF switch address bus have two stages of 3-state latches with two pairs
¯ and output enables OE
¯ signals. Enabling the first latch while
of independent latch LE
disabling the second one allows sequential preparation of new addresses for all four
RF switches without affecting on the current selection exposed to the switches. Then
the actual setting of the new addresses is performed by simultaneously enabling all
four banks.
Figure 4.11: Micro Controller
48
CHAPTER 5
MIMO GMTI: MATLAB SIMULATION
A Matlab simulation was written in order to compare the theoretical MIMO GMTI
model with the expected results from the hardware testbed. The hardware testbed
was described in the previous chapter in detail .
The Matlab simulation models the behavior of the testbed emulator for the specific
rooftop geometry that will be adopted in data collection. MIMO radar system with
N Rx and M Tx antennas will be placed on the roof a 30 meter high building looking
down to clutter cells at a range of 300 meters. The simulation supports two motion
emulation modes: switching Rx antennas only; or switching both Rx and Tx antenna
arrays simultaneously.
The simulator calculate an estimate of the clutter covariance matrix and its
rank. The covariance matrix is estimated from Monte Carlo simulation accomplished
through set of is orange reflectors with random amplitude and position. Random
position and amplitudes fort clutter cells are fixed during L pulses of a given CPI.
For each new CPI, clutter cell amplitudes and positions are randomized. At a given
pulse, each receiver adds up returns from all clutter cells illuminated by the four
transmitters. This is repeated for each pulse l ∈ {0, L − 1} as the simulator changes
the current position of the RX (and TX antennas). Clutter cells are chosen randomly
at different azimuth angles on an iso-range ring. For a clutter cell at a given azimuth,
49
the simulator uses small number of scatterers randomly positioned in range over one
range resolution bin.
In order to get reliable estimation of the rank of clutter covariance matrix number
of clutter cells have to be higher than the expected rank of the clutter covariance
matrix as given in (3.3.9). In the following we used 128 randomly selected clutter
cells uniformly distributed over the azimuth angle and 7 scattering centers per clutter
cell.
Two scenarios were considered:
• Scenario I: Stationary Tx array and moving Rx array
• Scenario II: Both TX and Rx arrays are moving
The simulation of stationary Tx array and moving Rx antennas configuration used
the next set of parameters:
The number of clutters = 64
Number of averages = 128
Number of Rx antennas = 4
Number of Tx antenna = 4
Number of pulses in CPI = 8
Tx/Rx antenna spacing ratio γ = 4
Motion coefficient β = 1
The expected rank of clutter covariance matrix for both scenarios is given by β(L −
1) + N + γ(M − 1) = 1(8 − 1) + 4 + 4(4 − 1) = 23.
The sorted eigenvalues of the estimated covariance matrix is given for the two
Scenarios in Figure 5.1 and Figure 5.2. For the first scenario we observe the eigenvalues drop off at the theoretical rank of 23 with a much sharper drop off starting
at eigenvalue 44. For the relatively close in distance of 300 meters the plane wave
50
Figure 5.1: Sorted eigenvalues of the estimated clutter covariance matrix for simulation Scenario I
51
assumption fails and there are small phase differences in the received signals not captured by the signal model derived using the single look angle derivation. This effect
is less pronounced for Scenario II when the TX and RX elements are kept at close
distance.
The expected rank of clutter covariance matrix is again
Figure 5.2: Sorted eigenvalues of the estimated clutter covariance matrix for simulation Scenario II
52
CHAPTER 6
FURTHER WORK
Results obtained in this work for nonorthogonal waveforms can be further extended
to the ultra wide band (UWB) signals. As was mentioned, reflectivity of an object
depends on frequency and time. UWB signals offer improved range resolution; however, the UWB signals violate the narrow-band assumptions underlying the system
analysis.
An interesting direction would be analysis of orthogonal frequency-division multiplexing (OFDM) waveforms, whereby each subcarrier yields a narrow-band signal with
independent scene response.
CDMA waveforms with short sequences can be a subject for further research in sensing with weakly correlated waveforms as well as methods for learning MIMO clutter
from samples that exploit structure (rank, geometry, etc).
Finally, six issues are recommended for consideration during the initial testing of
the system designed in this thesis: 1) The effective value of coupling between Tx and
Rx antennas and gain should be measured since they have strong impact on the total
system performance.
2) NF and gain figures of the lab designed LNA version need to be measured.
3) Electrical delay of all the RF switch sub-channels need to be measured and compensated.
4) LO sources should be synchronized and buffered. Mixers require +10dBm at LO
53
ports.
5) Single tone and two tones measurements for Tx and Rx should be conducted and
results compared to the designed performance values.
6) Timing synchronization should checked between switches and among addresses
generated by the MCM. 7) Crucial issue is synchronization of the waveforms coming
from AWGs. Both AWGs have to be synchronized using an external source.
54
REFERENCES
[1] Mark A. Richards, James A. Scheer and William A. Holm, Principles of Modern
Radar. SciTech Publishing 2010.
[2] J.Li and P. Stoica, MIMO radar signal processing, John Willey and Sons, 2009
[3] D.Bliss, K.Forsythe,S.Davis, G.Fawcett,D. Rabideau,L.Horowitz and S.Kraut,
”GMTI MIMO Radar”, International WD and D Conference, pp. 118-122, 2009.
[4] K. W. Forsythe and D. W. Bliss,”MIMO radar: concepts, performance enhancements, and applications”, in Signal Processing for MIMO Radar, J. Li and P.
Stoica, Eds. New York: Wiley, 2008.
[5] J.K.Kantor and D.W.Bliss,”Clutter covariance matrices for GMTI MIMO radar”,
Signals, Systems & Computers (ASILOMAR), 2010 Conference Record of the
Forty Fourth Asilomar Conference, Nov. 2010, pp. 1821-1826
[6] K. W. Forsythe and D. W. Bliss,”MIMO radar waveform constrains for GMTI”,
IEEE J.Sel. Topics Signal Proc.,vol.4, no. 1, pp.21-32, Feb. 2010
[7] K.W. Forsythe, D.W. Bliss, and G.S. Fawcett, ”Multiple-Input Multiple-Output
(MIMO) radar: performance issues”, Signals, Systems and Computers, 2004.
Conference Record of the Thirty-Eighth Asilomar Conference on, vol. 1, pp.
310315 Vol.1, Nov. 2004.
[8] D. W. Bliss and K. W. Forsythe, ”Multiple-input multiple-output (MIMO) radar
and imaging: degrees of freedom and resolution”, Conference Record of the
Thirty-Seventh Asilomar Conference on Signals, Systems & Computers, Pacific
Grove, Calif., vol. 1, pp. 5459, Nov. 2003.
[9] M.A. Richards, Fundamentals of Radar Signal Processing, McGraw-Hill, New
York, 2005.
[10] Chun-Yang Chen; Vaidyanathan, P.P.; , ”A Subspace Method for MIMO Radar
Space-Time Adaptive Processing,” Acoustics, Speech and Signal Processing,
2007. ICASSP 2007. IEEE International Conference on , vol.2, no., pp.II-925II-928, 15-20 April 2007
55
Appendix A
MATCHED FILTER
Received discrete time signal of length N collected till time moment n can be written
as
y n = [y[n] y[n − 1]
...
y[n − (N − 1)]]
(A.0.1)
(For simplicity, the subscript n will be discarded in the further notations.) Each
sample y[n] is a sum of signal and noise:
y[n] = s[n] + w[n],
n = 0, 1, ..., N − 1
(A.0.2)
where s is the desired noiseless signal, and w represents vector of i.i.d samples of
additive noise at the receiver input. We would like to find response of the filter that
maximizes the signal to noise ratio of the input signal. Input signal passes through
the input filter with impulse response of length N h = [h[0] h[1]
...
h[N − 1]].
The filter output is given by A.0.3
z = h T y = h T (ss + w ) = h T s + h T w
(A.0.3)
Power of each output filter sample is given by
hT y )∗ (h
hT y )T = (h
h∗ )T y ∗y T h = h H y ∗y T h
Po = |z|2 = z ∗ z T = (h
(A.0.4)
The average signal power
hT s )∗ (h
hT s )T ] = E[h
hH s ∗s T h ] = h H s ∗s T h
(Ps )av = E[(h
56
(A.0.5)
The average noise power
hT w )∗ (h
hT w )T ] = E[h
hH w ∗w T h ] = h H E[w
w ∗w T ]h
h = h H Σ wh (A.0.6)
(Pw )av = E[(h
w ∗w T ] is the noise covariance matrix. Then the signal to noise ratio
Where Σ w = E[w
can be written in terms of the above definitions as follow:
SN R =
(Ps )av
h H s ∗s T h
= H
(Pw )av
h Σ wh
(A.0.7)
Covariance matrix Σ w possesses the next properties:
• It is positive definite a T Σ wa > 0, ∀ a such that kaak2 6= 0
• Σ Tw = Σ ∗w
• ΣH
w = Σw
T
∗
• Σ −1
= Σ −1
w
w
To find impulse response of the filter that is optimal in terms of SNR the CouchySchwarz inequality will be used:
|ppH b |2 6 kppk2 kbbk2
(A.0.8)
The equality is achieved only when p = kbb for some real-valued k. Using positive
definite property of the covariance matrix it can be represented as a product of two
”square root” matrices: Σ w = A H A . Matrix A = QΛ 1/2Q −1 , where Q is matrix
composed of eigenvectors of Σ w and Λ = diag(λi ) is diagonal matrix with eigenvalues
of Σ w on the main diagonal.
AH )−1s ∗ , we get that p H b = (A
Ah )H (A
AH )−1s ∗ = h H s ∗ .
Assigning p = Ah and b = (A
Then
hH s ∗ )(h
hH s ∗ )H = h H s ∗s T h
|ppH b |2 = (ppH b )(ppH b )H = (h
57
(A.0.9)
Ah k2 k(A
AH )−1s ∗ k2 = (A
Ah )H (A
Ah )((A
AH )−1s ∗ )H ((A
AH )−1s ∗ )
h H s ∗s T h 6 kA
hH A H A h )(ssT A −1 (A
AH )−1s ∗ )
= (h
(A.0.10)
∗
hH Σ wh )(ssT Σ −1
= (h
w s )
hH Σ wh ) we receive the upper bond value for SNR that is
Dividing both sides on (h
achieved when p = kbb
hH s∗sT h
∗
6 sT Σ−1
SN R = H
w s
h Σ wh
(A.0.11)
Assigning k=1 in the p = kbb that leads to the optimal value of the matched filter
Ahopt = (A
AH )−1s∗
=⇒
(A.0.12)
−1
H −1 ∗
H
−1 ∗
A ) s = (A
A A) s =
h opt = A (A
58
∗
Σ −1
w s
Appendix B
RECEIVER LINK BUDGET
59
Figure B.1: Receiver link budget
60
Appendix C
AMPLIFIER NONLINEARITY
Output of an nonlinear amplifier can be modeled as [?]
Vout ≈ α1 Vin + α2 Vin2 + α3 Vin3
(C.0.1)
For single tone input signal :Vin = A cos (ωt)
Vout ≈ α1 A cos (ωt) + α2 (A cos (ωt))2 + α3 (A cos (ωt))3 + . . .
(C.0.2)
α32 A2 α22 A2
α32 A2
α22 A2 + 1+
α1 A cos (ωt) +
cos (2ωt) +
cos (3ωt) + . . .
≈
2
4α1
2
4
C.1
1-dB Compression Point
Input level where the gain drops by one dB
α2 A2 20 log α1 + 3 = 20 log|α1 | − 1
4
2 2
α A 20 log 1 + 3 = −1
4α1
α32 A2
= −0.11
4α1
r
α 1
A1dB = 0.145 α3
C.2
(C.1.1)
IP3 Intermodulation Point
For two tone input signal :Vin = A1 cos (ω1 t) + A2 cos (ω2 t)
Vout ≈ α1 (A1 cos (ω1 t) + A2 cos (ω2 t)) + α2 (A1 cos (ω1 t)
(C.2.1)
2
3
+ A2 cos (ω2 t)) + α3 (A1 cos (ω1 t) + A2 cos (ω2 t)) + . . .
61
Second Order Intermodulations
ω1 ± ω2 = α2 A1 A2 cos (ω1 + ω2 )t) + α2 A1 A2 cos (ω1 − ω2 )t)
Third Order Intermodulations
(C.2.2)
3α3 A21 A2
3α3 A21 A2
cos (2ω1 + ω2 )t) +
cos (2ω1 − ω2 )t)
4
4
3α3 A22 A1
3α3 A22 A1
cos (2ω2 + ω1 )t) +
cos (2ω2 − ω1 )t)
ω1 ± 2ω2 =
4
4
2ω1 ± ω2 =
For A1 = A2 = A, IP3 is defined as extrapolation point where power in 2ω1 −
ω2 and 2ω2 − ω1 sidebands is equal to power of fundamental signal:
3α A2 3 20 log IM3 = 20 log = 0dB
4α1
r
3α3 AIP3 = = A1dB [dB] + 9.6[dB])
4α1
(C.2.3)
IM3 is calculated based on IP3 measurement for given input A.
IM3 [dB] = 2(A[dB] − AIP3 [dB])
(C.2.4)
62