Post on 20-Mar-2018
LOW-COMPLEXITY DESIGN OF FREQUENCY HOPPING
CODES FOR MIMO RADAR FOR ARBITRARY DOPPLER
A thesis submitted in partial fulfilment of
the requirements for the degree of
Master of Science (by research)
in
Communication Systems and Signal Processing
by
Badrinath S.
200431002
badrinath@research.iiit.ac.in
Communications Research Center
INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY
GACHIBOWLI, HYDERABAD, A.P., INDIA - 500 032
January 2010
INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY
GACHIBOWLI, HYDERABAD, A.P., INDIA - 500 032
CERTIFICATE
It is certified that the work contained in this thesis, titled “Low-complexity design of
frequency hopping codes for MIMO radar for arbitrary Doppler” by Badrinath S, has
been carried out under my supervision and it is fully adequate in scope and quality as a
dissertation for the degree of Master of Science.
Date Prof V U Reddy (Advisor)
c© Copyright by Badrinath S 2010
All Rights Reserved
ii
The scientists from Franklin to Morse were clear thinkers and
did not produce any erroneous theories. The scientists of today
think deeply instead of clearly. One must be sane to think clearly.
But one can think deeply and be quite insane.
- Nikola Tesla
iii
Abstract
There has been a recent interest in the application of Multiple-Input Multiple-Output
(MIMO) communication concepts to radars. While traditional phased-array radars,
also referred to as Single-Input Multiple-Output radars, transmit coherent waveforms
from their many antenna elements, MIMO radars can transmit incoherent waveforms
from their transmitters, which offer additional benefits like spatial diversity and spa-
tial resolution. In the recent literature, optimization of orthogonal frequency hopping
waveforms for MIMO radars has been discussed. This optimization is based on a ‘cost
function’ derived from a newly formulated MIMO radar ambiguity function. Existing
literature however makes the assumption of small target Doppler.
In this thesis, we first extend the scope of this ambiguity function to large values
of target Doppler. We then introduce the concept of hit-matrix in the MIMO context.
This is based on the hit-array, which has seen extensive use in the context of frequency-
hopping waveforms for phased-array radars. We then propose new methods to obtain
optimal waveforms in both the large and small Doppler scenarios. Under the large
Doppler scenario, we propose the use of a cost function based on the hit-matrix which
offers a significantly lower computational complexity as compared to an ambiguity
based cost function, with no loss in code performance. In the small Doppler scenario,
we present an algorithm for directly designing the waveform from certain properties
which can be obtained from the ambiguity function in conjunction with the hit-matrix.
Finally, we address the problem of designing frequency-hopping waveforms which
yield a desired ambiguity function. This method is called “weighted optimization”
wherein we mask the cost function used in the heuristic search algorithm to reflect
the properties of the required ambiguity function.
iv
Acknowledgements
I would like to express my gratitude toward Prof. V Umapathi Reddy, my principal
advisor, for his support and guidance over the past two and a half years. Under his
tutelage, I have worked on a variety of areas - Ultra Wide-Band, signal detection and
classification in LPI radars, cognitive radio and most recently, MIMO radars. The
depth of knowledge I have gained in each of these domains is largely down to the em-
phasis he places on the processes of reading, understanding and precisely simulating
entire systems. His constant encouragement has contributed immensely to the way I
approach any given problem and I have, in a way, been taught a skill generally con-
sidered unteachable. His attention to detail has been a source of inspiration during
the course of my research assistantships. The relevance of each of the problems I have
worked on under him is probably what stands out the most, and I am grateful for the
opportunity to work on such topics.
This thesis was a part of a project sponsored by the LRDE, and I would like to
thank my project partners Anand Srinivas and Charvi Dhoot. Anand left a stable,
prestigious, well-paying but ultimately dull job in Qualcomm to follow his dream of
doing research and getting a Masters/Doctorate in the field of communications. A life
changing move like this is very rare to witness and I am glad to have had the privilege
of working with him. One of the hardest workers I know, I am certain he will realize
his dream of establishing a successful start-up company in his chosen field. The only
thing I can fathom him being unsuccessful at is his persistent but futile attempt at
speaking the Japanese language.
Charvi is without exception one of the brightest students I have worked with.
Her quick grasp of all topics of technical importance is probably one of her biggest
v
strengths. Her most astounding ability however, and I say this tongue-firmly-in-cheek,
is her uncanny knack of finding technical pdfs of all kinds - reports, journals, books
and other miscellaneous references - on the internet - something neither Anand nor
I seem particularly good at. I sincerely hope that Charvi accomplishes her goal of
leaving IIIT-Hyderabad in five years and finally figures out what she really wants to
do in life.
I am grateful to my labmates - A K Karthik, Chaitanya Vellampalli, Harinath
Reddy, Meenakshi Goyal, Nitin Jain, Prashanth Pai and VVM Niranjan Kumar for
their help and support.
I would also like to thank my friends Jimmy Narang and Sankalp Mulye for their
advice (most of it good, some tragically bad) at a time when I needed it the most.
My parents have been an ever-present support system and I am grateful for ev-
erything they have done for me over the past twenty-three years.
Debates will always rage in my mind about God, His/Her existence, and the role
such a being plays in our lives, but I am eternally thankful for life and everything it
offers.
vi
Contents
Abstract iv
Acknowledgements v
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 System Model and the MIMO Ambiguity Function 6
2.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 MIMO radar model . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Frequency-hopping waveforms . . . . . . . . . . . . . . . . . . 7
2.2 MIMO ambiguity function . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The hit-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Ambiguity function and the hit-matrix . . . . . . . . . . . . . . . . . 13
3 Waveform Design for Large Doppler 16
3.1 Algorithm for waveform design . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Waveform Optimization for Small Target Doppler 24
4.1 The hit-matrix under the small Doppler assumption . . . . . . . . . . 25
vii
4.2 Waveform design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Related discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Weighted Optimization 34
5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Conclusion 37
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Appendix A 39
A.1 Filling in frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.2 Re-labeling and the algorithm . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography 42
viii
List of Tables
4.1 Values of BT for which codes can be designed with the proposed algo-
rithm (M=4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Size of search space with M and Q for K = MQ/2 . . . . . . . . . . 31
ix
List of Figures
2.1 Transmitters and receivers in a MIMO radar. MF refers to the matched
filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The structure of frequency-hopping waveforms. . . . . . . . . . . . . 8
2.3 Cross ambiguity between sub-pulses, |Gm,m′,q,q′(τ, ν)| . . . . . . . . . 14
3.1 Decrease in g3(C) versus iterations of simulated annealing . . . . . . 19
3.2 Plot of the hit-matrix of a code obtained from simulated annealing
using g3(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 |Ω(τ, ν, 0, 0)| for the code whose hit-matrix is shown in Fig. 3.2. . . . 20
3.4 Empirical CDF of |Ω(τ, ν, f, f ′)| . . . . . . . . . . . . . . . . . . . . . 22
3.5 Magnitude below which 95% of samples of |Ω(τ, ν, f, f ′)| lie (the highest
peak is normalized to 0 dB) . . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Various cuts of |Ω(τ, ν, f, f ′)| for the code whose hit-matrix is shown
in Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Step 1 of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Step 2 of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Final CDFs of codes from the two methods with M = 4, Q = 10 and
K = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Graph showing performance of the algorithm with respect to simulated
annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 CDFs of the optimized codes with and without weighting in the spec-
ified region (M = 4, Q = 10, K = 80, lmax = 8, kmax = 2) . . . . . . . 36
x
Chapter 1
Introduction
MIMO radar is a recent evolution of radar that utilizes multiple transmitters and
receivers [3], [4]. Unlike a standard phased-array radar (also viewed as single-input
multiple-output or SIMO radar) , which transmits scaled versions of a single wave-
form, a MIMO radar system can transmit multiple probing signals via its antennas.
Jian Li and Petre Stoica summarize the superiority of the MIMO radar in sev-
eral fundamental aspects when compared to conventional SIMO radars in [4] . For
colocated transmit and receive antennas, the MIMO radar model has been shown to
offer higher resolution [16, 13], higher performance in detecting slowly moving targets
[17], better parameter identifiability [18], and direct applicability of adaptive array
techniques [19, 20]. The application of waveform optimization techniques to obtain
flexibility of transmit beampattern design has been described in [22, 23, 14].
MIMO radar waveforms can have any degree of coherence with each other, ranging
from complete coherence (in which case it becomes equivalent to a phased-array radar)
to complete incoherence (orthogonality). The choice of radar waveforms [5] plays
an important role in the resolution characteristics of the radar. The optimization of
radar waveforms for the phased-array radar focuses on obtaining a desirable ambiguity
function in terms of range and Doppler resolutions. On the other hand, MIMO radars
provide spatial resolution and spatial diversity in addition to range and Doppler
resolution.
1
CHAPTER 1. INTRODUCTION 2
1.1 Motivation
MIMO radar waveform design problems have been studied extensively in current lit-
erature [24, 14, 25, 7]. Waveform design methods can be broken into three categories:
1. Covariance matrix based design : The covariance matrix of the waveforms
is considered instead of the entire waveform and consequently, such design meth-
ods affect resolution only along the spatial domain. In [14], the objective behind
waveform design is to facilitate the radiation of power only along certain direc-
tions. The proposed method allows the signal power to be redistributed in the
spatial frequency space.
2. Ambiguity function based design : The ambiguity function of a given set
of MIMO waveforms provides information about its range, Doppler and spatial
frequency resolution. Such methods optimize the entire waveforms instead of
just their covariances and they involve optimization not only along the spa-
tial domain but also the range and Doppler domains. In [7], waveforms are
optimized so that a sharper radar ambiguity function can be obtained. Such
methods however make the assumption of point targets.
3. Prior information based design : Similar to the previous case, the entire
waveform is considered as opposed to just the covariance matrices. However, un-
like the ambiguity function based methods which aim to improve the resolutions
of point targets, in prior information based methods, we consider the detection
or estimation of extended targets. One key requirement is prior information
about the target and/or clutter impulse responses. In [3] the mutual informa-
tion between the received waveforms and the target impulse response has been
maximized by designing transmit waveforms. Other methods which fall under
this category such as [24] aim to design transmit waveforms which maximize
the Signal-to-Interference-plus-Noise-Ratio, which leads to an improvement in
detection performance.
Recent work by San Antonio and Daniel Furhmann [8] addresses the issue of
extending the conventional Woodwards ambiguity function to MIMO radars. The
CHAPTER 1. INTRODUCTION 3
derived ambiguity functions can simultaneously characterize the effects of array geom-
etry, transmitted waveforms and target scattering on resolution performance. Chen
and Vaidyanathan [6], [7] have built on the aforementioned MIMO ambiguity for-
mulation and dealt with the design of MIMO frequency-hopping codes based on the
optimization of this function. Frequency-hopping codes have been used in pulse com-
pression radars [9] because of their highly desired ambiguity properties. The design
of frequency-hopping codes for SIMO radars to obtain desired ambiguity functions
[10] has been well studied. The hit-array [11],[12] has also been extensively used for
waveform design in the SIMO context.
1.1.1 Problem formulation
The approach suggested by Chen and Vaidyanathan toward designing radar wave-
forms is to first parameterize these waveforms and then apply a heuristic search, such
as simulated annealing, to find a near-optimal set of parameters. The search algo-
rithm uses a ‘cost function’ that allows comparison of different parameter sets. Their
formulation assumes small target Doppler, which enables the decoupling of Doppler
from other parameters in the ambiguity function, resulting in a simple cost function.
This formulation, however, is inapplicable in the presence of large target Doppler.
This raises two critical questions
• In the presence of large target Doppler, the corresponding extension of the
aforementioned cost function becomes very computationally expensive. For
practical values of time-bandwidth product, performing a search over a space
as large as the code-matrix space (order of 1030 to 10100), while computing such
a cost function in each iteration becomes untenable. Is it possible to develop
an alternate, easily computable cost function without losing any performance?
• In the small Doppler scenario, simulated annealing has been performed over a
vast search space. Is it possible to formulate an algorithm which designs codes
instead of searching for them, without any loss of performance?
These are the two problems are addressed in this thesis. Our theoretical approach
substantiated with simulation results shows that solutions to these two problems have
CHAPTER 1. INTRODUCTION 4
been satisfactorily obtained [1, 2].
1.2 Contributions
As stated above, waveform design is a critical problem in the field of MIMO radar.
We approach this problem by expressing waveforms in terms of their parameters and
proceed to optimize them. The contributions of this thesis [1, 2] are listed below
1. The MIMO radar ambiguity function has been extended to large values of target
Doppler.
2. The hit-array formalism has been developed in the MIMO context, which we
now term the “hit-matrix”.
3. Cost functions are defined based on the hit-matrix and a search algorithm is
proposed for waveform design under the assumption of large target Doppler.
4. An algorithm is proposed for designing waveforms (as opposed to searching for
them) under the small Doppler case.
5. The method of weighted optimization is proposed, to facilitate better control
over sidelobes in a delay-Doppler subspace.
1.3 Thesis organization
This thesis is organized into six chapters. The first chapter provides a survey of
existing literature about MIMO radar waveform design as well as an overview of the
problem statement. In Chapter 2, we first define our system model and then extend
Chen’s [7] ambiguity function for large values of target Doppler. We also introduce
and develop the concept of hit-array [12] in the MIMO scenario, and call the resulting
formulation the “hit-matrix”. We then show the close correspondence between the
hit-matrix and the MIMO ambiguity function.
CHAPTER 1. INTRODUCTION 5
The next three chapters address the problem of designing suitable frequency-
hopping codes for different values of target Doppler. Waveform design for the large
Doppler scenario is explained in Chapter 3. We perform a heuristic search over the
code-matrix-space using a hit-matrix based cost function. Chapter 4 presents an algo-
rithm for waveform design under the small Doppler scenario. In this chapter, we make
certain observations about the ambiguity function and the hit-matrix and incorporate
them into an algorithm which directly designs waveforms. The performance is shown
to be very similar to Chen’s search-based algorithm. In Chapter 5, we introduce the
method of “Weighted optimization” and illustrate it with an example. This method
can be used to optimize ambiguity sidelobes in a specified region of the delay-Doppler
space.
Chapter 6 concludes the thesis. This is then followed by the bibliography.
Chapter 2
System Model and the MIMO
Ambiguity Function
2.1 System model
2.1.1 MIMO radar model
Consider a monostatic MIMO radar that contains M transmitters and N receivers
with their antennas configured as uniform linear arrays, as shown in Fig. 2.1. We
assume a point target and also that the target, transmitters and receivers lie in the
same 2-D plane. Let dT and dR represent the spacing between consecutive transmit-
ters and receivers respectively, and let γ = dT /dR. We define the spatial frequency of
the target as
f =dRsin(θ)
λ
where θ is the target angle with respect to the broadside direction and λ is the
wavelength of the RF carrier of the transmitted waveforms. Let τ and ν be the target
delay (which is a measure of target range) and Doppler frequency (a measure of target
velocity), respectively. The spatial frequency f corresponds to the angular location of
the target with respect to the arrays of the radar. Let um(t), m ∈ 0, · · · , M − 1
represent the M transmitter waveforms. Then, the waveform received at the nth
6
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION 7
Figure 2.1: Transmitters and receivers in a MIMO radar. MF refers to the matchedfilter.
receiver antenna can be expressed as [7]
yn(t)|τ,ν,f ≈
M−1∑
m=0
um(t − τ)ej2πνtej2πf(γm+n) (2.1)
2.1.2 Frequency-hopping waveforms
Frequency-hopping signals are good candidates for the radar waveforms because they
are easily generated and have constant modulus. The waveforms can be represented
as (see Fig. 2.2 )
um(t) =
L−1∑
l=0
φm(t − Tl)
where
φm(t) =
Q−1∑
q=0
ej2πcm,q∆fts(t − q∆t),
s(t) =
1 if 0 < t < ∆t
0 otherwise
Here, cm,q is the (m, q)th element of the matrix [C]M×Q and it can assume values
from the set 1, · · · , K, where K is the total number of frequency hops available.
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION 8
Figure 2.2: The structure of frequency-hopping waveforms.
As shown in Fig. 2.2, each transmitter waveform um(t) consists of a stream of L
identical pulses φm(t). Each pulse in turn contains Q constant amplitude frequency
subpulses each having width ∆t, and frequency cm,q∆f . Additionally, we impose the
following conditions for orthogonality [7]
∆f∆t = 1
cm,q 6= cm′,q, ∀m 6= m′, ∀q
Orthogonal waveforms result in a uniform beam pattern in all directions, which is a
key aspect of detection using MIMO radars. For fixed ∆t and ∆f , these waveforms
can be completely described by the code matrix
C =[
cm,q
]
M×Q
and the pulse spacing vector
~T =[
T0 T1 · · · TL−1
]
1×L
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION 9
The pulse spacing vector plays a role in shaping the Doppler resolution of the wave-
forms. In this thesis, we do not consider the optimization of this vector.
2.2 MIMO ambiguity function
The resolution of a radar system is determined by the response to a point target in
the matched filter output. This response can be characterized by a function called
the ambiguity function. The traditional Woodward ambiguity function for a SIMO
radar is given as
χ(τ, ν) =
∫ ∞
−∞
u(t)u∗(t + τ)ej2πνtdt (2.2)
In the above expression, τ and ν represent the delay and Doppler mismatch at the
receiver respectively. The ideal ambiguity function should be sharp around the region
of zero-mismatch, i.e. (τ, ν) = (0, 0). This idea has been extended to the MIMO case
in [7]. Consider the expression for the received signal in a MIMO radar given in (2.1).
Let (τ1, ν1, f1) represent the true parameters of a target, and let (τ2, ν2, f2) be the
assumed parameters at the receiver. The summed match filter output is given as
N−1∑
n=0
∫ ∞
−∞
(
yn(t)|τ1,ν1,f1
)(
yn(t)|τ2,ν2,f2)∗dt
=
(N−1∑
n=0
ej2π(f1−f2)n
)
×
(M−1∑
m=0
M−1∑
m′=0
∫ ∞
−∞
um(t − τ1)u∗m′(t − τ2)e
j2π(ν1−ν2)tej2π(f1m−f2m′)γdt
)
(2.3)
In the above expression, the second term corresponds to the ambiguity function when
only one receiver is present, while the first term brings out the effect of having multiple
receivers. To simplify the ambiguity function and the waveform design problem, the
first term can be decoupled from the above expression. The resulting expression is
termed the “MIMO ambiguity function”. We can now consider (τ1 − τ2) to be the
delay mismatch and (ν1−ν2) to be the Doppler mismatch, and rewrite the expression.
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION10
The MIMO radar ambiguity function is thus given as [7]
χ(τ, ν, f, f ′) =
M−1∑
m=0
M−1∑
m′=0
χm,m′(τ, ν)ej2π(fm−f ′m′)γ (2.4)
where τ and ν represent the Doppler and delay mismatch at the receiver, f represents
the target’s true spatial frequency, and f ′ represents the assumed spatial frequency
at the receiver, and χm,m′(τ, ν) represents the cross-ambiguity function between the
waveforms um(t) and um′(t).
χm,m′(τ, ν) =
∫ ∞
−∞
um(t)u∗m′(t + τ)ej2πνtdt (2.5)
For frequency-hopping waveforms (see Fig. 2.2 ), (2.5) becomes
χm,m′(τ, ν) =
L−1∑
l=0
L−1∑
l′=0
χφm,m′(τ + Tl − Tl′ , ν)ej2πνTl
where
χφm,m′(τ, ν) =
∫ Q∆t
0
φm(t)φ∗m′(t + τ)ej2πνtdt (2.6)
We assume no range folding(
i.e. |τ | <(
minl,l′(|Tl − Tl′|) − Q∆t)
)
due to which
χφm,m′(τ + Tl − Tl′, ν) = 0 for l 6= l′
and hence
χm,m′(τ, ν) = χφm,m′(τ, ν)
L−1∑
l=0
ej2πνTl (2.7)
χφm,m′(τ, ν) is the cross-ambiguity between two individual pulses of different transmit-
ters(
φm(t) and φm′(t))
and can be expanded as
χφm,m′(τ, ν) =
Q−1∑
q=0
Q−1∑
q′=0
χrect(τ + (q − q′)∆t, ν + (cm,q − cm′,q′)∆f)×
ej2π(
ν+(cm,q−cm′,q′)∆f)
q∆te−j2πcm′,q′∆fτ
(2.8)
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION11
where
χrect(τ, ν) =
∫ ∆t
0
s(t)s(t + τ)ej2πνtdt
=
(∆t − |τ |) sinc(
ν(∆t − |τ |))
ejπν(∆t−|τ |), if |τ | < ∆t
0, otherwise
(2.9)
represents the ambiguity function of the rectangular pulse s(t). In [7], the Doppler is
assumed to be small (ν∆t ≈ 0) due to which (2.8) reduces to
χφm,m′(τ, ν) =
Q−1∑
q=0
Q−1∑
q′=0
χrect(τ + (q − q′)∆t, (cm,q − cm′,q′)∆f)×
ej2π∆f(cm,q−cm′,q′)q∆te−j2πcm′,q′∆fτ
= χφm,m′(τ, 0)
(2.10)
We do not assume small Doppler in chapters 2 and 3, and work with (2.8). In
the remainder of this chapter, we describe the hit-matrix formalism and proceed to
waveform optimization for the large Doppler case in the next chapter.
2.3 The hit-matrix formalism
The hit-array has been introduced as a tool to analyze frequency hopping waveforms
in [12]. The central concept in this formulation is that of a ‘hit’, which occurs when
the received pattern has been shifted in the time-frequency space in such a way that
it overlaps with the original pattern at exactly one time-frequency position. In this
section, we develop the hit-matrix, which is applicable to frequency hopping codes
for MIMO radar under the large Doppler scenario. We define the hit-matrix for the
code matrix C as
H =[
hk,l
]
(2Q−1)×(2K−1)(2.11)
−Q < k < Q, −K < l < K; k, l ∈ Z
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION12
where
hk,l =
M−1∑
m=0
M−1∑
m′=0
Q−k−1∑
q=0
δ(cm,q − cm′,(q+k) + l) if k ≥ 0
M−1∑
m=0
M−1∑
m′=0
Q+k−1∑
q=0
δ(cm,q − cm′,(q−k) − l) otherwise
(2.12)
Here δ(.) refers to the Kronecker delta function. The hit-matrix H can also be written
as the sum of individual cross-hit-arrays Hm,m′ , as follows
H =M−1∑
m=0
M−1∑
m′=0
H(m, m′) (2.13)
where
H(m, m′) =[
hk,l(m, m′)]
(2Q−1)×(2K−1)(2.14)
and
hk,l(m, m′) =
Q−k−1∑
q=0
δ(cm,q − cm′,(q+k) + l) if k ≥ 0
Q+k−1∑
q=0
δ(cm,q − cm′,(q−k) − l) otherwise
(2.15)
As an example, consider the code matrix
C =
[
1 2 3
3 1 2
]
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION13
Here M = 2, Q = 3 and K = 3. The cross-hit-arrays for this code are
H(0, 0) =
1 0 0 0 0
0 2 0 0 0
0 0 3 0 0
0 0 0 2 0
0 0 0 0 1
H(1, 0) =
0 1 0 0 0
0 0 2 0 0
1 0 0 2 0
0 1 0 0 1
0 0 1 0 0
H(0, 1) =
0 0 1 0 0
1 0 0 1 0
0 2 0 0 1
0 0 2 0 0
0 0 0 1 0
H(1, 1) =
0 0 0 1 0
0 1 0 0 1
0 0 3 0 0
1 0 0 1 0
0 1 0 0 0
and the resulting hit-matrix is
H =
1 1 1 1 0
1 3 2 1 1
1 2 6 2 1
1 1 2 3 1
0 1 1 1 1
2.4 Ambiguity function and the hit-matrix
We now describe how the hit-matrix of a frequency-hopping code relates to its ambi-
guity function. Consider the MIMO radar ambiguity function in (2.4) which, in view
of (2.7) and (2.8), can be expressed as
χ(τ, ν, f, f ′) = Ω(τ, ν, f, f ′)
[
L−1∑
l=0
ej2πνTl
]
(2.16)
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION14
τν
2∆f2∆t
(τ, ν) = (−[q−q′]∆t , −[cm,q
− cm′,q′]∆f )
Figure 2.3: Cross ambiguity between sub-pulses, |Gm,m′,q,q′(τ, ν)|
where
Ω(τ, ν, f, f ′) =
[
M−1∑
m=0
M−1∑
m′=0
Q−1∑
q=0
Q−1∑
q′=0
Gm,m′,q,q′(τ, ν)ej2π(fm−f ′m′)γ
]
(2.17)
represents the ambiguity between two pulses and
Gm,m′,q,q′(τ, ν) = χrect
(
τ + (q − q′)∆t, ν + (cm,q − cm′,q′)∆f)
.
ej2π(
ν+(cm,q−cm′,q′ )∆f)
q∆te−j2πcm′,q′∆fτ(2.18)
is the cross-ambiguity between the qth sub-pulse of um(t) and the (q′)th sub-pulse
of um′(t). A plot of |Gm,m′,q,q′(τ, ν)| is shown in Fig. 2.3. The position of its peak
is (τpeak, νpeak) = (−(q − q′)∆t,−(cm,q − cm′,q′)∆f). Along the τ -axis, this function
drops to zero at a shift of ∆t from τpeak, and along the ν-axis, it falls by roughly 6
dB relative to the peak at a shift of ∆f from νpeak. Hence most of the energy of each
sub-pulse ambiguity function is located within an area of 4∆f∆t around (τpeak, νpeak).
For different values of (m, m′, q, q′), the shape of Gm,m′,q,q′(τ, ν) remains the same, and
only the position of its peak in the delay-Doppler space changes. The set of possible
CHAPTER 2. SYSTEM MODEL AND THE MIMO AMBIGUITY FUNCTION15
values of (τpeak, νpeak) is given as
(τpeak, νpeak) ∈ (k∆t, l∆f); k, l ∈ Z
−Q < k < Q, −K < l < K
The number of 4-tuples (m, m′, q, q′) for which Gm,m′,q,q′(τ, ν) peaks at the position
(τ0, ν0) is
P (τ0, ν0) =
M−1∑
m=0
M−1∑
m′=0
Q−τ0∆t
−1∑
q=0
δ
(
cm,q − cm′,(q+τ0∆t
) +ν0
∆f
)
if τ0 ≥ 0
M−1∑
m=0
M−1∑
m′=0
Q+τ0∆t
−1∑
q=0
δ
(
cm,q − cm′,(q−τ0∆t
) −ν0
∆f
)
otherwise
= hk,l| k=τ0∆t
, l=ν0
∆f
Thus each element of the hit-matrix corresponds to the number of functions Gm,m′,q,q′(τ, ν)
centered at the same point in the delay-Doppler space. The (MQ)2 different func-
tions Gm,m′,q,q′(τ, ν) for different values of the 4-tuple (m, m′, q, q′) are the building
blocks of the overall MIMO ambiguity function, and the distribution of the positions
of their peaks in the delay-Doppler space plays a key role in determining the shape
of the overall ambiguity function. The value of h0,0 corresponds to the height of the
mainlobe in the ambiguity function and is constant for a given code matrix size. The
values of hk,l outside of (k, l) = (0, 0) correspond to sidelobes in the ambiguity func-
tion centered at (τ, ν) = (k∆t, l∆f). The higher these values of hk,l, the higher will
be the corresponding sidelobes. Also, since the spatial frequency parameters f and
f ′ appear as complex exponential weights in the summation of (2.17), a reduction in
the values of the hit-matrix outside of (k, l) = (0, 0) will result in a corresponding
reduction in sidelobes along the spatial frequency dimensions as well.
Chapter 3
Waveform Design for Large
Doppler
We now describe how frequency-hopping codes can be optimized under the large
Doppler scenario to yield a desired ambiguity function. Since the second product
term in the RHS of (2.16) is not dependant on the choice of code matrix C, we
only concern ourselves with the optimization of the first term Ω(τ, ν, f, f ′). To apply
heuristic search algorithms like simulated annealing, we require a cost function that
allows the desirability of different codes to be compared. Following the formulation
of [7], a cost function for the large Doppler case is given as below
fp(C) =
∫ Q∆t
−Q∆t
∫ K∆f
−K∆f
∫ 1
γ
0
∫ 1
γ
0
|Ω(τ, ν, f, f ′)|pdfdf ′dνdτ
It can be shown that the height of the peak in Ω(τ, ν, f, f ′) at (τ, ν, f, f ′) = (0, 0, 0, 0) is
constant, and hence this cost function favors codes that have their sidelobes flattened
out over the delay, Doppler and spatial frequency dimensions. Increasing the value of
p increases the penalty on higher sidelobes. fp(C) can be evaluated for each code using
Riemann sums with a sufficient number of bins instead of integrations. However, this
results in a high computational complexity which increases as O(
M2Q2BT)
, where
BT is the time-bandwidth product of the code.
16
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 17
We propose the following alternative cost function based on the hit-matrix
gp(C) =
Q−1∑
k=−Q+1
K−1∑
l=−K+1
(hk,l)p (3.1)
Given that the hit-matrix contains a significant amount of information about the
nature of the ambiguity function, gp(C) will be highly correlated with fp(C). However,
the evaluation of gp(C) is far less computationally intensive than fp(C), and increases
only as O(M2Q2). This allows the heuristic search algorithms using this cost function
to rapidly traverse the code space, thereby allowing good codes to be found faster.
3.1 Algorithm for waveform design
We now describe how we apply simulated annealing using gp(C). We use a slightly
modified form of simulated annealing called quantum simulated annealing, which
allows faster convergence. The steps of the algorithm are as follows.
1. Randomly draw a code matrix C from 0, 1, · · · , K − 1MQ such that the code
is orthogonal, i.e. cm,q 6= cm′,q for m 6= m′.
2. Randomly draw j from 1, 2, · · · , ⌈J⌉, where J < MQ is the jump size.
3. Set C ′ = C, and repeat the following steps j times.
(a) Randomly draw m from 0, · · · , M − 1 and q from 0, · · · , Q − 1.
(b) Select k from 0, · · · , K − 1 with k 6= c′m,q, ∀m.
(c) Set c′m,q = k.
4. Randomly draw U from [0, 1].
5. If U < exp((gp(C) − gp(C′))/T ), then set C = C ′.
6. Set T = αT J = βJ with α ∈ (0, 1) and β ∈ (0, 1).
7. If a sufficiently small value of gp(C) has been obtained, terminate the algorithm.
Otherwise, return to step 2.
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 18
In the above algorithm, α represents the rate of decrease of temperature and β repre-
sents the rate of decrease of jump size J . T refers to the temperature, and a typical
set up could have T = 1 initially.
Consider as an example the optimization of frequency-hopping codes for (M, Q, K) =
(4, 6, 24). The total number of possible codes of this size is(
K!(K−M)!
)Q
= 2.8 × 1032.
We found the number of Riemann sum bins, required for reasonable accuracy in the
evaluation of fp(C) for this code size, to be 100 each along the delay and Doppler
dimensions, and 20 each along the two spatial frequency dimensions. Hence, while
fp(C) requires the computation and summation of 2.3 × 109 terms per iteration,
gp(C) only needs 576 computations per iteration, resulting in a significant decrease
in complexity.
3.2 Simulation results
We present a number of examples to demonstrate the effectiveness of the hit-matrix
based cost function in designing good frequency hopping codes. We have generated
codes of various sizes by simulated annealing using both fp(C) and gp(C) at p = 3.
The code parameters used were M = 4, Q ∈ 6, · · · , 10 and K ∈
MQ2
, MQ, 2MQ, 3MQ
for each value of Q. Other parameters used were ∆t = 1, ∆f = 1 and γ = 1. For
simulated annealing, we set the parameters T = 10, J = 12, α = 0.9 and β = 0.95.
First, we consider the code obtained for (M, Q, K) = (4, 6, 24) using g3(C). The
decrease in the cost with respect to the iterations of simulated annealing is shown in
Fig. 3.1. The final code obtained is
C =
20 2 24 5 6 1
22 14 4 24 1 20
18 1 12 6 23 22
11 24 9 2 17 14
(3.2)
A plot of the hit-matrix of this code is shown in Fig. 3.2 and a plot of |Ω(τ, ν, f, f ′)|
at (f, f ′) = (0, 0) is shown in Fig. 3.3. We observe that the ambiguity function of
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 19
0 100 200 300 400 500 600 700 8001.58
1.6
1.62
1.64
1.66
1.68
1.7
1.72
1.74
1.76x 10
4
Number of iterations
Val
ue o
f cos
t fun
ctio
n g 3(C
)
Figure 3.1: Decrease in g3(C) versus iterations of simulated annealing
the above code has a very sharp mainlobe along the delay and Doppler dimensions,
with 98% of the sidelobes lying below -10 dB with respect to the peak. Also note the
visual similarity between the plots of hit-matrix and ambiguity function.
Although it may appear that optimization using gp(C) involves just the time and
Doppler dimensions, it takes the spatial frequency into consideration as well. To show
this, we consider values of Ω(τ, ν, f, f ′) at different values of (τ, ν).
1. (τ, ν) = (0, 0)
The Ω function can be expressed from (2.18), (3.1) and (2.9) as follows -
Ω(0, 0, f, f ′) =M−1∑
m=0
M−1∑
m′=0
δm,m′ej2π(fm−f ′m′)γ
=M−1∑
m=0
ej2π(f−f ′)mγ
(3.3)
The above expression is independent of the code matrix, and thus the ambiguity
function in this formulation cannot be optimized along the spatial frequency
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 20
−5−4
−3−2
−10
12
34
5
−20
−10
0
10
20
0
5
10
15
20
k = τ/∆tl = ν/∆f
h(k,
l)
Figure 3.2: Plot of the hit-matrix of a code obtained from simulated annealing usingg3(C)
−4−2
02
46
−20
−10
0
10
20
0
5
10
15
20
τ/∆tν/∆f
|Ω(τ
,ν,0
,0)|
Figure 3.3: |Ω(τ, ν, 0, 0)| for the code whose hit-matrix is shown in Fig. 3.2.
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 21
domains at (τ, ν) = (0, 0).
2. (τ, ν) 6= (0, 0)
Noting that each peak, given by |Gm,m′,q,q′(τ, ν)|, corresponds to a hit which is
reflected in (2.18) and the ambiguity function consists of a weighted sum of the
hits, with the weights being of the form ej2π(fm−f ′m′)γ, optimizing the hit-matrix
is analogous to optimizing the upper bound on the ambiguity function along
the spatial frequency dimensions.
We may point out here that the ambiguity function has a very low magnitude
when (τ, ν) 6= (0, 0) for all mismatched values of f and f ′. To show this, various cuts
of |Ω(τ, ν, f, f ′)| are shown in Fig. 3.6 (at γ = 1) . We now compare various codes
obtained using fp(C) and gp(C). We take samples from the function |Ω(τ, ν, f, f ′)|
and plot their empirical cumulative distribution function (ECDF), which shows the
percentage of samples of |Ω(τ, ν, f, f ′)| which are less than specified magnitudes. The
highest peak has been normalized to 0 dB. Fig. 3.4 shows the ECDF curves for
various codes obtained for Q = 6. Note that use of either cost function yields codes
with similar ECDF curves.
In Fig. 3.5, we have plotted the magnitude below which 95% of the samples of
|Ω(τ, ν, f, f ′)| lie for different codes, as a function of the time bandwidth products
BT = (Q∆t × K∆f). Note that up to BT = 392, fp(C) and gp(C) both yield
codes with a similar number of undesirable peaks. The curve corresponding to fp(C)
could not be extended beyond BT = 392 because of the increasing computational
complexity. However, using gp(C), codes with BT as high as 1200 (and higher) could
be easily generated.
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 22
−15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 080
82
84
86
88
90
92
94
96
98
100
Em
piric
al C
DF
(%
)
Codes obtained using fp(C)
Codes obtained using gp(C)
(M,Q,K) = (4, 6, 12)
(M,Q,K) = (4, 6, 24)
(M,Q,K) = (4, 6, 48)
Magnitude (dB)
Figure 3.4: Empirical CDF of |Ω(τ, ν, f, f ′)|
0 200 400 600 800 1000 1200−16
−15
−14
−13
−12
−11
−10
−9
Time bandwidth product (BT = Q∆t×K∆f)
Mag
nitu
de (
dB)
belo
w
w
hich
95%
of s
ampl
es li
e
Codes obtained using gp(C)
Codes obtained using fp(C)
Figure 3.5: Magnitude below which 95% of samples of |Ω(τ, ν, f, f ′)| lie (the highestpeak is normalized to 0 dB)
CHAPTER 3. WAVEFORM DESIGN FOR LARGE DOPPLER 23
Figure 3.6: Various cuts of |Ω(τ, ν, f, f ′)| for the code whose hit-matrix is shown inFig. 3.2.
Chapter 4
Waveform Optimization for Small
Target Doppler
As derived in [7], for the small Doppler condition (ν∆t ≈ 0), and under the assump-
tion of no range folding, we have from (2.7) and (2.10)
χm,m′(τ, ν) ≈ χφm,m′(τ, 0)
L−1∑
l=0
ej2πνTl (4.1)
In this case, we can separate out the Doppler term from the expression (2.4). We can
write the ambiguity function from (2.4), (2.7) and (2.10) as follows.
χ(τ, ν, f, f ′) =
[
M−1∑
m=0
M−1∑
m′=0
χφm,m′(τ, 0)ej2π(fm−f ′m′)γ
]
.
[
L−1∑
l=0
ej2πνTl
]
(4.2)
which, in view of (2.16), becomes
χ(τ, ν, f, f ′) =
[
Ω(τ, 0, f, f ′)
]
.
[
L−1∑
l=0
ej2πνTl
]
(4.3)
Note from (4.2) that the φm(t) terms (corresponding to the waveform pulses),
which are contained in χφm,m′(τ, 0), do not affect the Doppler resolution. The objective
behind waveform design is to obtain a set of waveforms with a desirable MIMO
24
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER25
ambiguity function. In [7] a heuristic search (simulated annealing) is performed over
the space of all code-matrices, to acquire a code C which minimizes a cost function
fp(C).
In the SISO case, we expect the ambiguity function to be sharp about the point of
zero-mismatch, i.e. (τ, ν) = (0, 0). Similarly, in the MIMO scenario, we want the am-
biguity function to be sharp around the region of zero-mismatch, which corresponds
to values of the function Ω(τ, 0, f, f ′) over the line (τ, f, f ′)|τ = 0, f = f′
. As the
waveforms are assumed to be orthogonal, we can write
Ω(0, 0, f, f ′|f = f ′) =
M−1∑
m=0
M−1∑
m′=0
δm,m′ej2πfγ(m−m′) = M (4.4)
Thus Ω(τ, 0, f, f′
) at zero-mismatch is a constant value proportional to the number
of transmitters and is independent of the chosen code-matrix. The objective now is
to suppress all the peaks not lying on this line. Therefore, the code optimization
problem under the small Doppler scenario (as given in [7]) can be described as
minC
fp(C) subject to (4.5)
cm,q 6= cm′ ,q, for m 6= m′
and cm,q ∈ 1, 2, . . . , K
where
fp(C) =
∫ Q∆t
−Q∆t
∫ 1/γ
0
∫ 1/γ
0
|Ω(τ, 0, f, f ′)|pdfdf
′
dτ (4.6)
In the next section, we explain our proposed algorithm.
4.1 The hit-matrix under the small Doppler as-
sumption
In this scenario, the hit-matrix reduces from a (2Q − 1) × (2K − 1) matrix to a
(2Q − 1) × (1) array. We define the hit-matrix under small Doppler, Hsd as follows
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER26
Hsd =[
hk,0
]
(2Q−1)×(1)(4.7)
−Q < k < Q ; k ∈ Z
where hk,0 is given by
hk,0 =
M−1∑
m=0
M−1∑
m′=0
Q−|k|−1∑
q=0
δ(cm,q − cm′,(q+|k|)) (4.8)
and δ(.) refers to the Kronecker delta function.
Note that this hit-matrix will also have a close correlation with the ambiguity
function under the low Doppler scenario. Further, it is easy to see that h0,0 is a
constant equal to MQ and is independent of the values in the code matrix. Our
objective in code design can now be described by the following two conditions
1. Condition A: Within the constraints defining the code matrix (M, Q, ∆f, ∆t, K),
the sidelobe levels must be reduced. This means that the total number of hits
(corresponding to sidelobe levels), given by
S =
Q−1∑
k=−Q+1
hk,0 (4.9)
should be made as small as possible.
2. Condition B: The value of maxk;k 6=0 hk,0 is to be minimized, which requires
that the S peaks are spread out across the (2Q− 2) summation terms as evenly
as possible (excluding k = 0, as h0,0 is a fixed value). This will lead to a good
peak-to-sidelobe ratio, which is another key aspect in waveform design from
ambiguity functions.
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER27
4.2 Waveform design
From the structure of hk,0, we see that it is symmetric about k = 0. From hereafter,
any reference to hk,0 will correspond only to all positive values of k, i.e. k > 0. In our
proposed algorithm, we constrain the possible number of frequencies K to MQ/2.
This is a reasonable assumption, and is discussed further in the next section. The
problem of filling in the code matrix under this assumption, while simultaneously
minimizing the sum of hits hk,0, can now be stated as the following problem.
• Given MQ2/2 balls, with exactly Q balls having each of the MQ/2 possible
colours, what is the best way to choose MQ balls such that the number of
possible same-colour pairs is minimized?
The answer to this question is to choose balls of each colour twice. Consider an
example with MQ=6. This corresponds to 6 slots and 3 colours. In order to minimize
the number of possible same-colour pairs, we choose each of the three colours twice,
i.e., A, A, B, B, C, C where A, B, C correspond to colours. This leads us to three
possible pairs. Consider 6 slots filled up with A, A, B, C, C, C or any of the other
combinations. The number of possible same-colour pairs in this case is 4 - one A, A
pair and 3 C, C pairs. For a more detailed explanation, refer to Section A.1 in the
appendix.
In the above example, a same-colour pair is analogous to a ‘hit’. Thus, for such
codes with K = MQ/2, the total number of peaks S will be minimized when each
frequency occurs exactly twice in the code matrix. This satisfies Condition A.
Any code matrix with K = MQ/2 and with each frequency repeated twice will
have S = 2MQ. We know that all values in hk,0 are positive, h0,0 equals MQ and that
it is symmetric about k = 0 (from definition). Therefore, we have S ′ =∑k=Q−1
k=1 hk,0 =
K = MQ/2. To satisfy Condition B, the peaks in S ′ are to be spread out across all
the elements of hk,0 in as even a manner as possible. Thus, each element in hk,0, k 6= 0
should lie between ⌊ KQ−1
⌋ and ⌈ KQ−1
⌉. Define Nm = ⌊ KQ−1
⌋.
The algorithm has two steps, as shown in Figs. 4.1 and 4.2. The first step will
find a code matrix which has hk,0 = Nm, ∀k, k 6= 0. In those cases where KQ−1
is
not an integer, the code matrix will still have some unfilled positions. The second
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER28
Figure 4.1: Step 1 of the algorithm
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER29
Figure 4.2: Step 2 of the algorithm
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER30
step aims to fill the remaining slots, while constraining maxk,k 6=0 hk,0 to ⌈ KQ−1
⌉. While
the algorithm will always generate “good” codes, further minimization of the cost
function is possible by “relabeling” the frequency indices. When building the code
matrix, one possible solution is to label the frequencies sequentially, i.e. fill in the
code matrix from 1 to K. We have observed from extensive simulations that if this
order is randomized, it leads to a further slight reduction in the value of the cost
function in most cases. A typical run of the algorithm is illustrated in Section A.2 in
the appendix for both such labeling schemes.
As can be seen from the algorithm flowcharts, there is a “Reset” block, which
under certain conditions, discards the code matrix after a few iterations and attempts
to start designing the code matrix all over again. This case arises when, from a
partially filled code matrix, it is not possible to fill the remaining positions in such a
way that the resulting code matrix will generate orthogonal waveforms. However, the
probability of such an event occurring is very low. Through extensive simulations (for
various values of M and Q), we have observed that the probability of a code being
generated, with three or less resets is over 99%. When the algorithm successfully
reaches the “STOP” block, we have the desired code matrix which minimizes the
cost function.
4.3 Related discussion
For frequency-hopping waveforms, the parameters which can be controlled are -
M, Q, K, ∆t, ∆f. The above algorithm forces one of these parameters K to be
fixed to MQ/2. One possible limitation of this method is the loss of flexibility in
designing codes for a required time-bandwidth product, BT. Table 1 lists out all pos-
sible time bandwidth products between 0 and 400 for which codes can be designed
from the algorithm, with M = 4. Note that the bandwidth of the waveform is given
by K∆f , the duration of each pulse is Q∆t and hence the time-bandwidth product
is KQ∆t∆f .
In the proposed algorithm, we are constraining BT to values of the form MQ2∆t∆f
for which Q ≥ M , M, Q are natural numbers and ∆t∆f is an integer (condition for
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER31
orthogonal waveforms). The heuristic search for a general K can achieve TB products
of the form KQ∆t∆f for all K, Q being natural numbers. Table 2 shows that the
search space becomes prohibitively large, for increasing values of M and Q, further
reinforcing the advantages of our proposed algorithm.
Table 4.1: Values of BT for which codes can be designed with the proposed algorithm(M=4)
32 160 28850 162 29464 192 30072 196 32496 200 33898 216 350100 224 360128 242 384144 250 392150 256 400
Table 4.2: Size of search space with M and Q for K = MQ/2Q ↓ M → 4 5 6 7
6 1032 1040 1047 1055
8 1043 1053 1064 1074
10 1054 1067 1080 1092
12 1065 1080 1096 10110
4.4 Simulation results
We will now present simulation results to demonstrate the effectiveness of our pro-
posed algorithm. We have generated codes of various sizes using both methods, the
heuristic search proposed by [7], as well as our proposed algorithm. The code pa-
rameters used were M = 4, Q ∈ 6, · · · , 14 and K = MQ/2 for each value of Q.
We take the code matrices obtained from both methods for M = 4, Q = 10 and plot
the Emperical CDFs of their respective Ω functions. We compare these ECDFs with
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER32
−16 −14 −12 −10 −8 −6 −4 −2 080
82
84
86
88
90
92
94
96
98
100
Magnitude with respect to the peak (dB)
Em
piric
al C
DF
(%
)
Code from AlgorithmCode from AnnealingRandomly Generated Code
Figure 4.3: Final CDFs of codes from the two methods with M = 4, Q = 10 andK = 20
6 7 8 9 10 11 12 13 14
−13
−12
−11
−10
−9
−8
Values of Q
Mag
nitu
de w
ith r
espe
ct
to th
e pe
ak (
dB)
belo
ww
hich
95%
of t
he s
ampl
es li
e
Simulated AnnealingAlgorithm
Figure 4.4: Graph showing performance of the algorithm with respect to simulatedannealing
CHAPTER 4. WAVEFORM OPTIMIZATION FOR SMALL TARGET DOPPLER33
that obtained from a randomly generated orthogonal code matrix of the same size.
As seen from Fig. 4.3, both methods are similar in terms of performance and they
show a marked improvement over a randomly generated code.
In order to compare the performance of the two methods with increasing code size,
we have plotted the magnitude below which 95% of the samples of |Ω(τ, 0, f, f ′)| lie.
The lower the curve, the lower is the corresponding sidelobe level, which corresponds
to a better code. From Fig. 4.4, we see that the algorithm performance is either very
similar or slightly better than that of the heuristic search for all Q.
Chapter 5
Weighted Optimization
When optimizing frequency hopping codes using either fp(C) and gp(C), the aim is
to minimize the sidelobe levels over the entire range of delay-Doppler space , i.e.,
for −Q∆t < τ < Q∆t and −K∆f < ν < K∆f . However, it is possible to place
a higher importance on the minimization of sidelobe peaks in a sub-region of the
delay-Doppler space, by using a weighted cost function defined as follows
g′p(C) =
Q−1∑
k=−Q+1
K−1∑
l=−K+1
λk,l(hk,l)p (5.1)
where λk,l is the weight applied to the (k, l)th element of the hit-matrix.
5.1 Applications
We now discuss possible applications of such a weighted cost function.
Example 1
When the range of target Doppler is smaller than the overall bandwidth of the trans-
mitted signals(
max(|ν|) < K∆f)
, optimization can be performed for a subset of the
34
CHAPTER 5. WEIGHTED OPTIMIZATION 35
Doppler values using the weights
λk,l =
1 if 0 < |l| < lmax
0 otherwise
where lmax =⌈
max(|ν|)∆f
⌉
.
Example 2
Using this method, it is possible to find waveforms with their ambiguity sidelobes
constrained to particular regions of the the delay-Doppler space. Suppose we wish
to find codes with very few ambiguity sidelobes occurring in a region close to the
mainlobe. Consider the following mask
λk,l =
1 if 0 < |l| < lmax and 0 < |k| < kmax
0 otherwise
where lmax and kmax are bounds on the regions of delay-Doppler space. Thus, a
penalty is applied on sidelobes occurring inside the region of interest and the cost is
unaffected by sidelobes occurring outside this bounded region. To show the gain this
method could have, we now compare the final ambiguity functions obtained by using
gp(C) and g′p(C). In order to compare them, we consider the delay-Doppler subspace
of the corresponding ambiguity functions bounded by lmax and kmax and plot their
respective CDFs. The parameters for this simulation were M = 4, Q = 10, K = 80,
lmax = 8 and kmax = 2. Fig. 5.1 shows the gain achievable within the region of
interest.
CHAPTER 5. WEIGHTED OPTIMIZATION 36
−16 −15 −14 −13 −12 −11 −10 −9 −880
82
84
86
88
90
92
94
96
98
100
Magnitude with respect to the peak (dB)
Em
piric
al C
DF
(%
)
gp(C)
g’p(C)
Figure 5.1: CDFs of the optimized codes with and without weighting in the specifiedregion (M = 4, Q = 10, K = 80, lmax = 8, kmax = 2)
Chapter 6
Conclusion
In this thesis, we have extended the MIMO radar ambiguity function for orthogonal
frequency hopping waveforms, recently introduced in [7], to general values of Doppler
frequency. We have also presented the hit-matrix as an analysis tool for these wave-
forms. To enable the optimization of these waveforms under the large Doppler sce-
nario using simulated annealing, cost functions have been presented based on the
ambiguity function as well as the hit-matrix. The codes obtained using both cost
functions are shown to have similar performance based on their ECDF curves. The
hit-matrix based cost function has a significantly lower computational complexity, and
can be useful when searching for codes with high time-bandwidth products, where
using a ambiguity based cost function is infeasible.
Under the small Doppler scenario, existing literature proposes the use of a search
based algorithm to extract codes corresponding to a good ambiguity function. We
have made certain observations about waveform design from the ambiguity function
as well as the hit-matrix and proposed an algorithm which directly computes the
code matrix of a given size. It has been shown to perform as well as the heuristic
search proposed by Chen and Vaidyanathan [7]. The use of a weighted cost function
to optimize the ambiguity function within a sub-region of the delay-Doppler space
has also been explored. This method of “Weighted Optimization” also addresses the
problem of waveform design for intermediate Doppler. This thesis thus comprehen-
sively covers frequency hopping code design for MIMO radars for arbitrary values of
37
CHAPTER 6. CONCLUSION 38
target Doppler.
6.1 Future Work
The following are possible problems which could be explored in future work.
1. The pulse spacing vector mentioned in the system model could be optimized to
afford more control over the Doppler resolution of the waveforms.
2. The system model in this thesis puts a constraint on the orthogonality of the
waveforms. This is done so as to have equal energy radiated out in all the direc-
tions. A similar procedure of analysis could be carried out for non-orthogonal
waveforms. The effect of transmit and recieve beamforming could be analyzed
further. An algorithm built for designing non-orthogonal codes will facilitate
the radiation of energy in a desired direction. Our preliminary simulation and
analytical work in conjunction with existing literature [14] has suggested that
such beamforming should be possible by designing the covariance matrix of
the radar waveforms. This could be explored further, and algorithms could be
designed which afford more control over the spatial frequency resolution.
Appendix A
In this appendix, we provide a simple example to illustrate the logic used in arriving
at the algorithm in Section 4.2. As stated, we have M transmitter waveforms, Q
subpulses and K = MQ/2 frequencies. Consider the example of M = 2, Q = 3 and
K = 3.
A.1 Filling in frequencies
It was stated that given MQ slots and MQ/2 frequencies, the total number of hits
S =∑Q−1
k=−Q+1 hk,0 are minimized when each frequency is used exactly twice.
Let A, B, C be the three frequencies available. If each frequency is filled in twice,
one possible code matrix for an orthogonal set of waveforms is given by
C =
[
A A B
B C C
]
The above code matrix consists of frequencies of the form A, A, B, B, C, C. The
cross-hit-arrays and the hit-matrix for this code matrix (which are simply vectors of
length (2Q − 1) in the small doppler case) are shown below. The sum S is equal to
12.
39
APPENDIX A 40
k = (−2) (−1) (0) (1) (2)
hk,0(0, 0) = 0 1 3 1 0
hk,0(0, 1) = 0 0 0 0 1
hk,0(1, 0) = 1 0 0 0 0
hk,0(1, 1) = 0 1 3 1 0
hk,0 = 1 2 6 2 1
Suppose we filled in the code matrix with the frequencies A, A, A, B, C, C. One
possible code matrix for an orthogonal set of waveforms is given by
C =
[
A A A
B C C
]
The cross-hit-arrays and the hit-matrix for this code matrix are shown below.
Note that S = 14.
k = (−2) (−1) (0) (1) (2)
hk,0(0, 0) = 1 2 3 2 1
hk,0(0, 1) = 0 0 0 0 0
hk,0(1, 0) = 0 0 0 0 0
hk,0(1, 1) = 0 1 3 1 0
hk,0 = 1 3 6 3 1
Thus, we can see that S is minimized when each frequency is repeated twice.
A.2 Re-labeling and the algorithm
A comment has been made in Section 4.2 about the effect of relabeling on the cost
function. Different relabeling schemes are illustrated with an example . The algorithm
when configured with sequential frequency filling will proceed as follows (we start with
an empty code matrix).
Initial :
[
− − −
− − −
]
APPENDIX A 41
Step 1 :
[
1 − 1
− − −
]
Step 2 :
[
1 2 1
2 − −
]
Step 3 :
[
1 2 1
2 3 3
]
Note that a dash refers to an unfilled slot. The frequencies are filled in the order
1, 2, 3. If this order is randomized, the hit-matrix is unchanged, but the cost function
on average shows a slight reduction. When the frequencies are labeled in the order
2, 3, 1, one run of the algorithm proceeds as follows.
Initial :
[
− − −
− − −
]
Step 1 :
[
2 − 2
− − −
]
Step 2 :
[
2 3 2
3 − −
]
Step 3 :
[
2 3 2
3 1 1
]
While the effect of relabeling on the code matrix may not be significant for small
values of M, Q, K, this additional step can drastically alter code matrices for larger
values. Extensive simulations have shown that the performance of such random-
ordered filling in results in a slight improvement in performance of the codes.
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