LOW-COMPLEXITY DESIGN OF FREQUENCY HOPPING...

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

[email protected]

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


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


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.


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.


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


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


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)


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


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


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

]


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)


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


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.


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


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


−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.


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.


−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)


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


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.


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


Figure 4.1: Step 1 of the algorithm


Figure 4.2: Step 2 of the algorithm


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


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


−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


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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