# An Introduction to Sonar Systems Engineering

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## Book Description

**An Introduction to Sonar Systems Engineering**

Second Edition

Important topics that are fundamental to the understanding of modern-day sonar systems engineering are featured. Linear, planar, and volume array theory, including near-field and far-field beam patterns, beam steering, and array focusing, are covered. Real-world arrays such as the twin-line planar array and a linear array of triplets, which are solutions to the port/starboard (left/right) ambiguity problem associated with linear towed arrays, are examined in detail.

Detailed explanations of the fundamentals of side-looking (side-scan) and synthetic-aperture sonars are presented. Bistatic scattering with moving platforms is explored with derivations of exact solutions for the time delay, time-compression/time-expansion factor, and Doppler shift at a receiver for both the scattered and direct acoustic paths. Time-domain and frequency-domain descriptions, and the design of CW, LFM, and Doppler-invariant HFM pulses, are explained. Target detection in the presence of reverberation and noise is examined. Time-domain and frequency-domain descriptions of MFSK, MQAM, and OFDM underwater acoustic communication signals are also discussed.

Although the book is mathematically rigorous, it is written in a tutorial style. Many useful, practical design and analysis equations for both passive and active sonar systems are derived from first principles. No major steps in the derivation of important results are skipped – all assumptions and approximations are clearly stated. Particular attention is paid to the correct units for functions and parameters. Many figures, tables, examples, and practical homework problems at the end of each chapter are included to aid in the understanding of the material covered.

New to the Second Edition

- Chapter 15 Synthetic-Aperture Sonar

- Chapter 13, Section 13.3, The Rectangular-Envelope HFM Pulse

- Chapter 10, Section 10.7, Moving Platforms, was rewritten, which allowed for the elimination of Appendix 10C from the first edition

- New explanations/discussions were added to Subsections 1.2.1 and 1.3.1 in Chapter 1

- Appendix 1A was rewritten and the new Table 1A-1 was added to Chapter 1
- A solutions manual is available for adopting professors

## Table of Contents

Contents

Preface to the Second Edition xv

Preface from the First Edition xvii

1 Complex Aperture Theory – Volume Apertures – General Results 1

1.1 Coupling Transmitted and Received Electrical Signals to the

Fluid Medium 1

1.1.1 Transmit Coupling Equation 1

1.1.2 Receive Coupling Equation 4

1.2 The Near-Field Beam Pattern of a Volume Aperture 6

1.2.1 Transmit Aperture 6

Example 1.2-1 19

1.2.2 Receive Aperture 21

1.3 The Far-Field Beam Pattern of a Volume Aperture 28

1.3.1 Transmit Aperture 28

Example 1.3-1 31

1.3.2 Receive Aperture 33

Example 1.3-2 34

Problems 35

Appendix 1A 36

Appendix 1B Important Functions and Their Units at a Transmit and

Receive Volume Aperture 39

2 Complex Aperture Theory – Linear Apertures 41

2.1 The Far-Field Beam Pattern of a Linear Aperture 41

2.2 Amplitude Windows and Corresponding Far-Field Beam Patterns 44

2.2.1 The Rectangular Amplitude Window 45

2.2.2 The Triangular Amplitude Window 48

2.2.3 The Cosine Amplitude Window 52

2.2.4 The Hanning, Hamming, and Blackman Amplitude Windows 57

2.3 Beamwidth 63

Example 2.3-1 Vertical Beam Pattern 66

Example 2.3-2 Horizontal Beam Pattern 67

Example 2.3-3 68

2.4 Beam Steering 70

2.5 Beamwidth at an Arbitrary Beam-Steer Angle 73

2.6 The Near-Field Beam Pattern of a Linear Aperture 81

2.6.1 Aperture Focusing 84

2.6.2 Beam Steering and Aperture Focusing 85

Problems 86

viii Contents

Appendix 2A Transmitter and Receiver Sensitivity Functions of a

Continuous Line Transducer 90

Appendix 2B Radiation from a Linear Aperture 92

Example 2B-1 Transmitter Sensitivity Function and

Source Strength of a Continuous Line Source 95

Appendix 2C Symmetry Properties and Far-Field Beam Patterns 98

Appendix 2D Computing the Normalization Factor 100

Appendix 2E Summary of One-Dimensional Spatial Fourier

Transforms 102

3 Complex Aperture Theory – Planar Apertures 103

3.1 The Far-Field Beam Pattern of a Planar Aperture 103

3.2 The Far-Field Beam Pattern of a Rectangular Piston 106

Example 3.2-1 3-dB Beamwidths of the Vertical Far-Field

Beam Patterns of a Rectangular Piston 110

3.3 The Far-Field Beam Pattern of a Circular Piston 111

Example 3.3-1 3-dB Beamwidth of the Vertical Far-Field

Beam Pattern of a Circular Piston 117

3.4 Beam Steering 120

3.5 The Near-Field Beam Pattern of a Planar Aperture 122

3.5.1 Beam Steering and Aperture Focusing 125

Problems 126

Appendix 3A Transmitter and Receiver Sensitivity Functions of a

Planar Transducer 129

Appendix 3B Radiation from a Planar Aperture 132

Example 3B-1 Transmitter Sensitivity Function and

Source Strength of a Planar Transducer 135

Appendix 3C Computing the Normalization Factor 138

4 Time-Average Radiated Acoustic Power 141

4.1 Directivity and Directivity Index 141

4.2 The Source Level of a Directional Sound-Source 148

Problems 154

5 Side-Looking Sonar 157

5.1 Swath Width 157

5.2 Cross-Track (Slant-Range) Resolution 163

5.3 Along-Track (Azimuthal) Resolution 165

5.4 Slant-Range Ambiguity 169

5.5 Azimuthal Ambiguity 172

5.6 A Rectangular-Piston Model for a Side-Looking Sonar 175

5.7 Design and Analysis of a Side-Looking Sonar Mission 176

5.7.1 Deep Water 176

Example 5.7-1 Deep Water Mission 180

Contents ix

5.7.2 Shallow Water 183

Problems 188

6 Array Theory – Linear Arrays 191

6.1 The Far-Field Beam Pattern of a Linear Array 191

6.1.1 Even Number of Elements 192

Example 6.1-1 Two-Element Interferometer 198

Example 6.1-2 Dipole 200

Example 6.1-3 Cardioid Beam Pattern 203

6.1.2 Odd Number of Elements 207

Example 6.1-4 Axial Quadrupole 212

6.2 Common Amplitude Weights and Corresponding Far-Field

Beam Patterns 215

Example 6.2-1 Application of the Product Theorem 218

Example 6.2-2 Closed-Form Expression for the Array Factor for

Rectangular Amplitude Weights when N is Even 221

6.3 Dolph-Chebyshev Amplitude Weights 222

Example 6.3-1 228

6.4 The Phased Array – Beam Steering 231

Example 6.4-1 Steering the Null of a Dipole 233

6.5 Far-Field Beam Patterns and the Spatial Discrete Fourier Transform 235

6.5.1 Grating Lobes 239

Example 6.5-1 Spatial-Domain Sampling Theorem 243

6.6 The Near-Field Beam Pattern of a Linear Array 247

6.6.1 Beam Steering and Array Focusing 250

Example 6.6-1 Beam Steering and Focusing in the

Fresnel (Near-Field) Region 251

Problems 257

Appendix 6A Normalization Factor for the Array Factor for N Even

and Odd 261

Appendix 6B Transmitter and Receiver Sensitivity Functions of an

Omnidirectional Point-Element 264

Appendix 6C Radiation from an Omnidirectional Point-Source 266

Appendix 6D One-Dimensional Spatial FIR Filters 271

Appendix 6E Far-Field Beam Patterns and the Spatial Discrete

Fourier Transform for N Even 273

7 Array Gain 277

7.1 General Definition of Array Gain for a Linear Array 277

7.2 Acoustic Field Radiated by a Target 281

7.3 Total Output Signal from a Linear Array Due to the Target 287

7.3.1 FFT Beamforming for Linear Arrays 298

7.4 Total Output Signal from a Linear Array Due to Ambient Noise and

Receiver Noise 304

x Contents

7.5 Evaluation of the Equation for Array Gain 307

Problems 312

Appendix 7A Attenuation Coefficient of Seawater 313

Appendix 7B Fourier Transform, Fourier Series Coefficients,

Time-Average Power, and Power Spectrum via the DFT 315

8 Array Theory – Planar Arrays 319

8.1 The Far-Field Beam Pattern of a Planar Array 319

Example 8.1-1 Planar Array of Rectangular Pistons 324

Example 8.1-2 Planar Array of Circular Pistons 326

Example 8.1-3 Separable Complex Weights 328

Example 8.1-4 Tesseral Quadrupole 329

Example 8.1-5 Mainlobe in a Half-Space 333

Example 8.1-6 Concentric Circular Arrays 337

Example 8.1-7 Triplet – Cardioid Beam Pattern 340

Example 8.1-8 Linear Array in a Plane 345

8.2 The Phased Array – Beam Steering 347

Example 8.2-1 Twin-Line Planar Array 350

8.3 Far-Field Beam Patterns and the Two-Dimensional Spatial

Discrete Fourier Transform 357

8.4 The Near-Field Beam Pattern of a Planar Array 362

8.4.1 Beam Steering and Array Focusing 364

8.5 FFT Beamforming for Planar Arrays 366

Problems 375

Appendix 8A Two-Dimensional Spatial FIR Filters 378

Appendix 8B Normalization Factor for the Array Factor 379

9 Array Theory – Volume Arrays 381

9.1 The Far-Field Beam Pattern of a Cylindrical Array 381

9.1.1 The Phased Array – Beam Steering 387

Example 9.1-1 Beam Steering the Far-Field Beam Pattern

of a Stave 389

Example 9.1-2 Linear Array of Triplets 393

9.2 The Far-Field Beam Pattern of a Spherical Array 400

9.2.1 The Phased Array – Beam Steering 404

Problems 405

10 Bistatic Scattering 409

10.1 Target Strength 409

10.2 Computing the Scattering Function of an Object 421

10.3 Direct Path 424

10.4 Sonar Equations 426

10.4.1 Scattered Path 426

10.4.2 Direct Path 437

Contents xi

10.5 Broadband Solutions 442

10.5.1 Scattered Path 442

10.5.2 Direct Path 446

10.6 A Statistical Model of the Scattering Function 448

10.7 Moving Platforms 456

10.7.1 Scattered Path 456

Example 10.7-1 464

10.7.2 Direct Path 470

Problems 475

Appendix 10A Radiation from a Time-Harmonic, Omnidirectional

Point-Source 476

Appendix 10B Gradient of the Time-Independent, Free-Space, Green’s

Function 483

11 Real Bandpass Signals and Complex Envelopes 487

11.1 Definitions and Basic Relationships 487

11.1.1 Signal Energy and Time-Average Power 492

11.1.2 The Power Spectrum 495

11.1.3 Orthogonality Relationships 497

11.2 The Complex Envelope of an Amplitude-and-Angle-Modulated

Carrier 497

11.2.1 The Bandpass Sampling Theorem 503

11.2.2 Orthogonality Relationships 504

11.3 The Quadrature Demodulator 506

Problems 511

12 Target Detection in the Presence of Reverberation and Noise 515

12.1 A Binary Hypothesis-Testing Problem 515

12.2 The Signal-to-Interference Ratio 520

12.3 Probability of False Alarm and Decision Threshold 527

Example 12.3-1 Nonzero-Mean Reverberation Scattering

Function 535

12.4 Probability of Detection and Receiver Operating Characteristic

Curves 538

Example 12.4-1 Nonzero-Mean Target and Reverberation

Scattering Functions 546

Example 12.4-2 Receiver Operating Characteristic Curves –

Zero-Mean Target Scattering Function and No Reverberation

Return 549

Problems 553

Appendix 12A Mathematical Models of the Target Return and

Reverberation Return 554

Appendix 12B Derivation of the Denominator of the

Signal-to-Interference Ratio 565

xii Contents

Appendix 12C Table 12C-1 Marcum Q-Function Q(a, b) 575

Appendix 12D How to Compute Values for σ0 σ1 577

Appendix 12E 578

13 The Auto-Ambiguity Function and Signal Design 581

13.1 The Rectangular-Envelope CW Pulse 581

Example 13.1-1 Design of a Rectangular-Envelope, CW Pulse 590

13.2 The Rectangular-Envelope LFM Pulse 597

Example 13.2-1 Design of a Rectangular-Envelope, LFM Pulse 604

13.3 The Rectangular-Envelope HFM Pulse 612

13.3.1 First Equation Description 612

Example 13.3-1 Design of a Rectangular-Envelope,

HFM Pulse 618

13.3.2 Second Equation Description 624

Example 13.3-2 Alternate Design of a Rectangular-

Envelope, HFM Pulse 626

13.3.3 Doppler-Invariant Property of a HFM Pulse 629

13.3.4 Designing a HFM Pulse to Minimize Time-Delay

Estimation Error 632

Example 13.3-3 Time-Delay Estimation via

Cross-Correlation 633

Problems 636

14 Underwater Acoustic Communication Signals 639

14.1 M-ary Frequency-Shift Keying 639

14.1.1 Time-Domain Description 639

14.1.2 Frequency Spectrum and Bandwidth 641

14.1.3 Signal Energy and Time-Average Power 643

14.1.4 Orthogonality Conditions 646

14.1.5 Demodulation 647

Example 14.1-1 Gray-Encoded Quaternary FSK 650

14.2 M-ary Quadrature Amplitude Modulation 655

14.2.1 Time-Domain Description 655

Example 14.2-1 Gray-Encoded 8-QAM 659

14.2.2 Frequency Spectrum and Bandwidth 663

14.2.3 Signal Energy and Time-Average Power 665

Example 14.2-2 668

Example 14.2-3 Gray-Encoded 4-QAM 668

14.2.4 Demodulation 672

14.3 Orthogonal Frequency-Division Multiplexing 674

14.3.1 Time-Domain Description 674

14.3.2 Frequency Spectrum and Bandwidth 675

14.3.3 Signal Energy and Time-Average Power 679

Contents xiii

14.3.4 Demodulation 682

Example 14.3-1 Gray-Encoded QPSK 683

Problems 688

15 Synthetic-Aperture Sonar 691

15.1 Creating a Synthetic Aperture 691

15.2 Along-Track (Azimuthal) Resolution 697

15.3 Far-Field Beam Pattern of a Linear Synthetic Array 703

15.4 Slant-Range and Azimuthal Ambiguity 720

15.4.1 Multi-Element Synthetic-Aperture Sonar 722

Example 15.4-1 Multi-Element Synthetic-Aperture Sonar 726

15.5 Stripmap Synthetic-Aperture Sonar 726

Problems 728

Appendix 15A Rayleigh Beamwidth of the Horizontal, Far-Field

Beam Pattern of a Rectangular Piston 729

Appendix 15B 730

Appendix 15C 732

Appendix 15D 736

Bibliography

Index

## Author(s)

### Biography

Dr. Lawrence J. Ziomek is a Professor Emeritus with the Department of Electrical and Computer Engineering at the Naval Postgraduate School in Monterey, CA, where he was also a member of the Undersea Warfare Executive Committee and the Undersea Warfare Academic Group. With over 37 years of research and teaching experience, his research and teaching interests include underwater acoustics, sonar systems engineering and signal processing, Biosonar, communication theory, and signal detection and estimation theory. He received a B.E. degree in electrical engineering from Villanova University, a M.S.E.E. degree from the University of Rhode Island, and a Ph.D. degree in acoustics (underwater acoustics specialization) from The Pennsylvania State University, with a minor in electrical engineering.