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
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
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.
