^{1,a)}and Patrick J. Loughlin

^{2}

### Abstract

In this paper, the task of model-based transmit signal design for optimizing detection is considered. Building on past work that designs the spectral magnitude for optimizing detection, two methods for synthesizing minimum duration signals with this spectral magnitude are developed. The methods are applied to the design of signals that are optimal for detecting elastic objects in the presence of additive noise and self-noise. Elastic objects are modeled as linear time-invariant systems with known impulse responses, while additive noise (e.g., ocean noise or receiver noise) and acoustic self-noise (e.g., reverberation or clutter) are modeled as stationary Gaussian random processes with known power spectral densities. The first approach finds the waveform that preserves the optimal spectral magnitude while achieving the minimum temporal duration. The second approach yields a finite-length time-domain sequence by maximizing temporal energy concentration, subject to the constraint that the spectral magnitude is close (in a least-squares sense) to the optimal spectral magnitude. The two approaches are then connected analytically, showing the former is a limiting case of the latter. Simulation examples that illustrate the theory are accompanied by discussions that address practical applicability and how one might satisfy the need for target and environmental models in the real-world.

The authors wish to thank the anonymous reviewers for their thoughtful and constructive comments. This research was supported by the Office of Naval Research (Grant No. N00014-09-1-0448).

I. INTRODUCTION

II. BACKGROUND: OPTIMIZING SPECTRAL MAGNITUDE

A. The optimal spectral density

B. Example: Elastic target in colored noise and white reverberation

III. OPTIMIZING SPECTRAL PHASE: THE MINIMUM DURATION OPTIMAL TRANSMIT SIGNAL

IV. OPTIMIZING TEMPORAL CONCENTRATION

A. The concentration problem

B. The modified Slepian concentration problem

1. Mathematical formulation: Discrete-time/continuous-frequency

2. Mathematical formulation: Discrete-time/discrete-frequency

3. Mathematical formulation: Nonlinear program

C. Connection to the minimum duration solution

D. Examples

V. DISCUSSION

VI. CONCLUSION

### Key Topics

- Elasticity
- 12.0
- Acoustic noise
- 8.0
- Eigenvalues
- 8.0
- Active sonar systems
- 7.0
- Sonar
- 7.0

## Figures

Model of received signal , and associated detector . is the transmitted signal; is the impulse response of a random LTI filter that models channel interference induced by the transmit signal; is the LTI impulse response of the object to be detected; and represents ambient noise.

Model of received signal , and associated detector . is the transmitted signal; is the impulse response of a random LTI filter that models channel interference induced by the transmit signal; is the LTI impulse response of the object to be detected; and represents ambient noise.

Diagram of spherical target used in examples.

Diagram of spherical target used in examples.

(Color online) Example: Elastic target in colored noise and white reverberation. (Top) Spectrum of optimal signal based on the Resonance Scattering Theory (RST) model of the sphere (solid line with large dots), spectrum of optimal signal assuming point target (dashed-dotted line) LFM (dashed line), target spectrum (small dashed line with large dots), and additive noise spectrum (solid line), (bottom) ROC curves for optimal design based on the RST model of the sphere (solid line with large dots), optimal design based on point target assumption (dashed-dotted line), and LFM (dashed line).

(Color online) Example: Elastic target in colored noise and white reverberation. (Top) Spectrum of optimal signal based on the Resonance Scattering Theory (RST) model of the sphere (solid line with large dots), spectrum of optimal signal assuming point target (dashed-dotted line) LFM (dashed line), target spectrum (small dashed line with large dots), and additive noise spectrum (solid line), (bottom) ROC curves for optimal design based on the RST model of the sphere (solid line with large dots), optimal design based on point target assumption (dashed-dotted line), and LFM (dashed line).

(Color online) Minimum duration solution yielding narrow pulse width and high peak energy. (Main panel) Spectrogram with 20 dB dynamic range, (left panel) magnitude spectrum of optimal signal , (bottom panel) time series of optimal signal .

(Color online) Minimum duration solution yielding narrow pulse width and high peak energy. (Main panel) Spectrogram with 20 dB dynamic range, (left panel) magnitude spectrum of optimal signal , (bottom panel) time series of optimal signal .

(Color online) Block diagram showing the relationship between signal design approaches.

(Color online) Block diagram showing the relationship between signal design approaches.

(Color online) Example 1: Signal design results for . (Upper left) Time-domain signal from modified formulation (dashed-dotted line) and minimum duration solution (solid line). (Upper right) Optimal spectrum (solid line) and spectrum from modified solution (dashed-dotted line). (Bottom) ROC curves for optimal spectrum (solid line), spectrum from modified solution (dashed-dotted line) and LFM spectrum (dashed line). Note that in each of these three subfigures the minimum duration solution overlays closely with the modified Slepian solution since epsilon is small.

(Color online) Example 1: Signal design results for . (Upper left) Time-domain signal from modified formulation (dashed-dotted line) and minimum duration solution (solid line). (Upper right) Optimal spectrum (solid line) and spectrum from modified solution (dashed-dotted line). (Bottom) ROC curves for optimal spectrum (solid line), spectrum from modified solution (dashed-dotted line) and LFM spectrum (dashed line). Note that in each of these three subfigures the minimum duration solution overlays closely with the modified Slepian solution since epsilon is small.

(Color online) Example 2: Signal design results for . (Upper left) Time-domain signal from modified formulation (dashed-dotted line) and minimum duration solution (solid line). (Upper right) Optimal spectrum (solid line) and spectrum from modified solution (dashed-dotted line). (Bottom) ROC curves for optimal spectrum (solid line), spectrum from modified solution (dashed-dotted line) and LFM spectrum (dashed line). Note that in each of these three subfigures the minimum duration solution *does not* overlay closely with the modified Slepian solution since epsilon is *large*.

(Color online) Example 2: Signal design results for . (Upper left) Time-domain signal from modified formulation (dashed-dotted line) and minimum duration solution (solid line). (Upper right) Optimal spectrum (solid line) and spectrum from modified solution (dashed-dotted line). (Bottom) ROC curves for optimal spectrum (solid line), spectrum from modified solution (dashed-dotted line) and LFM spectrum (dashed line). Note that in each of these three subfigures the minimum duration solution *does not* overlay closely with the modified Slepian solution since epsilon is *large*.

(Color online) Magnitude spectrum of target (dashed-dotted line) for each shell thickness 1 cm (top), 5 cm (middle), and 10 cm (bottom). Solution to Eq. (39) (solid line) is given in each of the three panes, while the optimal magnitude spectrum based on only the 5 cm shell (dashed line) is given in the center pane.

(Color online) Magnitude spectrum of target (dashed-dotted line) for each shell thickness 1 cm (top), 5 cm (middle), and 10 cm (bottom). Solution to Eq. (39) (solid line) is given in each of the three panes, while the optimal magnitude spectrum based on only the 5 cm shell (dashed line) is given in the center pane.

(Color online) ROC curves assuming 1 cm shell (top), 5 cm shell (middle), or 10 cm shell (bottom) is insonified by optimal design based on 5 cm shell (dashed line), the solution to Eq. (39) , or a LFM signal with bandwidth . From these ROC curves we see that the performance loss that results from incomplete knowledge of the target is not as severe when using the solution to Eq. (39) .

(Color online) ROC curves assuming 1 cm shell (top), 5 cm shell (middle), or 10 cm shell (bottom) is insonified by optimal design based on 5 cm shell (dashed line), the solution to Eq. (39) , or a LFM signal with bandwidth . From these ROC curves we see that the performance loss that results from incomplete knowledge of the target is not as severe when using the solution to Eq. (39) .

## Tables

Material properties of shell and surrounding environment.

Material properties of shell and surrounding environment.

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