Information Theory and Radar Waveform Design

Mark R. bellSeptember 1993

Sofia FENNI

Outline:

- Introduction

problem and motivation

- Formulation of the problem

- Results and Algorithms

- Exemples and comparisons

- Summary

Formulation of the problem

Results and Algorithms Examples summary

Detecting targets

theorem 2

Random target impulse response g(t)

theorem 1

deterministic target impulse response h(t)

Identifying targets

problem

introduction

A- Target Impulse Response

TargetImpulse

response h(t)

Input e(t)

output v(t)Receiver filter

Impulse response r(t)

ys(t)

introduction

Results and Algorithms

Examples SummaryFormulation of

the problem

B- Optimal detection waveform

Given : - A target impulse response h(t) - stationary additive Gaussian noise n(t) with power spectral density Snn(f).

find : - waveform x(t) with total energy Ex - receiver-filter impulse response r(t) such that the signal-to-noise ratio (SNR)of the receiver output y(t) is maximized at time to.

Constraints: -Restrict the waveform x(t) such that it is zero outside the interval [-T/2, T/21 -x(t) with total energy Ex

introduction

Results and Algorithms

Examples SummaryFormulation of

the problem

C- Optimal estimation waveform

Given : a Gaussian target ensemble with random impulse response g(t) having spectral variance σ2

G(f)

find : - waveform x(t) that maximize the mutual information I(y(t);g(t)/x(t)) in additive Gaussian noise with one-sided power spectral density Pnn(f).

Constraints: -Restrict the waveform x(t) such that it is zero outside the interval [-T/2, T/21 -x(t) with total energy Ex , confined in (one-sided) frequency to W = [f0,f0+W]

introduction

Results and Algorithms

ExamplesFormulation of the problem Summary

A- Results on Detection Waveforms (theorem 1):

introduction Results and Algorithms Examples summar

yFormulation of

the problem

A- Results on Estimation Waveforms (theorem 2):

a)

c) The resulting maximum value Imax(y(t);g(t)/x(t)) :

b) A is found by solving the equation :

introduction Results and Algorithms Examples summar

yFormulation of

the problem

Example 1: Detection Waveforms the effect of various waveforms with identical energy on the output SNR:

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 1: Detection Waveforms

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 2: Detection Waveforms

- Problem: detecting a perfectly conducting metal sphere of radius a

-use two waveforms, both with unit energy :

1-pulse sinusoid waveform with its associated matched filter.

2- optimal waveform/receiver-filter pair

x(t) =βBT/2(t) cos(2πf0t)

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 2: Detection Waveforms

Comparison of the output SNR for the two resulting waveforms:

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 3: estimation Waveforms

examine the characteristics of the optimal transmitted signal’s spectrum and the amount of information obtained.-target at a range of 10 km. -Monostatic radar with an effective area Ae = 3 m²,

-Frequency interval : W = [f0, f0 + W] = [0.995 GHz, 1.005 GHz].constraints: -average power ranging from 1 W to 1000 W .

-observation times ranging from 10 μs to 100 ms.

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 3: estimation Waveforms

introduction Results and Algorithms

summary

Formulation of the problem Examples

Example 3: estimation Waveforms

introduction Results and Algorithms

summary

Formulation of the problem Examples

Comparison of Detection and Estimation Waveforms:

introduction Results and Algorithms

summary

Formulation of the problem Examples

Comparison of Detection and Estimation Waveforms:

-optimal target detection: put as much energy as possible into the mode of the target that gave the largest response, with respect to the noise.

- optimal estimation distributes the available energy in order to maximize the information obtained about the target.

introduction Results and Algorithms

summary

Formulation of the problem Examples

Idea: exploiting resonance phenomena to provide max SNR.

The maximum signal-to-noise ratio occurs when the mode of the target with the largest eigenvalue is excited by the transmitted waveform.

the shape of a radar signal, and not just its energy alone, can have a significant effect on extended target detection performance.

other scattering modes of the target may be useful for identifying or characterizing the target.

introduction Results and Algorithms

ExamplesFormulation of the problem Summary

Theorem 2 describes how to distribute the energy in such a way that the mutual information between the target ensemble and the received waveform is maximized.

the greater the mutual information, the better we would expect the radar’s classification

introductionResults and Algorithms ExamplesFormulation of

the problem Summary

#Any_ questions_?