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Information Theory and Radar Waveform Design Mark R. bell September 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 summaryintroduction

A- Target Impulse Response Target Impulse response h(t) Input e(t) output v(t) Receiver filter Impulse response r(t) y s (t) introducti on Results and Algorithms Examples Summary Formulation of the problem

B- Optimal detection waveform Given : - A target impulse response h(t) - stationary additive Gaussian noise n(t) with power spectral density S nn (f). find : - waveform x(t) with total energy E x - receiver-filter impulse response r(t) such that the signal-to-noise ratio (SNR)of the receiver output y(t) is maximized at time t o. Constraints: -Restrict the waveform x(t) such that it is zero outside the interval [-T/2, T/21 -x(t) with total energy E x introducti on Results and Algorithms Examples Summary Formulation of the problem

C- Optimal estimation waveform Given : a Gaussian target ensemble with random impulse response g(t) having spectral variance σ 2 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 P nn (f). Constraints: -Restrict the waveform x(t) such that it is zero outside the interval [-T/2, T/21 -x(t) with total energy E x, confined in (one-sided) frequency to W = [f 0,f 0 +W] introducti on Results and Algorithms Examples Formulation of the problem Summary

A- Results on Detection Waveforms (theorem 1): introduction Results and Algorithms Examples summary Formulation of the problem

A- Results on Estimation Waveforms (theorem 2): a) c) The resulting maximum value I max (y(t);g(t)/x(t)) : b) A is found by solving the equation : introduction Results and Algorithms Examples summary Formulation 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 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 A e = 3 m², -Frequency interval : W = [f 0, f 0 + 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 Examples Formulation 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 introduction Results and Algorithms Examples Formulation of the problem Summary

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