Two methods to deconvolve: L1-method using simplex algorithm and L2-method using least squares and parameter

If r(t) is the linear scattering response of an object to an excitation waveform e(t) , then r(t) = (e \ast h) (t) . One would like to deconvolve and solve for h(t) , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method rep...

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Bibliographic Details
Published in:IEEE transactions on antennas and propagation Vol. 32; no. 3; pp. 219 - 225
Main Author: Drachman, B.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.03.1984
Institute of Electrical and Electronics Engineers
Subjects:
Applied sciences
Convolution
Deconvolution
Exact sciences and technology
Filtering
Frequency domain analysis
Information, signal and communications theory
Least squares methods
Linear programming
Mathematics
Scattering parameters
Signal processing
Telecommunications and information theory
Vectors
Deconvolution
Least squares method
Simplex method
Wave scattering
Information processing
ISSN:0018-926X, 1558-2221
Online Access:Get full text
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Summary:If r(t) is the linear scattering response of an object to an excitation waveform e(t) , then r(t) = (e \ast h) (t) . One would like to deconvolve and solve for h(t) , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix form E \cdot H = R by the following problem. Minimize \|h_{1}\|+ \ldots + \|h_{n}\| subject to R - \lambda \leq E \cdot H \leq R + \lambda where \lambda is a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimize \parallel E \cdot H - R \parallel^{2} +\lambda \parallel H \parallel^{2} where again \lambda is chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data.
ISSN:0018-926X
1558-2221
DOI:10.1109/TAP.1984.1143312