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A NUMERICAL SOLUTION OF THE INVERSE CONVOLUTION PROBLEM
C.S. Jr. Williams, R.A. Hessemer
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Abstract
The convolution of the response of a linear system to unit impulse with an input time function of amplitude gives the output time function of amplitude. Frequently, the output function and response function are known and it is desired to find the input. The method described does so. The convolution integral is approximated by a sum and this result is fitted by least-squares to the known output. This leads to a set of equations linear in the ordinates of the unknown input function. It is of especial interest that the matrix of the coefficients of the linear equations is symmetrical. This means that the equations may be represented by a linear network and a solution readily obtained. Further, the coefficients along any diagonal (downward to the right) are the same; hence, the number of coefficients that must ae evaluated is small. The coefficients, as well as the right-hand constant terms, represent convolutions themselves. (auth)