The existence of a real, causal time function fN(t) whose auto¬correlation function (ACF) is f({r|) is proved. Such functions are called Autocorrelation-Invariant (A-l). Several construction tech¬niques and examples are given.nIt is shown that A-I functions have several properties of interest to the communications engineer. These include:n(a) %(t) is the right half of an ACF, providing a simple sufficiency test for a specified function to be an ACF.n(b) Associated with fN(t) and generated in the manner of the Laguerre functions, is an orthogonal set whose members are useful as basic functions in the design of orthogonal signalling waveforms with specified ACF's.n(c) fN(t+r) is the degenerate kernel of an integral equation whose N+l eigenvalues are real and unity in magnitude, and whose eigenfunctions span N, a finite -dimensional subspace of Hilbert space. This property allows several results in the characterization of time func¬tions in N.nThe Laguerre and Legendre functions of the first kind, two sets of A-I functions, are defined and discussed. A curious orthogonality property of any member of the former set, under time translations, is presented, giving rise to a conjecture concerning all A-I functions.
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