?ASONIC IMAGING 1, 121-135 (1979) IMPROVEMENT OF RANGE RESOLUTION BY SPECTRAL EXTRAPOLATION A. Papoulis’ and C. Chamzas Polytechnic Institute of New York Department of Electrical Engineering Farmingdale, New York 11735 Under various simplifying assumptions, the reflected signal y(t) in the interrogation of a substance by an ultrasonic wave is a convolution of the transmitted signal x(t) with a function h(t) that is related to the reflection coefficient of the medium in the direction of propagation. The function h(t) can, in principle, be determined by deconvolution. However, since the band B of the spectrum X(w) of x(t) is finite, the frequency components of h(t) outside B cannot be found reliably. In this paper, a method is presented for extrapolating X(w) beyond B. The resulting increase in resolution is limited only by the level of noise. The method is particularly effective if h(t) is a sum of impulses. Key words: Deconvolution; diagnostics; echoes; extrapolation; resolution; ultrasonic. 1. Intr oduc ti on Ultrasonic waves are used in metallurgy, in medicine, and in other areas to determine the structure of a medium. The medium is interrogated by a narrow beam (fig. 1) and the reflected signal y(t) is used to determine various properties of the medium, in particular, the location of its surface of discontinuity. Under various simplifying assumptions, the signal y(t) can be expressed as a convolution integral: y(t) = j-=x(t-T)h(T)d’r -m where x(t) is the transmitted signal (fig. 2a) and h(t) is a function related to the reflection coefficients of the medium in the direc- tion of propagation. The variable t is proportional to the distance along the beam. The assumptions leading to eq. (1) and the rela- tionship between h(t) and the parameters of the medium will not be considered here. If the medium consists of homogeneous layers, then h(t) is a sum of impulses as in figure 2b: 1 Address all correspondence to A. Papoulis.