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In this paper a method is presented to interpolate and extrapolate unequally spaced signals that are not bandlimited. This is achieved by utilizing the Gregory-Newton quadrature formula which circumvents aliasing without bandlimiting. Also with the GregoryNewton quadrature formula, it is possible to compute the Fourier spectrum of the unequally spaced signals, which is not bandlimited. Finally, the effect of noise on the Gregory-Newton quadrature formula is observed. Specifically, the quality of interpolation, extrapolation, and the spectrum as a function of various signal-to-noise ratios is observed. This may be a useful technique for computing unaliased spectra of unequally spaced damped sinusoidal signals having a signal-tonoise ratio greater than 20 dB. transform the interpolation polynomial instead of doing DFT of the unequally spaced samples to obtain the spectrum of the observed signal.Our computer simulation results show that the Gregory-Newton interpolation polynomial, which is represented by the unequally spaced sampled values, accurately approximates the true signal. Also our computer simulation examples show that the spectrum computed by our technique coincides with the true spectrum of the signal.