A new interpolation procedure is reported for fast Fourier transform (FFT) spectra of exponentially damped sinusoids multiplied by a family of sinα(X) windows. Using the rectangle (α0), the sine lobe (α1) and the Hanning (α2) windows we show interpolation formulas of the frequency, damping constant and amplitude determination. On the basis of these formulas, generalized interpolation formulas to which any integer value of α is applicable are derived. The formula of frequency determination is very simple and useful. An interpolated frequency is given from two ratios between just the largest three intensities on the spectral line shape in the discrete power spectrum. The interpolated frequency is used to determine the damping constant and amplitude. The characteristics of interpolation are examined for various integer values of α. Simulation results show that as a value of α increases, interpolation errors become remarkably reduced compared with those in Lorentzian interpolation corresponding to the case, α0. Thus a reasonably large value of α can be used in the generalized formulas for highly accurate interpolation.