A.~ n = HARMOMC NUMBER 255 Fig. 3. Magnitude of spectral components of (12) versus harmonic number using: (a) Sampled true Fourier transform, times 1 / T , (b) Original EF-FFT method [l], (c) De-aliased version of (b), (d) the same as (c) except that the spectral components of a ramp between t , -A t and I , has been subtracted.ponents of a ramp that rises to value U, between n = 199 and n = 200,where A t = T / N . The components to be subtracted are the sampled true Fourier transform of (13) times 1 / T , orThe result is shown in Fig. 3(d) and the maximum deviation of curve (d) from (a) is 0.43 dB. This remaining error is due to the piecewise linear approximation model not being perfect. IV. DISCUSSION A N D CONCLUSIONSIt has been shown that frequency aliasing errors introduced during the FFT analysis of step-like signals can be substantially decreased by a simple de-aliasing procedure. For example, the results shown in Fig. 2(c) were obtained with a de-aliased version of the original EF-FFT method, namely by postmultiplication of the original EF-FFT results with sinc' ( n / N ) . Of course de-aliasing with (9) does not mean removing all alias error because to do so would mean that the function is precisely known between sample points.Rather, use of (9) implies that a piecewise linear approximation is acceptable between data points. To obtain even more accurate final spectra it would be necessary to increase the sampling rate or to use a quadratic or higher order curve-fitting procedure in the dealiasing technique [ 5 ] , [6], [9].The attractive features of the aliasing error reduction method introduced here, compared to increasing the sampling rate, are that data reacquisition is not required, computer requirements are small, and the spectra are of high accuracy up to the Nyquist frequency. The applicability is limited to functions that can be modeled with linear transitions between data points, as opposed to step transitions unless the precise timing and shape of the step transitions are known. Since most data sets are comprised of samples from slowly varying analog signals, the de-aliasing procedure provides enhanced spectral accuracy with but minor additional mathematical complexity. 40, NO. 4. AUGUST 1991 775 REFERENCES [I] G. D. Cormack and J . 0. Binder, "The extended function fast Fourier transform (EF-FFT)," , "Enhanced spectral resolution FFT for step-like functions," IEEE Trans. Insrrum. Meas., vol. 40, pp. 34-36, Feb. 1991. J . Schutte, "New fast Fourier transform algorithm for linear system analysis applied in molecular beam relaxation spectroscopy," Rev. Sci. Insrrum. 52(3), pp. 400-404, Mar. 1981. S . Makinen, "New algorithm for the calculation of the Fourier transform of discrete signals," Rev. Sci. Instrum. 53(5), pp. 627-630, May 1982. S . Sorella and S . K. Ghosh, "Improved method for the discrete Fourier transform," Rev. Sei. Instrum. 55(12), pp. 1348-1352, August. 1984. M. Froeyen and L. Hellemans, "Improved algorithm for the discrete Fourier transform,'' Rev. Sci. Insrrum. 56( 12), pp. 23...