IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 5, MAY 2011 1579 A Fast, Simplified Frequency-Domain Interpolation Method for the Evaluation of the Frequency and Amplitude of Spectral Components Alessandro Ferrero, Fellow, IEEE, Simona Salicone, Senior Member, IEEE, and Sergio Toscani, Student Member, IEEE Abstract—The evaluation of the spectral components of a signal by means of discrete Fourier transform or fast Fourier transform algorithms is subject to leakage errors whenever the sampling frequency is not coherent with the signal frequency. Smoothing windows are used to mitigate these errors, and interpolation methods are applied in the frequency domain to reduce them further on. However, if cosine windows are employed, closed-form formulas for the evaluation of harmonic frequencies can be used only with the Rife–Vincent class I windows, while approximated formulas have to be used in other cases. In both cases, a high computation burden is required. This paper proposes a fast inter- polation method, independent of the window type and order, based on suitable lookup tables. Experimental results are reported, and the accuracy is discussed, proving that the method provides results as good as those obtained with other methods, without requiring the same high computation burden. Index Terms—Cosine windows, Fast Fourier Transform (FFT), frequency-domain interpolation, spectral analysis. I. I NTRODUCTION D ISCRETE FOURIER TRANSFORM (DFT) and fast Fourier transform (FFT) algorithms are widely employed to evaluate the frequency-domain components of signals in a variety of measurement applications. Although they are very effective and, in the case of the FFT, fast enough to be applied in real-time applications, they may become very inaccurate when the coherent sampling conditions are left because of the spectral leakage phenomena originated by the lack of coherence [1]. Suitable smoothing windows, particularly the so-called co- sine windows, originally proposed by Rife and Vincent [2], are considered the easiest-to-apply and most effective tool to mitigate leakage errors. They generally attain good accuracy in the evaluation of the amplitude of the frequency-domain com- ponents, but they cannot yet improve the accuracy in locating each spectral components on the frequency axis: The frequency resolution of the windowed DFT or FFT algorithm remains the Manuscript received June 18, 2010; revised September 10, 2010; accepted October 4, 2010. Date of publication November 22, 2010; date of current version April 6, 2011. The Associate Editor coordinating the review process for this paper was Dr. Wendy Van Moer. The authors are with the Dipartimento di Elettrotecnica, Politecnico di Milano, 20133 Milan, Italy (e-mail: alessandro.ferrero@polimi.it; simona. salicone@polimi.it; sergio.toscani@mail.polimi.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2010.2090051 same as the one of the nonwindowed algorithm, and the error in locating the spectral components might be as high as half the frequency resolution [1]. A better accuracy in frequency evaluation, and hence a fur- ther reduction of the leakage errors, can be obtained by applying suitable interpolation algorithms in the frequency domain, as initially proved by [2], [3]. These algorithms are very effective either when a single-tone signal is considered, or, in the more general case of multitone signals, if the employed window features a high attenuation of its sidelobes, thus minimizing the spectral interference or long- range leakage error [3]. This result can be best obtained if a high-order cosine window is used. Unfortunately, as it will be recalled in the next sections, the strict mathematical application of the frequency interpolation algorithm requires one to find the solution of a mathematical relationship; solution that can be found in strict closed form only with the class I Rife–Vincent windows [2]–[7]. Approximate solutions are available in the literature [7]–[9] for other high-order windows and different measurement ap- plications, although they are still generally based on complex formula that require a relatively high computation burden that might prevent their use in real-time applications, particularly when a wide bandwidth is required. This paper considers and discusses a much simpler method, originally proposed in [10], based on lookup tables that are preliminarily evaluated and stored for each employed window and that provide, through a simple linear interpolation, the correction factors necessary to evaluate the spectral component frequency and amplitude. After having described the proposed method, this paper dis- cusses its contribution to uncertainty and provides experimental results that confirm its effectiveness. II. THEORETICAL CONSIDERATIONS Let us consider a cosine weighing window of order L w(n)= L-1  l=0 c l · cos  2πln M  (1) where M is the total number of samples. 0018-9456/$26.00 © 2010 IEEE