###### Abstract  
The magnitude of the Discrete Fourier Transform (DFT) of a discrete-time signal has a limited frequency definition. Quadratic interpolation over the three DFT samples surrounding magnitude peaks improves the estimation of parameters (frequency and amplitude) of resolved sinusoids beyond that limit. Interpolating on a rescaled magnitude spectrum using a logarithmic scale has been shown to improve those estimates. In this article, we show how to heuristically tune a power scaling parameter to outperform linear and logarithmic scaling at an equivalent computational cost. Although this power scaling factor is computed heuristically rather than analytically, it is shown to depend in a structured way on window parameters. Invariance properties of this family of estimators are studied and the existence of a bias due to noise is shown. Comparing to two state-of-the-art estimators, we show that an optimized power scaling has a lower systematic bias and lower mean-squared-error in noisy conditions for ten out of twelve common windowing functions. acoustics; discrete Fourier transforms; frequency estimation; interpolation; signal analysis; sinusoidal modeling +
Footnote †: journal: Applied Mathematics and Physics Letters  
[PERSON]  
Center for Computer Research in Music and Acoustics (CCRMA), Department of Music, Stanford University, 660 Lomita Drive, Stanford, CA 94305-8180, USA; [EMAIL_ADDRESS]  
[PERSON]_ADDRESS]  
T.W.W. sinusoid--in practical situations the search will be repeated in each frame until all sinusoids of interest are detected and their parameters estimated [8]. Sinusoidal parameter estimation in the STFT domain is exploited heavily in multi-sinusoid analysis/synthesis methods [5; 9; 10; 11; 12]. For these systems, as well as fundamental frequency detection [13], audio coding, and music information retrieval (MIR), improving the estimation of sinusoidal parameters is very important. Early work presents interpolation methods only for signals windowed by the rectangular window or Rife-Vincent Class I windows. For the case of arbitrary windows, only a few methods have been presented. The state of the art for arbitrary windows is represented by the estimators of [PERSON] [14] and [PERSON] [15]. In [14], [PERSON] builds on previous magnitude spectrum interpolators [16; 17] derived for the Rife--incent Class I windows in order to build a first coarse estimator, whose error is compensated by an optimized polynomial remapping of the coarse estimated value to obtain a finer estimate, with the accuracy of the method controlled through the polynomial order of the remapping. In [15], [PERSON] proposes a method derived from his previous work on the rectangular window case [18], itself building on a previous complex spectrum interpolator [19], where a corrected interpolator is derived from the Taylor expansion of the signal's Discrete-Time Fourier Transform (DTFT) around the sinusoid frequency location. This work also proposes an iterative mechanism to update the correction factor using the previous estimate, further improving its accuracy. Another approach to this problem which is applicable to arbitrary windows involves parabolic interpolation over magnitude spectrum peaks [4]. In [12] it was shown that logarithmic scaling improves parabolic interpolation significantly. Inspired by this finding, we extend the basic parabolic interpolation approach by fitting a parabola to a _power-scaled_ magnitude spectrum--an approach we call the _XQIFFT_. Intuitively, this approach nonlinearly scales the shape of a window transform's main lobe so that it very closely resembles a parabola and parabolic fitting will be very accurate. This approach was introduced in [20], where a coarse brute-force approach was used to optimize a power scaling coefficient and the XQIFFT was shown to greatly outperform parabolic fitting on a linear or logarithmic scale [20] also contained a rudimentary noise analysis.