Composite Interpolated Fast Fourier Transform With the Hanning Window

The composite interpolated fast Fourier transform (IpFFT) with the Hanning window is investigated to decrease the estimation variance. This form of composite IpFFT makes use of four consecutive spectral lines around a spectral peak. These four consecutive lines render three estimators using the complex spectrum-based interpolation. The three estimators can be averaged to decrease the variance, and their weighting coefficients are established to produce the minimum variance. Theoretical analysis shows that this composite IpFFT has its best performance under noncoherent sampling conditions, achieving a minimum frequency variance of 1.8 folds of the Crame¿r-Rao bound (CRB). The worst case occurs in the situation of coherent sampling, with a maximum variance of 2.6 folds of the CRB. Theoretical analysis was validated by numerical experiments on three types of noise-contaminated signals, cisoids, real-valued sine waves, and double-tone complex-valued signals. The cisoid case shows that the empirical variance very well matches the theoretical expression. Insofar as simulation is concerned (signal-to-noise ratio down to -2 dB), the empirical variance deviating from the theoretical expression is no more than 25%. All three experiments show that the empirical variance of the IpFFT is almost independent of the initial phase, whereas the bias is the opposite.