An Accurate and Stable Sliding DFT Computed by a Modified CIC Filter [Tips & Tricks]

The sliding discrete Fourier transform (SDFT) is a popular algorithm used in nonparametric spectrum estimation when only a few frequency bins of an M-point discrete Fourier transform (DFT) are of interest. Although the classical SDFT algorithm described in [1] is computationally efficient, its recursive structure suffers from accumulation and rounding errors, which lead to potential instabilities or inaccurate output. Duda [2] proposed a modulated SDFT (mSDFT) algorithm, which has the property of being guaranteed stable without sacrificing accuracy, unlike previous approaches described in [1], [3], and [4]. However, all of these conventional SDFT methods presume DFT computation on a sample-by-sample basis. This is not computationally efficient when the DFT needs only to be computed every R(R > 1) samples. To address such cases when R-times downsampling is needed, Park et al. [5] proposed a hopping SDFT (HDFT) algorithm. Recently, Wang et al. [6] presented a modulated HDFT (mHDFT) algorithm, which combines the HDFT algorithm with the mSDFT idea to maintain stability and accuracy at the same time. In parallel, Park [7] updated the HDFT algorithm with its guaranteed stable modification called gSDFT, which exists only for certain M and L relationships.