Study on Single-bin Sliding DFT algorithms: Comparison, stability issues and frequency adaptivity
The standard method for spectrum analysis is the Discrete Fourier Transform(DFT), typically implemented using a Fast Fourier Transform (FFT) algorithm. However, certain applications require an on-line spectrum analysis only on a subset of M frequencies of an N-point DFT ( M< N ) . In such cases, the use of Single-bin Sliding DFT (Sb-SDFT) is preferred over the direct application of FFT. Along these lines, the most popular algorithms are the Sliding Discrete Fourier Transform (SDFT), the Sliding Goertzel Transform (SGT), the Modulated Sliding Discrete Fourier Transform (mSDFT), and the S. Douglas and J. Soh algorithm (D&S). Even though these methods seem to differ, they are derived from the conventional DFT using distinct approaches and properties. To better understand the advantages, limitations and similarities each of them have, this work thoroughly evaluates and compares the four Sb-SDFT methods. What is more, the direct application of these Sb-SDFTs may lead to inaccuracies due to spectral leakage and picket-fence effects, common pitfalls inherited by every DFT-based method. For this reason, a unified model of the Sb-SDFT methods is proposed, whose aim is to design a frequency adaptive control loop. This frequency adaptability allows to mitigate the problems associated with improper sampling frequency. By using this unified model, the election of the Sb-SDFT algorithm is independent of the controller design and all the methods are equivalent. Theoretical results are validated by simulations and a DSP implementation of the four frequency adaptive Single-bin Sliding DFT methods.