The Sliding DFT

The standard method for spectrum analysis in digital signal processing (DSP) is the discrete Fourier transform (DFT), typically implemented using a fast Fourier transform (FFT) algorithm. However, there are applications that require spectrum analysis only over a subset of the N center frequencies of an N-point DFT. A popular, as well as efficient, technique for computing sparse DFT results is the Goertzel algorithm that computes a single complex DFT spectral bin value for every N input time samples. This article describes a sliding DFT process whose spectral bin output rate is equal to the input data rate, on a sample-by-sample basis, with the advantage that it requires fewer computations than the Goertzel algorithm for real-time spectral analysis. In applications where a new DFT output spectrum is desired every sample, or every few samples, the sliding DFT is computationally simpler than the traditional radix-2 FFT. We’ll start our sliding DFT discussion by providing a brief review of the Goertzel algorithm and use its behavior as a yardstick to evaluate the performance of the sliding DFT technique. Following that, we will examine stability issues regarding the sliding DFT implementation as well as review the process of frequency-domain convolution to accomplish time-domain windowing. Finally, a modified sliding DFT structure is proposed that provides improved computational efficiency.