dsp tips & tricks - the sliding DFT

Abstract

The Sliding DFT T he 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. Eric Jacobsen and Richard Lyons "DSP Tips and Tricks" introduces practical tips and tricks of design and implementation of signal processing algorithms so that you may be able to incorporate them into your designs. We welcome readers who enjoy reading this column to submit their contributions.

Software implementation of the Recursive Discrete Fourier Transform

“…The beneficial property of the R-DFT compared to other resonator-based solutions (like Goertzel's [2] or Jacobsen's solutions [5]) is that the arithmetical inaccuracy of the resonator's pole does not cause divergence due to feedback [6]. As the poles of system are located on the unit circle, inaccuracies due to the finite numerical representation of these values may lead to divergence or convergence to zero.…”

Single Bin Sliding Discrete Fourier Transform

“…A very effective Sb-SDFT method for sample-by-sample DFT bin computation is the so-called sliding discrete Fourier transform (SDFT) technique [4]. Starting from Eq.…”

“…where X k ½n is calculated by phase shifting the sum of the previous X k ½nÀ1 with the difference between the current and delayed input sample, x½n and x½nÀN, respectively [4,5]. The complex output of the SDFT could be rewritten as:…”

Software and Hardware Solutions for Channel Estimation based on Cyclic Golay Sequences

“…To provide continuous spectrum calculation, the Sliding DFT (S-DFT) is proposed [8], which is a filter bank based on a Lagrange structure. The S-DFT uses a moving average filter which includes complex resonators.…”

“…Furthermore, the paper focuses on the application of the Recursive DFT (R-DFT) for signal processing in digital communication, which is a less known structure for calculating the DFT of a signal sequence in a real-time manner [6], [7]. In contrast with the conventional DFT filter structure [8], which can be unstable due to the quantization errors, this method is robust against stability issues. Until now the R-DFT structure was primary used for measurement purposes.…”