Sliding Spectrum Analysis

Sliding spectrum analysis

Emil Frenski, Member, IEEE

Vencislav Petrov, Dimitar Manolev

South-West University “Neofit Rilski”, Blagoevgrad, Bulgaria

Abstract: The standard method for spectrum analysis in DSP is the Discrete Fourier transform (DFT), typically implemented using a Fast Fourier transform (FFT) algorithm. The reconstruction of the time-domain signal is then performed by the IFFT (Inverse Fast Fourier transform) algorithm. The FFT calculates the spectral components in a window, on a block-by-block basis. If that window is move by one sample, it is obvious that most of the information will remain the same. This work describes sliding spectrum analysis techniques whose spectral bin output rates are equal to the input data rate, on a sample-by-sample basis. Here we review the basic ideas and describe Sliding Discrete Fourier transform implementation. A MATLAB program was written using this technique and validated. Future work includes computational cost analysis, synthesis issues and a viability study regarding the use of the algorithms for computing SDFT with high performance on graphics processing units (GPUs).

Keywords: Sliding Discrete Fourier transform (SDFT), Fast Fourier transform (FFT).

1. INTRODUCTION

The discrete Fourier transform (DFT), for transforming the time signal into its frequency domain counterpart, Is a popular signal analysis tool in science and engineering. Frequency and time are orthogonal. But some signals do have frequency components that change with time (for example speech having pitch that rises and falls over time). The solution to such problems is the sliding DFT algorithm [2]. With this method, the computational complexity for calculating DFT of each window is [pic 1] as compared to [pic 2] for standard DFT computation and [pic 3] for FFT.

We are proposing an approach to the implementation of a discrete sliding DFT where the spectrum of the signal is estimated by realizing the DFT through a bank of IIR Filters. The technique has all the advantages inherent to the IIR filters, including the capacity of doing sample by sample processing, and can estimate the spectrum at the exact frequencies of interest.

First we review the DFT approach followed by fast Fourier transform (FFT). The sliding DFT algorithm is then derived and proposed.

1. DFT and FFT

The discrete Fourier transform (DFT) plays an important role in the analysis and implementation of discrete-time signal-processing systems because its properties. The DFT is identical to samples of the Fourier transform. Computation of the N-point DFT corresponds to the computation of N samples of the Fourier transform at N equally spaced frequencies[pic 4] (bins^[1]), i.e. at N points on the unit circle in the z-plane. The DFT of finite – length sequence is [1]