Fast Fourier Transforms for Fast Fourier Transforms

Fast Fourier Transforms for Fast Fourier Transforms


Signal Types—Deterministic Biosignals (with or without Noise). Fast Fourier transform (FFT) is commonly used in analyzing the spectral content of any deterministic biosignal (with or without
noise). I will discuss the issue of estimating the spectrum of the signal under noisy conditions in the following subsections. Discrete Fourier transform (DFT) allows the decomposition of discrete time signals into sinusoidal components whose frequencies are multiples of a fundamental frequency. The amplitudes and phases of the sinusoidal components can be estimated using the DFT and is represented mathematically as


for a given biosignal x(n) whose sampling period is T with N number of total samples (NT is therefore the total duration of the signal segment). The spectrum X(k) is estimated at multiples of
fs /N, where fs is the sampling frequency. Fast Fourier transform (FFT) is an elegant numerical approach for quick computation of the DFT. However, users need to understand the resolution limitations (in relation to the signal length) and the effects of signal windowing on the accuracy of the estimated spectrum. In general, FFT does not work well for short-duration signals. The spectral resolution (or the spacing between ordinates of successive points in the spectrum) is directly proportional to the ratio of sampling frequency fs of the signal to the total number of points N in the signal segment. Therefore, if we desire a resolution of approximately 1 Hz in the spectrum, then we need to use at least 1-second duration of the signal  (number of points in 1-second segment = fs) before we can compute the spectral estimates using FFT. Direct application of FFT on the signal implicitly assumes that a rectangular window whose value is unity over the duration of the signal and zero at all other times multiplies the signal. However, multiplication in the time domain results in a convolution operation in the frequency domain between the spectrum of the rectangular window and the original spectrum of the signal x(n). Since the spectrum of the rectangular window is a so-called Sinc function consisting of decaying sinusoidal ripples, the convolved spectrum X(f ) can be a distorted version of the original spectrum of x(n). Specifically, spectral contents from one frequency component (usually the dominating spectral peak) tend to leak into neighboring frequency components due to the convolution operation. Therefore, it is often advisable to window the signals (particularly when one expects a dominating spectral peak adjacent to one with lower amplitude in the signal spectrum). Several standard windows, such as Hamming, Hanning, Kaiser, Blackman-Tukey, etc., are available in any modern signal-processing toolbox, each with its own advantages and disadvantages

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