FFT Tutorial

FFT Tutorial - FFT Tutorial Fourier series and Fourier...

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Unformatted text preview: FFT Tutorial Fourier series and Fourier transforms are powerful tools for determining the frequency content of time-varying signals. These tools allow you to move a signal from the time domain into the frequency domain. The inverse process allows you to move signals from the frequency domain back into the time domain. Representing a signal in either domain is entirely equivalent. That is, you can move from one domain to the other without losing any information. You might want to determine the frequency content of a signal for several reasons. For instance, if you knew all the frequencies a signal should contain, you could design a filter to remove unwanted frequencies (noise) from the signal. Another reason you might want to know the frequency content of a signal is that a change in frequency content may indicate a change in the system’s status. For instance, a patient’s mental state may be classified based on the predominant frequency components of the electroencephalogram (EEG). The frequency components from an electromyogram (EMG) may be used to determine if a muscle is about to undergo fatigue. Almost any text on signals and signal processing will have information on Fourier series and Fourier transforms. I will focus on trying to give an overview of Fourier theory and then show you how it might be used with some concrete examples using Matlab. Fourier theory essentially states that any periodic signal (or aperiodic signal if treated appropriately) can be represented as a weighted sum of sines and cosines. Euler’s identity gives a relationship between complex exponentials and sines and cosines; so you’ll often see Fourier theory described in terms of complex exponentials (usually more often than not). Matlab also uses the complex exponential form. Representing Fourier theory in terms of complex exponentials provides a more general and succinct mathematical expression, but personally I think it makes Fourier theory more difficult to understand. Just remember that regardless of whether the representation uses complex exponentials or sines and cosines, any periodic signal can be represented as a weighted sum of sines and cosines. Another confusing aspect of Fourier theory is that signals are treated slightly differently depending on whether they are periodic (Fourier series) or aperiodic (Fourier transforms) and whether they are continuous or discrete. Data in a computer will always be aperiodic and discrete (because it doesn’t repeat (finite in time) and because the data has been sampled). The Fourier transform for discrete data results in the frequency components being represented as a continuous variable, which is impossible to represent on a digital computer. Therefore, we will need to work with the discrete Fourier transform (DFT), which is the appropriate procedure when both the signal is discrete and the frequency components need to be discrete. The DFT can be thought of as a sampling of the Fourier transform for discrete data. thought of as a sampling of the Fourier transform for discrete data....
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FFT Tutorial - FFT Tutorial Fourier series and Fourier...

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