Fourier_Transforms_DFTs_FFTs - Fourier Transforms, DFTs,...

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Fourier Transforms, DFTs, and FFTs Author: John M. Cimbala, Penn State University Latest revision: 21 February 2008 Introduction In spectral analysis , our goal is to determine the frequency content of a signal . For analog signals, we use Fourier series , as discussed in a previous learning module. For digital signals, we use discrete Fourier transforms , as discussed in this learning module. Fourier transform (FT) The Fourier transform ( FT ) is a generalization of the Fourier series . Instead of the sines and cosines in a Fourier series, the Fourier transform uses exponentials and complex numbers ; instead of the summations in a Fourier series, the Fourier transform uses integrals . For a signal or function f ( t ), the Fourier transform is defined as () ( ) it fte d t ω −∞ = F , and the inverse Fourier transform is defined as () 1 2 tF e d f π −∞ = , where i is the unity imaginary number , defined as the square root of 1, 1 i =− , and is the range of angular frequencies associated with the signal – the frequency content of the signal. When working with time and the signal f ( t ), we are working in the time domain , and the variables are real . When working with angular frequency and the Fourier transform F ( ), we are working in the frequency domain , and F ( ) is complex . The FT is an analog tool – it is used for analyzing the frequency content of continuous signals. Discrete Fourier transform (DFT) We define the discrete Fourier transform ( DFT ) – a Fourier transform for a discrete (digital) signal. The DFT is a digital tool – it is used for analyzing the frequency content of discrete signals. The discrete Fourier transform is defined as ( ) ( ) 1 2 0 N ik f n t n Fkf fnte ΔΔ = Δ= Δ for k = 0, 1, 2, . .., N 1. Note that summation has replaced integration since discrete rather than continuous data are being examined. In the time domain , the relevant variables are: o N = total number of discrete data points taken. o T = total sampling time. o Δ t = time between data points, / tTN Δ = . o f s = sampling frequency, 1/ / s f tNT = . Note that integers n and k in the above definition of the DFT have values from 0 to N 1, not from 1 to N . For example, consider data sampled as in the plot to the right. Data are sampled discretely at a sampling frequency of 0.5 Hz. Starting at time t = 0, N = 4 data points are taken. Here, T = 8 s, Δ t = T / N = (8 s)/4 = 2 s, and f s = 1/ Δ t = 0.5 Hz. The discrete data used to calculate the DFT are those at t = 0, 2, 4, and 6 seconds. The data point at n = N (at t = T = 8 s) is not used . f ( t ) t (s) t = 0 n = 0 4 2 Δ t Data point not used 6 3 8 4 2 1 Data points used -3 -2 -1 0 1 2 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 t (s) f ( t ) Actual period of the signal T Consider a periodic signal that repeats itself every 0.1 s, as plotted.
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This note was uploaded on 07/23/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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Fourier_Transforms_DFTs_FFTs - Fourier Transforms, DFTs,...

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