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Unformatted text preview: 17 DYNAMICS CALCULATIONS USING THE TIME AND FREQUENCY DOMAINS 51 17 Dynamics Calculations Using the Time and Fre quency Domains 1. Don’t turn anything in on this part. The FFT (Fast Fourier Transform) is a standard method for decomposition of a signal into its harmonic components. You need to know a few specific items about it: (a) The length of the input signal x in the time domain  is N . The FFT itself doesn’t care about what is the time step between x n and x n +1 , although the spacing must be uniform; below, we’ll call it Δ t . Hence x 1 occurs at time 0, x 2 is at time Δ t , and so on, up to x N taken at time ( N 1)Δ t . (b) The formal definition of the transform and its inverse in MATLAB is X k = N X n =1 x n e i 2 π ( k 1)( n 1) /N , 1 ≤ k ≤ N ; x n = 1 N N X k =1 where i = √ 1. So the FFT takes a uniformlyspaced time signal x and makes a uniformlyspaced frequency signal X , of the same length. It is like a Fourier integral because of the exponential: e iz = cos z + i sin z . Furthermore, you notice that the FFT (the transform that makes X ) is a summation between the time domain signal x and an exponential with imaginary argument, which has unity magnitude. Thus, the size of the elements in X are roughly the size of x , multiplied by N . The inverse FFT has the factor 1 /N out front so as to cancel this effect out; ifft(fft(x)) = x; . Because of this arrangement, however, you can’t take the FFT of two signals, multiply them (to effect a convolution), take the inverse FFT, and expect to get the right scaling  each FFT induces a scaling of N, but the IFFT only corrects for one of them. Remembering this point will save you a lot of trouble in using the FFT in MATLAB! You can see this scaling for example with x = sin(0:0.1:100) ; X=fft(x);plot(abs(X)); . Here, you plot the absolute value ( abs() ) which is synonymous with magnitude, because X is complex. There is another scaling factor to notice, but this is easier; because the Fourier integral involves both negative and positive frequencies, it puts half the area on each side....
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 Spring '05
 GeorgeKocur
 Digital Signal Processing, LTI system theory, Fast Fourier transform

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