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# lecnotes7 - 7 Fourier transforms Except in special...

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7 Fourier transforms Except in special, idealized cases (such as the linear pendulum), the precise oscillatory nature of an observed time series x ( t ) may not be identified from x ( t ) alone. We may ask How well-defined is the the dominant frequency of oscillation? How many frequencies of oscillation are present? What are the relative contributions of all frequencies? The analytic tool for answering these and myriad related questions is the Fourier transform . 7.1 Continuous Fourier transform We first state the Fourier transform for functions that are continuous with time. The Fourier transform of some function f ( t ) is 1 F ( γ ) = f ( t ) e iσt d t 2 α −� Similarly, the inverse Fourier transform is 1 iσt d γ. f ( t ) = F ( γ ) e 2 α −� That the second relation is the inverse of the first may be proven, but we save that calculation for the discrete transform, below. 47

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7.2 Discrete Fourier transform We are interested in the analysis of experimental (or numerical) data, which is almost always discrete. Thus we specialize to discrete Fourier transforms . In modern data, one almost always observes a discretized signal x j , j = { 0 , 1 , 2 , . . ., n 1 } We take the sampling interval —the time between samples—to be Γ t . Then x j = x ( j Γ t ) . The discretization process is pictured as t x x(t) Δ t j−1 j j+1 A practical question concerns the choice of Γ t . To choose it, we must know the highest frequency, f max , contained in x ( t ). The shortest period of oscillation is T min = 1 /f max Pictorially, x t T min We require at least two samples per period. Therefore T min 1 = . Γ t 2 2 f max 48
The discrete Fourier transform (DFT) of a time series x j , j = 0 , 1 , . . ., n 1 is n 1 1 2 αjk x ˆ k = n x j exp i n k = 0 , 1 , . . ., n 1 j =0 To gain some intuitive understanding, consider the range of the exponential multiplier. k = 0 exp( i 2 αjk/n ) = 1 . Then 1 x ˆ 0 = x j n j Thus ˆ x 0 is, within a factor of 1 / n , equal to the mean of the x j ’s. This is the “DC” component of the transform. Question: Suppose a seismometer measures ground motion. What would x ˆ 0 = 0 mean? k = n/ 2 exp( i 2 αjk/n ) = exp( iαj ). Then ˆ x n/ 2 = 1 n j x j ( 1) j = x 0 x 1 + x 2 x 3 . . . Frequency index n/ 2 is clearly the highest accessible frequency. (19) (20) The frequency indices k = 0 , 1 , . . . , n/ 2 correspond to frequencies f k = k/t max , i.e., k oscillations per t max , the period of observation. Index k = n/ 2 then corresponds to n 1 1 f max = = 2 n Γ t t But if n/ 2 is the highest frequency that the signal can carry, what is the significance of ˆ x k for k > n/ 2? 49

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For real x j , frequency indicies k > n/ 2 are redundant , being related by x ˆ k = x ˆ n k where z is the complex conjugate of z (i.e., if z = a + ib, z = a ib ).
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