lecnotes7 - 7 Fourier transforms Except in special...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ).
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern