Fourier Series and Transforms

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Edward Walsh
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Fourier Series and Transforms Website is now online at: http://www.unc.

1 Fourier Series and Transforms Website is now online at: 9/2/08 Comp 665 Real and Special Signals 1

2 Discrete Exponen8al Func8on Discrete Convolu?on: Convolu?on with an exponen?al signal, : x(n) = e ω n k = y(n) = h(k)x(n k) y(n) = h(k)e ω(n k) k = = h(k)e ω k k = eωn That s not exactly the same definition of convolution that he used before but by commutivity it is identical If we define: then: H (e ω ) = k = Eigenvalue h(k)e ω k y(n) = H (e ω )e ω n Eigenvector 9/2/08 Comp 665 Fourier Series and Transforms 2

3 Sinusoids as Exponen8als Euler s Rela?on: Subs?tu?ng: e iωt = cos(ωt) + isin(ωt) y(n) = H (e iω )e iω n Interpreta?on: The response of a LSI system to a sinusoid input with frequency ω, is a scaled sinusoid of the same frequency h(n) What s the value of H(πn/12)? 9/2/08 Comp 665 Fourier Series and Transforms 3

4 Solving for H(e jω ) Recall from last?me that the real and imaginary components of a complex exponen?al can be equivalently interpreted as the magnitude and phase shiw of sinusoid H = Re(He jω ) 2 + Im(He jω ) 2 tan(ϕ) = Im(He jω ) Re(He jω ) 9/2/08 Comp 665 Fourier Series and Transforms 4

5 What we Know The input and the output x[n] = sin( πn πn ) y[n] = 0.3sin( + π ) Thus H = 0.3 ϕ = π 3 Re(H ) = H cos(ϕ) = 0.3( 1 2 ) Im(H ) = H sin(ϕ) = 0.3( 3 2 ) H πn 12 = 0.3( 1 2 ) + i0.3( 3 2 ) 9/2/08 Comp 665 Fourier Series and Transforms 5

6 Mul8ple Sinusoids h(n) 9/2/08 Comp 665 Fourier Series and Transforms 6

Fourier s Conjecture Joseph Fourier, an 18 th century French mathema?cian and physicist, claimed that any func?on of a variable, whether con?nuous or discon?

7 Fourier s Conjecture Joseph Fourier, an 18 th century French mathema?cian and physicist, claimed that any func?on of a variable, whether con?nuous or discon?nuous, could be expanded into a series of sinusoids with periods that are mul?ples of the variable Though not strictly correct, he is credited with inven?ng a decomposi?on of signals into series of sinusoids called their Fourier Series 9/2/08 Comp 665 Fourier Series and Transforms 7

8 Signals as Sums of Sinusoids How do we transform an arbitrary signal to a sum of sinusoids? N 1 X [k] = N 1 2πk i N x(n)e n k = 0,, N 1 n=0 X [k] = x(n) cos( 2πk n) N isin(2πk n) k = 0,, N 1 N n=0 [ ] Each term is just a dot product of a series with a complex sinusoid 9/2/08 Comp 665 Fourier Series and Transforms 8

9 Dot Products as Projec8ons The element wise sum of products of series elements or vector components is owen called the inner or dot product. a b A dot product can be interpreted as the length of one vector projected onto the other a b 9/2/08 Comp 665 Fourier Series and Transforms 9

10 Coordinates Coordinates are merely a series of projec?ons onto a specific set of vectors, each called a basis vector The same thing is going on when we Fourier transform b a signal, we project the original signal a (a point) onto an N dimensional basis x 2 x 1 9/2/08 Comp 665 Fourier Series and Transforms 10

11 Fourier Basis Func8ons 9/2/08 Comp 665 Fourier Series and Transforms 11

12 Inverse Mapping Once a signal is mapped from a series to a weighted sum of complex sinusoids, it can be mapped back to a series as follows: x[n] = 1 N N 1 X [k]e i 2πn k N n = 0,, N 1 k =0 Complex numbers X[k] represent the amplitude and phase of the sinusoidal components of the input "signal" x[n]. 9/2/08 Comp 665 Fourier Series and Transforms 12

13 Making Things Concrete Spatial Domain x[ ] Forward DFT Frequency Domain Re(X[ ]) Im(X[ ]) 0 N-1 N uniformly spaced samples Inverse DFT 0 N/2 N/2 + 1 coefficients (cosine amplitudes) 0 N/2 N/2 + 1 coefficients (sine amplitudes) 9/2/08 Comp 665 Fourier Series and Transforms 13

14 Some Context Why do we care? Mapping signals back and forth between spa?al and frequency domains simplifies analysis (convolu?on in par?cular) We have intui?on for periodic func?ons Provides a no?on of scale for characterizing signals Large scale = low frequency Small scale = high frequency Assumes that signals are periodic Perhaps they really are we can pretend they are outside of our domain of interest 9/2/08 Comp 665 Fourier Series and Transforms 14

15 Fourier Domain Proper8es Linearity ShiWing Symmetry a x[n] + b y[n] a X [k] + by [k] i 2πk N x[n + n 0 ] e n 0 X [k] Re(X [k]) = Re(X [N k]), k > 0 x[n],real Im(X [k]) = Im(X [N k]), k > 0 9/2/08 Comp 665 Fourier Series and Transforms 15

Graphically Observation: Each successive basis function represents a higher frequency sinusoid, that is related to the original signal s sampling rate by: period(x [k]) =

16 Graphically Observation: Each successive basis function represents a higher frequency sinusoid, that is related to the original signal s sampling rate by: period(x [k]) = 2πk N Once k>n/2 the number of samples per period are less than 1, and the k th basis aliases as one with lower frequency 9/2/08 Comp 665 Fourier Series and Transforms 16

17 More Proper8es Convolu?on Modula?on x[n] h[n] X [k]h [k] x[n] h[n] X [k] H [k] 9/2/08 Comp 665 Fourier Series and Transforms 17

Review of Linear System Theory

Review of Linear System Theory The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals. I assume the reader is familiar with linear algebra