Spectral Lines and Fourier Transform for Signal Analysis

1. Spectral Lines

2. Spectral Lines • Gives the frequency composition of the function • Amplitude, phase of sinusoidal components • Could provide information not found in time signal • E.g. Pitch, noise components • May help distinguish between signals • E.g speech/speaker recognition

3. Spectral Lines Example QUESTIONS -- DC Yes____ ao = ? No_____ ao = 0 Symmetry Even____ an = ? bn = 0 Odd____ an = 0 bn = ? Nether even nor odd ____ an = ? bn = ? Halfwave symmetry Yes_____ only odd harmonics No______ all harmonics Discontinuities Yes_____ proportional to1/n No______ proportional to1/n2 Note ? means find that variable. Comment on the general form of the Fourier Series coefficients [an and/or bn.] X X X X

4. Spectral Lines Example

5. Spectral Lines Example

6. Spectral Lines Example

7. The Fourier Transform • Applied to aperiodic signals • Could be continuous or discrete • Why Transform and not series? • Do we want continuous or discrete?

8. The Fourier Transform • Continuous Fourier Transform:

9. The Fourier Transform • The Discrete Fourier Transform:

10. Famous Fourier Transforms Sinusoid Delta function

11. Famous Fourier Transforms Gaussian Gaussian

12. Famous Fourier Transforms Sinc function Square wave

13. Famous Fourier Transforms Exponential Lorentzian

14. Fast Fourier Transform • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform • FFT principle first used by Gauss in 18?? • FFT algorithm published by Cooley & Tukey in 1965 • In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

15. FFT in matlab • Assign your time variables • t = [0:255]; • Assign your function • y = cos(2*pi*n/10); • Choose the number of points for the FFT (preferably a power of two) • N = 2048; • Use the command ‘fft’ to compute the N-point FFT for your signal • Yf = abs(fft(y,N)); • Use the ‘fftshift’ command to shift the zero-frequency component to center of spectrum for better visualization of your signals spectrum • Yf= fftshift(Yf); • Assign your frequency variable which is your x-axis for the spectrum • f = [-N/2:N/2-1]/N; - this is the normalized frequency symmetrical about f0 and about the y-axis • Plot the spectrum • plot(f, Yf)

16. FFT in matlab • Vary the fundamental frequency and see what happens

17. Spectrum of a signal