1. Where we’re going Speed, Storage Issues Frequency Space

2. Sine waves can be mixed with DC signals, or with other sine waves to produce new waveforms. Here is one example of a complex waveform: V(t) = A o + A 1 sin  1 t + A 2 sin  2 t + A 3 sin  3 t + … + A n sin  n t --- in this case--- V(t) = A o + A 1 sin  1 t Ao Ao A1 A1 Fourier Analysis Just an AC component superimposed on a DC component

3. More dramatic results are obtained by mixing a sine wave of a particular frequency with exact multiples of the same frequency. We are adding harmonics to the fundamental frequency. For example, take the fundamental frequency and add 3rd harmonic (3 times the fundamental frequency) at reduced amplitude, and subsequently add its 5th, 7th and 9th harmonics: Fourier Analysis, cont’d the waveform begins to look more and more like a square wave.

4. This result illustrates a general principle first formulated by the French mathematician Joseph Fourier, namely that any complex waveform can be built up from a pure sine waves plus particular harmonics of the fundamental frequency. Square waves, triangular waves and sawtooth waves can all be produced in this way. (try plotting this using Excel) Fourier Analysis, cont’d

5. Spectral Analysis Spectral analysis means determining the frequency content of the data signal Important in experiment design for determining sample rate, f s - sampling rate theorem states: f s  max f signal to avoid aliasing Important in post-experiment analysis - Frequency content is often a primary experiment result. Experiment examples: - determining the vibrational frequencies of structures - reducing noise of machines - Developing voice recognition software

6. Spectral analysis key points Any function of time can be made up by adding sine and cosine function of different amplitudes, frequencies, and phases. These sines and cosines are called frequency components or harmonics. Any waveform other than a simple sine or cosine has more than one frequency component.

7. Example Waveform 1000 Hz sawtooth, amplitude 2 Volts Fundamental frequency term Harmonic terms b 0 is the average value of the function over period, T Period, T =.001 sec

8. Fourier Coefficients, a n and b n These coefficients are simply the amplitude at each component frequency For odd functions [f(t)=-f(-t)], all b n = 0, and have a series of sine terms (sine is an odd function) For even functions [f(t)=f(-t)], all a n = 0, and have a series of cosine terms (cosine is an even function) For arbitrary functions, have a n and b n terms. Coefficients are calculated as follows:

9. An odd function (sine wave)

10. More odd functions Fundamental or First Harmonic Third HarmonicSine series or Pure imaginary amplitudes

11. An even function (cosine wave)

12. More even functions Fundamental or First Harmonic Second Harmonic Cosine series or Pure real amplitudes

13. Periodic, but neither even nor odd Cosine and sine series or Complex amplitudes

14. Sawtooth Fourier Coefficients Odd function so: Using direct integration or numerical integration we find the first seven a n ’s to be: We can plot these coefficients in frequency space: Our sawtooth wave is an ________ function. Therefore all ____ = 0

15. Start with a sine wave...

16. Add an odd harmonic (#3)...

17. Add another (#5)...

18. And still another (#7)...

19. Let’s transform a “Sharper” sawtooth Even or odd?

20. Frequency Domain Plot of Fourier Coefficients Get “power spectrum” by squaring Fourier coefficients

21. Construction of Sharp Sawtooth by Adding 1st, 2nd, 3rd Harmonic Third Harmonic First Harmonic Second Harmonic

22. Spectral Analysis of Arbitrary Functions In general, there is no requirement that f(t) be a periodic function We can force a function to be periodic simply by duplicating the function in time (text fig 5.10) We can transform any waveform to determine it’s Fourier spectrum Computer software has been developed to do this as a matter of routine. - One such technique is called “Fast Fourier Transform” or FFT - Excel has an FFT routine built in

23. Voice Recognition The “ee” sound

24. Voice Recognition (continued) The “eh” sound

25. Voice Recognition (continued) The “ah” sound

26. Voice Recognition (continued) The “oh” sound

27. Voice Recognition (continued) The “oo” sound