Complex Analysis in Fourier Transform Applications
Explore the practical applications of Fourier transforms through the lens of complex analysis. Delve into topics such as sound analysis, signal processing, differential equations, and spectroscopy. Understand how Fourier transforms can be used to analyze and interpret various real-world phenomena effectively.
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Presentation Transcript
Complex Analysis in Fourier Transform Applications By Brett Kassel and Matt Mulvehill 1
Overview Introduction to Fourier transforms Common Examples Applications of Fourier transforms Sound/Signal Analysis Program Conclusion 2
Example 4
Example cont. Fourier Transform Fourier Inverse Transform 5
Applications of Fourier Transforms Analysis of differential equations Similar to Laplace transforms Fourier transform spectroscopy NMR and MRI Quantum mechanics x p /2 Sound/ signal analysis 6
Sound What exactly is sound? Lots of vibrations What is a frequency? How fast the waveform oscillates What is a harmonic? A integer multiple of a fundamental frequency Its all a bunch of sines and cosines 7
Fourier Transform Sound Analysis Sound s waveform function Sound s frequencies Fourier Transform Amplitude vs. frequency Amplitude vs. time Fourier Inverse Transform 11
Program Create any audible wave equation: i.e. Put wave through Fourier transform & output graph of frequencies and amplitudes. i.e. 12
Conclusion Fourier Transforms Input = a function Output = a similar function in a different space Have neat real world applications Sound analysis 13
Thanks for listening References Boas, Mary L. "Fourier Series and Transforms. Mathematical Methods in the Physical Sciences. New York: Wiley, 1966. N. pag. Print. Neto, Joao. "Fourier Transform: A R Tutorial." N.p., Mar. 2013. Web. <http://www.di.fc.ul.pt/~jpn/r/ fourier/fourier.html>. "The Fourier Transform." Stanford University, n.d. Web. <http://web.stanford.edu/class/ee102/ lectures/fourtran>. "FFT (Fast Fourier Transform) Waveform Analysis." FFT (Fast Fourier Transform) Waveform Analysis. N.p., n.d. Web. <http://www.dataq.com/data-acquisition/general-education- tutorials/fft- fast-fourier-transform-waveform-analysis.html>. "Frequency-Domain And The Fourier Transform." Computer Musically Speaking Introduction to Digital Signal Processing (2009): 276-332. Web. "The Fourier Series and Fourier Transform." Circuit Analysis (n.d.): 102-23. Web. <http:// www.physik.uni-kl.de/fileadmin/beigang/Vorlesungen/WS_07_08/ Fourier_Transforms_Rick_Trebino.pdf>. 14
