Fourier theory made easy (?)

5*sin (2  4t) Amplitude = 5 Frequency = 4 Hz seconds A sine wave

5*sin(2  4t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second A sine wave signal

An undersampled signal

The Nyquist Frequency The Nyquist frequency is equal to one-half of the sampling frequency. The Nyquist frequency is the highest frequency that can be measured in a signal.

Fourier series Periodic functions and signals may be expanded into a series of sine and cosine functions

The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal)

The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform: close your eyes if you don’t like integrals

The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform:

A transform takes one function (or signal) and turns it into another function (or signal) The Discrete Fourier Transform: The Fourier Transform

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!

Famous Fourier Transforms Sine wave Delta function

Famous Fourier Transforms Gaussian

Famous Fourier Transforms Sinc function Square wave

Famous Fourier Transforms Sinc function Square wave

Famous Fourier Transforms Exponential Lorentzian

FFT of FID

Effect of changing sample rate

Lowering the sample rate: –Reduces the Nyquist frequency, which –Reduces the maximum measurable frequency –Does not affect the frequency resolution

Effect of changing sampling duration

Reducing the sampling duration: –Lowers the frequency resolution –Does not affect the range of frequencies you can measure

Effect of changing sampling duration

Measuring multiple frequencies

Some useful links –Fourier series java applet –Collection of demonstrations about digital signal processing –FFT tutorial from National Instruments –Dictionary of DSP terms 4FreeIndDecay.pdfhttp://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT 4FreeIndDecay.pdf –Mathcad tutorial for exploring Fourier transforms of free-induction decay –This presentation