Fourier Lab, Physics 218, Fall 2001, James Sethna

In this lab, we study the Fourier transform and Fourier series. Much more complete instructions are available in pdf format.

Setup

Come to computer lab (RUPH, Rockefeller B11 or possibly B06), get an account, get a password, and log in.

Copy the files for the assignment from my directory:

cp -r ~sethna/FourierLab .

cd FourierLab

Cosine Wave

To get started, let's try plotting cos(k x + delta) with our tools:

Wave32

Here we use only 32 points, and plot x from -10 to 10. Try various values of k: what value gives one wave in the interval? What does small k look like?

What value of k gives one wavelength per four plotted points (asterisks)? What happens if you input a number for k larger than this? (Hit "send", "OK", and "send"). This problem is called "aliasing": any time you evaluate the function only at points with finite spacing $\delta x$, you can't describe wavelengths that fall between the cracks.

Fourier Transform of Cosine Wave

Try the complex Fourier transform of the cosine wave:

FourierWave

On the left, you see the cosine wave (with 100 points plotted): on the right, you see its complex Fourier transform.

What's the Fourier transform of a cosine wave? Which curve is the real part? Which is the imaginary part? Vary delta. What value of delta gives a sine wave? What is the Fourier tranform of a sine wave?

See how the peak k is related to the wavelength. We're now using 100 points instead of 32: at what value of k should aliasing be important? Try typing in 20 for k, and scroll down to 10: watch the peaks shift past the maximum possible k in the Fourier transform. How is the maximum k in the Fourier tranform related to aliasing?

Try to fit an integer number of wavelengths into the interval (you can type in the slider box and hit "Send" if you like). Do the Fourier peaks look cleaner? The Fourier series in an interval L is more-or-less the same as a Fourier transform of the same function made periodic with period L: if the function isn't naturally periodic, the Fourier series gets this ringing phenomenon.

Fourier Transform of Gaussian

Try the Fourier transform of a Gaussian, or normal distribution, A exp(-(x-x0)2/2 sigma2), where A is chosen to keep the integral of the function exactly one (depending on sigma).

FourierPacket

Notice that the Fourier transform of a Gaussian is a Gaussian.

How is the width of the Fourier transform related to the width of the original Gaussian? Why?

How is the height of the Fourier transform related to the height of the original Gaussian? What does this have to do with the normalization A?

Pick a small width sigma so the Fourier transform is fairly broad. Shift x0. What happens? This is one of the typical forms for a wave packet.

Fourier Transform of White Noise

White noise is noise with roughly the same amplitude at all frequencies. Random noise is sound where the pressure at each time interval is a new random number, independently chosen from all the others.

FourierNoise

will take a random series of numbers and plot them and their power spectrum (the absolute square of the Fourier transform). You can see that random noise is white noise.

Try zooming into the random noise time series with the magnifying glass. (This doesn't work with the animations, but here you can do it.) Does it look random? Try zooming into the Fourier plot. Random? Random in, random out.

Try zooming in to the power spectrum near k=0. Why is it symmetric?

Fourier Transform Twice Gives Inversion

What happens when you Fourier tranform the Fourier transform?

FourierSquared

shows this for a random, smoothed function. How is this related to inversion?

Fourier Transform of Step Function (optional)

As you increase the number of Fourier coefficients, you converge to the step function. What happens near the step? This overshoot is called the Gibbs phenomenon.
Web page last modified: September 7, 2001

James P. Sethna, sethna@lassp.cornell.edu.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).