PHYCS 3730/6720 Lab Exercise

Reading and references:

Fourier transform handout HTML. or pdf

Answer file lab18.

Exercise 1. Maple Discrete Fourier Transforms

Here is an example Maple session for doing the Fourier transform of a cosine and producing a vector containing the power spectrum. You will need Maple version 9 or higher to do this exercise. In our labs you may need to run the command xmapleV9 & in a terminal window to get it.

with(DiscreteTransforms);
N := 128;
f := Vector(N,j->evalf[6](cos(40*Pi*(j-1)/N)),datatype=complex[8]);
F := FourierTransform(f,inplace=true);
P := Vector(128,i->Re(F[i]*conjugate(F[i])),datatype=float[8]);

The Vector command sets up a vector f of length N with components that sample the function f(t) = cos(w*t). The sampling is at discrete times t = (j-1) * Dt for t ranging from 0 to 127*Dt. The angular frequency is w = 40*Pi/(N*Dt). (Note that Dt is the time interval between sample points. Since we specify the angular frequency w in terms of Dt, it cancels out in f(t), so we don't need to give it a numeric value at this point. If it helps, you can think of it as one second.)

The FourierTransform command generates the Fourier transform and puts it in the vector F. The next line computes a real vector containing the squared norm of the Fourier components. Note that with Maple indexing the Fourier component for zero frequency appears in F[1] and its power appears in P[1]. More generally, in this notation we must use i = m+1 to translate from the indexing in the notes to Maple indexing.

To plot the power spectrum we can use

with(plots);
pointplot({seq([i,P[i]],i=1..128)});

In this first exercise, run the Maple commands to see the result. Then answer these questions.

Without looking at the Fourier transform, determine the frequency (not the angular frequency!) of the original signal. (Use t_j = (j-1) * Dt and write the cosine term as cos(2*Pi*v*t_j) Solve for v. (Your answer will be in terms of the sampling interval Dt.)
Where are the peaks in the power spectrum P[i]? That is, precisely at what values of i? You can check your reading of the plot by simply looking at individual values of P[i]. Or you can plot the power spectrum over a narrower range.
To convert Maple's indexing to those in our notes, we must take i = m + 1. Considering that the frequency values are related to the discrete index m through v = m/(N Dt), what is the frequency of the peak at the lower index i? (It should agree with the frequency of the signal.)
Why are there two peaks? See the discussion in the notes about identities for Fourier transforms of real data.

Exercise 2. Damped oscillation

Use Maple to construct the power spectrum for the signal f(t) = exp(-0.2*t/Dt)*cos(0.8*t/Dt) using the same discretization as above. That is, sample at t/Dt = j = 0,1,...,127. The line shape you get in power vs frequency is called a Lorentzian.

In your answer file give the Maple commands you used. Then answer these questions:

Where is the lower frequency peak in the power spectrum? Give the frequency in units of 1/Dt and compare it with the frequency of the cosine in the original signal.
Estimate the width of the peak at half its height and give its value in angular frequency (with the 2*Pi included). That width is related to the decay rate of the signal. The slower the decay, the narrower the width. If we write the exponential decay factor as exp(-Gamma/2*t). Your width should be equal to Gamma, which, in this case is 0.4/Dt. Is it?