1

variable star simulation.  you were given the following plot.

vsdat = Flatten[Import["variablestar.dat"]];

ListPlot[vsdat,Joined->True]

This is classic amplitude modulation.  From the data you can see that the carrier frequency is about 21/600 cycles per day and the modulation frequency is about 4/600 samples per day.  The sampling frequency is one sample per day.  So the nyquist frequency is 1/2 cycles per day.

fnyquist = .5;
df = .5/300;
fm = 4./600
fc = 21./600

frequencies = Table[ i df,{i,300}];
fouriervsdata = Abs[Fourier[vsdat]];

data = Table[{frequencies[[i]],fouriervsdata[[i]]},{i,300}];

ListPlot[data,Joined->True,PlotRange->{{0,.1},{0,200}}]

Plot[20 + 15Cos[2 Pi fm/2 t] Sin[2 Pi fc t],{t,0,600},BaseStyle->{FontFamily->"Helvetica",16}]

2

f[x_,a_] = Piecewise[{{a+x,-a<x<0},{a -x,0<x<a}}]

Plot[f[x,1],{x,-2,2},PlotRange->All,BaseStyle->{FontFamily->"Helvetica",16}]

g[k_] = FourierTransform[f[x,1],x,k]

Plot[g[x],{x,-10,10}]

3

x = {0,-1,1};
y = Fourier[x]

Chop[y]

ListPlot[Abs[y],Joined->True]

4

The periodogram converges to the power spectrum in the limit that the number of points goes to infinity.  The sampling thereom says that a bandlimited function is completely specified by its sampled points provided they are sampled at least as often as one over twice the maximum frequency.  A function is band limited if its Fourier transform is zero outside of some finite interval.  Aliasing is the production of spurrious frequencies that results from undersampling a function.  The sample frequency is the frequency at which samples are taken (or measured)  from the function.  If the time between samples is dt, then the sampling frequency is by definition one over dt.  The Nyquist frequency is half this:  1/(2 dt).

5

Sketch the power spectrum of  the following time series:

f[t_,dt_] =Exp[-(t-dt)^2] + Exp[-(t - 2 dt)^2]
g[ω_,dt_] = FourierTransform[f[t,dt],t,ω]

t0 = 5
Plot[f[t,t0],{t,0,50},PlotRange->All]
Plot[Abs[g[ω,t0]],{ω,-10,10},PlotRange->All]

6

Fourier Series

Clear[f,x]
f[x_] = Piecewise[{{1+x,-1<x<0},{1-x,0<x<1}}]
Plot[f[x],{x,-1,1}]

<<FourierSeries`

g[x_] = FourierTrigSeries[f[x],x,10,FourierParameters->{1,1/2}]

Plot[ Abs[g[x]-f[x]],{x,-.8,.8},PlotRange->All]

Integrate[g[x],{x,-1,1}]
Integrate[f[x],{x,-1,1}]

It turns out, that no matter how many terms in the FS you include the integral of the FS approximation on - 1, 1 always equals 1.  Why is this? Can you say anything about the rate of convergence of the Fourier Series?

7

The Fourier Transform of δ (t - t0) = 1 if and only if t0 = 0.  However the Power spectrum is always 1.  See the class notes.  The proof  that the Fourier Transform of 1 is a delta function is given on page 5 of the lecture notes from 10/29.