Discrete Wavelets with Applications

043 - Fourier Series

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
04/10/2008; last edits 02/15/2024.

SetDirectory[NotebookDirectory[]];
<<DW.wl

Taylor Polynomials

The package ' DW.m' contains the command ' Taylor'  and 'TaylorPlot':

?TaylorPlot

?Taylor

TaylorPlot[Sin[α],α,0,7,13]

TaylorPlot[ArcTan[t],t,0,1.2,10]

Taylor[ArcTan[x],x,0,12]

Fourier Sine and Cosine Series

f[t_]=t;
FourierSC[f[t],t,8]
FourierSCPlot[f[t],t,8]

f[t_]=Which[-π≤t≤0,0,0<t<π,1];
FourierSC[f[t],t,8]
FourierSCPlot[f[t],t,8]

FourierSC[1+y-Exp[y/2],y,12]
FourierSCPlot[1+y-Exp[y/2],y,12]

FourierSCPlot[1/π If[x>0,x,x+π] ,x]

FourierSCPlot[Sqrt[Abs[t]],t,6]

g[x_]=Which[-π≤x/π<.2,0,.2≤x/π<.5,Sin[8x],.5≤x/π< 0.7,0,0.7≤x/π<0.8, Sin[50x],0.8≤x/π≤1,0];

FourierSCPlot[g[x],x,12,PlotRange->All,PlotPoints->5000]

Fourier Series

FourierPlot[f[x],x]
FourierS[f[x],x]

Energy Spectrum

h[t_]= If[t>0,1,0] -1/2
FourierSC[h[t],t]
Plot[h[t],{t,-π,π}]
EnergySpectrum[h[t],t,20]

Fourier Filter Plot

The command below plots the absolute value of the complex Fourier expansion of a filter.

?FourierFilterPlot

FourierFilterPlot[{1,1}]