Least squares fit with functions 1,cos(2*pi*t),sin(2*pi*t),cos(4*pi*t),sin(4*pi*t)

Contents

Given periodic signal

N = 8;
t = (0:N-1)'/N;
u = t;                           % "sawtooth signal"

plot([t;t+1],[u;u],'o:')         % plot two periods
hold on

find coefficients of $e^{(0)},e^{(1)},e^{(-1)},e^{(2)},e^{(-2)}$

The discrete Fourier transform gives the vector $\hat u = [\hat u_0,\ldots ,\hat u_{N-1}]$.

four = @(v) fft(v)/length(v);    % define four, ifour
ifour = @(v) ifft(v)*length(v);

uh = four(u);                    % find discrete Fourier transform [uhat_0;...;uhat_7]

plot fitted curve

Note: $\hat u_0,\hat u_1,\hat u_2,\ldots,\hat u_{N-2},\hat u_{N-1}$ corresponds in Matlab to uh(1), uh(2), uh(3), ..., uh(N-1), uh(N).

Np = 1000;                       % use 1000 points for plotting g(t)
T = (0:Np-1)'/Np;

U1 = uh(1) + uh(2)*exp(2*pi*i*T) + uh(N)*exp(-2*pi*i*T);   % terms of freq 0,1
U2 = U1 + uh(3)*exp(4*pi*i*T) + uh(N-1)*exp(-4*pi*i*T);    % terms of freq 0,1,2

plot([T;T+1],[U1;U1])
plot([T;T+1],[U2;U2])
hold off
legend('given signal','fit with freq 0,1','fit with freq 0,1,2','Location','SE')


Published with MATLAB® R2018a