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Let's say I have an input signal

[tex] f(t,0)= \sum_n A_n cos(w_nt) [/tex]

And we know that

[tex] f(t,x)= \sum_n A_n cos(k_nx-w_nt) [/tex]

and we also have the relationship between k and w. We can find the coefficients A_n by taking a Fourier Transform of the relationship for f(0,t).

Here's my question. How is the sum for f(t,x) normalized? If this is unclear, below is an example:

I've been trying test cases to figure out what the relationship is by prescribing f(0,t) to be some easy function i.e.

[tex] f(t,0)= cos(\omega_1t) + 2cos(\omega_2t) [/tex]

Theoretically, we know that

[tex] \bar{f}(t,x)= cos(k_1x-\omega_1t) + 2cos(k_2x -\omega_2t) [/tex]

But if I did not know precisely what f(t,0) is and went through this numerically, I'd have

[tex] f^*(t,x)= \sum_n A_n cos(k_nx-\omega_nt) [/tex]

where

[tex]A_n=\frac{1}{length(f(0,t))}fft(f(t,0)) [/tex]

My problem is [tex] f^* \neq \bar{f} [/tex]. My question is, why is this the case? I think it comes down to normalizations but I'm not completely sure.

Any help would be appreciated,

Nick

PS

If you're interested, here's my code

close all;

g=10;

A= [1 .5];

tt=0:.5:2^5-.5;

omega=[1 .25];

kay=omega.^2/g;

signal = zeros(length(tt),1);

for i=1:length(tt);

signal(i,1) = sum(A.*cos(tt(i)*omega));

end

Fs= 50;

xx=1:.5:100;

yy=signal;

[YY,ff]=positiveFFT(yy,Fs); % this sets up our independent variable frequency as we want it

kk=(2*pi*ff).^2/g;

ETAETA=zeros(length(tt),length(xx));

for i =1:length(tt)

for j=1:length(xx)

ETAETA(j,i)=real(sum(YY'.*cos(kk*xx(j)-2*pi*ff*tt(i))));

end

end

figure (1)

imagesc(xx,tt,real(ETAETA));

set(gca,'YDir','normal');

caxis([-0.01 0.01])

colormap(copper);

colorbar;

xlabel('distance from wave maker'); ylabel('time');

shading('interp')

h = colorbar;

set(get(h,'YLabel'), 'String', 'SSH');

title('theoretical first order solution');

% %

% ETA3= zeros(length(xx),length(tt));

% for i=1:length(tt)

% for j=1:length(xx)

% ETA3(j,i)= sum(A.*cos(kay*xx(j)-omega*tt(i)));

% end

% end

% figure(2)

% imagesc(xx,tt,real(ETA3)');

% set(gca,'YDir','normal');

% caxis([-0.01 0.01])

% colormap(copper);

% colorbar;

% xlabel('distance from wave maker'); ylabel('time');

% shading('interp')

% h = colorbar;

% set(get(h,'YLabel'), 'String', 'SSH');

% title('theoretical first order solution');

where

positiveFFT.m is

function [X,freq]=positiveFFT(x,Fs)

N=length(x); %get the number of points

k=0:N-1; %create a vector from 0 to N-1

T=N/Fs; %get the frequency interval

freq=k/T; %create the frequency range

X=fft(x)/N; % normalize the data

%only want the first half of the FFT, since it is redundant

cutOff = ceil(N/2);

%take only the first half of the spectrum

X = X(1:cutOff);

freq = freq(1:cutOff);