# Matlab normalization using fft

1. Sep 22, 2011

### nickthequick

Hi,

Let's say I have an input signal

$$f(t,0)= \sum_n A_n cos(w_nt)$$

And we know that

$$f(t,x)= \sum_n A_n cos(k_nx-w_nt)$$

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.

$$f(t,0)= cos(\omega_1t) + 2cos(\omega_2t)$$

Theoretically, we know that

$$\bar{f}(t,x)= cos(k_1x-\omega_1t) + 2cos(k_2x -\omega_2t)$$

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

$$f^*(t,x)= \sum_n A_n cos(k_nx-\omega_nt)$$

where

$$A_n=\frac{1}{length(f(0,t))}fft(f(t,0))$$

My problem is $$f^* \neq \bar{f}$$. 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');
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');
% 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);