Identifying wave-number sign after applying 2-dimensional FFT

I am applying 2D FFT to identify the direction of of wave number (kx) and frequency (w). I am testing it with a simple wave propagating in +ve x-direction having form Acos(wt-kx). After applying 2D fft using inbuid functionn of MatLab (i.e fft2) I expect to peak in amplitude at [kx, w]. As FFT also gives information for -ve frequencies, I will get additional other peak at [-Kx, -w]. However, I am getting peak at [Kx,-w] and [-kx,w] which is not correct. Can you please share opinion on it. my program is given below

%%

dx=0.1; dt=0.1;

nx=64; nt=128;

dkx=2*pi/(nx*dx);

dw=2*pi/(nt*dt);

xt=1:nx;xt=xt*dx;

tt=1:nt;tt=tt*dt;

nxh=nx/2;

nth=nt/2;

%% generating frequency matrix

kx1=0:nxh;

kx1=kx1*dkx;

fkx1=flip(kx1);fkx1(nxh+1)=[]; kx1(nxh+1)=[];

kx1f=[0-fkx1 kx1];

w1=0:nth;

w1=w1*dw;

fw1=flip(w1);fw1(nth+1)=[];w1(nth+1)=[];

w1f=[0-fw1 w1];

%% generate wave with amplitude 1, wave number 1 and frequency 2

w=2; Kx=1;

A0=1;

for jj=1:nt

for ii=1:nx

xy(jj,ii)=A0*cos(w*dt*jj-Kx*ii*dx);

end

end

%% performing 2D fft

out=fft2(xy);

out=fftshift(out);

P=2*abs(out)/(nx*nt);

%% plotting data and fft2 output

subplot(1,2,1)

imagesc(xt,tt,xy);

axis xy

xlabel('x')

ylabel('time');

colorbar

caxis([-A0 A0]);

axis xy

subplot(1,2,2)

imagesc(kx1f,w1f,P);

xlabel('kx')

ylabel('w');

colorbar

axis xy

caxis([0 A0]);

xlim([-5 5])

ylim([-5 5])

Answer 1

[David Goodmanson]

HI Bharati,

WIthout going through the code in and detail (nice plot) I think the reason is the following. The cos wave is

(1/2) ( exp(i(k0x-w0t)) + exp(i(-k0x+w0t)) )

Take the first term, and and look at the continuous integral version of fft2. Not counting overall constants this is

Integral  exp(i(k0x-w0t)) exp(-ikx) exp(-iwt) dx dt
                             ^         ^

For the fft2, each of the exponential functions comes in with a minus sign as shown. The integral is

Integral  exp(i((k0-k)x-(w0+w)t)) dx dt

and the result is proportional to the product of two delta functions, delta(k0-k) delta(w0+w) which gives a 2d peak at (k0,-w0) which is what you get. Similarly the second term gives a peak at (-k0,w0).

The fft is set up with the minus sign so that a function like exp(+ik0x) gives a corresponding peak at +k0. But for the wave, w0 comes in with a negative sign and consequently the fft gives a peak at -w0. You could say that fft2 is not ideally set up to handle this situation.

[Bharati Kakad]

Thank you, David for your reply,

I am dealing with the simulation data in 2D, where I want to identify the wave direction correctly. Looking at fft2 output from my test code and your reply, can I conclude that MatLab fft2 function is not ideal to get the correct information about the wave direction?

[David Goodmanson]

Well, it's fine if you recognize what's going on and after you define w1f, add a line

w1f=[0-fw1 w1];
% wave has the form exp(+-i(k0x-w0t)) so after fft2, sign of resulting
% w variable must be reversed
w1f = -w1f;

with some kind of annotation like what is above. For me, if I made that change without annotation and took a look in a couple of years, that line would be very mysterious.