# Phase spectrum of 2D real function

I am trying to calculate the phase spectrum of a 2D function, $f(x,y)$, where $x$, $y$, and $f(x,y)$ are real, using MATLAB's fft2. Thus my operation is $f(x,y)\rightarrow F(k_x,k_y)$. My function is such that $k_x$ only has positive values and $k_y$ has both positive and negative values.

Since my function is real, I only need the one-sided spectrum. I can calculate the one-sided amplitude spectrum by doubling the spectrum where $k_x>0$. The result I get is exactly what I expect.

When it comes to the phase spectrum, I get a function that is not continuous. The phases appear to jump around with an approximate difference of $\pi$. Is this related to phase wrapping? How do I get a continuous phase plot in positive $k_x$?

More information on my image and code: Data can be found here

My signal in $x$ and $y$ looks like this:

figure; contour(y,x,eta_0)
xlabel('y')
ylabel('x')


The amplitude spectrum looks like this:

dx = mean(gradient(x));
Nx = 64;
Ny = 32;

Fsy = 1/dy;
Fsx = 1/dx;
kx_fft = 2*pi * Fsx/2 * linspace(0, 1, size(eta_0,1)/2+1);
ky_fft = 2*pi * Fsy/2 * linspace(0, 1, size(eta_0,2)/2+1);
ky_fft = [-fliplr(ky_fft(2:end-1)) ky_fft];

amp_xy = circshift(abs(fftshift(fft2(eta_0)))/Nx/Ny,-1,2);
figure; plot(ky,kx,amp_xy)


I know that my function does not contain any negative $k_x$ values and hence I can take the single-sided amplitude spectrum by doubling the positive $k_x$ values, to get this:

amp_xy = 2*amp_xy([Nx/2+1:Nx 1],:);
figure; imagesc(ky_fft,kx_fft,amp_xy);


The phase spectrum looks like this: phase_xy = circshift(angle(fftshift(fft2(eta_0))),-1,2); This clearly displays jumps from around zero (which is where I expect my phases to be) to $\pi$ and $-\pi$. Why is that and how can I avoid it?

To illustrate my point, I can take a cut through my signal at the centreline of $y$ and check the 1D amplitude and phase spectra. The 1D phase spectrum shows most of the phases close to zero, with the lowest frequencies having a phase close to $\pi$/$-\pi$ as I expected (from theory):

Now if I was to take a cut through the 2D phase spectrum where $k_y=0$, I see something similar, yet not continuous. It is as if I need to shift some phases by $\pi$ up or down:

In essence, the question is: how do I get a smooth 2D phase spectrum in positive values of $k_x$?

Edit: If I calculate the phase angle using atan, rather than angle, I get a smoothly varying phase plot. That is expected though, since atan limits angles to $-\pi$ to $\pi$ whereas angle uses atan2, which spans the entire circle.