Unwrapping atan vs. atan2

I can extract the phase of a complex number in Matlab using the atan() or atan2() functions.

atan() returns in interval limited to [-pi/2,pi/2], and atan2() returns in interval limited to [-pi,pi].

I want to see if I can unwrap the extracted phase in each case using the unwrap() function, but unwrap is only effective on phase extracted via atan2().

R = 1; % Magnitude
theta = linspace(0,6*pi,100); % (radians) Angle array
theta_atan = zeros(1,length(theta)); % Prellocate for calculation 
theta_atan2 = zeros(1,length(theta)); % Prellocate for calculation
X = zeros(1,length(theta)); %Prelloc. 
Y = zeros(1,length(theta)); %Prelloc.   

for i = 1:length(theta)
    X(i) = R*cos(theta(i)); % Real part
    Y(i) = R*sin(theta(i)); % Imaginary part
    theta_atan(i) = atan(Y(i)/X(i)); 
    theta_atan2(i) = atan2(Y(i),X(i)); 
end  

I plot the unwrapped extracted phase using each method:

figure(666)
plot(theta,unwrap(theta_atan));
hold on 
plot(theta,unwrap(theta_atan2));
legend('theta atan','theta atan2')
xlabel('input phase')
ylabel('extracted phase')

However, as you can see, unwrapping is only effective on the atan2() case. Even if I use unwrap(theta_atan, pi/2) (in this case unwrapping is based on increments of pi/2 instead of the default, pi), I fail to properly unwrap the atan() phase.

The second argument to unwrap is not the period of the input data, but the tolerance. The function always unwraps data assuming a 2π interval. That is, it wants to see that x(i)-x(i+1) is larger than the tolerance before unwrapping, and smaller after unwrapping. In the case of a tolerance of pi/2, if, for example, x(i)=0 and x(i+1)=3, the jump is larger than the tolerance, but adding or subtracting 2*pi to x(i+1) does not improve things.

One work-around would be to multiply the input by 2, and divide by 2 after unwrapping:

unwrap(theta_atan * 2) / 2

However, it is always best to use atan2 to obtain an angle.