unwrap

Define the $$x$$- and $$y$$-coordinates of a spiral with phase angle from 0 to $6\pi$. Plot the spiral.

t = linspace(0,6*pi,201);
x = t/pi.*cos(t);
y = t/pi.*sin(t);
plot(x,y)

Find the phase angle of the spiral from the $$x$$- and $$y$$-coordinates using the atan2 function. The atan2 function returns the angle values within the closed interval from $-\pi$ to $\pi$.

P = atan2(y,x);
plot(t,P)

Note that this plot has discontinuities. Use unwrap to eliminate the discontinuities. unwrap adds multiples of $\pm 2\pi$ when the phase difference between consecutive elements of P are greater than or equal to the jump threshold $\pi$ radians. The shifted phase angle Q lies in the interval from 0 to $6\pi$.

Q = unwrap(P);
plot(t,Q)

Shift the phase angle of a frequency response. The phase curve has two jumps. The first jump is 3.4250 radians between W = 3 and W = 3.4, and the second jump is 6.3420 radians between W = 5 and W = 5.4. Plot the phase curve.

clear; close all;
W = [0:0.4:3, 3.4:0.4:5, 5.4:0.4:7];
P = [-1.5723
     -1.5747
     -1.5790
     -1.5852
     -1.5922
     -1.6044
     -1.6269
     -1.6998
      1.7252
      1.5989
      1.5916
      1.5708
      1.5582
     -4.7838
     -4.8143
     -4.8456
     -4.8764
     -4.9002];
plot(W,P,'bo-')

Use unwrap to shift the phase angle using the default jump threshold $$\pi$$ radians. Plot the shifted phase curve. Both jumps are shifted since they are greater than the jump threshold $$\pi$$ radians.

plot(W,unwrap(P),'ro-')

Now shift the phase angle using a jump threshold of 5 radians. Plot the shifted phase curve. The first jump is not shifted since it is less than the jump threshold 5 radians.

plot(W,unwrap(P,5),'ro-')

Define a two-column matrix P that contains phase angles.

P = [0 7.07; 0.19 0.98; 6.67 1.18; 0.59 1.37; 0.78 1.56]

P = 5×2

         0    7.0700
    0.1900    0.9800
    6.6700    1.1800
    0.5900    1.3700
    0.7800    1.5600

The phase angles P(1,2) = 7.07 and P(3,1) = 6.67 have phase differences that are larger than $$\pi$$ compared to the rest of the data.

Unwrap the phase angles by first comparing the elements columnwise. Specify the dim argument as 1. Use the default jump threshold $$\pi$$ by specifying the second argument as [].

dim = 1;
P1 = unwrap(P,[],dim)

P1 = 5×2

         0    7.0700
    0.1900    7.2632
    0.3868    7.4632
    0.5900    7.6532
    0.7800    7.8432

To shift phase angles by rows instead of by columns, specify dim as 2 instead of 1.

dim = 2;
P2 = unwrap(P1,[],dim)

P2 = 5×2

         0    0.7868
    0.1900    0.9800
    0.3868    1.1800
    0.5900    1.3700
    0.7800    1.5600