To calculate the elliptical region \(}\), a relatively large number of independent waveforms is needed (\(187\le M\le 250\) in this report). First, \(}\) was transposed to a column vector of size \(M\times 1\) and centered about the origin on the complex plane by subtracting its complex mean value (\(\delta\)). Next, for computational purposes, \(}\) was changed from a \(M\times 1\) complex vector into an \(M\times 2\) matrix of real values, with the real and imaginary parts located in the first and second columns, respectively. The covariance matrix (unscaled) of the new 2-column matrix was then obtained by

$$}=\text\left(}\right)=\frac}}^} ,$$

(15)

where the superscript T indicates the matrix transpose operator. The covariance matrix \(}\) is a \(2\times 2\) matrix, with the diagonal elements being the variances and the off-diagonal elements being the covariances of the real and imaginary parts of \(}\). The eigenvalues and eigenvectors of the covariance matrix \(}\) were used to define an elliptical region (eigenvalues (\(}\)) represent the variance in the direction of the eigenvectors (\(}\))). The boundaries of the elliptical region were computed as follows:

$$}=}\sqrt}}}\right)}^/g,$$

(16)

where \(\upsilon\) is a \(2\times 2\) matrix of eigenvectors and \(\lambda\) is a \(2\times 2\) diagonal matrix of eigenvalues. The variable \(}\) is a \(2\times J\) matrix describing the Cartesian coordinates of \(J\) points on the unit circle. Each element in the first row is defined by \(\text(\theta )\) and each element in the second row is defined by \(\text(\theta )\), where \(\theta\) is a row vector of \(J=\text\) equally spaced radian phase values from 0 to \(2\pi\). The choice of the number of columns, \(J\), is not critical so long as it is large enough to give a reasonable estimation of the ellipse magnitude at any arbitrary phase value. In computing \(g\), the constant 0.385 is a scaling factor chosen so that the false positive rate obtained using \(}\) to determine overall significance (i.e., significance of magnitude and phase considered together) is \(\alpha =0.05\). Finally, the centering operation was undone (\(\delta\)) was added to \(}\)), and \(}\) was changed from a \(2\times J\) matrix into a \(J\times 1\) complex vector.

To determine whether \(}\) intersected the unit circle, \(}\), a vector of complex values was created, as described above. The angle of rotation of \(}\) was defined by

$$\vartheta =\text\left(\text}/\text}\right) ,$$

(17)

where arctan refers to the four-quadrant arctangent and \(}\) is the matrix of eigenvectors referred to in Eq. 16. \(}\) was shifted to the origin by subtracting the mean of \(}\) and then rotated by multiplying it by the complex exponential \(^\). The identical operation was applied to \(}\), and to the point \(1+0i\) on the complex plane, maintaining their geometrical relationships with \(}\). The shift and rotate operation transformed \(}\) to an ellipse centered at zero with major and minor axes aligned with the real and imaginary axes, respectively. In this form, \(}\) can be written in the standard form for an ellipse:

$$\frac^}^}+\frac^}^}=1 .$$

(18)

In this equation, \(a\) is half the length of the major axis of the ellipse and \(b\) is half the length of the minor axis (\(a\ge b\)), \(x\) refers to a (real) x-axis value, and \(y\) refers to a (imaginary) y-axis value. The quantities \(a\) and \(b\) are equal to \(\sqrt}}/g\). To compute overall significance, determination of whether \(}\) encompassed the point 1 + 0i on the complex plane was made by computing the left side of Eq. 18, with \(^\) and \(^\) replaced by the real and imaginary parts of the shifted and rotated value of 1 + 0i. If the computed value was < 1, then \(}\) encompassed the origin, and there was no significant shift. To compute magnitude significance, the left side of Eq. 18 was computed, with \(^\) and \(^\) replaced by the real and imaginary parts of the shifted and rotated elements of \(}\). If any of the computed values were < 1, then \(}\) intersected \(}\), and the magnitude shift was not significant. (Values < 1 fall inside the ellipse \(}\), values equal to 1 fall on the ellipse, and values > 1 fall outside the ellipse.) To compute phase significance, determination of whether \(}\) crossed the x-axis involved simply evaluating whether \(\text\left(\text\left(}\right)\right)<0<\text\left(\text\left(}\right)\right)\) was true. If any part of \(}\) crossed the x-axis, then phase change was not significant. Finally, there are some cases where the origin falls outside of \(}\), so that the overall change is significant; however, the magnitude change alone and the phase change alone are not significant by the above computations. In these cases, it is the combined effect of magnitude and phase together that yields a significant result. A MATLAB implementation of this test is available online.Footnote 1

An infinite number of magnitude and phase change combinations can result in a given value of ƍ. These possible combinations are defined by a circle on the complex plane centered on the point \(1+0i\) with a radius ƍ. For a given total change in the range 0 < ƍ < 2, there exist two phase changes (\(\theta <\pm \pi\)), either of which occurring alone (i.e., with no accompanying magnitude change) will result in that same given total change value. These equivalent phase changes for a given value of ƍ can be found by considering ƍ as the length of one leg of a triangle, as shown in Fig. 16. When the lengths of three sides of the triangle are known, the unknown angle (\(\theta\)) can be found by application of the Law of Cosines:

$$\theta =\text\left(\frac^+^-^}\right)$$

(19)

Because the triangle is located on the unit circle, the lengths of \(a\) and \(c\) are both 1, and the equation simplifies to

(20)

Calculation of equivalent phase change given total change (ƍ) using the law of cosines