Another look at recursive quadrature oscillators

In a recent post, we looked at which 2\times 2 Toeplitz real matrices T gave useful quadrature oscillators by the recurrence x_{n+1}=T x_n. There, we computed their eigenvalues and solved the recurrence in terms of them. Of course, there are many other ways to approach this problem. Here we look at another approach that gives a good geometric picture of what happens, can be applied to general 2\times 2 matrices, and may be used as a starting point for the n\times n case.

We are interested in characterizing the 2\times 2 matrices T such that if we consider the sequence defined by x_{n+1}=T x_n, x_0 \in \mathbb{R}^2 fixed, then its two components are sinusoids of constant phase difference 90º (or perhaps just approximately 90º).

We will denote by R(\omega) the matrix of rotation by \omega in \mathbb{R}^2:

R(\omega)=\begin{pmatrix}\cos\omega &-\sin\omega\\\sin\omega&\cos\omega\end{pmatrix}.

Reasoning as in the previous post, we see that the eigenvalues of T need to have modulus 1 and be complex conjugates. It follows that T is similar to a rotation matrix:

T = PR(\omega)P^{-1}.

Therefore

x_{n} = T^n x_0 = PR(\omega n)P^{-1}x_0.

In fact it makes sense and is worthy to consider the "continuous version" of the recurrence, which has the solution

x(t) = PR(\omega t)P^{-1}x_0,\quad t\in\mathbb{R}.

We denote by P=U\Sigma V^* the singular value decomposition of P. \Sigma is a diagonal matrix with strictly positive entries on the diagonal, and U and V are real orthogonal matrices. We note that V^*R(\omega t)V=R(\pm\omega t) depending on whether V preserves or reverses the orientation. Hence,

x(t) = U\Sigma R(\pm\omega t) \Sigma^{-1}U^*x_0.

We put y_0 = \Sigma^{-1}U^*x_0 and make the change of variables s = \pm\omega t. We have

x(s)=U\Sigma R(s)y_0.

Now we can clearly see geometrically what happens. As s varies along \mathbb{R}, R(s)y_0 traces the circumference of centre 0 and radius \|y_0\|. Then \Sigma stretches this circumference to produce an ellipse (except in the trivial case when \Sigma is a multiple of the identity, which implies that the T we started with was just a rotation matrix). The axes of this ellipse are the coordinate axes. Then U, which is a rotation and perhaps an orthogonal symmetry, just rotates this ellipse, so that its axes may no longer be the coordinate axes.

Therefore, for any x_0 \in \mathbb{R}^2\setminus\{0\}, the orbit x(s) traces an ellipse. Moreover, if we take two nonzero initial data x_0 and \widetilde{x}_0, their corresponding orbits X=\{x(s)\} and \widetilde{X}=\{\widetilde{x}(s)\}, seen as sets, are proportional: \widetilde{X} = (\|\widetilde{x}\|/\|x\|)X. So the initial data x_0 that we choose is not important.

Of course, the components of x(s) are sinusoids of angular frequency 1, because they are linear combinations of the the components of R(s)y_0, which are sinusoids of angular frequency 1 themselves. Hence,

x(s) = \begin{pmatrix}a\cos(s+\theta_0)\\ b\sin(s+\theta_0+\varphi)\end{pmatrix},

for some a> 0, b\neq 0, \theta_0 \in \mathbb{R} and \varphi\in (-\pi/2,\pi/2]. The curve traced by x(s) is an ellipse whose axes are rotated \varphi/2 with respect to the coordinate axes.

It follows that if we want the components of x(s) to be in quadrature, which amounts to \varphi=0, then the axes of the ellipse traced by x(s) should be the coordinate axes. This implies that U performs no rotation at all. It can only permute and change the sign of the vectors in the standard basis. Therefore, we may as well assume that U is the identity, because these simple transformations of the vectors in the standard basis don't make much difference.

We recall that T = U\Sigma R(\omega) \Sigma^{-1} U^*. Putting U = I, denoting the diagonal elements of \Sigma by \alpha,\beta > 0, and computing the matrix product, we see that

T=\begin{pmatrix}\cos\omega & -\frac{1}{\tau}\sin\omega\\ \tau\sin\omega & \cos\omega\\\end{pmatrix},

where \tau=\beta/\alpha. We see that the only matrices that produce a quadrature oscillator are Toeplitz: precisely those which we studied in the previous post.

Perhaps one is interested in matrices T that give oscillators which are almost in quadrature, meaning that \varphi is very small. Of course, this just means that U is essentially a rotation of very small angle, so the matrices T that satisfy this condition may be computed in a similar fashion.

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