# 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 $\mathbb{R}^2$:

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:

Therefore

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

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,

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

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,

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

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.