Generating a Sinusoid

Consider a rotating vector in the 2D Cartesian plane. Projections of this vector onto the horizontal and vertical axes yield sinusoids. Start with a vector along the horizontal axis.

$\large{\vec{v} = x \hat{\imath} + y \hat{\jmath}}$

Form a quadrature vector: $\large{\vec{v_q} = -y \hat{\imath} + x \hat{\jmath}}$

Add a scaled version of the quadrature vector to the original vector:

$\large{\vec{v_{shift}} = x \hat{\imath} + y \hat{\jmath} + \alpha( -y \hat{\imath} + x \hat{\jmath}) = (x - \alpha y)\hat{\imath} + (y + \alpha x)\hat{\jmath} }$

This results in a vector that is shifted by an angle $\theta$ and is also greater in magnitude. Suppose that the original vector $\vec{v}$ is a unit vector, as it is in the problem statement. Then we can divide the shifted vector by $\sqrt{1+\alpha^2}$ to re-normalize. Applying this routine iteratively results in the following:

$\large{x_k = \frac{x_{k-1} - \alpha y_{k-1}}{\sqrt{1+\alpha^2}} \\ y_k = \frac{y_{k-1} + \alpha x_{k-1}}{\sqrt{1+\alpha^2}}}$

To find the angular frequency, consider the difference quotient $\large{\frac{d\theta}{dt}}$ . Recall that $\vec{v}$ is a unit vector.

$\large{\frac{d\theta}{dt} = \frac{\theta_k - \theta_{k-1}}{t_k - t_{k-1}} = \frac{atan(\alpha)}{T}}$

Since the algorithm is run at 1000 Hz, $T = \frac{1}{1000}$ . Plugging in numbers (with $\alpha = \frac{1}{100})$ gives:

$\large{\frac{d\theta}{dt} = \frac{atan(\alpha)}{T} = \frac{atan(\frac{1}{100})}{\frac{1}{1000}} \approx \boxed{10}}$