Modified Coupled Form Resonator

Robust behavior in fixed-point implementations using short word lengths, including perfect stability in the undamped case, can be obtained by replacing the complex multiply in Equations (12-13) with a damped version of the so-called ``modified coupled form'' (MCF) sine oscillator [2]:

$$x(n+1) = r_1 [x(n) - \epsilon y(n)] + b_1 u(n)$$ (27)

$$y(n+1) = r_1 [\epsilon x(n+1) + y(n)] + b_2 u(n)$$ (28)

where $\epsilon = 2\sin(\theta_1/2)$, and nominally $b_1=1$ and $b_2=0$. This recursion is highly insensitive to round-off error. When excited by an impulse ( $u(n)=\delta(n)$) with no damping ($r_1=1$), the signals $[x(n),y(n)]$, viewed as coordinates in the complex plane, follow a fixed elliptical orbit over time. As $\epsilon\to0$, the ellipse approaches a perfect circle. The complex multiply algorithm of Equations (12-13), on the other hand, generates an exact circle for all frequencies in the absence of quantization errors.³The component sinusoids $x(n)$ and $y(n)$ remain samples of pure sinusoids at the same frequency, as in the complex multiply case. The elliptical orbit is due only to the components having a relative phase difference other than 90 degrees. The relative phase approaches 90 degrees as the oscillation frequency (pole angle) approaches zero.