Enhancement: the QUASAR system (Ding and Qian, JAES 1997)

There is a problem with the above: "exact" cubic phase interpolation can lead to large frequency deviations (see overhead figure). A better solution is to assume that some of the detected phases may have been in error (likely, given chirping effects and noise), and are less important to match than the general frequency trajectory:

$$\theta_m(t) = a_m + b_m t + c_m t^2$$

To implement this, a least squares formulation is needed. To do this, we use the second order spline model:

$$\theta(t) = \sum_{n=-2}^{N-1} \alpha^n B_n(t)$$

where $\alpha$ is the weight given to a particular "second order basis spine" $B$ (see overhead figure). So, we minimize the weighted squared error:

$$\epsilon = \lambda \sum_{n=0}^{N}[\theta(t_n) - \hat{\theta... ...1-\lambda) \sum_{n=0}^{N}[\omega(t_n) - \hat{\omega}^n]^2 T^2$$

$\lambda$ is relative importance of phase mismatch versus frequency mismatch. To more explicitly formulate the problem, we can reduce it to a least squares problem in the basis splines:

$$\epsilon = \lambda \sum_{n=0}^{N}[\theta(t_n) - \hat{\theta... ...1-\lambda) \sum_{n=0}^{N}[\omega(t_n) - \hat{\omega}^n]^2 T^2$$

Since it may be shown that

$$\theta(t_n) = \alpha^{n-2} B_{n-2}(t_n) + \alpha^{n-1} B_{n-1}(t_n)$$ (4)

$$= \ensuremath{\frac{1}{2}}(\alpha^{n-2} + \alpha^{n-1})$$ (5)

$$\omega(t_n) = \alpha^{n-2} B'_{n-2}(t_n) + \alpha^{n-1} B'_{n-1}(t_n)$$ (6)

$$= \ensuremath{\frac{1}{T}}(-\alpha^{n-2} + \alpha^{n-1}),$$ (7)

we can convert it to a problem in which we solve for the $\alpha$ values:

$$\epsilon = \lambda \sum_{n=0}^{N}\left[\ensuremath{\frac{1... ...n=0}^{N}[(-\alpha^{n-2} + \alpha^{n-1}) - \hat{\omega}^n T]^2$$

To solve for the $\alpha$ values, we take the partial derivatives of $\epsilon$ wrt each $\alpha$, and use the resulting equations to formulate a least-squares problem (a la EE263). Solving this problem gives the $\alpha$ coefficients that give the optimal frequency / phase trajectory in terms of the $\lambda$ we chose earlier.