Phase tracking

• Tracking linear frequency trajectory by means of adjusting $\hat{\omega}$ given constant and correct model amplitude $A$, and fixed boundary condintions for frequency $\Omega_0$, and phase $\phi_0$.

  
  

  $$\sum_n A^2(\cos((\Omega+\omega n)n+\phi) -\cos((\Omega_0+\hat{\omega} n)n+\phi_0))^2 = \hbox{Min}$$
  
  

• For $N >> 1$ and optimal $\hat{\omega}$ phase error evolution depend on initial frequency error $\Delta_\Omega$ (phase slope error), initial phase error $\Delta_\phi$, block size $N$, and offset relative to blocksize $x$ following
  $$\begin{eqnarray*} \left[\!\!\begin{array}{c} \Delta_W(x)\\ \Delta_\phi(x)\\ \e... ...in{array}{c} \Delta_W\\ \Delta_\phi\\ \end{array}\!\!\right]. \end{eqnarray*}$$
  
  
  
  
  

• Block size does not affect stability.