0.9 Carrier recovery  (Page 9/19)

Use the preceding code to “play with” the phase locked loop algorithm. How does $\mu$ affect the convergence rate? How does $\mu$ affect the oscillations in $\theta$ ? What happens if $\mu$ is too large (say $\mu =1$ )? Does the convergence speed depend on the value of the phase offset?

In pllconverge.m , how much filtering is necessary? Reduce the length of the filter. Does the algorithmstill work with no LPF? Why? How does your filter affect the convergent value of the algorithm?How does your filter affect the tracking of the estimates when $f_0\ne f_c$ ?

The code in pllconverge.m is simplified in the sense that the received signal rp contains just the unmodulated carrier. Implement a more realistic scenario by combining pulrecsig.m to include a binary message sequence, pllpreprocess.m to create rp , and pllconverge.m to recover the unknown phase offset of the carrier.

Using the default values in pulrecsig.m and pllpreprocess.m results in a $\psi$ of zero. [link] provided several situations in which $\psi \ne 0$ . Modify pllconverge.m to allow for nonzero $\psi$ , and verify the code on the cases suggested in [link] .

TRUE or FALSE: The optimum settings of phase recovery for a PLL operating on a preprocessed (i.e. squared and narrowlybandpass filtered at twice the carrier frequency) received PAM signal are unaffected by the channeltransfer function outside a narrow band around the carrier frequency.

Investigate how the PLL algorithm performs when the received signal contains pulse shaped 4-PAM data.Can you choose parameters so that $\theta \to \Phi$ ?

Many variations on the basic PLL theme are possible. Letting $u\left(k{T}_s\right)=r_p\left(k{T}_s\right)cos(2\pi f_{0}k{T}_s+\theta )$ , the preceding PLLcorresponds to a performance function of $J_{PLL}\left(\theta \right)=\text{LPF}\left\{u\left(k{T}_s\right)\right\}$ . Consider the alternative $J\left(\theta \right)=\text{LPF}\left\{u^2\left(k{T}_s\right)\right\}$ , which leadsdirectly to the algorithm This is sensible because $\theta$ that minimize $u^2\left(k{T}_s\right)$ also minimize $u\left(k{T}_s\right)$ .

which is

1. Modify the code in pllconverge.m to “play with” this variation on the PLL. Try a variety of initialvalues theta(1) . Are the convergent values always the same as with the PLL?
2. How does $\mu$ effect the convergence rate?
3. How does $\mu$ effect the oscillations in $\theta$ ?
4. What happens if $\mu$ is too large (say $\mu =1$ )?
5. Does the convergence speed depend on the value of the phase offset?
6. What happens when the LPF is removed (set equal to unity)?
7. Draw the corresponding error surface.