0.9 Carrier recovery  (Page 19/19)

To see the operation of the generalized PLL concretely, pllgeneral.m implements the system of [link] with $G_N\left(z\right)=2-\mu -2z$ and $G_D\left(z\right)=z-1$ . The implementation of the IIR filter follows the time domainmethod in waystofiltIIR.m . When the frequency offset is zero (when f0=fc ), theta converges to phoff . When there is a frequency offset (as in the default values with f0=fc+0.1 ), theta converges to a line. Essentially, this $\theta$ value converges to ${\theta }_1+{\theta }_2$ in the dual loop structure of [link] .

All of the PLL structures are nonlinear, which makes them hard to analyze exactly.One way to study the behavior of a nonlinear system is to replace nonlinear components with nearby linear elements.A linearization is only valid for a small range of values. For example, the nonlinear function $sin\left(x\right)$ is approximately $x$ for small values. This can be seen by looking at $sin\left(x\right)$ near the origin: it is approximately a line with slope one. As $x$ increases or decreases away from the origin, the approximation worsens.

The nonlinear components in a loop structure such as [link] are the oscillators and the mixers.Consider a non-ideal lowpass filter $F\left(z\right)$ with impulse response $f\left[k\right]$ that has a cutoff frequency below $2f_0$ . The output of the $\text{LPF}$ is $e\left[k\right]=f\left[k\right]*sin\left(2\theta \right[k]-2\Phi [k\left]\right)$ where $*$ represents convolution. With $\tilde{\Phi }\equiv \theta -\Phi$ and $\Phi \approx \theta$ , $sin\left(2\tilde{\Phi }\left[k\right]\right)\approx 2\tilde{\Phi }\left[k\right]$ . Thus the linearization is

In words, applying a $\text{LPF}$ to $sin\left(x\right)$ is approximately the same as applying the $\text{LPF}$ to $x$ , at least for sufficiently small $x$ .

There are two ways to understand why the general PLL structure in [link] works, both of which involve linearization arguments.The first method linearizes [link] and applies the final value theorem for $\mathcal{Z}$ -transforms. The second method is toshow that a linearization of the generalized single-loop structure of [link] is the same as a linearization of the dual-loop method of [link] , at least for certain values of the parameters. [link] shows that for certain special cases, the two strategies (the dual-loop and the generalized single-loop)behave similarly in the sense that both have the same linearization.

For further reading

• J.P. Costas, “Synchronous Communications," Proceedings of the IRE , pp. 1713–1718, Dec. 1956.
• L. E. Franks, “Carrier and Bit Synchronization in Data Communication—A Tutorial Review,” IEEE Transactions on Communications , vol. COM-28, no. 8, pp. 1107–1120, August 1980.