0.9 Carrier recovery  (Page 12/19)

In some applications, the Costas loop is considered a better solution than the standard PLL because it can be more robust in thepresence of noise.

Decision directed phase tracking

A method of phase tracking that works only in digital systems exploits the error between the received value and the nearestsymbol. For example, suppose that a $0.9$ is received in a binary $\pm 1$ system, suggesting that a $+$ 1 was transmitted. Then the difference between the $0.9$ and the nearest symbol $+$ 1 provides information that can be used to adjust the phase estimate. This method is called decision directed (DD)because the “decisions” (the choice of the nearest allowable symbol) “direct” (or drive) the adaptation.

To see how this works, let $s\left(t\right)$ be a pulse-shaped signal created from a message in which the symbols are chosenfrom some (finite) alphabet. At the transmitter, $s\left(t\right)$ is modulated by a carrier at frequency $f_c$ with unknown phase $\Phi$ , creating the signal $r\left(t\right)=s\left(t\right)cos(2\pi f_{c}t+\Phi )$ . At the receiver, this signal is demodulated by a sinusoid and then lowpass filtered to create

As shown in Chapter [link] , when the frequencies ( $f_0$ and $f_c$ ) and phases ( $\Phi$ and $\theta$ ) are equal, then $x\left(t\right)=s\left(t\right)$ . In particular, $x\left(k{T}_s\right)=s\left(k{T}_s\right)$ at the sample instants $t=k{T}_s$ , where the $s\left(k{T}_s\right)$ are elements of the alphabet. On the other hand, if $\Phi \ne \theta$ , then $x\left(k{T}_s\right)$ will not be a member of the alphabet. The difference between what $x\left(k{T}_s\right)$ is, and what it should be, can be used to form a performance function and hence a phase tracking algorithm.A quantization function $Q\left(x\right)$ is used to find the nearest element of the symbol alphabet.