0.9 Carrier recovery  (Page 16/19)

Indirect frequency estimation

Because the direct method of the previous section is unreliable, this section pursues an alternative strategy based on the observationthat the phase estimates of the PLL “converge” to a line that has a slope proportional to the difference between the actual frequencyof the carrier and the frequency that is assumed at the receiver.(Recall [link] and [link] .) The indirect method cascades two PLLs: the first finds this line(and hence indirectly specifies the frequency), the second converges to a constant appropriate for the phase offset.

The scheme is pictured in [link] . Suppose that the received signal has been preprocessedto form $r_p\left(t\right)=cos(4\pi f_{c}t+2\Phi )$ . This is applied tothe inputs of two PLLs. Or two SD phase tracking algorithms or two Costas loops, though in the latter casethe squaring preprocessing is unnecessary. The top PLL functions exactly as expected from previous sections: if the frequency of its oscillator is $2f_0$ , then the phase estimates $2{\theta }_1$ converge to a ramp with slope $2\pi (f_0-f_c)$ , that is,

where $b$ is the $y$ -intercept of the ramp. The ${\theta }_1$ values are then added to ${\theta }_2$ , the phase estimate in the lower PLL. The output of the bottom oscillator is

Effectively, the top loop has synthesized a signal that has the “correct” frequency for the bottom loop.Accordingly, ${\theta }_2\left(t\right)\to \Phi -b$ . Since a sinusoid with frequency $2\pi f_{0}t$ and “phase” ${\theta }_1\left(t\right)+{\theta }_2\left(t\right)$ is indistinguishable from a sinusoid with frequency $2\pi f_{c}t$ and phase ${\theta }_2\left(t\right)$ , these values can be used to generate a sinusoid that is aligned with $r_p\left(t\right)$ in both frequency and phase. This signal can then be used to demodulatethe received signal.

Some M atlab code to implement this dual PLL scheme is provided by dualplls.m .

The output of this program is shown in [link] . The upper graph shows that ${\theta }_1$ , the phase estimate of the top PLL, converges to a ramp. The middle plot shows that ${\theta }_2$ , the phase estimate of the bottom PLL, converges to a constant.Thus the procedure is working. The bottom graph shows the error between the preprocessed signal rp and a synthesized carrier carest . The parameters f0 , th1 , and th2 can then be used to synthesize a cosine wave that has the correct frequency and phase to demodulate the received signal.