Carrier synchronization

Direct Recovery Method#

The commonly used method is the Costas loop, and the principle diagram is as follows:

Assuming that the orthogonal carrier signal output by the NCO is:

$\begin{aligned}y_1&=\cos(w_ct+\theta)\y_2&=\sin(w_ct+\theta)\end{aligned}$

$\theta$ is the phase difference between the demodulation end NCO output carrier signal and the modulation end carrier signal, which is usually very small.
The modulated signal $m(t)cos(\omega_ct+\theta)$ multiplied by $y_1,y_2$ respectively yields:

$\begin{aligned}y_3&=m(t)cosw_ct\cos(w_ct+\theta)=\frac12m(t)[cos\theta+\cos(2w_ct+\theta)]\y_4&=m(t)cosw_ct\sin(w_ct+\theta)=\frac12m(t)[sin\theta+sin(2w_ct+\theta)]\end{aligned}$

After low-pass filtering, we get:

$\begin{aligned}y_5&=\frac12m(t)cos\theta\y_6&=\frac12m(t)sin\theta\end{aligned}$

Multiplying them together, we get the error signal:

$y_7=\frac{1}{8}m^2(t)\sin2\theta\approx\frac18m^2(t)2\theta=\frac14m^2(t)$

$y_7$ is used to control the phase control word and frequency control word of the NCO after loop filtering, ultimately reducing the carrier frequency difference between the NCO output carrier signal and the modulation signal to a small value.
The output $y_1=\cos(\omega_ct+\theta)$ of the NCO is the recovered demodulated carrier.

Loop Filter#

The main function of the loop filter is to filter out the high-frequency error signal output by the frequency discriminator and provide frequency control word error compensation.

The coefficient $C_1$ of the phase tracking module controls the phase performance of the loop filter output, and the coefficient $C_2$ of the frequency tracking module controls the frequency performance of the loop filter output. The theoretical calculation formulas are:

$\begin{aligned}C_1&=\frac{2\delta w_nT}{K_d}\C_2&=\frac{(w_nT)^2}{K_d}\end{aligned}$

$\delta$ represents the damping coefficient of the loop filter system, usually equal to 0.707.
$\omega_n$ represents the oscillation angular frequency of the loop filter system, and the calculation formula is:

\omega_n = \frac{8\delta B}{4\delta^2+1} \end{aligned}$$ $B$ represents the noise bandwidth of the loop filter, usually 1/10 or 1/100 of the symbol rate. The larger the value, the larger the frequency capture range of the loop, the faster it enters the lock state, the more noise it generates, and the more it is affected by noise. Therefore, the frequency and phase jitter of the loop output also increase. $T$ represents the update time period of the frequency control word of the NCO. $K_d$ is the loop gain, which is related to the gain $K_p$ of the frequency discriminator and the gain $K_0$ of the NCO. $K_p$ is generally set to 0.03, and $K_0$ is calculated as: $$K_0=\frac{f_c}{2^N\times f_s},\\ f_c is the driving clock of the NCO, f_s is the sampling clock of the NCO output, and N is the bit width of the NCO phase accumulator.$$ The most important thing in the design of the loop filter is the selection of the loop bandwidth. ### NCO Assuming the frequency control word is S, then $\frac{S}{2^N}=\frac{f_{out}}{f_s}$ ### Improved Costas ![image.png](ipfs://QmSup36AP68sTAWAMW2nRua9M7BC7dAJ54sRKtS7E1ybDG)