Digital Communication - Synchronization

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Introduction

There are three kinds of distortion that can cause degradation in signal quality during the demodulation process:

  • Frequency shift ($\Delta f$): This is caused by factors such as the Doppler effect and channel properties.

  • Time delay ($\tau$): This refers to the delay in transmission.

  • Phase shift ($\Delta \phi$): This is the combined effect of frequency shift and time delay.

The mathematical representation of the received signal considering these three distortions is as follows:

$$
r(t) = A v(t - \tau) \cos [2 \pi (f_c + f_d)(t - \tau) + \theta) + n(t)],
$$

which is equivalent to

$$
r(t) = A v(t - \tau) \cos [2 \pi (f_c + f_d) t+ \phi) + n(t)], \quad
\phi = \theta - 2 \pi (f_c + f_d) \tau.
$$

In order to improve the quality of the signal, it is necessary to estimate $f_d$, $\tau$, and $\phi$. This estimation process is referred to as synchronization.

The estimation for the carrier phase is known as carrier phase synchronization and is implemented using a phase-locked loop (PLL).

The estimation for the transmission delay is called symbol time synchronization and is implemented using a delay-locked loop (DLL).

Effect of Synchronization Errors

We will start with a BPSK system.

$$
r(t) = \pm A p_T (t - \tau) \cos [2 \pi (f_c + f_d) (t - \tau) + \theta] + n(t) \
= \pm A p_T (t - \tau) \cos [2 \pi (f_c + f_d) t + \phi] + n(t).
$$

The probability of the bit error is

$$
P_b = Q(\alpha \sqrt{\frac{2 E_b}{N_0}}),
$$

where

$$
\alpha = \frac{1}{T} \int^{T + \min(\tau, \hat{\tau})}_{\max(\tau, \hat{\tau})} \cos [2 \pi (f_d - \hat{f_d}) t + (\phi - \hat{\phi})] dt,
$$

$\alpha \leq 1$ and the equality only holds when $\hat{f_d} = f_d$ and $\hat{\phi} = \phi \mod 2 \pi$. Therefore, the lower bound of the bit error is $Q(\sqrt{\frac{2 E_b}{N_0}})$.

If $|f_d - \hat{f_d} \ll \frac{1}{T}|$, which means the frequency deviation is much less than the carrier frequency, then

$$\alpha \approx (1 - \frac{|\Delta \tau|}{T}) \cos (\Delta \phi).$$

Therefore, the only significant terms for the bit error are the time delay and the phase shift error.