The Fundamental Problem of Synchronization

Phase jumps at every zero crossing from modulating data onto the carrier phase for a QPSK waveform

We have seen in the effect of phase rotation that the matched filter outputs do not map back perfectly onto the expected constellation, even in the absence of noise and no other distortion. Unless this rotation is small enough, it causes the symbol-spaced optimal samples to cross the decision boundary and fall in the wrong decision zone. And even for small rotations, relatively less amount of noise can cause decision errors in this case, i.e., noise margin is reduced. In fact, for higher-order modulation, the rotation becomes even worse because the signals are closely spaced with each other for the same Tx power. Similarly, sampling the incoming Rx signal anywhere other than the zero-ISI instants causes inter-symbol interference (ISI) among neighbouring symbols. A synchronization unit is vital to ensure the proper functionality of a wireless communication system. We explain the fundamental problem of synchronization through the help of the phase rotation impairment.

The task of a phase synchronization unit in a Rx is to estimate this phase offset and de-rotate (since the rotation of a complex number is defined with respect to anticlockwise fashion, de-rotation implies rotating the input in a clockwise direction) the matched filter outputs by this estimate either in a feedforward or a feedback manner. In such a way, the carrier at the Rx can be said to become synchronized with the carrier used at the Tx. We need to analyze the Rx phase as follows.

Breakdown of the Rx phase


The modulated signal arriving at the Rx consists of two different phase components:

  1. Unknown phase shifts arising due to modulating data occurring at symbol rate R_M. For example, in BPSK modulation, phase changes by 0^\circ or 180^\circ at the boundary of each symbol interval T_M. For QPSK, there are four different phase shift possibilities, i.e., 45^\circ, 135^\circ, -135^\circ, or -45^\circ. A similar argument holds for Quadrature Amplitude Modulated (QAM) signals as well.
  2. Unknown phase difference between Tx and Rx local oscillator, \theta_\Delta.

Imagine a QPSK signal input directly into a black box designed to estimate and compensate for the phase and frequency of a sinusoid (a PLL for example). Depending on its design parameters, the mechanism implemented in the black box will try to lock onto the incoming phase within a specific time duration. However, that phase is jumping around by 0^\circ, \pm 90^\circ or 180^\circ at every symbol boundary due to modulating data, never allowing our mechanism to converge. This is the fundamental problem every synchronization subsystem needs to address and is illustrated in Figure below.

Phase jumps at every zero crossing from modulating data onto the carrier phase for a QPSK waveform

We are faced with two options in this situation. Either transmit the synchronization signal in parallel to the data signal that costs increased power and/or bandwidth, or some procedure needs to be invoked to remove the modulation induced phase shifts in the data signal as a result of which

[for a phase offset] the output should become a constant complex number whose phase is our unknown parameter, and
[for a frequency offset] the output should become a simple complex sinusoid whose frequency is our unknown parameter.

Then, either a feedforward (one-shot) estimator or a feedback mechanism (a PLL) can detect this unknown phase. We can either utilize this observation directly to come up with some intuitive phase estimators, or employ our master algorithm — the correlation — to this problem and see where it leads. Interestingly, it turns out that the derivation of phase estimators through the maximum correlation process actually leads to the same intuitively satisfying results. Maximum correlation is a result of a procedure termed as maximum likelihood estimation. Almost all ad hoc synchronization algorithms can be derived through different approximations of the maximum likelihood estimation. For Gaussian noise under some constraints, maximum likelihood and maximum correlation are one and the same thing.

Example


Suppose that the training symbols a_I[m] and a_Q[m] are known (a data-aided system), then their angle can be subtracted from the angle of the matched filter output at each symbol time as

(1)   \begin{equation*}         \hat\theta_\Delta[m] = \measuredangle~ \frac{z_Q(mT_M)}{z_I(mT_M)} - \measuredangle ~\frac{a_Q[m]}{a_I[m]}     \end{equation*}

The intuition this approach is simple: the phase difference between received and expected symbols is the remaining phase offset.

Phase difference between known symbols and downsampled matched filter outputs can be utilized to estimate

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