Equations related to decision

1. it has been assumed that the carrier phase is known exactly. An approach to
estimating and tracking the carrier phase can be motivated by minimizing the mean-
squared error with respect to the parameters of the phase sequence generated by the
receiver’s Carrier Recovery block. Suppose this sequence has the form
ϕn = ωcnT + θ
where ωc is the carrier frequency and θ is a fixed unknown phase offset. Replacing
hR,m by
θ in (15.8), we find that the derivative of the mean-squared error with respect to the
phase
offset is

2. Remember that the baseband error is
(nT) = cn−nd− ˜σ(nT) = cn−nd−σ+(nT)e−j(ωcnT+θ)
Therefore,
On replacing ˜(nT) by cn−nd − ˜σ(nT), the following alternative formula for the derivative
is obtained:
This derivative has an interesting physical interpretation. Let
cn−nd = Rcejβc and ˜σ(nT) = Rσejβσ

3. Let be the polar form representations for these two complex numbers. Remember that the
equalized baseband output sample ˜σ(nT) is supposed to be a close approximation to the
ideal symbol cn−nd . Then
m[ cn−nd ˜σ(nT)] = RcRσ sin(βσ − βc)
so

4. This has the same sign as the phase error between the ideal constellation point cn−nd
and the equalized baseband received point ˜σ(nT) if the phase error is not too large, and
is nearly a linear function of the error for small phase errors.
This phase error measure can be used in a phase-locked loop to iteratively adjust θ so
the baseband equalized points are aligned in angle with the ideal constellation points.
Changing θ by some angle has the effect of rotating the baseband equalized points by
the negative of the angle.

5. can be replaced by Rc in (15.48 ). These observations suggest adjusting θ according to
the formula
where k1 is a small positive constant. A practical realization for a second-order
carrier tracking loop based on this equation and including carrier frequency
offset tracking is shown in Figure 15.4. First, an approximation to the phase error
is computed from the baseband equalizer output sample ˜σ(nT) by the formula
During initial startup, a known symbol sequence is often transmitted and the Ideal
Reference
Generator in the receiver replicates these symbols. After the equalizer and carrier
tracking
loop have converged, the outputs of the Slicer are substituted for the known
sequence and the system operates in the decision directed mode. The phase
estimate generated by the lower part of the block diagram is
ϕn+1 = ˆϕn + ωcT + k1Δθ(n) + ψ(n)
where
ψ(n) = ψ(n − 1) + k2Δθ(n)

6. Notice that ωcT is the nominal change in the carrier phase angle between symbols.
When Δθ(n) is zero for all n and the z−1 delay elements are initially cleared, the phase
generated is
The philosophy behind the carrier tracking loop is to increment the phase angle
predicted for the next symbol instant, ˆ ϕn + ωcT, by a small fraction, k1, of the current
phase error estimate Δθ(n). In addition, a fraction, k2, of the phase error is accumulated
to measure any bias caused by a frequency offset, and added to the phase increment.
The system is a second order phase-locked loop similar in behaviour to the ones
discussed in previous chapters. It will track a constant phase and frequency offset with
zero final error. The ratio k1/k2 should be in the order of 100 for good transient
response.