  In this paper, we consider an OFDM radio link corrupted by oscillator phase
noise in the receiver, namely the problem of estimating and compensating for
the impairment. To lessen the computational burden and delay incurred onto the
receiver, we estimate phase noise using only scattered pilot subcarriers, i.e.,
no tentative symbol decisions are used in obtaining and improving the phase
noise estimate. In particular, the phase noise estimation problem is posed as
an unconstrained optimization problem whose minimizer suffers from the
so-called amplitude and phase estimation error. These errors arise due to
receiver noise, estimation from limited scattered pilot subcarriers and
estimation using a dimensionality reduction model. It is empirically shown
that, at high signal-to-noise-ratios, the phase estimation error is small. To
reduce the amplitude estimation error, we restrict the minimizer to be drawn
from the so-called phase noise geometry set when minimizing the cost function.
The resulting optimization problem is a non-convex program. However, using the
S-procedure for quadratic equalities, we show that the optimal solution can be
obtained by solving the convex dual problem. We also consider a less complex
heuristic scheme that achieves the same objective of restricting the minimizer
to the phase noise geometry set. Through simulations, we demonstrate improved
coded bit-error-rate and phase noise estimation error performance when
enforcing the phase noise geometry. For example, at high
signal-to-noise-ratios, the probability density function of the phase noise
estimation error exhibits thinner tails which results in lower bit-error-rate.
