The paper deals with carrier recovery based on pilot symbols in single-carrier systems. Wiener's method is used to determine the optimal unconstrained filter in estimation of phase noise assuming that a sequence of equally spaced pilot symbols is available. Our analysis allows to capture two effects that are not considered in the existing literature: the impact of aliasing due to sampling of the phase noise sequence at the pilot rate and the cyclostationary nature of the estimate hence of its performance. Experimental results are derived also for the case, where the filter is constrained to the cascade of two moving averages. These results show that, in the considered example, the mean-square phase error of the constrained filter is within 0.35 dB from the MSE of the optimal filter.

The performance of single-carrier microwave radio systems operating at high spectral efficiency is often limited by the phase noise introduced by local oscillators that generate the sinusoid used for up/down-conversion. As the number of constellation points grows, the S-curve of the conventional decision-directed phase detector used in carrier recovery based on phase-locked loop becomes narrow and narrow. As a consequence, the receiver becomes less and less able to recover the phase noise that affects the incoming carrier, up to the appearance of the phenomenon of cycle slip, that definitively degrades performance. To overcome this limit, pilot symbols can be periodically introduced in the data sequence to improve the robustness of the receiver in estimating the carrier phase, as proposed in [

In the present paper, we consider the optimization of the filter that produces the phase estimate at symbol rate from the noisy observation of the phase noise at pilot rate. One crucial point in pilot-aided carrier recovery is the design of the pilot rate. The pilot rate should be kept as small as possible to maintain low the overhead due to the pilot symbols. However, a too small pilot rate induces substantial aliasing in the sequence of sampled phase noise, compromising the performance of carrier recovery. The novelty of the present contribution is the consideration of aliasing in the design of the carrier recovery mechanism and in the evaluation of its performance.

The paper is organized as follows. In Section

Consider a passband signal of power

Using the pilot symbols the receiver produces the zero-padded sequence

A remark about the implementation of the pilot filter is in order. A common approach in feedforward synchronization is that of

Block diagram of carrier recovery. The block ZP produces a zero-padded sequence at symbol rate by introducing

Given a discrete-time sequence

From (

For

Let

The complex exponentials appearing in (

The model of phase noise considered in this section is the popular

where

To derive specific results, we put

Figure

MSE versus

Figure

Normalized noise bandwidth of the

versus SNR, for

In Figure

MSE versus SNR. Solid line: computed optimal filter. Dashed line: computed suboptimal filter. Crosses: simulation, optimal filter. Squares: simulation, suboptimal filter.

Figure

Impulse response of the planar filter for

The performance of pilot-aided carrier recovery has been analyzed, and the optimal planar filter has been derived taking aliasing into account. The main contribution of the paper is that of having captured the important role of aliasing in the design of pilot rate. Compared to the previous literature [