Analysis of a Gradient Algorithm for Simultaneous Passband Equalization and Carrier Phase Recovery

A two-dimensional receiver structure has been proposed, incorporating two innovations: passband equalization, which mitigates intersymbol interference, and data-directed carrier recovery and demodulation following equalization, which enables compensation of carrier frequency offset and phase jitter, but does not require transmission of a separate pilot tone with the data signal. The receiver is fully adaptive; the adjustment of the equalizer tap coefficients and of the estimate of the current channel phase shift is based on a gradient algorithm for jointly minimizing the mean squared error with respect to those parameters. In this paper, we analyze the dynamic behavior of the deterministic gradient algorithm (where channel parameters entering into the gradient expression are assumed known in advance). The corresponding estimated gradient algorithm (where these parameters are initially unknown) has previously been studied experimentally, but is not treated here. The first part of the present study concerns system start-up (or transient) response when the channel's phase shift is fixed. Examination of the analytical solution leads to the qualitative conclusion that, if the equalizer tap adaptation coefficient β is small relative to the phase-tracking coefficient α, the added phase estimation feature does not strongly affect the start-up behavior of the passband equalizer under typical operating conditions. Indeed, if the equalizer tap coefficients all start at zero, their evolution in the deterministic gradient algorithm is completely unaffected by the phase-tracking loop. The second situation analyzed is the steady-state response of the system to a constant carrier frequency offset. In this case, the phase-tracking loop is found to reduce the resulting rate of rotation of the equalizer taps to about β/(α + β) of the original frequency offset. As a result, the degradation in system mean squared error due to frequency offset is typically quite small. The final analysis is of the response of a linearized version of the receiver structure to sinusoidal phase jitter. When the channel's linear