Phase Error Statistics in Costas Loops Due to Channel Distortion and Noise

This paper treats the problem of phase recovery loop performance loss due to a nonideal RF channel. The linear channel under consideration is assumed to suffer from a variety of impairments such as bandwidth limitation, hardware imperfections, and asymmetry of its transfer function. The phase error probability density function of a first-order Costas loop is presented. It has been determined that the phase error is biased in general, and the condition for eliminating this bias is pointed out. To avoid computational complexity and elude computer programming, a simple method is presented for hand computation of the ISI intensity in the loop. Channel bandwidth reduction also introduces phase jitter which can be controlled by proper signal spectrum shaping. For example, it is possible to communicate with a spectral efficiency of 1 bit/s/Hz without degrading the loop performance. Examples have been presented to examine a few practical problems.