I and Q analysis of a Costas phase-locked loop with mismatched filters

There exist a wide variety of communication systems in operation that utilize different modulation bandwidths. The first purpose of this paper is to analyze the phase-error variance of a Costas phase-locked loop (PLL) when there is a mismatch between the modulation bandwidth and the filter bandwidth in the receiver. One such example occurs in signal intelligence (SIGINT) operations where the receiver does not possess perfect knowledge of the parameters of the received signal. The second purpose of this work is to analyze the Costas circuit when perfect phase-lock is not achieved. That is, to ascertain the fraction of signal and noise power that leaks from the in-phase channel into the quadrature channel, and the fraction of noise power that leaks from the quadrature channel into the in-phase channel. While the phase-error variance is already well known for the Costas PLL with matched filters, it is not a trivial extension to the case when there is a filter mismatch in the system