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Thus far, analytical expressions have not been developed for the output signal-to-noise ratio of a polarity coincidence correlator when the noise is correlated between the input channels. Approximate expressions are developed here for the output signal-to-noise ratios for two types of cross-correlator systems- one being the polarity coincidence correlator with a hard limiter in each input channel and the other variation of this correlator having a hard limiter in only one input channel. The approximation scheme is based upon the known output signal-to-noise characteristics of a limiter and that of the standard analog correlator with no limiters. The derived expressions are applicable for the case of deterministic sine waves plus narrow-band Gaussian noise in each input channel, where the noise between channels may be correlated For the case of uncorrelated noise between input channels, good agreement is found between the approximate formulas and known exact results. Finally, asymptotic expressions are developed for certain limiting cases of input signal-to-noise ratios having extreme values in one or both input channels.