A compromise equalizer design incorporating performance invariance

We give a solution to the problem of designing a fixed compromise equalizer for use in transmission systems involving an ensemble of random channels. The signal and noise spectra, along with the second-order statistics of the channel ensemble, are used to find the equalizer characteristic that minimizes the mean-square distortion between the equalizer output and a scaled version of the transmitter output. The key departure from previous work is that the criterion better captures practical performance invariance; specifically, the cost function incorporates the insensitivity of a well-designed demodulator to any amplitude scaling or time delay introduced by a particular channel. After demonstrating that the optimum equalizer shape is related to the principal eigenfunction of a normalized channel correlation function, we consider several special cases that give further insight into the properties of the solution. We find that the equalizer amplitude is attenuated over those frequencies where the signal-to-noise or signal-to-channel-variance ratios are small. The analysis confirms the standard engineering practice of inverting the average channel in the absence of noise and when the variance of the channel characteristics is small.