The frequency responses of the receiving channels of a radio astronomy synthesis array and, in particular, the variations between them for different antennas are factors that limit the accuracy and sensitivity of the instrument. A response that approximates a rectangular passband as closely as possible is advantageous from considerations of maximizing sensitivity within a given frequency band and minimizing the smearing of detail in the outer regions of a map. Tolerances on slopes, ripples, and other bandpass distortions can be determined by examining their effects with regard to loss in sensitivity and the introduction of errors in the assignment of complex gain factors for individual antennas by using calibration observations. These calibration errors are generally larger in amplitude than in phase and their effect is usually more serious than the loss in sensitivity. Velocity dispersion in waveguide transmission systems and the accuracy in adjustment of compensating time delays also affect the phase responses of the signal channels, and examination of these effects also leads to limits on system design parameters. Implementation of the tolerances derived, including some details of the filter specifications, is discussed for the very large array radio telescope of the National Radio Astronomy Observatory. Appendices contain an algorithm for performing a least squares solution for antenna gains and an analysis of the variation of the signal-to-noise ratio over the channels of a spectral line correlator system.