18 that the input to the matched filter r t can be expressed as a heisenberg representation with impulse response f u03c4 u03bd plus noise the output of the matched filter then has two contributions the last term is the contribution of noise which we will denote v u03c4 u03bd a g r u03bd u03c4 u03bd the first term on the right hand side is the matched filter output to the noiseless input comprising of a superposition of delayed and modulated versions of the transmit pulse we next establish that this term can be expressed as the twisted convolution of the two dimensional impulse response f u03c4 u03bd with the cross ambiguity function or two dimensional cross correlation of the transmit and receive pulses theorem 1 fundamental time frequency domain channel equation if the received signal can be expressed as then the cross ambiguity of that signal with the receive pulse g tr t can be expressed as recall from 19 that f u03c4 u03bd h u03c4 u03bd x n m that is the composite impulse response is itself a twisted convolution of the channel response and the modulation symbols substituting f u03c4 u03bd from 19 into 22 we obtain the end to end channel description in the time frequency domain where v u03c4 u03bd is the additive noise term eq 25 provides.