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Phase Shift Measurement and Phase Equalizer

I am attempting to characterise a phase shift in a channel so that I
can compensate it at the receiver end using a phase equalizer. I need a
method which produces a phase shift curve with more measurements than a
Lissajous method produces. I was thinking possibly a phase shift
oscillator may be the way to go with each stage producing a 45 degree
shift, but I am unsure how to use this. I want to define the frequency
and calculate the resulting shift. Also, I am not sure how to use an
equalizing filter. I would probably use an all pass filter but how can
I define where the curve goes?

Please help. I am mightily perplexed!!!