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Consider a phase-locked loop consisting of a multiplier,

Consider a phase-locked loop consisting of a multiplier,

Consider a phase-locked loop consisting of a multiplier, loop filter, and voltage controlled oscillator (VCO). Let the signal applied to the multiplier input be a PSK signal defined by s(t) = Ac cos [2pfct + kpm(t)] where kp is the phase sensitivity, and the data signal m(t) takes on the value + 1 for binary symbol 1 and – 1 for binary symbol 0. The VCO output is r(t) = Ac sin [2pfct + ?(t)]
(a) Evaluate the loop filter output, assuming that this filter removes only modulated components with carrier frequency 2fc.
(b) Show that this output is proportional to the data signal m(t) when the loop is phase locked, that is, ?(t) = 0.

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