As part of a project I am doing, I have to investigate how having a carrier frequency offset would affect the constellation pattern of the received signal.This question has got me baffled because to me a carrier frequency offset should not affect the constellation pattern because any offset would be dealt with on the receiver side.I might not be understanding some concepts very clearly.
ie. On Transmit side we transmit
$$ s(t) = x(t)\cos(2 \pi (f_c + f _\phi)) $$$$ s(t) = x(t)\cos(2 \pi (f_c + f _\phi) t) $$
then on receive side
$$ r(t) = 2 \text { LPF } (s(t) \cos(2 \pi (f_c + f_\phi)) ) = x(t) $$$$ r(t) = 2 \text { LPF } (s(t) \cos(2 \pi (f_c + f_\phi)) t) = x(t) $$
So regardless of the offset we have on the carrier frequency there should no be no effect on the constellation diagram.Any help will be appreciated?