Relationship between Carrier Frequency Offset and the Constellation Diagram

Question

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

$$ s(t) = x(t)\cos(2 \pi (f_c + f _\phi) t) $$

then on receive side

$$ r(t) = 2 \text { LPF } (s(t) \cos(2 \pi (f_c + f_\phi)) t) = x(t) $$

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?

Answer 1

I think that you misunderstand the problem. The actual problem is that the frequency offset is unknown to the receiver. I won't give away the answer since this appears to be a homework project, but I'll try to help you by giving a hint. First, imagine what would happen if there was an unknown carrier phase offset at the receiver. How would this affect the constellation? If you know the answer to that question, then think of a carrier frequency offset as a continuing change in phase offset. So how would this affect the constellation?

Answer 2

Simplified Explanation

Assume a MPSK constellation: $$ c_m = e^{(j2\pi m/M + j\pi/4)} $$ where $m = 0, ..., M-1$ and $M$ can be $4, 8, 16$ etc. Now the carrier offset will appear as another complex exponential $$ x(k) = c_m e^{(j2\pi \Delta f k)} $$ Now by definition frequency $\Delta f$ is a changing phase. That means this offset will start changing the phase of the MPSK constellation. As a result I'll see a rotation constellation instead of a static once. My decision boundaries expect a nice static constellation. This will mess up the BER curves beyond recongnition.