# Why does fine CFO cause rotation of the constellation?

Just thinking about it, why does fine CFO cause rotation only? I only see diagrams that show rotation of the constellation.

We assume this is a complex magnitude frequency domain plot

In order for rotation to occur, the complex magnitude would have to stay the same when CFO occurs but the complex argument differ, but clearly, if fine CFO occurs here, it will be sampling a point on the envelope that isn't at the magnitude of one of the points on the constellation after gain equalisation, but inbetween. Especiallly as the magnitudes of the subcarriers can differ if using 16QAM like below.

This does not reflect the typical constellation I see:

CFO is modeled like this $$x(t) = s(t) e^{j2\pi f_{\text{CFO}}t}$$. This is a phase rotation only and does not effect the magnitude. Convince yourself of it by breaking $$e^{j2\pi f_{\text{CFO}}t}$$ up as it magnitude ($$= 1$$) and its phase ($$=2\pi f_{\text{CFO}}$$). This is all happening in the time domain. If you plot the CFO-free signal and the CFO-corrupted signal magnitude over the top of each other, you should see little to no difference.

This is different from plotting the CFO-free spectrum and the CFO-corrupted spectrum. From the Fourier properties, multiplying by a complex exponential in time will result in a shift in frequency, so the CFO-corrupted spectrum will be shifted by $$f_{\text{CFO}}$$.

## Edit

After seeing the comments, I think I understand the confusion better now. I'm using the term "sampling" to refer to taking a continuous time signal and sampling it. In a receiver, you do it at the symbol period to get a nice clean constellation back.

From what I can tell, the OP uses sampling to mean something else (not wrong just something else). In OFDM, the receiver does a FFT, which is also sampling but in frequency. The CFO introduces a shift in frequency so when the receiver goes to do the FFT, it is going to sample at points where you are getting a component from the main subcarrier and also other subcarriers (ICI).

To answer the question, CFO does not effect the magnitude of the signal (only the phase) and it does effect the magnitude of the spectrum at the subcarrier sampling points.

• I'm not sure I understand. The only way I've seen CFO is in diagrams like this: i.imgur.com/XD42QS0.png where It's a shift in the frequency domain due to mismatch of local oscillator, which means that all the sampling instants are offset, which could mean it would sample where I said (at like 3.5 instead of 4 or 2 on the sum of the sincs), so those bins could be in between and not just rotated on the constellation Nov 2, 2020 at 19:28
• Okay, it's because the magnitudes are actually in different directions isn't it so it's not a sum. I forget that about the complex magnitude plot Nov 2, 2020 at 19:35
• Well actually, the imaginary frequency domain is going to be a bunch of sincs at the amplitudes of the imaginary parts on the constellations and the real frequency domain is going to be a bunch of sincs at the amplitudes of the real parts on the constellations. They are both shifted so the magnitude of both changes and the overall complex magnitude changes. So yeah. I still don't understand Nov 2, 2020 at 22:22
• Turns out I'm right. CFO cause 2 things. ICI, which I describe, which causes magnitude change and a phase change and CPE, which causes an identical phase rotation in all subcarriers. The diagram that i sent must have had very fractional CFO for the effect I described to not take place to such a degree. This diagram makes a lot more sense: i.imgur.com/LgvwWpR.png Nov 3, 2020 at 2:07
• I will add that the CPE appears to be from the phase noise of the oscillator being convolved with the subcarriers Nov 3, 2020 at 4:33