# Why does carrier frequency offset (CFO) cause carrier phase error (CPE)?

I've already asked 2 questions on this but there is one discrepancy I cannot reconcile and is the crux of the problem:

\begin{align} \newcommand{\prefix}{\frac1N\sum\limits_{n=0}^{N-1}} R_l &= \prefix r[n]e^{-j2\pi \frac lN n}\\ &= \prefix\sum\limits_{i=0}^{N-1}X_i\, e^{j2\pi \frac iN n} \, e^{-j2\pi \frac lN n} \, e^{j2\pi \frac {\varepsilon}N n}\\ &= X_l \prefix e^{j2\pi \frac {\varepsilon}N n} + \sum\limits_{n=0}^{N-1}\sum\limits_{i=0, i\ne l}^{N-1} X_i e^{j2\pi \frac{i-l+\varepsilon}N n}\\ &= \alpha X_l \underbrace{\color{blue}{e^{j\pi\varepsilon\frac{N-1}N}}\phantom{\hspace{-7.5em}\sum\limits_{i=0, i\ne l}^{N-1} X_i e^{j2\pi \frac{i-l+\varepsilon}N n}}}_\text{CPE} + \underbrace{ \color{darkgreen}{ \sum\limits_{n=0}^{N-1}\sum\limits_{i=0, i\ne l}^{N-1} X_i e^{j2\pi\frac{i-l+\varepsilon}N n} } }_ {\text{ICI}_l} \\ \end{align}

Here it quite clearly shows how the original frequency values get rotated by the CFO frequency i.e. $$X_l$$ is attenuated and phase rotated and the other subcarrier quantities at that frequency is the ICI

But I cannot reconcile this with what CFO actually does, which is shift the frequency domain. The frequency domain should consist of sincs in the imaginary and real domains at the heights of the constellation points.

Imagine the constellation point $$2j+2$$ as $$X_l$$: When this is shifted, the result of the DFT will be a sampling the point of the cross in the frequency domain. This does not change the phase, only the magnitude, because it will now be $$1.7j+1.7$$ at that frequency as shown.

I just cannot reconcile this latter scenario to show the phase rotation of the first scenario, when the imaginary and real part are of the same magnitude, there is no phase change

• A frequency offset is a linear phase versus time, so the phase offset will occur and be time dependent Nov 28 '20 at 14:33