Synchronisation

The carrier frequency offset $\Delta f_{\text{offset}}$ is the difference between the transmitter and receiver local oscillator frequencies. In OFDM, it is normalised to the subcarrier spacing:

$$\epsilon = \frac{\Delta f_{\text{offset}}}{\Delta f}$$

The CFO is decomposed into an integer part $\epsilon_I = \lfloor \epsilon \rceil$ and a fractional part $\epsilon_F = \epsilon - \epsilon_I$, where $|\epsilon_F| \leq 0.5$.

• Integer CFO $\epsilon_I$: shifts the subcarrier indices by $\epsilon_I$ positions (cyclic shift in frequency), causing subcarrier misalignment but no ICI.

• Fractional CFO $\epsilon_F$: destroys orthogonality and causes inter-carrier interference between all subcarriers.

ICI occurs when subcarrier orthogonality is broken, causing each subcarrier to interfere with all others. With a normalised fractional CFO $\epsilon_F$, the received signal on subcarrier $k$ after DFT becomes:

$$Y[k] = \underbrace{S(\epsilon_F)\, H[k]\, X[k]}_{\text{desired}} + \underbrace{\sum_{\substack{m=0 \\ m \neq k}}^{N-1} S(m - k + \epsilon_F)\, H[m]\, X[m]}_{\text{ICI}} + W[k]$$

where the ICI coefficient is:

$$S(p) = \frac{\sin(\pi p)}{N \sin(\pi p/N)}\, e^{j\pi p(N-1)/N}$$

The desired signal is attenuated by $|S(\epsilon_F)| < 1$, and ICI from all other subcarriers is added.

A symbol timing offset $\delta$ (in samples) means the receiver's DFT window starts $\delta$ samples away from the ideal position.

• If $|\delta| \leq N_{\text{cp}} - L + 1$ (within the CP margin), the timing error causes only a linear phase rotation across subcarriers: $Y[k] \to Y[k] e^{-j2\pi k\delta/N}$, which can be absorbed into the channel estimate.

• If $|\delta| > N_{\text{cp}} - L + 1$, the DFT window extends outside the ISI-free region, causing ISI and ICI.