OFDM principles

Orthogonal Frequency Devision Multiplexing (OFDM) is the choice of LTE/LTE-A due to its efficiency and simplicity using modulation and demodulation stages based on IFFT and FFT. Mathematically, the $k$-th OFDM symbol can be written as $$ x_{k}(t) = \sum_{n=0}^{N_{\textrm{sub}}-1} s_{k,n} \textrm{rect}\left(t-kT\right) e^{j\frac{2\pi}{T}nt}, $$ where

- $n$ denotes the subcarrier and $k$ denotes symbol index, respectively
- $N_{\textrm{sub}}$ is the total number of subcarriers
- $\rm{rect}\left(\cdot\right)$ is a rectangular pulse with the symbol period of $T$
- $s_{k,n}$ is the data symbol (e.g. QAM) of $n$-th subcarrier at $k$-th time interval

To allow the transition period of channel, cyclic prefix (CP) is added between consecutive symbols. The inserted CP serves as a guard time between symbols which protects against ISI. In addition, it preserves the orthogonality between subcarriers after passing a channel provided the CP duration is longer than the channel transition period.