OFDM principles

Orthogonal Frequency Division Multiplexing (OFDM) [1], [2] is a commonly used modulation scheme in many communication standards. Mathematically, the $k$-th OFDM symbol can be written as $$x_{k,m}=\frac{1}{\sqrt{N_{\mathrm{sub}}}}\cdot\sum_{n=0}^{N_{\mathrm{sub}}-1}s_{k,n}\cdot e^{\mathrm{j}2\pi\frac{nm}{N_{\mathrm{sub}}}}$$ where

• $k$ denotes the symbol index
• $m$ denotes the time index
• $n$ denotes the subcarrier index
• $N_{\textrm{sub}}$ is the total number of subcarriers
• $s_{k,n}$ is the data symbol (e.g. QAM) of the $n$-th subcarrier at the $k$-th time interval

The time domain signal $x_{k,m}$ can be efficiently computed by the inverse fast fourier transformation (IFFT) of length $N_\mathrm{sub}$ $$x_{k,m}=\sqrt{N_\mathrm{sub}}\cdot\operatorname{IFFT}_{N_{\mathrm{sub}}}\left\{ s_{k,n}\right\}$$ or in matrix-vector representation $$\mathbf{x}_{k}=\mathbf{F}^{\mathrm{H}}\cdot\mathbf{s}_{k}$$ where $$\mathbf{F}^{\mathrm{H}}=\frac{1}{\sqrt{N_{\mathrm{sub}}}}\cdot\begin{pmatrix}1 & 1 & 1 & \cdots & 1\\ 1 & u & u^{2} & \cdots & u^{N_{\mathrm{sub}}-1}\\ 1 & u^{2} & u^{4} & \cdots & u^{2(N_{\mathrm{sub}}-1)}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & u^{N_{\mathrm{sub}}-1} & u^{2(N_{\mathrm{sub}}-1)} & \cdots & u^{(N_{\mathrm{sub}}-1)^{2}} \end{pmatrix} \qquad \mathrm{with} \qquad u=e^{\mathrm{j}\frac{2\pi}{N_{\mathrm{sub}}}}$$