Editing Orthogonal frequency-division multiplexing (section)

==Mathematical description==
[[File:N-OFDM.jpg|thumb|500 px|Subcarriers system of OFDM signals after [[FFT]]]]
If <math>N</math> subcarriers are used, and each subcarrier is modulated using <math>M</math> alternative symbols, the OFDM symbol alphabet consists of <math>M^N</math> combined symbols.

The [[low-pass equivalent]] OFDM filter is expressed as:
:<math>
  \nu(t) = \sum_{k=0}^{N-1} X_k e^{j2\pi kt/T},\quad 0 \le t < T,
</math>
where <math>\{X_k\}</math> are the data symbols, <math>N</math> is the number of subcarriers, and <math>T</math> is the OFDM symbol time. The subcarrier spacing of <math display="inline">\frac{1}{T}</math> makes them orthogonal over each symbol period; this property is expressed as:
:<math>\begin{align}
        &\frac{1}{T}\int_0^{T}\left(e^{j2\pi k_1 t/T}\right)^*
\left(e^{j2\pi k_2t/T}\right)dt \\
  {}={} &\frac{1}{T}\int_0^{T} e^{j2\pi\left(k_2 - k_1\right)t/T}dt = \delta_{k_1 k_2}
\end{align}</math>

where <math>(\cdot)^*</math> denotes the [[complex conjugate]] operator and <math>\delta\,</math> is the [[Kronecker delta]].

To avoid intersymbol interference in multipath fading channels, a guard interval of length <math>T_\text{g}</math> is inserted prior to the OFDM block. During this interval, a ''cyclic prefix'' is transmitted such that the signal in the interval <math>-T_\text{g} \le t < 0</math> equals the signal in the interval <math>(T - T_\text{g}) \le t < T</math>. The OFDM signal with cyclic prefix is thus:
:<math>\nu(t) = \sum_{k=0}^{N-1}X_k e^{j2\pi kt/T}, \quad -T_\text{g} \le t < T</math>

The low-pass signal filter above can be either real or complex-valued. Real-valued low-pass equivalent signals are typically transmitted at baseband—wireline applications such as DSL use this approach. For wireless applications, the low-pass signal is typically complex-valued; in which case, the transmitted signal is up-converted to a carrier frequency <math>f_\text{c}</math>. In general, the transmitted signal can be represented as:
:<math>\begin{align}
  s(t) &= \Re\left\{\nu(t) e^{j2\pi f_c t}\right\} \\
       &= \sum_{k=0}^{N-1}|X_k|\cos\left(2\pi \left[f_\text{c} + \frac{k}{T}\right]t + \arg[X_k]\right)
\end{align}</math>