Orthogonal Frequency-division Multiplexing - Mathematical Description

Mathematical Description

If sub-carriers are used, and each sub-carrier is modulated using alternative symbols, the OFDM symbol alphabet consists of combined symbols.

The low-pass equivalent OFDM signal is expressed as:

\ \nu(t)=\sum_{k=0}^{N-1}X_k e^{j2\pi kt/T}, \quad 0\le t<T,

where are the data symbols, is the number of sub-carriers, and is the OFDM symbol time. The sub-carrier spacing of makes them orthogonal over each symbol period; this property is expressed as:

\begin{align} &\frac{1}{T}\int_0^{T}\left(e^{j2\pi k_1t/T}\right)^* \left(e^{j2\pi k_2t/T}\right)dt \\ = &\frac{1}{T}\int_0^{T}e^{j2\pi (k_2-k_1)t/T}dt = \delta_{k_1k_2} \end{align}

where denotes the complex conjugate operator and is the Kronecker delta.

To avoid intersymbol interference in multipath fading channels, a guard interval of length is inserted prior to the OFDM block. During this interval, a cyclic prefix is transmitted such that the signal in the interval equals the signal in the interval . The OFDM signal with cyclic prefix is thus:

The low-pass signal 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 . In general, the transmitted signal can be represented as:

\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 t + \arg\right) \end{align}

Read more about this topic:  Orthogonal Frequency-division Multiplexing

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