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

Famous quotes containing the words mathematical and/or description:

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)

    “Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.”
    —Paul Tillich (1886–1965)