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:

    “All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.”
    —Roger Bacon (c. 1214–c. 1294)

    “I was here first introduced to Joe.... He was a good-looking Indian, twenty-four years old, apparently of unmixed blood, short and stout, with a broad face and reddish complexion, and eyes, methinks, narrower and more turned up at the outer corners than ours, answering to the description of his race. Besides his underclothing, he wore a red flannel shirt, woolen pants, and a black Kossuth hat, the ordinary dress of the lumberman, and, to a considerable extent, of the Penobscot Indian.”
    —Henry David Thoreau (1817–1862)