Editing Orthogonal frequency-division multiplexing (section)

==Wavelet-OFDM==

OFDM has become an interesting technique for power line communications (PLC). In this area of research, a wavelet transform is introduced to replace the DFT as the method of creating orthogonal frequencies. This is due to the advantages wavelets offer, which are particularly useful on noisy power lines.<ref name=WaveletOFDM>{{cite book |author=S. Galli |author2=H. Koga |author3=N. Nodokama |title=2008 IEEE International Symposium on Power Line Communications and Its Applications |chapter=Advanced signal processing for PLCS: Wavelet-OFDM |at=pp. 187–192 |date=May 2008 |doi=10.1109/ISPLC.2008.4510421 |url=https://www.researchgate.net/publication/4330667|isbn=978-1-4244-1975-3 |s2cid=12146430 }}</ref>

Instead of using an IDFT to create the sender signal, the wavelet OFDM uses a synthesis bank consisting of a <math>N</math>-band transmultiplexer followed by the transform function

:<math> F_n(z) = \sum_{k=0}^{L-1} f_n(k) z^{-k}, \quad 0 \leq n < N </math>

On the receiver side, an analysis bank is used to demodulate the signal again. This bank contains an inverse transform

:<math> G_n(z) = \sum_{k=0}^{L-1} g_n(k) z^{-k}, \quad 0 \leq n < N </math>

followed by another <math>N</math>-band transmultiplexer. The relationship between both transform functions is

:<math>\begin{align}
  f_n(k) &= g_n(L - 1 - k) \\
  F_n(z) &= z^{-(L-1)} G_n * (z - 1)
\end{align}</math>

An example of W-OFDM uses the Perfect Reconstruction Cosine Modulated Filter Bank (PR-CMFB)<ref>{{cite journal |last1=Koilpillai |first1=R. D. |last2=Vaidyanathan |first2=P. P. |title=Cosine-modulated FIR filter banks satisfying perfect reconstruction |journal=IEEE Transactions on Signal Processing |date=April 1992 |volume=40 |issue=4 |pages=770–783 |doi=10.1109/78.127951|bibcode=1992ITSP...40..770K }}</ref> and Extended Lapped Transform (ELT)<ref>{{cite journal |last1=Malvar |first1=Henrique |title=Extended lapped transforms: properties, applications, and fast algorithms |journal=IEEE Transactions on Signal Processing |date=November 1992 |volume=40 |issue=11 |pages=2703–2714 |doi=10.1109/78.165657|bibcode=1992ITSP...40.2703M }}</ref><ref>{{cite book |last1=Malvar |first1=Henrique |title=Signal Processing with Lapped Transforms |date=November 1991 |publisher=Artech House |location=Norwood, MA |isbn=9780890064672 |url=https://us.artechhouse.com/Signal-Processing-with-Lapped-Transforms-P354.aspx}}</ref> is used for the wavelet TF. Thus, <math>\textstyle f_n (k)</math> and <math>\textstyle g_n (k)</math> are given as

:<math>\begin{align}
  f_n (k) &= 2 p_0(k) \cos \left[ \frac{\pi}{N}\left(n + \frac{1}{2}\right)\left(k - \frac{L-1}{2}\right) - (-1)^{n} \frac{\pi}{4} \right] \\
  g_n (k) &= 2 p_0(k) \cos \left[ \frac{\pi}{N}\left(n + \frac{1}{2}\right)\left(k - \frac{L-1}{2}\right) + (-1)^{n} \frac{\pi}{4} \right] \\
   P_0(z) &= \sum_{k=0}^{N-1} z^{-k} Y_k\left(z^{2N}\right)
\end{align}</math>

These two functions	are their respective inverses, and can be used to modulate and demodulate a given input sequence. Just as in the case of DFT, the wavelet transform creates orthogonal waves with <math>\textstyle f_0</math>, <math>\textstyle f_1</math>, ..., <math>\textstyle f_{N-1}</math>. The orthogonality ensures that they do not interfere with each other and can be sent simultaneously. At the receiver, <math>\textstyle g_0</math>, <math>\textstyle g_1</math>, ..., <math>\textstyle g_{N-1}</math> are used to reconstruct the data sequence once more.

===Advantages over standard OFDM===

W-OFDM is an evolution of the standard OFDM, with certain advantages.

Mainly, the sidelobe levels of W-OFDM are lower. This results in less ICI, as well as greater robustness to narrowband interference. These two properties are especially useful in PLC, where most of the lines aren't shielded against EM-noise, which creates noisy channels and noise spikes.

A comparison between the two modulation techniques also reveals that the complexity of both algorithms remains approximately the same.<ref name=WaveletOFDM/>