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Daubechies-12-Wavelets-funktioner vises i illustrationen. Wavelet-skaleringsfunktionerne ender på "L" - fx: LLL. Basen af disse funktioner (undtagen basisfunktioner (fx LLL) med integrale forskellig fra nul) kan fx anvendes i Wavelet-OFDM. Basisfunktionerne er forskudt noget for, at man lettere kan se dem. Selvom basisfunktionerne lægges oven i hinanden - i samme kompakt støttede tidsinterval (summeres sammen) - kan de i den resulterende summerede funktion med matematisk garanti hver især detekteres, grundet deres indbyrdes matematiske ortogonalitet.[1]
Daubechies-12-Wavelets-funktionernes i frekvensfunktionsrummet.
Benyttet Wavelet filterbank-træ. g er "lavpas" (midlingsfilter) - detekterer Wavelet-skaleringsfunktioner (dukker op i koefficienterne). h er "højpas" (fluktuationsfilter) - detekterer Wavelet-funktioner (dukker op i koefficienterne). Resultatet af de tre filterbank-filtreringer i tre niveauer, er otte underkanaler (output/koefficienter) fås til højre). De otte Daubechies-12-Wavelets-funktioner med rette translation oven i hinanden fra øverste illustration - summeret sammen tidsværdi for tidsværdi, vil give otte koefficienter forskellig fra nul; netop én i hver af de otte underkanaler. (Hvis flere underkanaler ønskes, fortsættes blot med flere filterbank-niveauer.) For hver filterbank filtrering (fra venstre mod højre) fordobles koefficienternes tilhørende basisfunktionslængde (og hermed symbollængden).

Wavelet-modulation er en modulationsteknik, der benytter Wavelet-funktioner til at repræsentere dataene, som skal formidles.


Wavelet-modulation kan relativt let benyttes, så den også formidler mange underkanaler samtidigt ligesom OFDM - og kaldes så Wavelet-OFDM, WOFDM[2], - Wavelet Packet Modulation eller WPM.[3] I WOFDM erstattes de fase og amplitude modulerede OFDM-bærebølger med ortogonale (og typisk statiske) Wavelet-skalaer. Hvis der er Wavelet-skalaer fra flere Wavelet-skalaniveauer kan man kalde det fraktalmodulation eller hierarkisk modulation.

En fordel ved WOFDM er, at guard-intervallet bliver overflødigt med de rette valgte ortogonale Wavelet-funktioner, da disse både er ortogonale med Wavelets fra samme familie - i samme tidsinterval - og for ortogonal tidsforskudte tider.[1][4][5]

Nogle af fordelene ved Wavelet-OFDM, med de rette valgte Wavelet-funktioner, er:[2][6][7][8]

Kompleks Wavelet-OFDM

Wavelet-OFDM kan i stedet for reelle Wavelet-funktioner, anvende (totræs) komplekse wavelet-funktioner (DTCWT) - og kan så kaldes Dual-Tree-Complex-Wavelet-modulation eller DTCW-modulation. Ifølge en artikel skrevet af Mohamed H. M. Nerma, Nidal S. Kamel og Varun Jeoti er fordelen, at signal-til-støj-forholdet kravet kan sænkes ca. 2 dB i forhold til WOFDM og OFDM.[9]

Dual-Tree-Complex-Wavelet repræsentation hævdes at kunne være en af de bedste Wavelet-repræsentations måder at opløse signaler/funktioner i 1D, 2D osv. inkl. med rotationsinvarians i fx 2D - ulempen er redundansen.[10]

Dynamisk Wavelet-modulationsskala valg

Et af formålene med Wavelet-modulation kombineret med fraktalmodulation er, at kunne formidle data ved flere forskellige hastigheder over en radiokanal med ukendt beskaffenhed.[11] Hvis radiokanalen ikke er brugbar ved fx en høj bithastighed, hvilket betyder at signalet ikke kan modtages (for mange bitfejl) under de givne forhold, kan signalet blive sendt ved en lavere bithastighed hvor signal-til-støj forholdet er højere. - Og omvendt - hvis signal-til-støj forholdet viser sig at være højt under de givne forhold, kan der skiftes til en højere bithastighed.


  1. ^ a b Wavelet University: Wavelets 101 - Wavelets for Dummies (Non-Mathematicians) Citat: "...If a single symbol of data were sent (the equivalent of the modulated and summed wavelets covering all of the available bands), then it would create a time limited signal as below...Through careful design and selection of phases and timing, these Wavelet Symbols can be overlapped in time; the spacing between successive symbols need only be a fraction of the width, or length, of the symbol it self..."
  2. ^ a b IEEE 802.16 Broadband Wireless Access Working Group Citat: "...[pdf-side 3]...The following figure illustrates a comparable WOFDM power spectral density, illustrating the additional 35dB of stop-band, allowing much superior adjacent band rejection as compared to an OFDM PHY layer (Power Spectrum Magnitude in dB versus Frequency in MHz)...[pdf-side 6]...The PHY overhead for WOFDM is less than that of Fourier based OFDM modulations. OFDM Guard Intervals of 20% or more are typical for wireless thus giving Wavelet OFDM an advantage of roughly 20% in BW efficiency...[pdf-side 6]...In addition, pilot tones are not necessary for wavelets so that in comparison systems like 802.11a / HiperLan2 uses 4 out of 52 sub-bands for pilots which gives WOFDM another 8% advantage over typical OFDM implementations...[pdf-side 8]...The performance of wavelets in Gaussian noise is nominally the same as QAM and OFDM. However, wavelet modulations are intrinsically a convolutional modulation and further improvements appear possible for the “raw” modulation (i.e. before adding FEC). The effects of multi-path Rayleigh fading are negligible for wavelets when the mean delay is less that 10 % of the symbol time..."
  3. ^ "Wavelet Packet Modulation for Wireless Communications. Antony Jamin, Petri Ma ̈ho ̈nen" (PDF). Arkiveret fra originalen (PDF) 12. december 2015. Hentet 15. april 2015.
  4. ^ Wavelet OFDM Performance in Flat Fading Channels. Marius Oltean Arkiveret 15. april 2015 hos Wayback Machine Citat: "...Abstract – This paper represents an investigation of the wavelet based multi-carrier modulation performance in flat fading channels. The fading envelope is distributed according to a Rayleigh probability density function....Besides its incontestable advantages, OFDM presents some well known drawbacks as: diminished spectral efficiency because of the cyclic prefix (CP) overhead, slow decay of the out-of-band side-lobes, high sensitivity to time and frequency synchronization, increased peak-to- average-power ratio [1,2]....The authors in [5] have shown that wavelet-based OFDM (WOFDM) has better spectral efficiency, is simpler and at least as rapid as OFDM in practical implementations......"
  5. ^ "Performance Analysis of Wavelet OFDM (WOFDM) By Asma Latif, Dr. N.D. Gohar: Performance Analysis (pdf-side 14)" (PDF). Arkiveret fra originalen (PDF) 10. november 2012. Hentet 15. april 2015.
  6. ^ Performance Comparison of Wavelet and FFT Based Multiuser MIMO OFDM over Wireless Rayleigh Fading Channel Arkiveret 15. april 2015 hos Wayback Machine Citat: "...We can have all the benefits of OFDM even if we replace traditional sinusoid carriers of the fourier based OFDM with suitable wavelets. Wavelet based systems have been shown to have better immunity to impulse and narrowband noises than Fourier OFDM [3, 4] also the interference power can to a large extent, be mitigated [5]...Also the performance of equalization in wavelet system is better than conventional OFDM [6]. Wavelet packet modulation will have much lower side lobes in transmitted signals which reduce the inter-carrier interference (ICI) and narrowband interference (NBI) [7]...This article compares the performances of wavelet and FFT based OFDM systems in multiuser environment in terms of error performance and through an extensive computer simulation it is shown that wavelet outperforms the FFT based OFDM...."
  7. ^ 2008: Multy (multi?) binary turbo coded wofdm performance in flat rayleigh fading channels Citat: "...V. CONCLUSIONS AND FURTHER WORK. In this paper we evaluate the BER performance of MBTC associated with two multicarrier modulation techniques: OFDM and its wavelet based version, in a flat fading channel. The WOFDM provides better results than OFDM, especially for high values of the Doppler shift parameter, which quantifies the time variability of the channel. Furthermore, the coded WOFDM performance shows no significant dependency on the wavelets mother used for DWT computation. A surprising result of our simulations is that, unlike for the un-coded system, the BER and FER performance for the MBTC-WOFDM is better for higher values of the Doppler shift..."
  8. ^ Wavelet University: Wavelets 201 - QAM OFDM Wavelets - A Relative Performance Comparison
  9. ^ " OFDM Based on Complex wavelet Transform" (PDF). Arkiveret fra originalen (PDF) 15. april 2015. Hentet 15. april 2015.
  10. ^ Ivan W. Selesnick, Richard G. Baraniuk, and Nick G. Kingsbury: The Dual-Tree Complex Wavelet Transform Citat: "...trouble in paradise: four problems with real solution: complex wavelets...The second school seeks a redundant representation, with both wr(t) and wi(t) individually forming orthonormal or biorthogonal bases. The resulting CWT is a 2x redundant tight frame [26] in 1-D, with the power to overcome the four shortcomings...In this article, we will focus on a particularly natural approach to the second, redundant type of CWT, the dual-tree approach, which is based on two FB trees and thus two bases [55], [57]...From Figure 3, we see that we can reach quite close to the ideal even with quite short filters...As a result, the dual-tree CWT comes very close to mirroring the attractive properties of the Fourier transform, including a smooth, nonoscillating magnitude (see Figure 1); a nearly shift-invariant magnitude with a simple near-linear phase encoding of signal shifts; substantially reduced aliasing; and directional wavelets in higher dimensions. The only cost for all of this is a moderate redundancy: 2x redundancy in 1-D (2^d for d-dimensional signals, in general)...In the following, we describe three methods for FIR dual-tree filter design. Fast implementations of some of these filters have been recently described in[1]...MATLAB software for the dual-tree complex wavelet transform (and related algorithms) is available at the following locations on the web: Arkiveret 22. april 2015 hos Wayback Machine,, and"
  11. ^ Wavelet Modulation in Gaussian and Rayleigh Fading Channels, Manish J. Manglani, (Masters thesis)

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