Geometry and Shannon Capacity

28/08/2013
OAI : oai:www.see.asso.fr:2552:5065
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Geometry and Shannon Capacity

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Geometry and Shannon Capacity Philippe Jacquet Bell Labs Poisson shot models on uniform maps • Most theoretical models assume: – Wireless device uniformly distributed on with density – F Baccelli, B Blaszczyszyn, "Stochastic geometry and wireless networks," 2009 - 2 R  MISO model • emitters transmit independent flows toward an access point z. – Each flow is noise for the other flows – Taking Shannon for Gaussian (noiseless case):  ,,,1 izz  z  zi Hzperbit )( )( 1log)( 2                        i ij j i zs zs EC  MISO capacity • Emitters distributed like Poisson of density • Signal attenuation and i.i.d. fading • Theorem: • In presence of noise  i i i zz F zs  )(  2for  2log2 )(   C                         i ij j i zsB zs EC )( )( 1log)( 2 2log2 )(lim    C Generalized Poisson shot model • Devices uniformly distributed in >D – MISO capacity theorem • P. Jacquet, "Shannon capacity in poisson wireless network model," 2009 D R 2log )( D C    Uniform Poisson models are not realistic • The world is fractal – Fractal cities Fractal models • Fractal maps – from fractal generators Xxx corps1 Xxx corps2 Xxx corps3 Xxx corps4 X brigade Bernadotte Davout Lanne Augerau Napoléon X brigade1 X brigade2 X brigade3 X brigade4 III regiment II bataillon1 II bataillon2 II bataillon3 II bataillon1 II bataillon2 II bataillon3 I compagnie1 I compagnie2 I compagnie3 I compagnie4 section section1 section2 section3 section4 escouade1 escouade2 escouade3 escouade4 MISO capacity theorem extended to fractal maps • Poisson shot model on fractal map – Eg Sierpinsky triangle – Fractal dimension – Extension of MISO capacity theorem (access point in the fractal map) 2log 3log Fd )(log 2log )(    P d C F  Small periodic fluctuations of mean Present contribution • Simple proof of MISO capacity theorem on uniform map • Extension of MISO capacity theorem on fractal maps (Cantor maps). The energy field theorem • Total energy received • Differential form: the energy field theorem – Independent of fading, etc.  i izsS )()(                 2 2 ) )( )( 1log)( dx S xs EC        2 22 )(log))()((log)( dxSxsSEC   )(log)(    SE d d C   z x Exists with proba 2 dx MISO capacity proof (i.i.d. fading) • Space contraction – By arbitrary factor – increases energy • Equivalent to density increase – By factor )()(   SaS   a D a /1 )()( /1   SaaS D   ))1((loglog))((log SE D SE     2log ))((log)( 2 D SE d d C       z  zi Fractal maps Cantor maps 2 log 2log 2  a dF 2 log 4log 2  a dF a a MISO Capacity in fractal map • Access point at left corner – Contraction by factor – Density increases – a a )()(   SaS    4 ))((loglog))4((log  SEaSE  )(loglog 4log log ))((log  Q a SE  periodic )(log' 4log log ))((log)(    Q a SE d d C  Periodic of Mean zero MISO Capacity in fractal maps • Access point in the fractal map in position z – ),( zC  )(log 4log log )),(( 2  P a zCE  Small periodic fluctuations of mean Numerical analysis • Alternative expression – With – Allows exact analysis via Fourier transform: – Fixed access point – Amplitude of order     0 )( )()(     d efC f    )()1()( )( xdeEf F xs   4,3.0  a 3 10 Numerical analysis • Random access points – Only theoretical upper and lower bounds – Simulations show small amplitudes  Fd  Generalization and perspectives • Processing gain • Ring invariant fading • Environmental noise • More sophisticated fractal maps • Self similar maps )( )( 1log)( 2                        i ij j i zs zs GEC 