Comment calculer la capacité Shannon-Hartley C=B*log2(1+SNR)

Calculez capacité maximale canal Shannon-Hartley C=B*log2(1+S/N) bit/s. Claude Shannon Bell Labs 1948 + Hartley 1928. WiFi 20 MHz SNR 25 dB=166 Mbps, 5G NR 100 MHz=663 Mbps, GSM 200 kHz, LoRa IoT 125 kHz France Orange.

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Comment calculer la capacité d'un canal de communication: théorème fondamental de théorie information formule par Claude Shannon (USA 1916-2001) Bell Labs Murray Hill NJ 1948, base sur travail Ralph Hartley (USA 1888-1970) Bell Labs 1928. Formule fondamentale: C = B * log2(1 + S/N), ou C capacité maximale théorique canal (bits/s), B largeur bande (Hz), S puissance signal (W), N puissance bruit (W), S/N rapport signal-bruit SNR. Equivalence dB: SNR_linear = 10^(SNR_dB/10). Exemples: telephone POTS 3,4 kHz SNR=30 dB capacité=33,9 kbps; WiFi 20 MHz SNR=25 dB cap=166 Mbps; 5G NR 100 MHz SNR=20 dB cap=663 Mbps; satellite GEO 36 MHz SNR=10 dB cap=125 Mbps; IoT LoRa 125 kHz SNR=-5 dB cap=37 kbps. Outil essentiel pour: design modems fibre optique DSL ADSL VDSL FTTH (Orange Paris-Issy France SFR Bouygues Telecom Free Iliad), planification 4G LTE 5G NR cellular (Orange SFR Free Bouygues), design WiFi router IEEE 802.11 ax/be Wi-Fi 6/7, communication satellite Eutelsat Paris leader Europe, IoT LoRaWAN (Sigfox Labege France originaire LPWAN), modulation adaptative BPSK QPSK 16QAM 64QAM 256QAM 1024QAM 4096QAM, deep space NASA Voyager Pioneer LDPC Turbo coding, fibre optique cohérent QAM modems Nokia Bell Labs Espoo Finlande (former Lucent).

Formules capacité Shannon-Hartley

Théorème Shannon-Hartley (1948): C = B*log2(1+S/N). Pour S/N>>1: C ~ B*log2(S/N). Pour S/N<<1: C ~ B*S/(N*ln2). Efficacite spectrale eta = C/B = log2(1+SNR) bit/s/Hz. WiFi 802.11ax 1024QAM eta_pratique ~10 bit/s/Hz, Shannon limit théorique eta_max~12 bit/s/Hz a SNR 35 dB.

Conversion dB-linear SNR: SNR_dB = 10*log10(SNR_linear). 0 dB=1, 10 dB=10, 20 dB=100, 30 dB=1000.

Efficacite spectrale modulation: BPSK 1 bit/symbol, QPSK 2, 16QAM 4, 64QAM 6, 256QAM 8, 1024QAM 10, 4096QAM 12 bits/symbol. SNR threshold (BER 10^-6 AWGN): BPSK 11 dB, 16QAM 18 dB, 256QAM 30 dB. LDPC turbo gain 5-10 dB - originaire francais Claude Berrou ENST Bretagne 1993.

Bandwidth standards: POTS 3,4 kHz, AM 9 kHz, FM 200 kHz, GSM 200 kHz, UMTS 5 MHz, LTE 1,4-20 MHz, 5G NR sub-6 GHz 5-100 MHz, 5G mmWave 50-400 MHz, WiFi 20-320 MHz, satellite C-band 36 MHz, fibre optique DWDM 50 GHz.

Noise floor thermique: N = -174 dBm/Hz + 10*log10(B_Hz). B=20 MHz noise=-101 dBm. France: LNE Laboratoire National Essais Paris référence calibration.

Histoire et contexte

Claude Shannon (USA 1916-2001) Bell Labs Murray Hill NJ depuis 1941. Publia 1948 "A Mathematical Theory of Communication" Bell Sys Tech J 27:379+623. Ralph Hartley (USA 1888-1970) Bell Labs 1928 "Transmission of Information" Bell Sys Tech J 7:535. Harry Nyquist (Suede-USA 1889-1976) Bell Labs 1928 sampling theorem. Robert Fano (Italie-USA 1917-2016) MIT 1949. David Huffman (USA 1925-1999) MIT 1952 Huffman coding optimal. Robert Gallager (USA 1931-) MIT 1962 LDPC codes. Andrew Viterbi (Italie-USA 1935-) UC San Diego 1967 Viterbi algorithm, Qualcomm fondateur 1985. Claude Berrou (France 1951-) ENST Bretagne 1993 Turbo codes (avec Glavieux Thitimajshima) approaching Shannon limit 0,5 dB - originaire francais Telecom Bretagne. Erdal Arikan (Turquie 1958-) Bilkent 2009 Polar codes 5G NR. France telecom: Orange Paris-Issy France Telecom historique operateur principal national, SFR Numericable Paris (Altice), Bouygues Telecom Velizy Paris, Free Iliad Paris-Levallois disruptive low-cost. France industrie telecom: Alcatel-Lucent Paris-Boulogne-Billancourt (Nokia 2016 acquisition) infrastructure 5G, Thales Paris-La Defense electronique embarquee defense. Industrie fibre optique: Prysmian Group Milano Italia leader globale (CEI cavo elettrico industriale), Sumitomo Tokyo Japan, Furukawa Tokyo. Fibre optique France leader pos: Saint-Gobain Paris-La Defense leader mondial verres optiques, Nexans Paris-La Defense cables.

Operateurs satellite: Eutelsat Paris France leader Europe historique 36 satellites, Inmarsat London UK, Iridium McLean VA LEO 66 satellites, Starlink SpaceX. ESA-CNES Kourou Guyane Francaise base lancement Ariane 6 Vega-C 5°N latitude. CNES Centre National Etudes Spatiales Toulouse Paris-La Defense agence spatiale francaise. Galileo ESA Toulouse 30 satellites European GNSS. CNRS LIMSI Orsay laboratoire fluidique. ESPCI ParisTech Telecom Paris Palaiseau ingenieurs telecom. ENS Paris ENS Lyon Polytechnique Palaiseau INSA Lyon Centrale Lyon ECL ecoles ingenieurs. France radio: Radio France Paris RFI France Inter, NRJ NRJ Group Paris.

Tableau capacité Shannon-Hartley systèmes

  • POTS 3,4 kHz SNR 30 dB: C=33,9 kbps
  • AM radio 9 kHz: C=74,8 kbps
  • FM stereo 200 kHz: C=3,3 Mbps
  • GSM 200 kHz: C=815 kbps
  • UMTS 5 MHz: C=17,3 Mbps
  • LTE 20 MHz: C=133 Mbps
  • 5G NR 100 MHz: C=663 Mbps
  • 5G mmWave 800 MHz: C=4 Gbps
  • WiFi 802.11n 20 MHz: C=166 Mbps
  • WiFi 802.11ax 160 MHz: C=1,6 Gbps
  • WiFi 802.11be 320 MHz: C=3,72 Gbps
  • Satellite C-band 36 MHz: C=125 Mbps
  • IoT LoRa 125 kHz: C=37,4 kbps
  • NB-IoT 200 kHz: C=29,6 kbps
  • Sigfox 100 Hz UNB: C=346 bps
  • Fibre optique 50 GHz DWDM: C=498 Gbps par canal
  • BPSK: 1 bit/symbol
  • QPSK: 2 bit/symbol
  • 16QAM: 4 bit/symbol
  • 64QAM: 6 bit/symbol
  • 256QAM: 8 bit/symbol
  • 1024QAM: 10 bit/symbol
  • 4096QAM: 12 bit/symbol

Exemples numériques detailles

Exemple 1 - POTS V.90 modem. POTS analogique 3,4 kHz SNR 30 dB. C=33,9 kbps. Application: Orange Paris France Telecom historique centrales numériques, phase out POTS France 2025-2030 vers all-IP fibre.

Exemple 2 - WiFi 802.11n 2,4 GHz routeur. B=20 MHz SNR 25 dB. C=166 Mbps Shannon limit. Realite 802.11n MIMO 2x2: 144 Mbps PHY. Application: Orange Livebox 6 Wi-Fi 6, SFR Box, Bouygues Bbox, Free Box V8.

Exemple 3 - 5G NR sub-6 GHz France. B=100 MHz banda n78 (3,3-3,8 GHz). SNR 20 dB. C=666 Mbps Shannon. Realite 5G NR MIMO 4x4 256QAM: 940 Mbps DL theoretical, 200-400 Mbps réel. Application: Orange France 3,5 GHz n78 banda principale, SFR Free Bouygues Telecom mid-band.

Exemple 4 - Satellite C-band transponder. Eutelsat Hot Bird 13E broadcast B=36 MHz, SNR 10 dB QPSK 3/4 DVB-S2. C=125 Mbps Shannon. Realite DVB-S2: 81 Mbps net. Application: Eutelsat Paris France leader Europe historique satellite, ESA-CNES Kourou Guyane base lancement Ariane.

Exemple 5 - IoT LoRa 868 MHz UE. Capteur LoRa CSS 868 MHz, B=125 kHz, SNR -5 dB. C=49,3 kbps Shannon. Realite SF7-SF12: 5,47 kbps a 300 bps. Application: smart meters utilities, Sigfox Labege France originaire LPWAN UNB ultra-narrow-band, smart cities Paris Lyon Marseille.

Limites et considerations

  1. Capacité théorique maximale: codage FEC infini, retard illimite.
  2. SNR bruit additif Gaussien (AWGN).
  3. Bandwidth -3 dB cutoff: filtres raised cosine.
  4. Efficacite spectrale réelle limit hardware.
  5. Rate distortion theorem source-coding.
  6. MIMO multiplexing M*M antennes multiplie capacité.
  7. Nyquist sampling f_sampling >= 2*B_max.
  8. Eb/N0 normalise differente C/N.
  9. Fading channels Rayleigh outage 5-10 dB less.
  10. Limites quantiques h*f minimum photon/bit.

Erreurs courantes

  1. Confondre SNR linear avec SNR dB.
  2. Calculer capacity bits/Hz vs bits/s.
  3. Negliger modulation order.
  4. Pretendre capacity = Shannon limit.
  5. Confondre C/N avec Eb/N0.
  6. Negliger FEC overhead rate.
  7. Pretendre log2 valide tout SNR.
  8. Confondre bandwidth -3 dB avec noise bandwidth.
  9. Pretendre bit-rate illimite.
  10. Negliger delay codage LDPC turbo ms.

Applications numerotees

  1. France telecom 5G: Orange Paris-Issy historique, SFR Numericable, Bouygues Telecom Velizy, Free Iliad Paris-Levallois.
  2. Fibre optique France: Saint-Gobain Paris-La Defense leader mondial verres optiques, Nexans Paris-La Defense cables, Prysmian Italie distribution France.
  3. WiFi planning: Cisco USA, Aruba HPE, Orange Livebox 6 Wi-Fi 6, SFR Box Free Bouygues routers domestiques.
  4. Satellite Eutelsat Paris: leader Europe historique satellite 36 satellites, ESA-CNES Kourou Guyane base lancement.
  5. IoT LoRa Sigfox: Sigfox Labege France originaire LPWAN UNB ultra-narrow-band, smart meters utilities.
  6. France industrie: Alcatel-Lucent Paris-Boulogne-Billancourt (Nokia 2016) infrastructure 5G, Thales Paris-La Defense.
  7. Modulation Turbo codes: Claude Berrou ENST Bretagne 1993 originaire francais, ENSL Lyon, Telecom Paris Palaiseau.
  8. Deep space ESA: ESA-CNES Kourou Guyane base, CNES Toulouse Paris-La Defense agence spatiale francaise.
  9. Universites théorie information: ENSL Lyon, Telecom Paris Palaiseau, Polytechnique Palaiseau, Centrale Lyon ECL.
  10. Educational: ITU-T Geneva, ETSI Sophia Antipolis France standards, AETIC France ingenieurs telecom.

Support

Pour questions techniques: info@calcoi.com. Outil didactique théorie information preliminaire. Pour recherche professionnelle Shannon CE 1948, Hartley RVL 1928, Cover-Thomas 'Éléments of Information Theory' 2006 Wiley, Proakis 'Digital Communications' 2007.

Mis à jour en juillet 2026

Questions fréquentes

Que mesure le théorème Shannon-Hartley?

Théorème Shannon-Hartley (Claude Shannon Bell Labs 1948 + Ralph Hartley 1928) mesure capacité maximale théorique canal communication: C = B*log2(1+S/N) bits/s. B bandwidth Hz, S/N rapport signal-bruit SNR linear. Exemples: POTS 3,4 kHz SNR 30 dB C=33,9 kbps, WiFi 20 MHz SNR 25 dB C=166 Mbps, 5G NR 100 MHz SNR 20 dB C=663 Mbps, IoT LoRa 125 kHz SNR -5 dB C=37 kbps. Applications: design modems fibre optique DSL ADSL VDSL FTTH, cellular 4G LTE 5G NR Orange Paris-Issy SFR Free Bouygues Telecom France, WiFi 802.11 ax/be, satellite Eutelsat Paris leader Europe, IoT LoRaWAN Sigfox Labege France originaire LPWAN.

Difference SNR linear vs SNR dB?

SNR linear adimensionnel. SNR dB = 10*log10(S/N). SNR_linear = 10^(SNR_dB/10). 0 dB=1, 10 dB=10, 20 dB=100, 30 dB=1000. Erreur commune: utiliser SNR dB directement formule log2(1+SNR_linear). Exemple: 30 dB -> 1000, log2(1001)=9,97 bits/s/Hz, PAS log2(31)=4,95.

Qu'est-ce que l'efficacite spectrale?

Efficacite spectrale eta = C/B = log2(1+SNR) bit/s/Hz. Limites pratiques: WiFi 802.11ax 1024QAM eta~10 bit/s/Hz, Wi-Fi 7 4096QAM eta~12 bit/s/Hz, LTE 256QAM eta~5,5, 5G NR 256QAM eta~6, fibre optique cohérent QAM-16K eta~14. Shannon limit théorique SNR 30 dB eta~10.

Comment choisir modulation correcte?

Modulation adaptative basee SNR canal: SNR35 dB 1024QAM 4096QAM. Standards: LTE-A 256QAM, 5G NR 256QAM, WiFi 802.11ax 1024QAM, Wi-Fi 7 4096QAM. Turbo codes Claude Berrou ENST Bretagne 1993 originaire francais.

Qu'est-ce que Eb/N0 vs SNR?

Eb/N0 énergie par bit/spectral noise density (dB). Conversion: Eb/N0_dB = SNR_dB - 10*log10(B/Rb), Rb bit-rate. WiFi SNR 25 dB BW 20 MHz Rb 144 Mbps: Eb/N0=33,6 dB. BER 10^-6 AWGN threshold: BPSK 11 dB, QPSK 11, 16QAM 18, 64QAM 24, 256QAM 30, 1024QAM 36. LDPC turbo Shannon limit BPSK ~0,2 dB Eb/N0.

Comment ameliorer capacité réelle?

Augmenter capacité: 1) bandwidth plus grand B; 2) SNR plus haut (haute Pt antenne gain low noise NF); 3) MIMO multiplexing antennes multiples M*M; 4) modulation avancee 256QAM 1024QAM 4096QAM; 5) FEC LDPC turbo polar codes closing Shannon limit 0,5-2 dB - turbo codes originaire francais Berrou ENST Bretagne 1993; 6) cooperative relays mesh networks. France: Orange 5G UWB 1,4 Gbps 5G C-band 3,7 GHz 2024.

Limites fondamentales Shannon-Hartley?

Limite Shannon-Hartley fondamental: C maximale théorique avec codage optimal infinitely long delay. Real systems achievable 0,5-2 dB sous limit. Exceptions: 1) AWGN assumption: real channels Rayleigh Rician fading -5-10 dB; 2) bandwidth -3 dB; 3) MIMO improves capacity not bandwidth; 4) lossy compression source bandwidth; 5) quantum limit photon h*f minimum. Future: post-Shannon network coding, Polar codes 5G NR Arikan 2009, optical CW high-photon FOC fibre optique communication.

Formule

C = B*log2(1+S/N) bit/s
eta = C/B = log2(1+SNR) bit/s/Hz
SNR_linear = 10^(SNR_dB/10); SNR_dB = 10*log10(SNR_linear)
For S/N<<1: C ~ B*S/(N*ln2) = 1.44*B*S/N
For S/N>>1: C ~ B*log2(S/N) = B*(SNR_dB/3.01)
Eb/N0_dB = SNR_dB - 10*log10(B/Rb) - normalized per bit
N = -174 dBm/Hz + 10*log10(B_Hz) at T=290 K
BPSK 1, QPSK 2, 16QAM 4, 64QAM 6, 256QAM 8, 1024QAM 10, 4096QAM 12 bit/symbol
BPSK 11 dB Eb/N0, 16QAM 18 dB, 64QAM 24 dB, 256QAM 30 dB, 1024QAM 36 dB (BER 10^-6 AWGN)
WiFi 20 MHz SNR 25 dB=166 Mbps, 5G NR 100 MHz SNR 20 dB=663 Mbps, GSM 200 kHz=815 kbps
Shannon CE 1948 'A Mathematical Theory of Communication' Bell System Technical Journal 27:379-423 + 27:623-656; Hartley RVL 1928 'Transmission of Information' Bell Sys Tech J 7:535-563; Cover TM Thomas JA 2006 'Elements of Information Theory' 2nd ed Wiley; Proakis JG Salehi M 2007 'Digital Communications' 5th ed McGraw-Hill

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