Noise in radio communications. Matjaž Vidmar

Transcription

1 Noise in radio communications Matjaž Vidmar LSO, FE, Ljubljana,

2 List of figures: Noise in radio communications 1 - The dispute between famous scientists 2 - Noise spectral density 3 - Black-body thermal radiation 4 - Received thermal-noise power 5 - Thermal equilibrium 6 - Natural noise sources 7 - Natural sky noise 8 - Sun-noise example 9 - Receiver signal-to-noise ratio 10 - Chain noise temperature 11 - Amplifier noise figure 12 - Relationship F<->T 13 - Attenuator noise 14 - G/T figure of merit 15 - Noise of active components 16 - Noise parameters 17 - Example: the sensitivity of a GSM phone 18 - Change of S/N versus F 19 - Receiver-sensitivity measurement 20 - Hot/cold method 21 - Noise-figure meter 22 - Bit-Error Rate BER calculation 23 - BER <-> S/N table for BPSK 24 - BER for different modulations 25 - Forward Error Correction (FEC) 26 - Oscillator phase noise 27 - Leeson's equation 28-1/f noise 29 - Resonator quality Q 30 - Phase-Locked Loop (PLL) 31 - Effects of phase noise 32 - Phase noise without approximations 33 - Width of Lorentzian spectral line 34 - Noise as test signal 35 - Cryptographic-key source 36 - Noise cryptography 37 - LFSR pseudo-random sequences 38 - Use of pseudo-random sequences

3 Fifth Solvay International Conference on Electrons and Photons (October 1927). The leading figures Albert Einstein and Niels Bohr disagreed: Albert Einstein: God does not play dice! Niels Bohr: Einstein, stop telling God what to do! In telecommunications random signals are called noise. Noise impairs the performance of any communication link. Noise is a macroscopic description of quantum effects! 1 - The dispute between famous scientists

4 log N 0 noise spectral density Thermal noise: P N =Δ f k B T Boltzmann constant: k B J/K N 0 =k B T T =293K WHITE NOISE ELECTRONICS/RADIO f 100GHz Shot noise: P N =Δ f h ν Planck constant: h Js N 0 /k B =11000K P N =Δ f N 0 Δ f bandwidth N 0 =h ν ν=231thz (λ 0 =1.3μ m) OPTICS ν 30THz BLUE NOISE 2 - Noise spectral density log f log ν frequency

5 Black body Γ=0 da' Thermal radiation dp Solid angle dω Spectral brightness B f ( f,t )= Planck law B f ( f,t )= 2 h f 3 dp df da' d Ω 3 Black-body thermal radiation c 0 2 h f 1 k e T B 1 Free space ϵ 0,μ 0 c 0 = m/s m/s Radio h f k B T Rayleigh Jeans approximation B f ( f,t ) 2 k B T f 2 c 0 2 = 2 k B T λ 2

6 B f = 2 k B T λ 2 Black body Γ=0 da' Single polarization dp N = 1 dp 2 B f Δ f da' Δ Ω r Free space ϵ 0,μ 0 Lossless antenna η=1 A eff (Θ,Φ) Bandpass filter Δ f N da '=r 2 d Ω P N = A ' Δ Ω= A eff (Θ,Φ) 1 2 B Δ f da ' Δ Ω= 1 f 4 π 2 2k BT (Θ, Φ) λ 2 T (Θ, Φ) F (Θ,Φ) 2 d Ω 4π P N =Δ f k B F (Θ,Φ) 2 d Ω 4 π 4 Received thermal-noise power r 2 = λ2 D(Θ,Φ) 4 π r 2 = Δ f r 2 d Ω λ 2 F (Θ, Φ) 2 r 2 F (Θ*, Φ*) 2 d Ω* 4π r 2 4 π λ 2 F (Θ,Φ) 2 F (Θ*,Φ *) 2 d Ω* =Δ f k B T (Θ, Φ) F (Θ,Φ) 4π = 2 d Ω F (Θ,Φ) 2 d Ω 4 π

7 Black body Γ=0 T 1 0 B f = 2 k B T 1 λ 2 T 1 = R Antenna radiation resistance! Antenna beam F (Θ,Φ) P N 1 =Δ f k B T 1 P N 2 =Δ f k B T 2 Lossless antenna η=1 Bandpass filter Δ f T 2 0 R is NOT a property of a lossless antenna! R T =0 T 1 0 P N 1 =Δ f k B T 1 R R R T 0 U NU Neff = 4 R Δ f k B T Bandpass filter Δ f P N 2 =Δ f k B T 2 T Thermal equilibrium

8 Cold sky T 10K Γ=0 * Sun * T 10 6 K * * * * * * = T (Θ, Φ) F (Θ,Φ) 2 d Ω 4π F (Θ,Φ) 2 d Ω 4 π Lossless antenna η=1 Vegetation T 290K Γ 0 Ground soil T 290K Γ 0 Lake Γ 1 Mirror! 6 Natural noise sources

9 +attenuation Milky way B Earth's atmosphere 7 Natural sky noise

10 * * Sun * Cold sky T N 10K T S * Ω S * = T S Ω S F (Θ,Φ) 2 d Ω+T N 4π Ω S F (Θ, Φ) 2 d Ω D= 4 π F (Θ MAX,Φ MAX ) 2 F (Θ,Φ) 2 d Ω 4 π Ω A F (Θ,Φ) 2 d Ω 4 π T S Ω S D 4 π 106 K srd π srd +T N +10K α S mrd F (Θ MAX, Φ MAX ) F (Θ,Φ) 0 α S 476K+10K=486K T S 10 6 Example: D=20dBi=100 f =2GHz Ω S =2 π[1 cos(α S /2)] π α S 2 / srd 8 Sun-noise example Ω A 4 π D =0.126srd Ω S F (Θ,Φ)=0 outside beam Lossless antenna η=1

11 Signal Lossless antenna η=1 T S amplifier noise temperature reduced to the input! Amplifier Bandpass filter Δ f P S ' =G P S P S G T RX G P S ( S = P ' S N )output P N ' Natural thermal noise Apparent noise at the input P N =Δ f k B ( +T RX ) 9 Receiver signal-to-noise ratio Amplifier thermal noise P N ' =G Δ f k B ( +T RX ) ( S P S = N )output Δ f k B ( +T RX ) k B J/K T 0 =290K k B T 0 10log dbm/ Hz 1mJ Different noise sources are not correlated, therefore noise powers or noise temperatures are summed!

12 Lossless antenna η=1 Amplifier # 1 Amplifier # 2 Amplifier # 3 P S ' =G 3 G 2 G 1 P S Bandpass filter Δ f P S G 1 T 1 G 2 T 2 G 3 T 3 T RX G=G 1 G 2 G 3 P N ' =Δ f k B [G 3 G 2 G 1 ( +T 1 )+G 3 G 2 T 2 +G 3 T 3 ] P N ' =G 3 G 2 G 1 Δ f k B ( +T RX ) T RX =T 1 + T 2 G 1 + T 3 G 1 G P S G e T e G e T e Infinite chain of identical amplifiers G e T e T RX = T e 1 1 G e 10 Chain noise temperature

13 Lossless antenna η=1 P S ( S = P S N )input P N P N =Δ f k B Amplifier G F T RX P N ' =G Δ f k B ( +T RX ) Bandpass P filter Δ f S ' =G P S ( S = P ' S N )output P N ' Nonsense ( S A property of definition N )input Δ f k an amplifier of the F = = B = T + RX =1+ T RX G P S can not be a noise figure: function of! ( S N )output P S G Δ f k B ( +T RX ) Sensible definition F =1+ T RX T T 0 =290K T RX =T 0 (F 1) Logarithmic units F db =10 log 10 F =10 log 10( 1+T RX T 0 ) ( F db T RX =T ) Amplifier noise figure

14 F db =10 log 10( 1+ T RX 290K) Noise temperature in degrees K Noise figure in db T RX =290K( 10 F db 10 1) 12 Relationship F<->T T 0 =290K

15 P S T RX ' F ' Attenuator a R 2 R 1 T R R 3 P S a Amplifier G F T RX G P S a Bandpass filter Δ f P S ' = G P S a Lossless antenna η=1 a +T R( 1 1 a) [ P N ' =G Δ f k B a +T R( 1 1 a) +T ] RX T RX ' =T R (a 1)+a T RX F '=1+ T RX ' T 0 =1+ T R T 0 (a 1)+a T RX T 0 Frequent case T R T 0 =290K F ' a+a T RX T 0 =a( 1+ T ) RX T =a F 0 F db ' a db +F db 13 Attenuator noise ( S = P ' S N )output P N ' = P S Δ f k B [ +T R (a 1)+a T RX ] Attenuator examples T R T 0 =290K F ' a F F db ' a db +F db (1) lossy antenna a db = 10log 10 η (2) lossy transmission line a db (3) lossy bandpass filter a db (4) passive mixer loss a db

16 Satellite transmitter P TX Transmit antenna G TX Free space radio link P S =P TX G TX G RX ( λ 4 π r ) 2 Transmitter Receiver Receiving ground station (G /T )= G RX ( +T RX ) [K 1 ] (G/T ) db/ K =10 log 10 G RX 1K ( +T RX ) [db/k] (G/T ) db/ K =G RX db 10 log 10 +T RX 1K r ( S =P N TX G TX 1 )output Δ f k ( λ B [db/k] Receive antenna G RX P S System ( S P S = N )output Δ f k B ( +T RX ) Amplifier G T RX G RX ) 2 4 π r ( +T RX ) Bandpass filter Δ f 14 G/T figure of merit Receiving ground station

17 Amplifier device Vacuum tube with control grid (triode, pentode) Vacuum tube with speed modulation (klystron, TWT) Parametric amplifier (room temperature) Si BJT, JFET or MOSFET (room temperature) GaAs FET or HEMT (room temperature) GaAs FET ali HEMT (liquid-nitrogen 77K) Si or GaAs MMIC amplifier 15 Noise of active components Gain G [db] Noise tempe rature T RX [K] Noise figure F db [db] Operational amplifier

18 Linear description [S ]= [ S 11 S 21 S 12 S 22] U N Source Γ g (antenna) Input # 1 I N [S] Output # 2 Description of noise properties U N, I N F MIN,Γ O, r N Amplifier (transistor) with uncorrelated noise sources U N, I N F =F MIN +4 R N Γ g Γ O 2 Γ Z K (1 Γ g 2 ) 1+Γ O 2=F g Γ O 2 MIN +4r N (1 Γ g 2 ) 1+Γ O 2 F MIN lowest noise figure at Γ g =Γ O i n linear units (not i n db!) Γ O optimum source reflectivity for F MIN (unrelated t o [S ]!) r N = R N Z K normalized noise resistance (usually Z K =50Ω) 16 Noise parameters Low-noise devices are usually NOT unconditionally stable amplifiers!

19 Non directional antenna P S =? T 0 =290K P N Amplifier G F db =5dB T RX =T 0 ( 10 F db 10 1) =290K ( )=627K Bandpass filter Δ f =200kHz k B J/K ( S 10dB N )db ( S ( S ) N )db N = P N =Δ f k B ( +T RX )=200kHz J /K (290K+627K)= W ( P S =P S ) N N = W 10= W P S dbm =10log 10 P S 1mW = 106dBm Simplified calculation exclusively i n case T 0 =290K P S dbm (S / N ) db +(Δ f ) db Hz +(k B T 0 ) dbm/ Hz +F db k B T 0 (k B T 0 ) dbm /Hz =10log 10 1mJ 174dBm/Hz (Δ f ) db Hz =10log 10( 1Hz) Δ f =53dB Hz P S dbm 10dB+53dB Hz 174dBm/ Hz+5dB= 106dBm 17 Example: the sensitivity of a GSM phone

20 * * Two different receivers #1 and #2: F 1 =1dB T RX1 =75K P TX Cold sky T N 10K F 2 =0.5dB T RX2 =35K Δ F db =F 1 F 2 =0.5dB * Δ( S =10log N )db 10[ +T RX2 +T RX1] * Vegetation T 290K Δ( S =10log N )db 10[ 20K+75K ] 20K+35K =2.37dB Ground soil T 290K 20K P S Amplifier G T S Bandpass filter Δ f 18 Change of S/N versus F Receiving ground station

21 Signal generator P O 10mW ~ Oscillator R 2 T R R 1 R 3 290K Attenuator a db Shielding >150dB! P S Amplifier G F T RX Bandpass filter Δ f Receiver under test ( S N )output 50dB<a db <150dB Coupling via radiation? Additional requirements for the test source (signal generator) for sensitivity measurements of radio receivers: (1) Shielding>150dB (2) T R = =T 0 =290K (1) Required S/N before demodulation? (2) Required S/N after demodulation? (3) Required BER? 19 Receiver-sensitivity measurement

22 Cold resistor T 1 Hot resistor T 2 R R Switch Amplifier under test G F T RX Bandpass filter Δ f P 1 =G Δ f k B (T 1 +T RX ) Power meter P P 2 =G Δ f k B (T 2 +T RX ) The unknowns G Δ f k B cancel i n the Y ratio! Resistor type Y = P 2 P 1 = T 2 +T RX T 1 +T RX T RX = T 2 Y T 1 Y 1 F db =10 log 10[ 1+ T 2 Y T 1 (Y 1) T 0] 20 Hot/cold method T 0 =290K Antenna into cold sky Liquid N 2 cooled R Antenna into absorber R at room temperature Light-bulb filament as R Ionized gas as R Avalanche breakdown Temperature ~20K ~77K ~290K ~290K ~2000K ~10 4 K ~10 6 K

23 28V 1kHz Noise head Device under test G D T D F Mixer T 2 / T 1 X G M T M Δf IF amplifier Detector MC Usually T 1 T 0 =290K ENR=10 log 10 (T 2 /T 1 ) Calibration ~ LO Test receiver Y, T D, F Two measurements without calibration: Y = P 2 P 1 = T 2 +T D +T M /G D T 1 +T D +T M /G D T D = T Y T 2 1 T M Y 1 G D F db =10 log 10[ 1+ 1 T 0 ( 21 Noise-figure meter known G D T Y T 2 1 Y 1 T M G D)] G M Δ f uncertain! Four measurements with calibration: (1) P 1 =G M G D Δ f k B (T 1 +T D +T M /G D ) (2) P 2 =G M G D Δ f k B (T 2 +T D +T M /G D ) (3) P 3 =G M Δ f k B (T 1 +T M ) (4) P 4 =G M Δ f k B (T 2 +T M ) Solve 4 equations for 4 unknowns: T D, G D, T M an d (G M Δ f k B )

24 0 Decision threshold 0 ^ U S Q 1 ^ U S Gaussian distribution of probability density of the in-phase a and quadrature jb noise components ^ U N ^ U N 2 = a 2 + b 2 =2σ 2 Example BPSK a U^ N =a+ jb b p(a)= 1 σ 2 π e p(b)= 1 σ 2 π e 22 Bit-Error Rate (BER) calculation 1 I U ^ P 1 0 = S a 2 2 σ 2 b 2 2 σ 2 p(a)da P = 0 1 U ^ S p(a)da Symmetric threshold : P 1 0 =P 0 1 =BER BER= U ^ S 1 π ^ U N 2 erfc(x)= 2 e a 2 π x ^ U N 2 da e u 2 du BER= 1 2 erfc ( ^ U S ^ U N 2 ) P S =α U ^ S 2 P N =α ^ U N 2 BER= 1 2 erfc ( P S P N )

25 (S/N) db BER (S/N) db BER BER (S/N) db BER (S/N) db -5dB -4dB -3dB -2dB -1dB -0dB 1dB 2dB 3dB 4dB 5dB 6dB 7dB 23.6% 18.6% 15.9% 13.1% 10.4% 7.9% 5.7% 3.8% 2.3% 1.3% 0.6% 0.24% dB 9dB 10dB 11dB 12dB 13dB 14dB 15dB 16dB 17dB 18dB 19dB 20dB % 10% 3% 1% 0.3% 0.1% dB -0.8dB 2.5dB 4.3dB 5.8dB 6.8dB 7.7dB 8.4dB 9.1dB 9.6dB 10.1dB 10.5dB 11dB dB 11.7dB 12dB 12.3dB 12.6dB 13.1dB 13.5dB 13.9dB 14.3dB 14.7dB 15dB 15.3dB 15.6dB (S/N) db BER (S/N) db BER BER (S/N) db BER (S/N) db 23 BER <-> S/N table for BPSK

26 0.5 Bit-Error Rate BER BER= 1 2 erfc S N Shannon C=Δ f log 2( 1+ P ) S N 0 Δ f log 10 (ln 2)= 1.6dB Shannon Δf BPSK C/Δf 1bit QPSK C/Δf 2bit 16-QAM C/Δf 4bit 64-QAM C/Δf 6bit dB 9.5dB 7.4dB -1.6dB -10dB -5dB 0dB 5dB 10dB 15dB 20dB 25dB 24 BER for different modulations 10log 10 (S/N)

27 Bit-Error Rate BER Shannon lim Δ f C=C = P S N 0 ln 2 MESSAGE FEC CODER 0.1 Unprotected BPSK /QPSK MESSAGE PARITY log 10 (ln 2)= 1.6dB Shannon Δf 0.5dB Turbo 3Δf 4dB BER= 1 2 erfc S N NASA deep space 2.3 Δ f -1.6dB -10dB -5dB 0dB 5dB 10dB 10log 10 (S/N) CORRUPTED! NOISY COMMUNICATION CHANNEL FEC DECODER MESSAGE CORRECTED! 25 Forward Error Correction (FEC)

28 ~ + U Nin Equivalent noise source T R T 0 =290K T R +T G T 0 F Spectrum log F(ω) Thermal noise Phase noise 26 Oscillator phase noise Resonator T R L Steady state oscillation R Cu H (ω) Carrier P 0 (ω 0 ) Phase noise Δ ω ω 0 =1/ L C C Thermal noise H (ω 0 ) G=1 R out Amplifier H (ω) G= Q L = ω 0 L Σ R ω G R in F U Nout =U Nin +H (ω) G U Nout T G Σ R P 0 Σ R+ jω L+ 1 j ωc U Nin U Nout = 1 H (ω) G Σ R=R out +R Cu +R i n 1 1+ j2q Δ ω L ω0 U Nout U Nin (1+ ω 0 j 2Q L Δ ω) [ P Nout P 1+ Nin ( Amplitude an d phase noise [ P Nout P 1+ ( Nin Valid at P Nout P 0 ω 0 ω)2] 2Q L Δ f 0 2Q L Δ f )2]

29 Normalized phase noise spectral density log L(Δ f ) [dbc/hz ] Valid at L(Δ f ) Δ f 1 Saturation removes amplitude noise P ϕ =P Nout /2 L(Δ f )= 1 P 0 dp ϕ df Phase noise only = 1 2 [ 1+ ( f 0 2Q L Δ f dp Nin df P 0 carrier )2 =N 0 =k B (T R +T G ) k B T 0 F ] k BT 0 F ( P 1+ f ) C 0 Δ f [Hz 1 ] power 1/f noise α(δ f ) 3 L(Δ f ) dbc/ Hz =10log [ 1+ ( f 0 2Q L Δ f )2 ] k BT 0 F ( P 1+ f f ) C 0 Δ 1Hz 1/ f noise α(δ f ) 2 Simplified phase noise L(Δ f ) 1 8 ( f 0 Q L Δ f )2 k B T 0 F P 0 f C f 0 2 Q L Thermal moise 27 Leeson's equation Offset f rom carrier log Δ f

30 1/ f noise usually does not have a clear explanation! Noise figure log F ' Surface devices Si-MOSFET GaAs-MESFET GaAlAs-HEMT Noise figure log F ' 1/ f noise Mixing Saturated amplifier F ' Upcon verted noise Δ f 1 Transistor F ' i n oscillator circuit! 1/ f noise Flicker noise Pink noise Thermal noise Δ f Carrier f 0 Thermal noise f Bulk devices Si-BJT Si-JFET F '=F( 1+ f C f ) increased W hite thermal noise LF noise! 28 1/f noise f C 1kHz f C 1MHz Frequency log f

31 The loaded resonator quality Q L defines the oscillator phase noise! L(Δ f )= 1 2 [ 1+ ( f 0 2 Q L Δ f )2 ] k B T 0 F ( P 1+ f f ) C 0 Δ Variable-frequency oscillators Q L RC VCO ~1 BWO tube ~1 Varactor-tuned LC VCO YIG (Y 3 Fe 5 O 12 ) oscillator Oscillator f 0 f 0 L(Δ f ) Frequency multiplier f N The phase noise multiplies with the square of the frequency multiplication factor! 29 Resonator quality Q f 0 N L(Δ f ) N 2 Fixed-frequency oscillators Q L RC multivibrator ~1 LC resonator Cavity resonator Ceramic dielectric resonator AT-cut quartz crystal (fundamental mode) AT-cut quartz crystal (third/fifth overtone) Electro-optical delay line ($) ~10 5 (noisy!) Sapphire dielectric resonator ($$$) ~ Red HeNe LASER ~10 8

32 Normalized phase noise spectral density log L(Δ f ) [dbc/ Hz ] Reference (crystal ) f REF X f N Phase comparator Divider f N f VCO Control inside Δ f loop Filter Δ f loop Loop delay? Δ f loop f REF Reference phase noise N 2 Free running VCO phase noise Thermal noise 30 Phase-Locked Loop (PLL) Offset f rom carrier log Δ f

33 Example QPSK Q σ ϕ = ±σ ϕ f MAX 2 f MIN L(Δ f )d Δ f f MAX =Δ f modulation I f MIN =Δ f carrier recovery Modulation constellation rotation Spectrum log F( f ) σ f = deviation=±σ f f 0 Residual FM f MAX 2 f MIN Δ f 2 L(Δ f )d Δ f Usual choice f MAX =3kHz f MIN =50Hz (S / N ) speech f Spectrum log F( f ) P 0 Δ f 2 P i =P 0 L(Δ f )d Δ f Δ f 1 u(t) σ t = 1 2 π f 0 2 f MAX f MIN L(Δ f )d Δ f f 0 Δ f 1 Δ f 2 P i f Adjacent-channel interference 31 Effects of phase noise f MAX f clock Clock jitter ±σ t f MIN =Δ f clock recovery t

34 log L(Δ f ) [dbc/hz ] Usually f MIN f HW i n radio communications! Spectrum F( f ) 2 f 0 L(Δ f )d Δ f =1 f HW XTAL Crystal oscillator Thermal noise f HW LC Limit L(Δ f ) Δ f =1 α(δ f ) 3 LC oscillator f 0 2 Q XTAL α(δ f ) 2 f 0 2 Q LC 3dB f 0 2 f HW Oscillator spectral line Device f MAX Drawing not t o scale! Planck f Rayleigh Jeans f C 32 Phase noise without approximations log Δ f

35 Spectrum F( f ) 2 3dB f 0 L(Δ f )= 2 f HW Not t o scale! C 2 f HW = f FWHM Lorentzian spectral line +Δ f 2 f Flat thermal noise can be neglected : device f MAX o r Planck law LC oscillator 1/ f L(Δ f )= 1 8 ( f 0 Q L)2 noise can be neglected 2 f HW 1 T F +Δ f 2 k B 0 P 0 Lorentzian line in Leeson's equation f 0 L(Δ f )d Δ f =1 L(Δ f )d Δ f = 1 8 ( f 0 Q L)2 k B T 0 F P 0 2 f HW 1 +Δ f d Δ f = 2 = 1 8 ( f 0 Q L)2 k B T 0 F P 0 [ 1 arctan Δ f f HW Δ f = f HW ]Δ f = = k B T 0 F 8 P 0 ( f 0 Q L)2 π f HW f HW = π k T F ( f B P 0 Q L)2 Example f 0 =3GHz Q L =10 P 0 =0.1mW F =10dB f HW =14Hz f FWHM =28Hz 33 Width of Lorentzian spectral line C= k T F ( f B 0 0 = 8 P 0 Q L)2 f HW π L(Δ f )= f HW /π 2 +Δ f 2 f HW

36 Noise source W hite noise F i n (ω) =const. Bandpass filter F out (ω) linear device under test H (ω) Spectrum analyzer Test source Noise source Bandpass filter Δ ω F i n (ω) Amplifier F out (ω) nonlinear device under test Spectrum analyzer log F i n (ω) log F out (ω) Δ ω 34 Noise as test signal ω Nonlinearity IMD IMD ω 0 ω 0 ω

37 Natural sources of random signals: Thermal noise Shot noise Avalanche breakdown Radioactive decay... f clock Δ f Clock Address counter Noise source Amplifier Δ f S&H A/D Memory Shield from interference Sampling Quantization Interference is not random! Interference might be intentional! 35 Cryptographic-key source Arbitrary-length cryptographic key: Password (rather short key...) Keys for DES, AES etc One-time pad (very long but unbreakable key!)

38 MESSAGE FEC CODER P TX =P MIN Amplification without noise not possible! MESSAGE FEC DECODER TX MESSAGE RX Sender FEC DECODER Receiver Transmission path with low attenuation: optical fiber or radio in Fresnel region RX RX TX Eavesdropper #1 Eavesdropper #2 FEC DECODER MESSAGE FEC CODER TX 36 Noise cryptography FEC adds delay!

39 Divider p( x)=1+x 5 + x 9 Period T Clock Shift register D Q T D Q T D Q T D Q T D Q T 37 LFSR pseudo-random sequences EXOR gate Primitive polynomial p(x)=1+ x l +x m max sequnce length N =2 m 1 2 m-1 ones and 2 m-1-1 zeros arranged in groups of 1X m ones, m-1 zeros 1X m-2 ones and zeros 2X m-3 ones and zeros 4X m-4 ones and zeros... 2 m-5 groups 111 and m-4 groups 11 and 00 2 m-3 individual 1 and 0 D Q T LFSR Linear-Feedback Shift Register Autocorrelation( τ) 2 m 1 (2 m 1)T 1 D Q T τ D Q T Spectrum F( f ) 2 m 1 lines D Q T f CLOCK =1/T Sequence m=9 N =511 Out 2 f CLOCK Sound and appear as white noise!

40 Two-valued autocorrelation with a single very pronounced peak: - synchronization headers for data frames - spreading sequences in CDMA - accurate time transfer in radio navigation (GPS, GLONASS) LFSR pseudo-random sequences are of NO cryptographic value: algorithm Berlekamp-Massey 1969 LFSR sequences are the result of pure mathematics that does not appear anywhere else in nature! Perfect frequency spectrum of uniformly-spaced lines and simple generation/checking: - test sequences for all kinds of telecommunication links - data scrambling (randomization) as part of line coding Peak-to-average power ratio: LFSR: P MAX P 1 Noise : P MAX P 38 Use of pseudo-random sequences How to present ourselves to the inhabitants of a neighbor galaxy? How to find out that they look for us?