Next Generation of wireless communication: some questions concerning - - PowerPoint PPT Presentation

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Next Generation of wireless communication: some questions concerning reliability. Laurent Clavier (laurent.clavier@telecom-lille.fr)

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First I would like to thanks several people who contributed to these reflexions: Gareth, Ido, Nourddine, Malcolm, François… and many others. And a special thanks to Tomoko… and Yoko.

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Connected things… Where are we going?

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IoT and M2M…

Forecasts – 50 billions of connected objects (5.1010) in 2020 1000 billions (1012)

(depending on the studies).

If each object is powered by an AA battery (~15 kJ), we need 0,75 peta Joules (0,75 1015 or 750 thousand billions of Joules)… 15 peta Joules  it means one 1GW nuclear reactor during ~9 days 180 days

…. So what?

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Let’s have a closer look…

Performing some measurements

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Sensor Microcontroller Microcontroller Communication Low-Power Transceiver Data Storage – External Memory Data Storage – External Memory NODE MEASUREMENT PLATFORM Serial Port to PC Differential Amplifiers

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Time (seconds)

Classification of the energy measurements

4.74 4.76 4.78 4.8 4.82 4.84 4.86 5 10 15 20 25 30 35 40 45 50

Current (mA)

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Energy versus interference

In addition to the energy measurement, we also make some interference measures characterized by the RSSI The X-MAX protocol allows to try 3 successive transmissions Our approach allows

• to take into account the energy consumed by retransmission
• to take into account the listening periods
• to take into account the lost ACKs
• …

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Energy versus interference

Energy used per packet (mJ) RSSI (dBm)

In an anechoïc chamber, introducing interferers (all with IEEE802.15.4 – 1, 2 or 3 interferers, highly actives)

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Energy versus interference

Energy used per packet (mJ) RSSI (dBm)

In an anechoïc chamber, introducing interferers (all with IEEE802.15.4 – 1, 2 or 3 interferers, highly actives)

No interferer High probability of success Low level of energy needed per packet

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Energy versus interference

Energy used per packet (mJ) RSSI (dBm)

In an anechoïc chamber, introducing interferers (all with IEEE802.15.4 – 1, 2 or 3 interferers, highly actives)

2 to 3 interferers Increase of lost packets (1) 1 transmission red (2) 2 black (3) 3 green (4) Lost packets blue Increase in consumption

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In a laboratory, introducing interferers (all with IEEE802.15.4 – 1, 2 or 3 interferers, highly actives) Real radio channel, uncontrolled sources of interference

Energy used per packet (mJ) RSSI (dBm)

Energy versus interference

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Reliability of a transmission for a given level of interference

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Mean consumption per packet for a given level of interference (transmitter)

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Lifetime of a Zigbee node

Anechoïc chamber Laboratory

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Agenda

• The problem
• Radio channel
• Interference

Some questions concerning reliability.

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Agenda

• The problem
• Radio channel
• Interference

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An environment

What is the question?

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What is the question?

To be monitored

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Nodes communicate to gather the information…

What is the question?

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Let’s have a look on one single receiver 𝒛 𝒖 = 𝒊 𝒖 ∗ 𝒚 𝒖 + 𝒋(𝒖) + 𝒐(𝒖)

Transmitted signal Received signal Thermal noise Radio channel impulse response Interference

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Two problems

(a) The radio channel… 𝑧 𝑢 = 𝑏0𝑦 𝑢 − 𝑢0 + 𝑏1𝑦 𝑢 − 𝑢1 + ⋯ + 𝑏𝑜𝑦 𝑢 − 𝑢𝑜 = ℎ 𝑢 ∗ 𝑦 𝑢

Distance Shadowing – large scale fading Multipath – small scale fading; inter symbol interference

Source: x(t) Destination: y(t) Reflected paths

What is a good model for h(t)?

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Two problems

(b) Interference… 𝑗 𝑢 =

𝑗=1 𝑜

ℎ𝑗 𝑢 ∗ 𝑦𝑗 𝑢

Source: x(t) Destination: y(t)

What is a good model for i(t)?

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Two problems

(b) Interference… Another way to address the problem is to consider that interference is a signal

Source: x(t) Destination: y(t)

𝒁 = 𝑰. 𝒀 + 𝑶 Distributed MIMO – Massive MIMO

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We keep on the simple antenna receiver. (a) Channel modeling issues (b) Interference modeling issues 𝑧 𝑢 = ℎ 𝑢 ∗ 𝑦 𝑢 + 𝑗(𝑢) + 𝑜(𝑢)

Source: x(t) Destination: y(t)

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Agenda

• The problem
• Radio channel
• Interference

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Agenda

• The problem
• Radio channel (statistical model)
• Background / Previous works
• Measurements and new approach
• Some results
• Space evolution…
• Interference

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x(t) Channel y(t) 𝑧 𝑢 = 𝑃𝑢𝑢 𝑦 𝑢 = 𝑦 𝑡 𝐿1 𝑢, 𝑡 𝑒𝑡 So… to be known, traditionally:

• We work with second order statistics
• The signal resolution is much less than the multipath resolution

We introduce t such that 𝐿1 𝑢, 𝑢 − τ = ℎ(𝑢, τ) and we take 𝑡 = 𝑢 − τ: 𝑧 𝑢 = 𝑦 𝑢 − 𝜐 ℎ 𝑢, 𝜐 𝑒𝜐 Impulse response

The channel as a linear transformation

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 t Amplitude Multipaths

Impulse response – check your glasses...

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• T

T 2T 0.5 1 t Amplitude

Symbol duration T>>1

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• T

T 2T 0.5 1 t Amplitude Symbol 1

Symbol duration T>>1

The impact of the previous symbol at the sampling instant is very low.

BUT we want higher data rates… so we increase the bandwidth.

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• T

T 2T 3T 0.5 1 t Amplitude

When sampling at 3T for recovering the 4th symbol, inter-symbol interference arises due to those late paths that carries the 1st symbol.

Symbol duration T<<1

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Impulse responses Four random variables / vectors: number of paths, delays, amplitudes, phases

h(t) x(t) y(t)

 

 



 

 

1 N k k k

h t t   t

40 80 0.1 0.2 Peak detection Delay (ns) 0.1 0.2 Measured Response Delay (ns) 50 100

Modelling the channel as a linear filter

Bandwidth 2GHz Central frequency 58 GHz

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Turin (Modified Poisson) Suzuki, Hashemi (D-k) Saleh et Valenzuela (Two Poisson) [ Used by IEEE 802.15.3a ] Other approaches found in literature : Weibull, Non stationary Poisson, -stable processes.

Delays

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Continuous D-K model (Turin)

K.l(t) l(t) Mean arrival rate delay D

Discrete D-K (Suzuki, Hashemi)

20 40 60 0.5 1

Saleh Valuenzela Based on clusters and… two Poisson

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Amplitudes/Phases

Rayleigh, Rice, Nakagami… Gamma, lognormal, … Delay and amplitude modeling are difficult Data pre-processing is complex

Delays

Besides those statistical models rely on second order statistics Turin (Modified Poisson) Suzuki, Hashemi (D-k) Saleh et Valenzuela (Two Poisson) [ Used by IEEE 802.15.3a ] Other approaches found in literature : Weibull, Non stationary Poisson, -stable processes.

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Checking the WSSUS property…

US: Uncorrelated Scatterers Paths that arrive at different time are not correlated. (Stationary in frequency domain) WSS: Wide Sense Stationary In a given area, second order statistical properties do not change. Correlation between two measures at different times t1 and t2 (or different locations d1 and d2) only depends on differences Dt = t2-t1 (or Dd = d2-d1). WSSUS hypothesis

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 We determine RH (transfer function autocorrelation), meaning

• n the different measurement positions with Dd = 0.

RH(f) should be independent of f. US property is not verified in our measurement environment. A determinism is present due to the geometry of the room. Phases of close paths (“distinguished” because UWB) are correlated.

57 59

• 5

5 dB 58 RH(f) (for Df = 0) GHz

Checking the US property

𝑆𝐼 𝑔, 𝑔 + ∆𝑔 = 𝔽𝑒 𝐼 𝑔; 𝑒 𝐼∗ 𝑔 + ∆𝑔; 𝑒

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T 1 2 3 4 5 6 7 8 10 9 14 15 16 17 18 19 20 21 22 23 25 24 26 11 12 13

3 m 4 m 5 m 6 m 7 m Transmitter A few l Around 8l Around 60l > 100l

20 40 60 80 100 0.2 0.4 0.6 0.8 1 20 40 60 80 100 0.2 0.4 0.6 0.8 10 20 40 60 80 100 0.2 0.4 0.6 0.8 1

Checking the WSS property

 We determine RH meaning on frequencies with Df = 0.

Stationary areas (l = 5 mm)

𝑆𝐼 𝑒, 𝑒 + ∆𝑒 = 𝔽𝑔 𝐼 𝑒; 𝑔 𝐼∗ 𝑒 + ∆𝑒; 𝑔

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To conclude…

Tapped delay line is not adapted High frequency

• Stationary areas are very small
• Difficulty to evaluate the power delay profile

Ultra Wide Band:

• High temporal accuracy makes difficult the impulse

response meaning (fast change in the path time of arrival)

• The « tap » number is very important (160 coefficients for

a 2 GHz band and a 100ns impulse response)

• « Taps » are correlated
• Rice and Rayleigh distribution are not adapted anymore.

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Agenda

• The problem
• Radio channel (statistical model)
• Background / Previous works
• Measurements and new approach
• Some results
• Space evolution…
• Interference

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We measured the 60 GHz channel transfer function with a network analyzer in order to characterize it, model it and run digital communication simulations: Measures from 57 to 59 GHz 1,25 MHz step 26 positions 250 measures per position

T 1 2 3 4 5 6 7 8 10 9 14 15 16 17 18 19 20 21 22 23 25 24 26 11 12 13

3 m 4 m 5 m 6 m 7 m Transmitter

Measured impulse response Amplitude (dB) Delay (ns)

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H(w,.) may be defined by a random measure x defined on R. What is its nature ?

 

,. H w Random process Random measure

 





 dt ,. t h ejtw

 



 l

l x  d ejw H(w,.) is the inverse Fourier transform of h(t)

h(t,.) or H(w,.) Input Output

New approach based on Stable processes

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• Existing models are based on Poisson process

-stable distribution can be seen as a mixture of Poisson processes.

• Properties of measured data

Important variability Infinite variance Skewed long tail distribution

What made us think about -stable process ?

2500 2 N° d’échantillon |H(w)| 250 500 0.02 0.03 58 GHz Nbre d’échantillons Variance

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 

  



 

  



 

 l w  w  D

l   w w w w d e ,. ' H ,. H ' , C

' j

Second order moments are not defined so we work on

• covariation. For harmonizable Symmetric  Stable (SS)

functions, we can write under some conditions (Cambanis): The spectral measure 

• is unique
• characterizes the stochastic process H(w,.)
• can be determined by the Inverse Fourier

Transform of the covariation function

How then can we characterize the transfer function ?

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ℎ 𝑢, . ~ 𝜈 ℝ 𝑑𝛽

1 𝛽 𝑗=1 +∞

𝛿𝑗Γ

𝑗 −1 𝛽 𝑇𝑗 1 + 𝑘𝑇𝑗 2 𝜀 𝑢 − 𝜑𝑗

How do we generate the impulse response ?

Independent Rademasher variables P(g=1|v)=P(g=-1|v)=0.5 Gamma distributed of rate i Arrival time of Poisson process Random variable with PdF 𝜈 𝜈 ℝ Independent RV uniformly distributed on the unit circle

We can decompose H(w,.) in Lepage series:

Inverse Fourier Transform Thresholding: we limit the infinite sum. Γ𝑗

−1

𝛽 becomes very small when i increases.

𝐼 𝜕, . ~ 𝜈 ℝ 𝑑𝛽

1 𝛽 𝑗=1 +∞

𝛿𝑗Γ

𝑗 −1 𝛽𝑓𝑘𝜕𝜑𝑗 𝑇𝑗 1 + 𝑘𝑇𝑗 2

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Agenda

• The problem
• Radio channel (statistical model)
• Background / Previous works
• Measurements and new approach
• Some results
• Space evolution…
• Inteference

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Result example – simulated on the left and measured on the right.

0.5 1 25 50 0.5 1 delay(ns) Amplitude 0.5 1 40 80 ns 0.2 0.6 1 40 80 ns 0.2 0.6 1 40 80 ns 0.2 0.6 1

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5 10 15 0.1 0.2 0.3 Measures Modèle Modèle (250 mesures)

Result example – Delay spread PdF Good fit. Suitable for the whole room.

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Agenda

• The problem
• Radio channel (statistical model)
• Background / Previous works
• Measurements and new approach
• Some results
• Space evolution…
• Inteference

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Space evolution – only some thoughts

Being able to characterize and model this space evolution is important to optimise the communications. The radio channel changes due to mobility

• r simply to the different

location in space of different objects.

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Spatial evolution model

 

  

  p x x x

H H f H , ,

1



Channel state Previous channel states Noise Problems: statistical nature of H. dimension of H. 𝑔 . = 𝔽 𝐼𝑦+1 𝐼𝑦, … , 𝐼𝑦−𝑞

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Dimension of H  AR modelling of the transfer function

     

i , x q k k i x k p k i k x k i 1 x

H f H  w   w  w             

    

58 58,5 59 59,5 60

• 50
• 40
• 30

Frequency (GHz) abs(H) 58 58,5 59 59,5 60

• 0.1

0.1 Frequency (GHz) Real(H)

Spatial evolution model

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Parameter estimation. We propose a framework: semi-parametric single index models We need more measurements to evaluate our model. Validity of the model Is the function f existing? Is it realistic to introduce a frequency dependency? Solutions are proposed but asymptotic behaviours have to be studied.

Spatial evolution model

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Channel models – some conclusions

-stable random processes are good candidates Needs for further validation (data) How can we say if the fit is good? Time to generate Better interpretation of the model is still need Space/time evolution

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Agenda

• The problem
• Radio channel
• Interference

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Agenda

• The problem
• Radio channel
• Interference
• Previous works
• Dependence

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What is interference?

Collection of other transmitters using the same frequency band at the same time (from the same networks but not necessarily; it can be Zigbee and WIFI for instance). All the contributions add at the destination. 𝑍 =

𝑗=1 𝑂

𝑆𝑗

−𝜏 2𝐵𝑗𝑑𝑗𝑓𝑘𝜚𝑗 = 𝑗=1 𝑂

𝑍

𝐽,𝑗 + 𝑘 𝑗=1 𝑂

𝑍

𝑅,𝑗 Channel PHY layer Delay

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Common approach: interference is modelled by a Gaussian random process. Problem:  when the number of interferers is large but there are dominant interferers In many cases, the interference pdf exhibits a heavier tail than what is predicted by the Gaussian model.  impulsive interference: Middleton Class A, Gaussian- mixture, generalized Gaussian, Laplace, -stable...

Brief state of the art

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Ψ𝑍 𝜕𝐽, 𝜕𝑅 = log 𝔽 𝑓𝑘𝜕𝐽𝑍𝐽,𝑗+𝑘𝜕𝑅𝑍𝑅,𝑗 = −𝜇𝜌 𝜕𝐽2 + 𝜕𝑅2

2 𝜏𝔽 𝑩𝒅 4 𝜏

+∞ 𝐾1 𝑦 𝑦

4 𝜏

𝑒𝑦 = −𝛿 𝜕 𝛽

We derive the characteristic function

Interference distribution?

𝑍 =

𝑗=1 𝑶

𝑺𝑗

−𝜏 2𝑩𝑗𝒅𝑗𝑓𝑘𝝔𝑗 = 𝑗=1 𝑂

𝑍

𝐽,𝑗 + 𝑘 𝑗=1 𝑂

𝑍

𝑅,𝑗

-stable distribution

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• 30
• 15

15 30 Time Amplitude -stable noise (=1.5, =0.5)

• 30
• 15

15 30 Time Amplitude Gaussian noise (=2, =1)

• 5

5 0.25 0.5 Probability density function

• 5

5 0.01 0.02 Zoom on tails Heavy tail

Modelling Interference

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           b i h y b i h y . 1 : H . 1 : H

1

 

h y, max arg ˆ x P x

x



MAP approach: ML approach:

   





 

K k k h x y x x x

y f f x

k k

1 , ,

argmax argmax ˆ y

h y

 



 

 

K k k k k B I x

h x x h y f x

1

, argmax ˆ

Finally: Detection of the transmitted bit (in the binary case): Interference + thermal noise Can be applied to a large number of situations: UWB, MIMO, Cooperation and even more as soon as some redundancy is used.

Impact on Receivers

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• 5 0 5

5

• 5

Sample 2 Sample 1

Possible transmitted symbols Received samples

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• 5 0 5

5

• 5

Gaussian case Sample 2 Sample 1

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-stable (=1.5 ; g=0.5) + Gaussian (s²=0.25)

• 5 0 5

5

• 5

r2 r1

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10-4 10-3 10-2 10-1

BER comparison of different receivers for NIR=0dB, =1.5. 1/g (dB) BER

p-norm Soft-limiter NIG MME Linear combiner Myriad receiver 4 6 8 10 12 14 16 18

26/30

Medium impulsiveness

Performance

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Channel capacity

Result : we obtained lower and upper bounds for an additive - stable channel: Theorem: a lower bound to the capacity of an additive symmetric -stable channel subject to 𝔽 𝑌 ≤ 𝑏 is: 𝐷 ≥ 1 𝛽 log2 1 + 𝑏 𝔽 𝑂

𝛽

Exact bound obtained numerically (Blahut Arimoto algorithm) Extension to an isotropic -stable additive interference

• introduce some dependence between the real and

imaginary part

• written as a sub-Gaussian random process

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Agenda

• The problem
• Radio channel
• Inteference
• Previous works
• Dependence

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RV Shaping filters Phase shift RV Source RV Channel

𝑧 𝑗 𝑢 =

𝑙=0 𝑜

𝐵𝑙

𝑗 𝑓𝑘𝜒𝑙

𝑗

𝑆𝑗

−𝛿 2

𝑌𝑙

𝑗 𝑑𝑙 𝑗 𝑢

More details on the received interference

In the end – lots of random variable, leading to usual uncorrelated assumptions

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time Y(t)

Noise realisation

x fY(x)

Noise pdf

x1 x2

Two dependent samples

x2 x1

Two independent samples

SIMO case with or without correlated channels

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time Y(t)

Noise realisation

x fY(x)

Noise pdf

x1 x2

Joint representation of two repetitions

UWB case

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Traditionally

• Finite second order moments
• Correlation coefficient is an adapted concordance measure

𝜍𝑌,𝑍 = 𝐹 𝑌 − 𝜈𝑌 𝐹 𝑍 − 𝜈𝑧 𝜏𝑌𝜏𝑍 But:

• Not adapted to impulsive interference (and especially -stable

distributions)

• Do not allow to model tail dependence (simultaneous strong samples

in the same vector) As a consequence we are interested in other dependence models allowing more flexible concordance measures.

How to introduce dependence?

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Multivariate -stable models exist… but are difficult to handle (intractable distribution function). As an alternative, we keep the marginal behavior of the stable random variables and propose a copula to model the dependence structure We can define a copula as follows. Consider a random vector 𝑌 ∈ 𝑆𝑒 with a continuous distribution F. Then to X one can associate a d-copula 𝐷: 0,1 𝑒 → 0,1 defined by: 𝐺 𝑦1, 𝑦2, … , 𝑦𝑒 = 𝐷 𝐺

1 𝑦1 , … , 𝐺𝑒 𝑦𝑒

Marginal (1D) distributions of 𝑌𝑗

Copulas

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Sklar’s theorem (for the 2D case but can be extended to any dimensions): Let X and Y be random variables with distribution functions F and G and joint distribution H. There exists a copula C such that for all (𝑦, 𝑧) ∈ ℝ × ℝ one has: 𝐼 𝑦, 𝑧 = 𝐷 𝐺 𝑦 , 𝐻 𝑧 If F and G are continuous then C is unique. Otherwise, C is uniquely determined on 𝑆𝑏𝑜 𝐺 × 𝑆𝑏𝑜 𝐻 . Conversely, if C is a copula and F and G are distribution functions, then the function H previously defined is a joint distribution function with margins F and G.

Copulas

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 



 

 

K k k k k B I x

h x x h y f x

1

, argmax ˆ

Back on receiver strategies

We have to define some robust strategies for detection or error correction. It assumes at least independent samples…. So we need to go one step back to the multidimensional distribution…

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Binary case – transmitted symbol 𝑡 ∈ −1, +1 Two symbols received 𝑧1 = 𝑡 + 𝑗1, 𝑧2 = 𝑡 + 𝑗2 . Log-likelihood ratio: Λ 𝑧1, 𝑧2 = log ℙ 𝑧1, 𝑧2 𝑡 = +1 ℙ 𝑧1, 𝑧2 𝑡 = −1 Let 𝑔 be the joint density of 𝑗1, 𝑗2 then, Λ 𝑧1, 𝑧2 = log 𝑔 𝑧1 − 1, 𝑧2 − 1 𝑔 𝑧1 + 1, 𝑧2 + 1

If 𝐺 𝑦, 𝑧 = 𝐷 𝐺𝑗 𝑦 , 𝐺𝑗 𝑧 Then 𝑔 𝑦, 𝑧 = 𝑔

𝑗 𝑦 𝑔 𝑗 𝑧 𝑑 𝐺𝑗 𝑦 , 𝐺𝑗 𝑧

Where 𝑑 𝑦, 𝑧 =

𝜖2𝐷 𝜖𝑦𝜖𝑧 𝑦, 𝑧 is the density of the copula.

Back on receiver strategies

USR – CNRS 3380

STM Workshop - July 2016

In the following, we assume:

• Archimedean copulas; C 𝑣, 𝑤 = 𝜚−1 𝜚 𝑣 + 𝜚 𝑤

𝜚 . is the generator of the copula, continuous and convex function with 𝜚 1 = 0

• Cauchy Marginals: 𝑔 𝑦 =

𝜀 𝜌 𝜀2+ 𝑦−𝜈 2 , with 𝜈 = 0

Back on receiver strategies

Λ 𝑧1, 𝑧2 = log 𝑔 𝑧1 − 1, 𝑧2 − 1 𝑔 𝑧1 + 1, 𝑧2 + 1 = log 𝑔

𝑗1 𝑧1 − 1 𝑔 𝑗2 𝑧2 − 1 𝑑 𝐺𝑗1 𝑧1 − 1 𝐺𝑗2 𝑧2 − 1

𝑔

𝑗1 𝑧1 + 1 𝑔 𝑗2 𝑧2 + 1 𝑑 𝐺𝑗1 𝑧1 + 1 𝐺𝑗2 𝑧2 + 1

= Λ⊥ 𝑧1, 𝑧2 + Λ𝑑 𝑧1, 𝑧2

Divided in an independent and a dependent term

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STM Workshop - July 2016

Independent case / Cauchy marginals

It can be reduced to 𝑧1 + 𝑧2 𝑧1𝑧2 + 𝜀2 + 1 > 0 in the Cauchy independent case

Λ⊥ 𝑧1, 𝑧2 = log 𝑑 𝐺𝑗1 𝑧1 − 1 𝐺𝑗2 𝑧2 − 1 𝑑 𝐺𝑗1 𝑧1 + 1 𝐺𝑗2 𝑧2 + 1

𝐺𝑗 𝑦 is the Cauchy CdF: 𝐺𝑗 𝑣 =

1 𝜌 arctan 𝑣−𝜈 𝜀

+

1 2

Λ⊥ 𝑧1, 𝑧2 = log 𝑔 𝑧1 − 1 − log 𝑔 𝑧1 + 1 + log 𝑔 𝑧2 − 1 + log 𝑔 𝑧2 + 1

USR – CNRS 3380

STM Workshop - July 2016

Dependent case / Cauchy marginals - Clayton

𝐷𝜄 𝑣, 𝑤 = 𝑣−𝜄 + 𝑤−𝜄 − 1

− 1 𝜄

𝑑𝜄 𝑣, 𝑤 = 1 + 𝜄 𝑣𝑤 − 𝜄+1 𝑣−𝜄 + 𝑤−𝜄 − 𝜄+2

Λ𝐷 𝑦, 𝑧 = 1 + 𝜄 log 𝐺𝑗 𝑧1 + 1 𝐺𝑗 𝑧2 + 1 − log 𝐺𝑗 𝑧1 − 1 𝐺𝑗 𝑧2 − 1 + 2 + 1 𝜄 log 𝐺𝑗 𝑧1 + 1 −𝜄 + 𝐺𝑗 𝑧2 + 1 −𝜄 − 1

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STM Workshop - July 2016

Dependent case / Cauchy marginals - Gumbel

𝐷𝜄 𝑣, 𝑤 = exp − − log 𝑣 𝜄 + − log 𝑤 𝜄

1 𝜄

𝑑𝜄 𝑣, 𝑤 = 𝐷𝜄 𝑣, 𝑤 − log 𝑣 𝜄 + − log 𝑤 𝜄

1 𝜄−2

𝜄 − 1 + − log 𝑣 𝜄 + − log 𝑤 𝜄

1 𝜄

− log 𝑣 𝜄−1 + − log 𝑤 𝜄−1 𝑣𝑤

Λ𝐷 𝑦, 𝑧 = − log 𝐺

𝑗 𝑧1 + 1 𝜄 + − log 𝐺 𝑗 𝑧2 + 1 𝜄 − − log 𝐺 𝑗 𝑧1 − 1 𝜄 − − log 𝐺 𝑗 𝑧2 − 1 𝜄

+ 1 𝛽 + 2 log 𝐺

𝑗 𝑧1 − 1 −𝜄 + 𝐺 𝑗 𝑧2 − 1 −𝜄 − 2

𝐺

𝑗 𝑧1 + 1 −𝜄 + 𝐺 𝑗 𝑧2 + 1 −𝜄 − 2

+ log 𝜄 − 1 + 𝜚0 𝐺

𝑗 𝑧1 − 1 𝜄 + 𝜚0 𝐺 𝑗 𝑧2 − 1 𝜄 1 𝜄

𝜄 − 1 + 𝜚0 𝐺

𝑗 𝑧1 + 1 𝜄 + 𝜚0 𝐺 𝑗 𝑧2 + 1 𝜄 1 𝜄

+ log 𝐺

𝑗 𝑧1 − 1 𝜄−1 − 1

𝐺

𝑗 𝑧2 − 1 𝜄−1 − 1

𝐺

𝑗 𝑧1 + 1 𝜄−1 − 1

𝐺

𝑗 𝑧2 + 1 𝜄−1 − 1

+ log 𝐺

𝑗 𝑧1 − 1 𝐺 𝑗 𝑧2 − 1

𝐺

𝑗 𝑧1 + 1 𝐺 𝑗 𝑧2 + 1

USR – CNRS 3380

STM Workshop - July 2016

Dependent case / Cauchy marginals - Franck

𝐷𝜄 𝑣, 𝑤 = − 1 𝜄 log 1 + 𝑓−𝜄𝑣 − 1 𝑓−𝜄𝑤 − 1 𝑓−𝜄 − 1 𝑑𝜄 𝑣, 𝑤 = 𝜄𝑓−𝜄 𝑣+𝑤 𝑓−𝜄 − 1 + 𝑓−𝜄𝑣 − 1 𝑓−𝜄𝑤 − 1 𝑓−𝜄𝑣 − 1 𝑓−𝜄𝑤 − 1 𝑓−𝜄 − 1 + 𝑓−𝜄𝑣 − 1 𝑓−𝜄𝑤 − 1 − 1 Λ𝐷 𝑦, 𝑧 = log

𝜄𝑓−𝜄 𝐺𝑗 𝑧1−1 +𝐺

𝑗 𝑧2−1

𝑓−𝜄 − 1 + 𝑓−𝜄𝐺

𝑗 𝑧1−1 − 1

𝑓−𝜄𝐺

𝑗 𝑧2−1 − 1

+ log 𝑓−𝜄𝐺𝑗 𝑧1−1 − 1 𝑓−𝜄𝐺𝑗 𝑧2−1 − 1 𝑓−𝜄 − 1 + 𝑓−𝜄𝐺𝑗 𝑧1−1 − 1 𝑓−𝜄𝐺𝑗 𝑧2−1 − 1 − 1 − log 𝜄𝑓−𝜄 𝐺𝑗 𝑧1+1 +𝐺𝑗 𝑧2+1 𝑓−𝜄 − 1 + 𝑓−𝜄𝐺𝑗 𝑧1+1 − 1 𝑓−𝜄𝐺𝑗 𝑧2+1 − 1 − log 𝑓−𝜄𝐺𝑗 𝑧1+1 − 1 𝑓−𝜄𝐺𝑗 𝑧2+1 − 1 𝑓−𝜄 − 1 + 𝑓−𝜄𝐺𝑗 𝑧1+1 − 1 𝑓−𝜄𝐺𝑗 𝑧2+1 − 1 − 1

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

Clayton copula (=1) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

Clayton copula (=1) and Cauchy marginals S1(0,0.05,0)

What difference does it make?

Λ⊥ Λ𝑑 Λ⊥ + Λ𝑑

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

Clayton copula (=1) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

1 2 3

• 2.6
• 2.4
• 2.2
• 2
• 1.8

 log10(BER) Gaussian receiver Cauchy receiver Copula receiver Clayton copula (=1) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

Gumbel copula (=3) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

Gumbel copula (=3) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

What difference does it make?

Gumbel copula (=3) and Cauchy marginals S1(0,0.05,0)

USR – CNRS 3380

STM Workshop - July 2016

1 2 3 4 5

• 2.6
• 2.4
• 2.2
• 2
• 1.8

Gumbel copula and Cauchy marginals S1(0,0.05,0)  log10(BER) Gaussian receiver Cauchy receiver Copula receiver

What difference does it make?

USR – CNRS 3380

STM Workshop - July 2016

Frank copula (=4) and Cauchy marginals S1(0,0.05,0)

What difference does it make?

USR – CNRS 3380

STM Workshop - July 2016

Frank copula (=4) and Cauchy marginals S1(0,0.05,0)

What difference does it make?

USR – CNRS 3380

STM Workshop - July 2016

Frank copula (=4) and Cauchy marginals S1(0,0.05,0)

What difference does it make?

USR – CNRS 3380

STM Workshop - July 2016

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

• 2.6
• 2.4
• 2.2
• 2
• 1.8

Frank copula and Cauchy marginals S1(0,0.05,0)  log10(BER) Gaussian receiver Cauchy receiver Copula receiver

What difference does it make?

USR – CNRS 3380

STM Workshop - July 2016

Thanks for your attention….