Information Rates for Phase Noise Channels Luca Barletta - - PowerPoint PPT Presentation

Department of Information, Electronics and Bioengineering Politecnico di Milano

Information Rates for Phase Noise Channels

Luca Barletta Politecnico di Milano luca.barletta@polimi.it ESIT 2019 Antibes, July 15th - 19th, 2019

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Joint collaboration with: Gerhard Kramer (Technische Universitaet Muenchen) Stefano Rini (National Chiao Tung University)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A Classic Communication Scheme

Source

Shaping filter Channel transfer function Matched filter Channel encoder Channel decoder

Sink

PS

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The AWGN Channel

Source

Shaping filter Matched filter Channel encoder Channel decoder

Sink

PS

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The AWGN Channel

Source

Shaping filter Matched filter Channel encoder Channel decoder

Sink

PS

Yk = Xk + Wk

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Department of Information, Electronics and Bioengineering Politecnico di Milano

The Actual AWGN Channel

Source

Shaping filter Matched filter? Channel encoder Channel decoder

Sink

PS

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Department of Information, Electronics and Bioengineering Politecnico di Milano

The Actual AWGN Channel

Source

Shaping filter Matched filter? Channel encoder Channel decoder

Sink

PS

Phase noise processes Θtx and Θrx

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Department of Information, Electronics and Bioengineering Politecnico di Milano

The Actual AWGN Channel

Source

Shaping filter Matched filter? Channel encoder Channel decoder

Sink

PS

Phase noise processes Θtx and Θrx Sampling at kTsymb is no longer optimal!

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Phase noise arises due to Imperfections in the oscillator circuits at the transceivers

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Phase noise arises due to Imperfections in the oscillator circuits at the transceivers

Even for high-quality oscillators: if the continuous-time waveform is processed by long filters at the receiver (e.g., long symbol time, OFDM systems), the phase uncertainty accumulates!

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Phase noise arises due to Imperfections in the oscillator circuits at the transceivers

Even for high-quality oscillators: if the continuous-time waveform is processed by long filters at the receiver (e.g., long symbol time, OFDM systems), the phase uncertainty accumulates!

Fast and slow fading effects in wireless environments

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Phase noise arises due to Imperfections in the oscillator circuits at the transceivers

Even for high-quality oscillators: if the continuous-time waveform is processed by long filters at the receiver (e.g., long symbol time, OFDM systems), the phase uncertainty accumulates!

Fast and slow fading effects in wireless environments Nonlinear propagation effects in fiber-optic commun. (amplitude mod. is converted into phase mod. − → phase-noise strength depends also on signal amplitude)

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Questions: Models for phase noise channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Questions: Models for phase noise channels Impact of phase noise on channel capacity

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Phase Noise Channels

Questions: Models for phase noise channels Impact of phase noise on channel capacity Signal and code design

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Representation of continuous-time waveforms

Assume input waveform {X(t)}T

t=0 is square integrable:

X(t) ∈ L2[0, T] − → T |X(t)|2dt < ∞

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Representation of continuous-time waveforms

Assume input waveform {X(t)}T

t=0 is square integrable:

X(t) ∈ L2[0, T] − → T |X(t)|2dt < ∞ Let {φm(t)}m be a complete orthonormal basis of L2[0, T], i.e. T φm(t)φn(t)⋆dt = 1 m = n m = n

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Representation of continuous-time waveforms

Assume input waveform {X(t)}T

t=0 is square integrable:

X(t) ∈ L2[0, T] − → T |X(t)|2dt < ∞ Let {φm(t)}m be a complete orthonormal basis of L2[0, T], i.e. T φm(t)φn(t)⋆dt = 1 m = n m = n X(t) =

∞

• m=1

Xmφm(t), Xm = T X(t)φm(t)⋆dt

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Representation of continuous-time waveforms

Assume input waveform {X(t)}T

t=0 is square integrable:

X(t) ∈ L2[0, T] − → T |X(t)|2dt < ∞ Let {φm(t)}m be a complete orthonormal basis of L2[0, T], i.e. T φm(t)φn(t)⋆dt = 1 m = n m = n X(t) =

∞

• m=1

Xmφm(t), Xm = T X(t)φm(t)⋆dt Equivalent representations: {X(t)}T

t=0 ⇐

⇒ X1X2 · · ·

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Mutual information for random waveforms

The average mutual information between {X(t)}T

t=0 and

{Y (t)}T

t=0 is [Gallager, 1968]

I

• {X(t)}T

t=0 ; {Y (t)}T t=0

• = lim

n→∞ I (X1 · · · Xn ; Y1 · · · Yn)

(if it exists)

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Mutual information for random waveforms

The average mutual information between {X(t)}T

t=0 and

{Y (t)}T

t=0 is [Gallager, 1968]

I

• {X(t)}T

t=0 ; {Y (t)}T t=0

• = lim

n→∞ I (X1 · · · Xn ; Y1 · · · Yn)

(if it exists) Xm = T X(t)φm(t)⋆dt, Ym = T Y (t)φm(t)⋆dt

• L. Barletta — Information Rates for Phase Noise Channels

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Example: mutual information for a phase noise channel

Input-output relation: Y (t) = X(t)ejΘ(t) + W (t)

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Example: mutual information for a phase noise channel

Input-output relation: Y (t) = X(t)ejΘ(t) + W (t) Choose an incomplete orthonormal basis as φm(t) = 1 √ ∆ rect t − m∆ + ∆/2 ∆

• ,

m = 1 . . . n, ∆ = T/n

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Example: mutual information for a phase noise channel

Input-output relation: Y (t) = X(t)ejΘ(t) + W (t) Choose an incomplete orthonormal basis as φm(t) = 1 √ ∆ rect t − m∆ + ∆/2 ∆

• ,

m = 1 . . . n, ∆ = T/n Ym = T Y (t)φm(t)⋆dt = m∆

(m−1)∆

Y (t) √ ∆ dt = Xm m∆

(m−1)∆

ejΘ(t) ∆ dt + Wm

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Example: mutual information for a phase noise channel

Discretized model: Ym = Xm m∆

(m−1)∆

ejΘ(t) ∆ dt + Wm

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Example: mutual information for a phase noise channel

Discretized model: Ym = Xm m∆

(m−1)∆

ejΘ(t) ∆ dt + Wm Both amplitude fading and phase noise!

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Example: mutual information for a phase noise channel

Discretized model: Ym = Xm m∆

(m−1)∆

ejΘ(t) ∆ dt + Wm Both amplitude fading and phase noise! Commonly used discrete-time phase noise channel model: Ym = XmejΘm + Wm, Θm = Θ((m − 1)∆)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Example: mutual information for a phase noise channel

Discretized model: Ym = Xm m∆

(m−1)∆

ejΘ(t) ∆ dt + Wm Both amplitude fading and phase noise! Commonly used discrete-time phase noise channel model: Ym = XmejΘm + Wm, Θm = Θ((m − 1)∆) How different are the two models in terms of capacity?

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

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Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

It depends on the statistics of {Θ(t)}

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Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

It depends on the statistics of {Θ(t)} In general, as small as possible

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

It depends on the statistics of {Θ(t)} In general, as small as possible

Oversampling helps! ∆ ց = ⇒ I (X1 · · · Xn ; Y1 · · · Yn) ր

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

It depends on the statistics of {Θ(t)} In general, as small as possible

Oversampling helps! ∆ ց = ⇒ I (X1 · · · Xn ; Y1 · · · Yn) ր

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled channel model

YmL+ℓ = Xm FmL+ℓ + Wm, ℓ = 1, . . . , L m = 1, . . . , n FmL+ℓ = (mL+ℓ)∆

(mL+ℓ−1)∆

ejΘ(t) ∆ dt, ∆ = Tsymb L

… …

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Receivers with finite time resolution

Tx Integrate and dump

Tx

Detector

How small should ∆ be?

It depends on the statistics of {Θ(t)} In general, as small as possible

Oversampling helps! ∆ ց = ⇒ I (X1 · · · Xn ; Y1 · · · Yn) ր

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Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process Variance increases: Θ(t) ∼ N(0, γ2t)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process Variance increases: Θ(t) ∼ N(0, γ2t) Samples are not independent: E [Θ(t)Θ(s)] = γ2 min{t, s}

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process Variance increases: Θ(t) ∼ N(0, γ2t) Samples are not independent: E [Θ(t)Θ(s)] = γ2 min{t, s} Process with memory!

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise

Widely used for modeling oscillators Also known as Brownian motion: Θ(t) = Θ(0) + γ t Z(t′)dt′ where Z is a white Gaussian process Variance increases: Θ(t) ∼ N(0, γ2t) Samples are not independent: E [Θ(t)Θ(s)] = γ2 min{t, s} Process with memory! Oversampling, i.e. L > 1, increases information rates

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise channel

Define Θk = Θ((k − 1)∆) and Nk ∼ N(0, 1): Θk = Θk−1 + γ √ ∆Nk

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Wiener phase noise channel

Define Θk = Θ((k − 1)∆) and Nk ∼ N(0, 1): Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ (mL+ℓ)∆

(mL+ℓ−1)∆

ej(Θ(t)−ΘmL+ℓ) ∆ dt + WmL+ℓ

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Wiener phase noise channel

Define Θk = Θ((k − 1)∆) and Nk ∼ N(0, 1): Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ (mL+ℓ)∆

(mL+ℓ−1)∆

ej(Θ(t)−ΘmL+ℓ) ∆ dt + WmL+ℓ Contour plot of the unnormalized fading pdf for ∆ = 6 and γ = 1. (Y. Wang et al.,

TCOM 2006, vol. 54, no. 5)

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled Discrete-time Wiener Phase Noise Channel

Let us study the capacity of a simpler model Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ + WmL+ℓ

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled Discrete-time Wiener Phase Noise Channel

Let us study the capacity of a simpler model Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ + WmL+ℓ Capacity under an average power constraint: lim

T→∞

1 T T |X(t)|2 dt ≤ P

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled Discrete-time Wiener Phase Noise Channel

Let us study the capacity of a simpler model Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ + WmL+ℓ Capacity under an average power constraint: E 1 T T |X(t)|2 dt

• ≤ P
• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled Discrete-time Wiener Phase Noise Channel

Let us study the capacity of a simpler model Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ + WmL+ℓ Capacity under an average power constraint: E 1 T T |X(t)|2 dt

• ≤ P

= ⇒ E

• |Xm|2

≤ P∆

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Oversampled Discrete-time Wiener Phase Noise Channel

Let us study the capacity of a simpler model Θk = Θk−1 + γ √ ∆Nk YmL+ℓ = XmejΘmL+ℓ + WmL+ℓ Capacity under an average power constraint: E 1 T T |X(t)|2 dt

• ≤ P

= ⇒ E

• |Xm|2

≤ P∆ C(P, ∆, γ) = lim

n→∞

sup

FXm: E[|Xm|2]≤P∆

1 nI (X1 · · · Xn ; Y1 · · · Yn)

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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How to compute capacity

Challenges: Channel with memory

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How to compute capacity

Challenges: Channel with memory How to deal with oversampling?

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How to compute capacity

Challenges: Channel with memory How to deal with oversampling? Unknown optimal input distribution

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How to compute capacity

Challenges: Channel with memory How to deal with oversampling? Unknown optimal input distribution We will see how to: Get rid of the memory

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How to compute capacity

Challenges: Channel with memory How to deal with oversampling? Unknown optimal input distribution We will see how to: Get rid of the memory Compute an upper bound to capacity

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Department of Information, Electronics and Bioengineering Politecnico di Milano

How to compute capacity

Challenges: Channel with memory How to deal with oversampling? Unknown optimal input distribution We will see how to: Get rid of the memory Compute an upper bound to capacity We assume iid Xm’s with Xm ∼ U[0, 2π)

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

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A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

• (DPI) ≤ 1

n

n

• m=1

I

• X n

1 , ΘmL+1; Ym | Ym−1 1

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

• (DPI) ≤ 1

n

n

• m=1

I

• X n

1 , ΘmL+1; Ym | Ym−1 1

• = 1

n

n

• m=1

I

• X n

1 ; Ym | ΘmL+1, Ym−1 1

• + I
• ΘmL+1; Ym | Ym−1

1

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

• (DPI) ≤ 1

n

n

• m=1

I

• X n

1 , ΘmL+1; Ym | Ym−1 1

• = 1

n

n

• m=1

I

• X n

1 ; Ym | ΘmL+1, Ym−1 1

• + I
• ΘmL+1; Ym | Ym−1

1

• (Markov) = 1

n

n

• m=1

I (Xm; Ym | ΘmL+1) + I

• ΘmL+1; Ym | Ym−1

1

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

• (DPI) ≤ 1

n

n

• m=1

I

• X n

1 , ΘmL+1; Ym | Ym−1 1

• = 1

n

n

• m=1

I

• X n

1 ; Ym | ΘmL+1, Ym−1 1

• + I
• ΘmL+1; Ym | Ym−1

1

• (Markov) = 1

n

n

• m=1

I (Xm; Ym | ΘmL+1) + I

• ΘmL+1; Ym | Ym−1

1

• (Stationary) = I (X1; Y1 | ΘL+1) + 1

n

n

• m=1

I

• ΘmL+1; Ym | Ym−1

1

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A capacity upper bound

Define X n

1 = X1 · · · Xn

1 nI (X n

1 ; Yn 1) = 1

n

n

• m=1

I

• X n

1 ; Ym | Ym−1 1

• (DPI) ≤ 1

n

n

• m=1

I

• X n

1 , ΘmL+1; Ym | Ym−1 1

• = 1

n

n

• m=1

I

• X n

1 ; Ym | ΘmL+1, Ym−1 1

• + I
• ΘmL+1; Ym | Ym−1

1

• (Markov) = 1

n

n

• m=1

I (Xm; Ym | ΘmL+1) + I

• ΘmL+1; Ym | Ym−1

1

• (Stationary) = I (X1; Y1 | ΘL+1) + 1

n

n

• m=1

I

• ΘmL+1; Ym | Ym−1

1

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A capacity upper bound

Assumption: iid Xm’s with Xm ∼ U[0, 2π)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Assumption: iid Xm’s with Xm ∼ U[0, 2π) I

• ΘmL+1; Ym | Ym−1

1

• (DPI) ≤ I
• ΘmL+1; ΘmL+1 ⊕ Xm ⊕ γ

√ ∆NmL+1

• Ym−1

1

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Assumption: iid Xm’s with Xm ∼ U[0, 2π) I

• ΘmL+1; Ym | Ym−1

1

• (DPI) ≤ I
• ΘmL+1; ΘmL+1 ⊕ Xm ⊕ γ

√ ∆NmL+1

• Ym−1

1

• = 0
• L. Barletta — Information Rates for Phase Noise Channels

27/33

Department of Information, Electronics and Bioengineering Politecnico di Milano

A capacity upper bound

Assumption: iid Xm’s with Xm ∼ U[0, 2π) I

• ΘmL+1; Ym | Ym−1

1

• (DPI) ≤ I
• ΘmL+1; ΘmL+1 ⊕ Xm ⊕ γ

√ ∆NmL+1

• Ym−1

1

• = 0

1 nI (X n

1 ; Yn 1) ≤ I (X1; Y1 | ΘL+1)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Polar decomposition

I (X1; Y1 | ΘL+1) = I (|X1|, X1; Y1 | ΘL+1) = I (|X1|; Y1 | ΘL+1)

• amplitude mod.

+ I ( X1; Y1 | ΘL+1, |X1|)

• phase mod.
• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Polar decomposition

I (X1; Y1 | ΘL+1) = I (|X1|, X1; Y1 | ΘL+1) = I (|X1|; Y1 | ΘL+1)

• amplitude mod.

+ I ( X1; Y1 | ΘL+1, |X1|)

• phase mod.

Different bounding techniques are used for the two terms

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Polar decomposition

I (X1; Y1 | ΘL+1) = I (|X1|, X1; Y1 | ΘL+1) = I (|X1|; Y1 | ΘL+1)

• amplitude mod.

+ I ( X1; Y1 | ΘL+1, |X1|)

• phase mod.

Different bounding techniques are used for the two terms

Reveal all phase noise samples to the receiver: amplitude mod.

• n AWGN channel

C ≤ 1 2 log

• 1 + P

2

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Polar decomposition

I (X1; Y1 | ΘL+1) = I (|X1|, X1; Y1 | ΘL+1) = I (|X1|; Y1 | ΘL+1)

• amplitude mod.

+ I ( X1; Y1 | ΘL+1, |X1|)

• phase mod.

Different bounding techniques are used for the two terms

Reveal all phase noise samples to the receiver: amplitude mod.

• n AWGN channel

Application of the I-MMSE formula to the phase mod. term

C ≤ 1 2 log

• 1 + P

2

• +log(2π)+1

2 log

• P∆
• 1 +

4 γ2P∆2 − 1

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Degrees of freedom

Define the degrees of freedom as D(α) = lim

P→∞

C(P, ∆ = P−α, γ) log(P)

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Degrees of freedom

Define the degrees of freedom as D(α) = lim

P→∞

C(P, ∆ = P−α, γ) log(P)

Full DoF Amplitude Mod. Phase Mod.

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Outline

1

Motivation

2

From continuous to discrete time

3

Finite Resolution Receivers

4

Capacity bounds

5

Conclusions

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed Even with infinite time resolution CPN < CAWGN

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed Even with infinite time resolution CPN < CAWGN

For any γ > 0! Even for very high-quality oscillators!

• L. Barletta — Information Rates for Phase Noise Channels

31/33

Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed Even with infinite time resolution CPN < CAWGN

For any γ > 0! Even for very high-quality oscillators!

Conjectures:

• L. Barletta — Information Rates for Phase Noise Channels

31/33

Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed Even with infinite time resolution CPN < CAWGN

For any γ > 0! Even for very high-quality oscillators!

Conjectures:

Simplifying the model by discarding the amplitude fading is too much

• L. Barletta — Information Rates for Phase Noise Channels

31/33

Department of Information, Electronics and Bioengineering Politecnico di Milano

Conclusions

The capacity pre-log depends on the growth rate of receiver’s time resolution ∆ If ∆ ∼ 1/ √ P, an asymptotic capacity pre-log of 0.75 can be achieved, but not surpassed Even with infinite time resolution CPN < CAWGN

For any γ > 0! Even for very high-quality oscillators!

Conjectures:

Simplifying the model by discarding the amplitude fading is too much The fundamental tension between additive noise and phase noise limits the degrees of freedom

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Thanks for your attention!

• L. Barletta — Information Rates for Phase Noise Channels

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Department of Information, Electronics and Bioengineering Politecnico di Milano

Derivation of the average power constraint

lim

T→∞

1 T T |X(t)|2 dt = E 1 T T |X(t)|2 dt

• = E

  1 T T

• n
• m=1

Xm

L

• ℓ=1

φmL+ℓ(t)

• 2

dt   (orthogonality) = E

• 1

T

n

• m=1

L

• ℓ=1

(mL+ℓ)∆

(mL+ℓ−1)∆

|Xm|2 ∆ dt

• = Ln

T E

• |Xm|2

= 1 ∆E

• |Xm|2

≤ P

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