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 1/33

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 2/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Outline Motivation 1 From continuous to discrete time 2 Finite Resolution Receivers 3 Capacity bounds 4 Conclusions 5 L. Barletta — Information Rates for Phase Noise Channels 3/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Outline Motivation 1 From continuous to discrete time 2 Finite Resolution Receivers 3 Capacity bounds 4 Conclusions 5 L. Barletta — Information Rates for Phase Noise Channels 4/33

Department of Information, Electronics and Bioengineering Politecnico di Milano A Classic Communication Scheme Channel Shaping Source encoder filter Channel transfer function Channel Matched Sink PS filter decoder L. Barletta — Information Rates for Phase Noise Channels 5/33

Department of Information, Electronics and Bioengineering Politecnico di Milano The AWGN Channel Channel Shaping Source encoder filter Matched Channel Sink PS decoder filter L. Barletta — Information Rates for Phase Noise Channels 6/33

Department of Information, Electronics and Bioengineering Politecnico di Milano The AWGN Channel Channel Shaping Source encoder filter Matched Channel Sink PS decoder filter Y k = X k + W k L. Barletta — Information Rates for Phase Noise Channels 6/33

Department of Information, Electronics and Bioengineering Politecnico di Milano The Actual AWGN Channel Channel Shaping Source encoder filter Channel Matched Sink PS decoder filter? L. Barletta — Information Rates for Phase Noise Channels 7/33

Department of Information, Electronics and Bioengineering Politecnico di Milano The Actual AWGN Channel Channel Shaping Source encoder filter Channel Matched Sink PS decoder filter? Phase noise processes Θ tx and Θ rx L. Barletta — Information Rates for Phase Noise Channels 7/33

Department of Information, Electronics and Bioengineering Politecnico di Milano The Actual AWGN Channel Channel Shaping Source encoder filter Channel Matched Sink PS decoder filter? Phase noise processes Θ tx and Θ rx Sampling at kT symb is no longer optimal! L. Barletta — Information Rates for Phase Noise Channels 7/33

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 L. Barletta — Information Rates for Phase Noise Channels 8/33

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 8/33

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 L. Barletta — Information Rates for Phase Noise Channels 8/33

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) L. Barletta — Information Rates for Phase Noise Channels 8/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Phase Noise Channels Questions: Models for phase noise channels L. Barletta — Information Rates for Phase Noise Channels 9/33

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 9/33

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 9/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Outline Motivation 1 From continuous to discrete time 2 Finite Resolution Receivers 3 Capacity bounds 4 Conclusions 5 L. Barletta — Information Rates for Phase Noise Channels 10/33

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: � T X ( t ) ∈ L 2 [0 , T ] − | X ( t ) | 2 dt < ∞ → 0 L. Barletta — Information Rates for Phase Noise Channels 11/33

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: � T X ( t ) ∈ L 2 [0 , T ] − | X ( t ) | 2 dt < ∞ → 0 Let { φ m ( t ) } m be a complete orthonormal basis of L 2 [0 , T ], i.e. � 1 � T m = n φ m ( t ) φ n ( t ) ⋆ dt = 0 m � = n 0 L. Barletta — Information Rates for Phase Noise Channels 11/33

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: � T X ( t ) ∈ L 2 [0 , T ] − | X ( t ) | 2 dt < ∞ → 0 Let { φ m ( t ) } m be a complete orthonormal basis of L 2 [0 , T ], i.e. � 1 � T m = n φ m ( t ) φ n ( t ) ⋆ dt = 0 m � = n 0 ∞ � T � X ( t ) φ m ( t ) ⋆ dt X ( t ) = X m φ m ( t ) , X m = 0 m =1 L. Barletta — Information Rates for Phase Noise Channels 11/33

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: � T X ( t ) ∈ L 2 [0 , T ] − | X ( t ) | 2 dt < ∞ → 0 Let { φ m ( t ) } m be a complete orthonormal basis of L 2 [0 , T ], i.e. � 1 � T m = n φ m ( t ) φ n ( t ) ⋆ dt = 0 m � = n 0 ∞ � T � X ( t ) φ m ( t ) ⋆ dt X ( t ) = X m φ m ( t ) , X m = 0 m =1 Equivalent representations: { X ( t ) } T t =0 ⇐ ⇒ X 1 X 2 · · · L. Barletta — Information Rates for Phase Noise Channels 11/33

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] � � { X ( t ) } T t =0 ; { Y ( t ) } T n →∞ I ( X 1 · · · X n ; Y 1 · · · Y n ) I = lim t =0 (if it exists) L. Barletta — Information Rates for Phase Noise Channels 12/33

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] � � { X ( t ) } T t =0 ; { Y ( t ) } T n →∞ I ( X 1 · · · X n ; Y 1 · · · Y n ) I = lim t =0 (if it exists) � T � T X ( t ) φ m ( t ) ⋆ dt , Y ( t ) φ m ( t ) ⋆ dt X m = Y m = 0 0 L. Barletta — Information Rates for Phase Noise Channels 12/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Example: mutual information for a phase noise channel Input-output relation: Y ( t ) = X ( t ) e j Θ( t ) + W ( t ) L. Barletta — Information Rates for Phase Noise Channels 13/33

Department of Information, Electronics and Bioengineering Politecnico di Milano Example: mutual information for a phase noise channel Input-output relation: Y ( t ) = X ( t ) e j Θ( t ) + W ( t ) Choose an incomplete orthonormal basis as � t − m ∆ + ∆ / 2 � 1 √ φ m ( t ) = rect , m = 1 . . . n , ∆ = T / n ∆ ∆ L. Barletta — Information Rates for Phase Noise Channels 13/33