Information-Energy Capacity Region for SWIPT Systems with Power Amplifier Nonlinearity Ioannis Krikidis Dpt. of Electrical and Computer Engineering IRIDA Research Centre for Communication Technologies University of Cyprus E-mail: krikidis@ucy.ac.cy IEEE ISIT’20, Los Angeles, USA I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 1 / 21
Outline Background/Problem formulation 1 Information-energy capacity region 2 Numerical results 3 Conclusion/Future work 4 I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 2 / 21
SWIPT/Motivation (1) Massive Connectivity: Low-power devices: - 2017: 5 Billions phones - e.g., state-of-the-art sensors 10 µ Watt - 2035: Trillions IoT devices Low-power Koomey's law Computations per Microjoules electrical efficiency of computing doubled every 1.5 years Year I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 3 / 21
SWIPT/Motivation (2) Simultaneous Wireless Information and Power Transfer (SWIPT). - Downlink: Simultaneous (same waveform) information and power transfer. Energy Flow Information Flow Co-located Information & Energy Receivers Separate Information & Energy Receivers Fundamental information-energy trade-off - Input distribution , waveform design, beamforming design etc (Tx). - Receiver structure/signal processing techniques (Wireless devices). Information-energy region: all the achievable information and energy tuples ( R , E ) under a given transmit power constraint P . I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 4 / 21
Problem Formulation Signals with high PAPR boost the wireless energy harvesting e.g., multisine signals, chaotic signals etc. - High-power amplifier (HPA) nonlinearities. Information-energy capacity region - Linear information transfer channel. - Non-linear power transfer channel. - Average power (AP) and Peak power (PP) constraints. Information-Energy Capacity Region - HPA nonlinearities. I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 5 / 21
Outline Background/Problem formulation 1 Information-energy capacity region 2 Numerical results 3 Conclusion/Future work 4 I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 6 / 21
System model (1) h I IR x [ k ] HPA h E Transmitter ER Time-slotted real-valued transmission. Channel fading gains h I and h E are flat and fixed over all time slots and perfectly known at TX; AWGN channel. PAM signal x ( t ) = � ∞ k = −∞ x [ k ] p ( t − kT ) , with rectangular p ( t ) . x [ k ] i.i.d real random variable X with CDF F X ( x ) . AP: E [ X 2 ] ≤ σ 2 x , PP: | X | ≤ A , where A is the peak amplitude. I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 7 / 21
System model (2) h I IR x [ k ] h E HPA Transmitter ER y ( t ) = h I ˆ x ( t ) + n ( t ) (information channel) � − ( y − h I d ( x )) 2 � 1 p ( y | x ) = 2 π exp (transition probability) √ 2 I 0 ( Bh E | ˆ � � Energy harvesting ∝ E X | ) (power channel). Solid state power amplifier (SSPA) model i.e., ˆ X = d ( X ) . 1 =1 =2 =5 =80 0.5 Output Voltage [Volts] r d ( r ) = 2 β , 1 � 2 β � 1 A s 0 Non-linear � regime � 0.5 r 1 + 0 A s -0.5 -0.5 A 0 -1 -5 0 5 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 Input Voltage [Volts] I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 8 / 21
Maximize energy transfer We consider firstly the case where the IR is not present/active. I 0 ( Bh E | ˆ � � ( P 1 ) max E X | ) F X E [ X 2 ] ≤ σ 2 s.t. x | X | ≤ A . Proposition 1 The maximum average harvested energy and the associated mass point distribution are given by E max = pI 0 � Bh E d ( λ ) � + ( 1 − p ) , where If A 2 ≤ σ 2 x , we have p = 1, λ = A , and mass point distribution Π A = 1 2 ( δ − A + δ A ) . x ≤ A 2 and g ( x ) ց ( σ x , A ) , we have p = 1, λ = σ x , and Π A = 1 If σ 2 2 ( δ − σ x + δ σ x ) . σ 2 σ 2 � σ 2 � x ≤ A 2 and g ( x ) ր ( σ x , A ) , we have p = If σ 2 A 2 , λ = A , and Π A = x 2 A 2 ( δ − A + δ A ) + x 1 − x δ 0 . A 2 σ 2 x ≤ A 2 and the function g ( x ) ր ( σ x , A ′ ) and g ( x ) ց ( A ′ , A ) , we have p = If σ 2 A ′ 2 , λ = A ′ , and mass x σ 2 � σ 2 � δ 0 , with A ′ ≈ A s for β ≫ 1, x x point distribution Π A = 2 A ′ 2 ( δ − A ′ + δ A ′ ) + 1 − A ′ 2 1 where δ x is the Dirac measure (point mass) concentrated at x , and g ( x ) = x 2 [ I 0 ( Bh E d ( x )) − 1 ] . � ∞ p ( y | x j ) � j p ( y | x j ) p j dy where p j = P [ X = x j ] ; Π A = � I min = j p ( y | x j ) p j log 2 j p j δ x j −∞ � is a binary/ternary distribution (low computation complexity). I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 9 / 21
Maximize information capacity We consider the case where the target of the system is to maximize the Shannon information capacity under both AP and PP constraints. ( P 2 ) max I ( X ; Y ) F X E [ X 2 ] ≤ σ 2 s.t. x | X | ≤ A , - Channel output Y = h I ˆ X + N with ˆ X = d ( X ) . � A � ∞ � � p ( y | x ) - I ( X ; Y ) = −∞ p ( y | x ) log 2 dydF X . − A p ( y ; F X ) - Solution: The optimal input probability function F X is unique, finite and discrete [Smith, 1971]. I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 10 / 21
Information-energy capacity region (3) (P2) can be reformulated by the following convex opt. problem: I � � m � n p ij ( P 3 ) max j = 1 p ij p j log 2 i = 1 � n k = 1 p ik p k p p p E [ X 2 ] ≤ σ 2 s.t. x | X | ≤ A 1 ⊤ p p p p � 0 , 1 1 p p = 1 , 1 where 1 1 denotes a vector with ones, p ij = P ( Y = y i | X = x j ) , and p = [ p 1 , p 2 , . . . , p n ] ⊤ . p p p ∗ is the solution to (P3), the maximum mutual information becomes p - If p equal to I max = � m � n p ij j = 1 p ij p ∗ j log 2 k . � n i = 1 k = 1 p ik p ∗ - In case that ER is active, the average energy harvested is written as E min = � n j = 1 p ∗ j I 0 ( Bh E d ( x j )) . Remark 1 If d ( A ) ≤ A ≈ 1 . 665 (peak output amplitude) and A 2 ≤ σ 2 x , there is not a trade-off between information/energy and the same input distribution (i.e., equiprobable binary with mass points at ± A ) maximizes both information and energy transfer simultaneously. I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 11 / 21
Information-energy capacity region (4) Then, we consider the case where both ER and IR are active/present. The IE-CR is defined as � � C ( σ 2 ( I , E ) : I ≤ I max , E ≤ E max , E [ x 2 ] ≤ σ 2 x , A ) = x , | X | ≤ A . - If I ≤ I min , the maximum average harvested energy is given by the input distribution that achieves the rate tuple ( I min , E max ) ; see (P1). - If E ≤ E min , the maximum information rate is given by the input distribution that achieves the rate tuple ( I max , E min ) ; see (P3). - The other points of the boundary I min ≤ I ≤ I max and E min ≤ E ≤ E max ; see (P3) with the extra constraint E min ≤ E [ I 0 ( Bh E | ˆ X | )] ≤ E max (convex problem). I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 12 / 21
Digital predistortion - Compansate the non-linear HPA effects and linearize the non-saturation regime of HPA. - If d ( r ) is deterministic and known at the transmitter, an ideal PD corresponds to the function q ( r ) i.e., A s , If r ≥ A s , d − 1 ( r ) = r 2 β , If − A s < r < A s , q ( r ) = 2 β � 1 � 1 − ( r As ) − A s , If r ≤ − A s . - the information energy capacity region is given by (P1), (P2), (P3). the AP constraint is replaced by E [ q ( x ) 2 ] ≤ σ 2 x . 1 HPA’s output is equal to ˆ X = d ( q ( X )) . 2 Remark 2 It is worth noting that r ≥ d ( r ) and therefore PD penalizes the AP constraint (increases transmit power), while it facilitates the objective functions in (P1)-(P3). I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 13 / 21
Outline Background/Problem formulation 1 Information-energy capacity region 2 Numerical results 3 Conclusion/Future work 4 I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 14 / 21
Numerical results (1) −3 3 x 10 0.5 2.5 0.4 2 0.3 A s =10, β =1 g(x) p(x) A s =10, β =10 1.5 0.2 A s =10, β =80 1 0.1 A s =100, β =10 0.5 0 5 10 15 −16 −10 −7 0 7 10 16 (a) x (b) x 0.8 1 0.8 0.6 0.6 p(x) p(x) 0.4 0.4 0.2 0.2 0 0 −16 −10 0 10 16 −16 −10 0 10 16 (c) x x (d) Figure: (a) The function g ( x ) for different parameters of the SSPA model; we also assume A = 16, B = 0 . 1, and σ 2 x = 49, (b) Input distribution for g ( x ) ց ( σ x , A ) , (c) Input distribution for g ( x ) ր ( σ x , A s ) and g ( x ) ց ( A s , A ) , and (d) Input distribution for g ( x ) ր ( σ x , A ) . I. Krikidis (ECE/UCY) IEEE ISIT’20, Los Angeles, USA 21-26 June, 2020 15 / 21
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