Timely Estimation Using Coded Quantized Samples Ahmed Arafa 1 Karim - PowerPoint PPT Presentation

Timely Estimation Using Coded Quantized Samples Ahmed Arafa 1 Karim Banawan 2 Karim G. Seddik 3 H. Vincent Poor 4 1 Department of Electrical and Computer Engineering, University of North Carolina at Charlotte 2 Department of Electrical Engineering, Alexandria University 3 Electronics and Communications Engineering Department, American University in Cairo 4 Electrical Engineering Department, Princeton University International Symposium on Information Theory June 2020

Age of Information (AoI) 70 °F

Age of Information (AoI) 70 °F observer

Age of Information (AoI) °F 70 °F observer receiver

Age of Information (AoI) °F 70 °F observer receiver

Age of Information (AoI) °F 70 °F observer receiver Data is now fresh.

Age of Information (AoI) °F 72 °F observer receiver Data has now aged.

Age of Information (AoI) Evolution Curve Typical age of information evolution versus time curve: age d 2 d 1 d 3 0 t 1 t 1 + d 1 t 2 t 2 + d 2 t 3 t 3 + d 3 T time d 1 d 2 d 3 t i : measurement (and transmission) time of i th update. d i : service time for i th update; typically a random quantity.

Age of Information (AoI) Evolution Curve Typical age of information evolution versus time curve: age d 2 d 1 d 3 0 t 1 t 1 + d 1 t 2 t 2 + d 2 t 3 t 3 + d 3 T time d 1 d 2 d 3 1 Long Term Average AoI = lim sup T E [ A T ] T →∞

Connection to Remote Estimation Using AoI presumes the monitored process is frequently varying. MMSE may be more suitable for slowly varying processes. AoI is inherently embedded in MMSE estimates. Other variant metrics have also been studied; we focus on MMSE. Related works: [ Nar-Basar (CDC ’14)] [ Chakravorty-Mahajan (ISIT ’15)] [ Gao-Aykol-Basar (CDC ’15)] [ Yun-Joo-Eryilmaz (CDC ’18)] [ Ayan-Vilgelm-Klugel-Hirche-Kellerer (ICCPS ’19)] [ Mitra-Richards-Bagchi-Sundaram (ACC ’19)] [ Chakravorty-Mahajan (T-AC to appear)] [ Sun-Polyanskiy-UysalBiyikoglu (T-IT to appear)] [ Ornee-Sun (arXiv) ’19] [ Huang-Liu-Shirvanimoghaddam-Li-Vucetic (arXiv ’19)] [ Maatouk-Kriouile-Assaad-Ephremides (arXiv ’19)] [ Ramirez-Erkip-Poor (arXiv ’19)] [ Bastopcu-Ulukus (arXiv ’19)]

Connection to Remote Estimation Weiner process estimation [ Sun-Polyanskiy-UysalBiyikoglu (T-IT) to appear] Communication channel is perfect (distortion-free); introduces random delays. Two sampling schemes: signal-independent and signal-dependent. MMSE = AoI in case of signal-independent sampling; threshold policy on AoI. MMSE � = AoI in case of signal-dependent sampling; threshold policy on signal. Ornstein-Uhlenbeck (OU) process estimation [ Ornee-Sun (arXiv ’19)] Similar results; MMSE = g (AoI) in case of signal-independent sampling. This work: effects of distortion and coding over noisy channels . . . We focus on OU processes.

Connection to Remote Estimation Weiner process estimation [ Sun-Polyanskiy-UysalBiyikoglu (T-IT) to appear] Communication channel is perfect (distortion-free); introduces random delays. Two sampling schemes: signal-independent and signal-dependent. MMSE = AoI in case of signal-independent sampling; threshold policy on AoI. MMSE � = AoI in case of signal-dependent sampling; threshold policy on signal. Ornstein-Uhlenbeck (OU) process estimation [ Ornee-Sun (arXiv ’19)] Similar results; MMSE = g (AoI) in case of signal-independent sampling. This work: effects of distortion and coding over noisy channels . . . We focus on OU processes.

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling.

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling. MMSE quantizer represents X S i as ˜ X S i using ℓ bits: X S i = ˜ X S i + Q S i

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling. MMSE quantizer represents X S i as ˜ X S i using ℓ bits: X S i = ˜ X S i + Q S i Channel coding schemes: Infinite Incremental Redundancy ( IIR ): n -bit codewords, increment if needed. Fixed Redundancy ( FR ): fixed n -bit codewords, repeat with new samples if needed.

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling. MMSE quantizer represents X S i as ˜ X S i using ℓ bits: X S i = ˜ X S i + Q S i Channel coding schemes: Infinite Incremental Redundancy ( IIR ): n -bit codewords, increment if needed. Fixed Redundancy ( FR ): fixed n -bit codewords, repeat with new samples if needed. Decoding consumes fixed β time units for processing and feedback.

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling. MMSE quantizer represents X S i as ˜ X S i using ℓ bits: X S i = ˜ X S i + Q S i Channel coding schemes: Infinite Incremental Redundancy ( IIR ): n -bit codewords, increment if needed. Fixed Redundancy ( FR ): fixed n -bit codewords, repeat with new samples if needed. Decoding consumes fixed β time units for processing and feedback. Successfully decoded messages are used to construct an MMSE estimate.

System Model transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder channel ℓ -bit MMSE quantizer + IIR/FR coding β processing time feedback (ACK/NACK) OU process evolution: σ X t = X s e − θ ( t − s ) + e − θ ( t − s ) W e 2 θ ( t − s ) − 1 , √ t ≥ s 2 θ Sensor acquires the i th sample at time S i ; signal-independent sampling. MMSE quantizer represents X S i as ˜ X S i using ℓ bits: X S i = ˜ X S i + Q S i Channel coding schemes: Infinite Incremental Redundancy ( IIR ): n -bit codewords, increment if needed. Fixed Redundancy ( FR ): fixed n -bit codewords, repeat with new samples if needed. Decoding consumes fixed β time units for processing and feedback. Successfully decoded messages are used to construct an MMSE estimate. Tradeoff between distortion and timeliness.

System Model: Channel Delays transmitter receiver X t : OU process X t : MMSE estimate ˆ sensor n -bit channel encoder channel decoder ℓ -bit MMSE quantizer channel + IIR/FR coding β processing time feedback (ACK/NACK) D i : reception time of the i th successfully decoded message. For the IIR coding scheme, each message is eventually decoded correctly: D i = S i + Y i Y i is a random delay incurred due to IR bits added until success: Y i = nT b + β + r i ( T b + β ) , r i ∈ { 0 , 1 , 2 , . . . } � �� � ¯ n Channel is memoryless; Y i ’s are i.i.d. ∼ Y . For the FR coding scheme, not every message is decoded: D i = S i , M i + nT b + β � �� � ¯ n M i is the number of transmission attempts between ( i − 1)th and i th successes. Channel is memoryless; M i ’s are i.i.d. ∼ M (geometric).

MMSE Estimates transmitter receiver ˆ X t : OU process X t : MMSE estimate sensor n -bit channel encoder channel decoder ℓ -bit MMSE quantizer channel + IIR/FR coding β processing time feedback (ACK/NACK) For the IIR coding scheme: X t = ˜ ˆ X S i e − θ ( t − S i ) , D i ≤ t < D i +1 mse ( t , S i ) = σ 2 � � 1 − 2 − 2 ℓ � e − 2 θ ( t − S i ) � 1 − 2 θ � h ℓ ( t − S i ) , D i ≤ t < D i +1 For the FR coding scheme: X S i , Mi e − θ ( t − S i , Mi ) , X t = ˜ ˆ D i ≤ t < D i +1 mse ( t , S i , M i ) = h ℓ ( t − S i , M i ) , D i ≤ t < D i +1 In both schemes, the MMSE is an increasing functional of AoI: t − S .