Joint State Sensing and Communication: Theory and Applications Mari Kobayashi A joint work with Björn Bissinger, Giuseppe Caire, Lorenzo Gaudio, Hassan Hamad, Gerhard Kramer
Introduction V2P V2I real-time traffic safety alerts traffic flow control V2N V2V collision avoidance real-time traffic cooperative adaptive cruise infotainment control, platooning cloud services Future high-mobility networks must ensure both connectivity and real-time adaptation . A key-enabler is the ability to continuously track the dynamically changing environment, “state”, and react accordingly by exchanging information. M. Kobayashi ESIT April 15, 2019 1 / 85
Example: Joint Radar and Vehicular Communication state estimation data & decoding encoding feedback (reflection) data-carrying signal transmitter receiver The spectrum crunch encourages to use sensing and communication in the same frequency bands (e.g. IEEE S band shared between LTE and radar). One vehicle wishes to track the “state” (velocity, range) and simultaneously convey a message (safety/traffic-related). M. Kobayashi ESIT April 15, 2019 2 / 85
Outline of my talk Part I: Preliminaries ◮ Introduction ◮ Channels with feedback Part II: Joint state sensing and communication ◮ A single-user case ◮ A two-user multiple access channel Part III: Vehicular applications ◮ Joint radar and V2X communication ◮ Performance analysis with multi-carrier modulation M. Kobayashi ESIT April 15, 2019 3 / 85
Part I: Preliminaries M. Kobayashi ESIT April 15, 2019 4 / 85
Feedback in our daily life system reference input System Control + system measured output output Sensor Feedback enables a system to improve its capability by taking benefits from the response of actions and incorporating it into the design. Closed-loop control, rather than open-loop control without feedback. M. Kobayashi ESIT April 15, 2019 5 / 85
Example 1: Thermostat Invented by Albert Butz in 1886, giving a birth to “Honeywell”. Objective: keep the temperature constant in a room. ◮ Reference: desired temperature ◮ Control: switch on/off of boiler ◮ Sensor: measures the temperature M. Kobayashi ESIT April 15, 2019 6 / 85
Example 2: Cruise control of a car Invented by Peerless and first commercialized for “Chrysler Imperial” in 1958. Objective: maintain speed whether up hill or down ◮ Reference: desired speed ◮ Control: accelerate or not ◮ System: a channel with some disturbance (wind, hill). ◮ Sensor: measures the speed. M. Kobayashi ESIT April 15, 2019 7 / 85
Examples in communication standards Hybrid Automatic Request Control (HARQ) ◮ included in High Speed Downlink/Uplink Packet Access (HSD/UPA) and LTE. ◮ based on ACK/NACK feedback from users. ◮ enables to improve error probability. Closed-loop MIMO ◮ included in LTE ◮ based on channel estimated at users. ◮ A base station choose appropriate directions (precoder) to enhance data rate. M. Kobayashi ESIT April 15, 2019 8 / 85
Feedback in communications X Y Decoder Encoder Channel ˆ W W processing/ Z channel Feedback enables a communication system to improve capacity, reliability or simplify encoding. Types of feedback. ◮ Output feedback: Z = Y . ◮ State feedback: estimated channel state given Y (processing). ◮ Geralized feedback: Z is any causal function of Y (no processing). In information theory, feedback can be noise-free and even non-causal. M. Kobayashi ESIT April 15, 2019 9 / 85
Feedback doesn’t increase capacity of a memoryless channel 1 Y i − 1 ˆ X i W Y i W Enc Dec p ( y | x ) The capacity of a memoryless channel with and without feedback is C = max P X I ( X ; Y ) achieved by Random encoding: to convey a message w ∈ [1 : 2 nR ] , choose x n ( w ) from randomly and independently generated 2 nR sequences. w such that ( x n ( ˆ w ) , y n ) are jointly Joint typicality decoding: choose ˆ typical. 1C. Shannon, “The zero error capacity of a noisy channel,” IRE Trans. Information Theory, vol. 2, no. 3, 1956. M. Kobayashi ESIT April 15, 2019 10 / 85
Converse: prove that we cannot transmit at R > C . nR = H ( W ) = I ( W ; Y n ) + H ( W | Y n ) ≤ I ( W ; Y n ) + nǫ n n � I ( W ; Y i | Y i − 1 ) + nǫ n = i =1 n � I ( W, Y i − 1 : Y i ) + nǫ n ≤ i =1 n � I ( W, Y i − 1 , X i : Y i ) + nǫ n = i =1 n � = I ( X i : Y i ) + nǫ n i =1 M. Kobayashi ESIT April 15, 2019 11 / 85
Error probability for the channel w/o feedback A decoder makes an error if one of the following events occurs. E 1 = { ( X n (1) , Y n ) / E 2 = { ( X n ( w ) , Y n ) ∈ T , ∀ w � = 1 } ∈ T } , Union bound P ( E ) = P ( E 1 ∪ E 1 ) ≤ P ( E 1 ) + P ( E 2 ) where by law of large number lim n →∞ P ( E 1 ) = 0 and we have 2 nR � P (( X n ( w ) , Y n ) ∈ T ) P ( E 2 ) ≤ w =2 2 nR � 2 − n ( I ( X ; Y ) − ǫ ) ≤ joint typicality lemma w =2 = 2 − n ( C − R − ǫ ) M. Kobayashi ESIT April 15, 2019 12 / 85
Well-known results on output feedback 6 Feedback improves reliability of a memoryless channel The capacity of a two-user Gaussian multiple access channel (MAC) with feedback The capacity of a two-user erasure MAC An achievable rate region of a two-user Gaussian broadcast channel (BC) An achievable rate region of a Gaussian network with more than two users 2 3 Tight bounds for a two-user Gaussian interference channel 4 Upper bounds of the K -user Gaussian MAC using dependence balance bounds 5 2G. Kramer, “Feedback strategies for white Gaussian interference networks,” IEEE Trans. Inf. Theory, vol. 48, no. 6, 2002. 3Ardestanizadeh et al., “Linear-feedback sum-capacity for Gaussian multiple access channels”, IEEE Trans. Inf. Theory, vol. 58, no.1 2012 4C. Suh and D. Tse, “Feedback capacity of the Gaussian interference channel to within 2 bits”, IEEE Trans. Inf. Theory, 2011 5E.Sula, “Sum-Rate Capacity for Symmetric Gaussian Multiple Access Channels with Feedback”, ISIT’2018 6A. El Gamal and Y.-H. Kim, Network Information Theory, Cambridge University Press, 2011. M. Kobayashi ESIT April 15, 2019 13 / 85
A Gaussian channel: Schalkwijik and Kailath A Gaussian channel Y i = X i + B i with B i ∼ N (0 , 1) and the input � n subject to 1 i =1 E [ | X i | 2 ] ≤ P . n Recursively send an estimation error seen by receiver. √ P 2 − nR ∆ = 2 p 0 √ − P P B 0 X 0 Y 0 X 0 = θ ( w ) Y 0 = θ ( w ) + B 0 X 1 = γ 1 B 0 Y 1 = X 1 + B 1 X 2 = γ 2 ( B 0 − E [ B 0 | Y 1 ]) Y 2 = X 2 + B 2 . . . . . . X n = γ n ( B 0 − E [ B 0 | Y n − 1 ]) Y n = X n + B n Receiver estimates ˆ θ ( w ) = Y 0 − E [ B 0 | Y n ] = θ ( w ) + B 0 − E [ B 0 | Y n ] M. Kobayashi ESIT April 15, 2019 14 / 85
Error probability of Schalkwijik-Kailath ’s scheme Orthogonality property implies that error B 0 − E [ B 0 | Y i ] is independent of Y i for each i . The output sequence is i.i.d. Gaussian Y i ∼ N (0 , 1 + P ) . Write mutual information in two ways (exercise!): n � I ( B 0 ; Y n ) = I ( B 0 ; Y i | Y i − 1 ) = . . . i =1 = n ∆ 2 log(1 + P ) = C ( P ) . I ( B 0 ; Y n ) = h ( B 0 ) − h ( B 0 | Y n ) = 1 1 2 log var ( B 0 | Y n ) M. Kobayashi ESIT April 15, 2019 15 / 85
Error probability of Schalkwijik-Kailath ’s scheme The estimate at receiver ˆ θ ∼ N ( θ ( w ) , 2 − 2 nC ( P ) ) . 2 = 2 − nR √ The decoder makes an error if | θ − ˆ θ ( w ) | > ∆ P for any w ∈ [1; 2 nR ] . The error probability is bounded by θ (1) | > 2 − nR √ � � | θ − ˆ P e = Pr P � ∞ = 2 Q (2 n ( C − R ) √ 1 2 π e − t 2 / 2 dt P ) with Q ( x ) = √ x � 2 � � − 2 2 n ( C − R ) P 1 2 π e − x 2 / 2 ≤ π exp with Q ( x ) ≤ √ 2 For R < C ( P ) , the error probability decays doubly exponentially ! M. Kobayashi ESIT April 15, 2019 16 / 85
Multiple Access Channel (MAC) without feedback X 1 i Encoder 1 W 1 Y i P Y | X 1 X 2 Receiver W 1 , ˆ ˆ W 2 Encoder 2 W 2 X 2 i Two transmitters wish to convey messages W 1 , W 2 to the receiver, respectively. The capacity region of MAC w/o feedback is the convex hull of the union of 7 R 1 ≤ I ( X 1 ; Y | X 2 ) R 2 ≤ I ( X 2 ; Y | X 1 ) R 1 + R 2 ≤ I ( X 1 , X 2 ; Y ) 7 An alternative expression is to use a time-sharing random variable Q . M. Kobayashi ESIT April 15, 2019 17 / 85
Multiple Access Channel (MAC) without feedback I ( X 2 ; Y | X 1 ) I ( X 2 ; Y ) I ( X 1 ; Y ) I ( X 1 ; Y | X 2 ) Random encoding: to convey a message w k ∈ [1 : 2 nR k ] , choose x n k ( w k ) from randomly and independently generated 2 nR k sequences. Successive interference decoding ◮ Find the unique message ˆ w 1 such that ( x n w 1 ) , y n ) ∈ T . 1 ( ˆ ◮ Then, find the unique message ˆ w 2 such that ( x n w 1 ) , x n w 2 ) , y n ) ∈ T . 1 ( ˆ 2 ( ˆ M. Kobayashi ESIT April 15, 2019 18 / 85
MAC with output feedback Y i − 1 X 1 i Encoder 1 W 1 Y i P Y | X 1 X 2 Receiver W 1 , ˆ ˆ W 2 Encoder 2 W 2 X 2 i Y i − 1 Encoder 1 sends X 1 i = f 1 i ( W 1 , Y i − 1 ) . Thanks to the feedback, two 1 symbols ( X 1 i , X 2 i ) can be correlated. Correlation enables to reduce the multiuser interference and increase the sum rate. ◮ Successive refinement of error seen by receivers. ◮ A common message to be decoded by both encoders. M. Kobayashi ESIT April 15, 2019 19 / 85
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