Lossy coding of a time-limited piece of a band-limited white - PowerPoint PPT Presentation

Lossy coding of a time-limited piece of a band-limited white Gaussian source Youssef Jaffal, Ibrahim Abou-Faycal American University of Beirut Department of Electrical and Computer Engineering Email: yaj03@mail.aub.edu 2020 IEEE International Symposium on Information Theory 21-26 June 2020 – Los Angeles, California, USA Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 1 / 23

Motivation 1 Motivation Prolate Spheroidal Wave Functions (PSWFs) 2 3 System Model 4 Bounds Lower Bound Upper Bound 5 Asymptotic analysis as T → ∞ Numerical Results 6 7 Summary Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 2 / 23

Motivation Rate Distortion Function of Band-limited Sources For a real W -Hz band-limited “white” Gaussian source: � Q � R Shannon = W log 2 bits per seconds D Q : the average power of the source D : the average mean-squared error distortion level How to derive this equation? 1- Transform the problem to a discrete one using the Shannon-Nyquist sampling theorem 2- Apply the rate distortion function for the discrete-time stationary Gaussian source However: This requires compression at once of an asymptotically infinite duration piece from the source Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 3 / 23

Motivation Practical Setup Independently encode adjacent T -seconds short-duration pieces − 3 T − T T 3 T X ( t ) 2 2 2 2 . . . . . . . . . . . . . . . ✻ . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . X − 1 ( t ) X 0 ( t ) X 1 ( t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Then: Sampling the short duration sub-functions is lossy : 1 → Solution: make use of Prolate Spheroidal Wave Functions (PSWFs) Deriving non-asymptotic results for the equivalent discrete-time 2 problem: → Solution: Derive bounds in the finite-blocklength regime Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 4 / 23

Prolate Spheroidal Wave Functions (PSWFs) 1 Motivation Prolate Spheroidal Wave Functions (PSWFs) 2 3 System Model 4 Bounds Lower Bound Upper Bound 5 Asymptotic analysis as T → ∞ Numerical Results 6 7 Summary Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 5 / 23

Prolate Spheroidal Wave Functions (PSWFs) Properties of PSWFs Define c = 2 WT > 0 . The PSWFs { ϕ c,i ( t ) } i ∈ N satisfy: � ∞ 1- ϕ c,i ( t ) ϕ c,l ( t ) dt = δ il , ∀ i, l ∈ N −∞ � T/ 2 2- ϕ c,i ( t ) ϕ c,l ( t ) dt = λ c,i δ il , ∀ i, l ∈ N − T/ 2 � � Dϕ c,i ( t ) 3- The normalized time-limited PSWFs √ form a CON set λ c,i i ∈ N The Eigenvalues { λ c,i } satisfy: 10 0 ∞ � λ c,i = 2 WT = c 100,i 1- 100,i i =0 10 -5 For any γ > 0 , c →∞ λ c,c (1+ γ ) = 0 lim 10 -10 c →∞ λ c,c (1 − γ ) = 1 . lim 84 88 92 96 100 104 108 112 116 i Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 6 / 23

System Model 1 Motivation Prolate Spheroidal Wave Functions (PSWFs) 2 3 System Model 4 Bounds Lower Bound Upper Bound 5 Asymptotic analysis as T → ∞ Numerical Results 6 7 Summary Youssef Jaffal, Ibrahim Abou-Faycal (American University of Beirut Department of Electrical and Computer Engine ISIT 2020 7 / 23