ISIT 2013, Istanbul July 9, 2013 Minimax Universal Sampling for Compound Multiband Channels Yuxin C Chen, A Andrea G Gol oldsmi smith, Yon onina Eldar Eldar Stanfor S ord Un Universi sity Technion on
Capacity of Undersampled Channels Point-to-point channels N ( f ) Analog Channel H ( f ) Message Message Encoder Decoder y ( t ) x ( t ) C. E. Shannon Issue: wideband systems preclude Nyquist-rate sampling! Issu
Capacity of Undersampled Channels Point-to-point channels N ( f ) Analog Channel H ( f ) Message Message Encoder Decoder y ( t ) x ( t ) C. E. Shannon Issu Issue: wideband systems preclude Nyquist-rate sampling! Sub-Nyquist sampling well explored in Signal Processing Landau-rate sampling, compressed sensing, etc. Objective metric: MSE H. Nyquist
Capacity of Undersampled Channels Point-to-point channels N ( f ) Analog Channel H ( f ) Message Message Encoder Decoder y ( t ) x ( t ) C. E. Shannon Issu Issue: wideband systems preclude Nyquist-rate sampling! Sub-Nyquist sampling well explored in Signal Processing Landau-rate sampling, compressed sensing, etc. Objective metric: MSE H. Nyquist Question: which sub-Nyquist samplers are optimal in terms of CAPACITY ?
Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor
Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor The channel-optimized sampler (op optimi mized f for or a a si singl gle c channel) (1) a filter bank followed by uniform sampling (2) a single branch of and modulation and filtering with t = n ( mT ) s uniform sampling s 1 t ( ) y 1 n [ ] ( t ) η t = n ( mT ) s y i [ n ] h ( t ) x ( t ) s i ( t ) t = n ( mT ) s s m ( t ) y m [ n ]
Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor The channel-optimized sampler (op optimi mized f for or a a si singl gle c channel) (1) a filter bank followed by uniform sampling (2) a single branch of and modulation and filtering with t = n ( mT ) s uniform sampling s 1 t ( ) y 1 n [ ] ( t ) η t = n ( mT ) s y i [ n ] h ( t ) x ( t ) s i ( t ) Suppresses Aliasing t = n ( mT ) s No need to use non-uniform sampling grid! s m ( t ) y m [ n ]
Universal Sampling for Compound Channels The channel-optimized sampler suppresses aliasing What if there are a collection of channel realizations?
Universal Sampling for Compound Channels The channel-optimized sampler suppresses aliasing What if there are a collection of channel realizations? Un Universa sal ( channel-b -blind ) ) Sampling ---- A sampler is typically integrated into the hardware ---- Need to operate independently of instantaneous realization
Sub-optimality of Channel-optimized Samplers Consider 2 possible channel realizations ... ⋯⋯ Effective channel (a) Effective channel gain gain (b)
Sub-optimality of Channel-optimized Samplers Consider 2 possible channel realizations ... ⋯⋯ Effective channel (a) Effective channel gain gain (b) Fa Far f from op om optima mal! optimal sampler for (a) (a) Effective channel gain Effective channel gain
Sub-optimality of Channel-optimized Samplers Consider 2 possible channel realizations ... ⋯⋯ Effective channel (a) Effective channel gain gain (b) Fa Far f from op om optima mal! optimal sampler for (a) (a) Effective channel gain Effective channel gain No si o singl gle l linear sa samp mpler c can ma maximi mize c capacity f for or a all r realization ons! s! Qu Quest stion on: how to design a universal sampler robust to different channel realizations
Robustness Measure: Minimax Capacity Loss Consider a channel state s and a sampler Q : Capacity Loss:
Robustness Measure: Minimax Capacity Loss Consider a channel state s and a sampler Q : Capacity Loss: Minimax Capacity Loss: accounting for all channel states s s
Robustness Measure: Minimax Capacity Loss Consider a channel state s and a sampler Q : Capacity Loss: Minimax Capacity Loss: optimize over a large class of samplers accounting for all channel states s s -- Minimax Sampler
Minimax Universal Sampling Nyquist-rate Capacity Capacity Capacity under Minimax Sampler State: s
Minimax Universal Sampling Nyquist-rate Capacity Capacity minimax capacity loss Capacity under Minimax Sampler State: s － A sampler that minimizes the worse-case capacity loss due to universal sampling
Minimax Universal Sampling Nyquist-rate Capacity Capacity minimax capacity loss Capacity under Minimax Sampler Sampler that maximizes compount channel capacity State: s -- A sampler that maximizes compound channel capacity － A sampler that minimizes the worse-case capacity loss due to universal sampling
Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k k ou out of of n n su subbands a are a active.
Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k k ou out of of n n su subbands a are a active.
Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k k ou out of of n n su subbands a are a active. m- branch sampling with modulation and filtering: q 1 ( t ) t = n ( mT s ) ⊗ y 1 n [ ] S 1 ( f ) F 1 ( f ) y 1 ( t ) ( t ) η q i ( t ) t = n ( mT s ) x ( t ) y i [ n ] h ( t ) ⊗ F i ( f ) S i ( f ) y i ( t ) r ( t ) q m ( t ) t = n ( mT s ) ⊗ F m ( f ) S m ( f ) y m [ n ] y m ( t )
Converse: Landau-rate Sampling ( α = β ) Hertz obeys: Theor orem ( Con onverse se ): The minimax capacity loss per H
Converse: Landau-rate Sampling ( α = β ) Hertz obeys: Theor orem ( Con onverse se ): The minimax capacity loss per H At high SNR and large n, minimax capacity loss determined by subband uncertainty
Converse: Landau-rate Sampling ( α = β ) Theorem ( Converse ): The minimax capacity loss per Hertz obeys: Key observation for the proof :
Converse: Landau-rate Sampling ( α = β ) Theorem ( Converse ): The minimax capacity loss per Hertz obeys: Key observation for the proof : The minimax sampler achieves equivalent loss across all channel states
Achievability: Landau-rate Sampling ( α = β ) Determi minist stic optimization is NP-hard (non-convex).
Achievability: Landau-rate Sampling ( α = β ) Determi minist stic optimization is NP-hard (non-convex). q 1 ( t ) mpling Hop ope: random sa om samp ⊗ y 1 [ l ] LPF y 1 ( t ) Fourier t Fou transf sfor orm of m of p period odic ( t ) η q i ( t ) se sequence i is a s a sp spike-t -train x ( t ) h ( t ) ⊗ y i [ l ] LPF y i ( t ) r ( t ) q m ( t ) ⊗ y m [ l ] LPF y m ( t )
Achievability: Landau-rate Sampling ( α = β ) Determi minist stic optimization is NP-hard (non-convex). q 1 ( t ) mpling Hop ope: random sa om samp ⊗ y 1 [ l ] LPF y 1 ( t ) Fourier t Fou transf sfor orm of m of p period odic ( t ) η q i ( t ) sequence i se is a s a sp spike-t -train x ( t ) h ( t ) ⊗ y i [ l ] LPF y i ( t ) r ( t ) q m ( t ) ⊗ y m [ l ] LPF y m ( t ) A sampling system is called independent random sampling if the coefficients of the spike-train are independently and randomly generated.
Achievability: Landau-rate Sampling ( α = β ) q 1 ( t ) random sa om samp mpling ⊗ y 1 [ l ] LPF y 1 ( t ) à random modulation coefficients ( t ) η q i ( t ) x ( t ) h ( t ) ⊗ y i [ l ] LPF y i ( t ) r ( t ) q m ( t ) ⊗ y m [ l ] LPF y m ( t ) orem ( Achievability Achievability ): The capacity loss per H Hertz under Theor mpling is independent r random sa om samp with probability exceeding
Implications: Landau-rate Sampling ( α = β ) Theor orem ( Con onverse se ): orem ( Achievability Achievability ): Under independent r random sa om samp mpling (with Theor zero mean and unit variance), with exponentially high probability,
Implications: Landau-rate Sampling ( α = β ) Theor orem ( Con onverse se ): orem ( Achievability Achievability ): Under independent r random sa om samp mpling (with Theor zero mean and unit variance), with exponentially high probability, Random sampling is Minimax Sharp concentration – exponentially high probability
Implications: Landau-rate Sampling ( α = β ) Theor orem ( Con onverse se ): orem ( Achievability Achievability ): Under independent r random sa om samp mpling (with Theor zero mean and unit variance), with exponentially high probability, Random sampling is Minimax Sharp concentration – exponentially high probability Un Universa sality p phenome omena: A large class of distributions can work! -- Gaussian, Bernoulli, uniform ⋯ No need for i.i.d. randomness -- can be a mixture of Gaussian, Bernoulli, uniform ⋯
Capacity Loss for Multiband Channels Nyquist-rate Capacity Capacity minimax capacity loss Capacity under Minimax Sampler State: s
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