Minimax Universal Sampling for Compound Multiband Channels

Transcription

1 ISIT 2013, Istanbul July 9, 2013 Minimax Universal Sampling for Compound Multiband Channels Yuxin Chen, Andrea Goldsmith, Yonina Eldar Stanford University Technion

2 Capacity of Undersampled Channels Point-to-point channels Message Encoder Analog Channel N( f ) H ( f ) x y Decoder Message Issue: wideband systems preclude Nyquist-rate sampling! C. E. Shannon

3 Capacity of Undersampled Channels Point-to-point channels Message Encoder Analog Channel N( f ) H ( f ) x y Decoder Message Issue: wideband systems preclude Nyquist-rate sampling! C. E. Shannon Sub-Nyquist sampling well explored in Signal Processing Landau-rate sampling, compressed sensing, etc. Objective metric: MSE H. Nyquist

4 Capacity of Undersampled Channels Point-to-point channels Message Encoder Analog Channel N( f ) H ( f ) x y Decoder Message Issue: wideband systems preclude Nyquist-rate sampling! C. E. Shannon 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?

5 Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor

6 Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor The channel-optimized sampler (optimized for a single channel) (1) a filter bank followed by uniform sampling (2) a single branch of and modulation and filtering with uniform sampling η s 1 ( t) t = n( mts t = n( mts ) ) y 1 [ n] x h s i y i [n] t = n( mts ) s m y m [n]

7 Prior work: Channel-specific Samplers Consider linear time-invariant sub-sampled channels Preprocessor The channel-optimized sampler (optimized for a single channel) (1) a filter bank followed by uniform sampling (2) a single branch of and modulation and filtering with uniform sampling η s 1 ( t) t = n( mts t = n( mts ) ) y 1 [ n] Suppresses Aliasing x h s i y i [n] No need to use non-uniform sampling grid! s m t = n( mts ) y m [n]

8 Universal Sampling for Compound Channels The channel-optimized sampler suppresses aliasing What if there are a collection of channel realizations?

9 Universal Sampling for Compound Channels The channel-optimized sampler suppresses aliasing What if there are a collection of channel realizations? Universal (channel-blind) Sampling ---- A sampler is typically integrated into the hardware ---- Need to operate independently of instantaneous realization

10 Sub-optimality of Channel-optimized Samplers (a) Consider 2 possible channel realizations... Effective channel gain Effective channel gain (b)

11 Sub-optimality of Channel-optimized Samplers (a) Consider 2 possible channel realizations... Effective channel gain Effective channel gain (b) optimal sampler for (a) Far from optimal! Effective channel gain Effective channel gain

12 Sub-optimality of Channel-optimized Samplers (a) Consider 2 possible channel realizations... Effective channel gain Effective channel gain (b) optimal sampler for (a) Far from optimal! Effective channel gain Effective channel gain No single linear sampler can maximize capacity for all realizations! Question: how to design a universal sampler robust to different channel realizations

13 Robustness Measure: Minimax Capacity Loss Consider a channel state s and a sampler Q : Capacity Loss:

14 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

15 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 -- Minimax Sampler

16 Minimax Universal Sampling Capacity Nyquist-rate Capacity Capacity under Minimax Sampler State: s

17 Minimax Universal Sampling Capacity Nyquist-rate Capacity minimax capacity loss Capacity under Minimax Sampler State: s A sampler that minimizes the worse-case capacity loss due to universal sampling

18 Minimax Universal Sampling Capacity Nyquist-rate Capacity minimax capacity loss Capacity under Minimax Sampler Sampler that maximizes compount channel capacity -- A sampler that maximizes compound channel capacity A sampler that minimizes the worse-case capacity loss due to universal sampling State: s

19 Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k out of n subbands are active.

20 Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k out of n subbands are active.

21 Focus on Multiband Channel Model A class of channels where at each time only a fraction of bandwidths are active. k out of n subbands are active. m-branch sampling with modulation and filtering: q 1 t = n(mt s ) x h η r F 1 ( f ) F i ( f ) q i q m S 1 ( f ) S i ( f ) y 1 y i t = n(mt s ) t = n(mt s ) y 1 [ n] y i [n] F m ( f ) S m ( f ) y m y m [n]

22 Converse: Landau-rate Sampling (α=β) Theorem (Converse): The minimax capacity loss per Hertz obeys:

23 Converse: Landau-rate Sampling (α=β) Theorem (Converse): The minimax capacity loss per Hertz obeys: At high SNR and large n, minimax capacity loss determined by subband uncertainty

24 Converse: Landau-rate Sampling (α=β) Theorem (Converse): The minimax capacity loss per Hertz obeys: Key observation for the proof :

25 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

26 Achievability: Landau-rate Sampling (α=β) Deterministic optimization is NP-hard (non-convex).

27 Achievability: Landau-rate Sampling (α=β) Deterministic optimization is NP-hard (non-convex). Hope: random sampling Fourier transform of periodic sequence is a spike-train x h η r q 1 q i q m LPF LPF y 1 y i y 1 [l] y i [l] LPF y m y m [l]

28 Achievability: Landau-rate Sampling (α=β) Deterministic optimization is NP-hard (non-convex). Hope: random sampling Fourier transform of periodic sequence is a spike-train x h η r q 1 q i q m LPF LPF y 1 y i y 1 [l] y i [l] LPF y m y m [l] A sampling system is called independent random sampling if the coefficients of the spike-train are independently and randomly generated.

29 Achievability: Landau-rate Sampling (α=β) q 1 random sampling à random modulation coefficients x h η r q i q m LPF LPF LPF y 1 y i y m y 1 [l] y i [l] y m [l] Theorem (Achievability): The capacity loss per Hertz under independent random sampling is with probability exceeding

30 Implications: Landau-rate Sampling (α=β) Theorem (Converse): Theorem (Achievability): Under independent random sampling (with zero mean and unit variance), with exponentially high probability,

31 Implications: Landau-rate Sampling (α=β) Theorem (Converse): Theorem (Achievability): Under independent random sampling (with zero mean and unit variance), with exponentially high probability, Random sampling is Minimax Sharp concentration exponentially high probability

32 Implications: Landau-rate Sampling (α=β) Theorem (Converse): Theorem (Achievability): Under independent random sampling (with zero mean and unit variance), with exponentially high probability, Random sampling is Minimax Sharp concentration exponentially high probability Universality phenomena: 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

33 Capacity Loss for Multiband Channels Capacity Nyquist-rate Capacity minimax capacity loss Capacity under Minimax Sampler State: s

34 Capacity Loss for Multiband Channels Capacity Nyquist-rate Capacity minimax capacity loss Capacity under Minimax Sampler State: s Minimax sampling yields equivalent capacity loss over all possible channel realizations when SNR and n are large!

35 Converse: Super-Landau Sampling (α>β) Theorem (Converse): The minimax capacity loss per Hertz obeys: Capacity gain due to oversampling is

36 Achievability: Super-Landau Sampling (α>β) q 1 Gaussian sampling à Gaussian modulation coefficients x h η r q i q m LPF LPF LPF y 1 y i y m y 1 [l] y i [l] y m [l]

37 Achievability: Super-Landau Sampling (α>β) q 1 Gaussian sampling à Gaussian modulation coefficients x h η r q i q m LPF LPF LPF y 1 y i y m y 1 [l] y i [l] y m [l] Theorem (Achievability): If α+β<1, then the capacity loss per Hertz under i.i.d. Gaussian random sampling is with probability exceeding

38 Implications: super-landau sampling (α=β, α+β<1) Theorem (Converse): The minimax capacity loss per Hertz obeys: Theorem (Achievability): Under i.i.d. Gaussian random sampling, with exponentially high probability Gaussian sampling is Minimax! Sharp concentration: exponentially high probability Universality phenomena not shown We have only shown the results for i.i.d. Gaussian sampling

39 Concluding Remarks " Minimax Capacity Loss -- A new metric to characterize robustness against different channel realizations -- For multiband channels, it depends only on undersampling factor and sparsity ratio " The power of random sampling -- Near-optimal in an overall sense (minimax) -- Large random samplers behave in deterministic ways (sharp concentration + universality) " A Non-Asymptotic analysis of random channels

40 Full-Length Paper Y. Chen, A. J. Goldsmith, and Y. C. Eldar, Minimax Capacity Loss under Sub-Nyquist Universal Sampling, submitted to IEEE Trans Info Theory, arxiv.org/abs/ , April 2013,