Minimax Universal Sampling for Compound Multiband Channels Yuxin C - - PowerPoint PPT Presentation

Minimax Universal Sampling for Compound Multiband Channels

Yuxin C Chen, A Andrea G Gol

• ldsmi

smith, Yon

• nina Eldar

Eldar

S Stanfor

• rd Un

Universi sity Technion

• n

ISIT 2013, Istanbul July 9, 2013

Capacity of Undersampled Channels

— Point-to-point channels

Issu Issue: wideband systems preclude Nyquist-rate sampling!

• C. E. Shannon

) ( f H

) ( f N

) (t x

) (t y

Analog Channel € € Encoder € € Message € € Message € € Decoder € €

Capacity of Undersampled Channels

— Point-to-point channels

Issu 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

) ( f H

) ( f N

) (t x

) (t y

Analog Channel € € Encoder € € Message € € Message € € Decoder € €

Capacity of Undersampled Channels

— Point-to-point channels

Issu 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?

) ( f H

) ( f N

) (t x

) (t y

Analog Channel € € Encoder € € Message € € Message € € Decoder € €

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 — The channel-optimized sampler (op

• ptimi

mized f for

• r 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

uniform sampling

Preprocessor €

) (t h ) (t η ) (t x

) (

1 t

s

) (t si ) (t sm

) (

s

mT n t = ) (

s

mT n t = ) (

s

mT n t =

] [

1 n

y ] [n yi

] [n ym

Prior work: Channel-specific Samplers

— Consider linear time-invariant sub-sampled channels — The channel-optimized sampler (op

• ptimi

mized f for

• r 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

uniform sampling

— No need to use non-uniform sampling grid!

Preprocessor €

) (t h ) (t η ) (t x

) (

1 t

s

) (t si ) (t sm

) (

s

mT n t = ) (

s

mT n t = ) (

s

mT n t =

] [

1 n

y ] [n yi

] [n ym

— Suppresses Aliasing

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 ...⋯⋯ (a)

Effective channel gain

(b)

Effective channel gain

Sub-optimality of Channel-optimized Samplers

Consider 2 possible channel realizations ...⋯⋯

• ptimal sampler for (a)

(a) (a)

Fa Far f from op

• m optima

mal!

Effective channel gain

(b)

Effective channel gain Effective channel gain Effective channel gain

Sub-optimality of Channel-optimized Samplers

— No si

• singl

gle l linear sa samp mpler c can ma maximi mize c capacity f for

• r a

all r realization

• ns!

s!

Consider 2 possible channel realizations ...⋯⋯

• ptimal sampler for (a)

(a) (a)

Fa Far f from op

• m optima

mal!

Effective channel gain

(b)

Effective channel gain Effective channel gain Effective channel gain

— Qu

Quest stion

• n: 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:

accounting for all channel states s s

Minimax Capacity Loss:

Robustness Measure: Minimax Capacity Loss

— Consider a channel state s and a sampler Q :

Capacity Loss:

accounting for all channel states s s

• ptimize over a large class of samplers

Minimax Capacity Loss:

• - Minimax Sampler

Minimax Universal Sampling

Capacity

Nyquist-rate Capacity Capacity under Minimax Sampler

State: s

Minimax Universal Sampling

Capacity

Nyquist-rate Capacity Capacity under Minimax Sampler

State: s minimax capacity loss

－ A sampler that minimizes the worse-case capacity loss due to universal sampling

Minimax Universal Sampling

Capacity

Nyquist-rate Capacity Sampler that maximizes compount channel capacity Capacity under Minimax Sampler

State: s minimax capacity loss

• - 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

• ut of
• f 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

• ut of
• f 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

• ut of
• f n

n su subbands a are a active. m-branch sampling with modulation and filtering:

) (t h ) (t η S1( f ) Si( f )

Sm( f )

t = n(mTs) t = n(mTs) t = n(mTs)

] [

1 n

y

] [n yi

ym[n]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t) yi(t) ym(t)

F

1( f )

F

i( f )

F

m( f )

) (t x

Converse: Landau-rate Sampling (α=β)

Theor

• rem (Con
• nverse

se): The minimax capacity loss per H

Hertz obeys:

Converse: Landau-rate Sampling (α=β)

Theor

• rem (Con
• nverse

se): The minimax capacity loss per H

Hertz obeys:

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).

— Hop

• pe:

random sa

• m samp

mpling

) (t h ) (t η LPF

y1[l]

yi[l]

ym[l]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t)

yi(t)

ym(t)

) (t x LPF LPF

Fou Fourier t transf sfor

• rm of

m of p period

• dic

se sequence i is a s a sp spike-t

• train

Achievability: Landau-rate Sampling (α=β)

— Determi

minist stic optimization is NP-hard (non-convex).

— Hop

• pe:

random sa

• m samp

mpling

) (t h ) (t η LPF

y1[l]

yi[l]

ym[l]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t)

yi(t)

ym(t)

) (t x LPF LPF

A sampling system is called independent random sampling if the coefficients of the spike-train are independently and randomly generated. Fou Fourier t transf sfor

• rm of

m of p period

• dic

se sequence i is a s a sp spike-t

• train

Achievability: Landau-rate Sampling (α=β)

) (t h ) (t η LPF

y1[l]

yi[l]

ym[l]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t)

yi(t)

ym(t)

) (t x LPF LPF

random sa

• m samp

mpling à random modulation coefficients

Theor

• rem (Achievability

Achievability): The capacity loss per H

Hertz under

independent r random sa

• m samp

mpling is

with probability exceeding

Implications: Landau-rate Sampling (α=β)

Theor

• rem (Achievability

Achievability): Under independent r random sa

• m samp

mpling (with zero mean and unit variance), with exponentially high probability,

Theor

• rem (Con
• nverse

se):

Implications: Landau-rate Sampling (α=β)

Theor

• rem (Achievability

Achievability): Under independent r random sa

• m samp

mpling (with zero mean and unit variance), with exponentially high probability,

Theor

• rem (Con
• nverse

se):

— Sharp concentration – exponentially high probability — Random sampling is Minimax

Implications: Landau-rate Sampling (α=β)

Theor

• rem (Achievability

Achievability): Under independent r random sa

• m samp

mpling (with zero mean and unit variance), with exponentially high probability,

Theor

• rem (Con
• nverse

se):

— Sharp concentration – exponentially high probability

— Un

Universa sality p phenome

• mena:

— 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⋯

— Random sampling is Minimax

Capacity Loss for Multiband Channels

Capacity

Nyquist-rate Capacity Capacity under Minimax Sampler

State: s minimax capacity loss

Capacity Loss for Multiband Channels

Capacity

Nyquist-rate Capacity Capacity under Minimax Sampler

State: s minimax capacity loss

Minimax sampling yields equivalent capacity loss over all possible channel realizations when SNR and n are large!

Converse: Super-Landau Sampling (α>β)

Theor

• rem (Con
• nverse

se): The minimax capacity loss per H

Hertz obeys:

— Capacity gain due to oversampling is

Achievability: Super-Landau Sampling (α>β)

) (t h ) (t η LPF

y1[l]

yi[l]

ym[l]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t)

yi(t)

ym(t)

) (t x LPF LPF

Gaussi ssian sa samp mpling g à à G Gaussi ssian mod modulation

• n c

coe

• efficients

Achievability: Super-Landau Sampling (α>β)

) (t h ) (t η LPF

y1[l]

yi[l]

ym[l]

⊗

q1(t)

⊗

qi(t)

⊗

qm(t) r(t)

y1(t)

yi(t)

ym(t)

) (t x LPF LPF

Gaussi ssian sa samp mpling g à à G Gaussi ssian mod modulation

• n c

coe

• efficients

Theor

• rem (Achievability

Achievability): If α+β<1, then the capacity loss per H

Hertz under

i.i.d i.i.d. G Gaussi ssian r random sa

• m samp

mpling is

with probability exceeding

Implications: super-Landau sampling ( (α=β, α+β<1)

Theor

• rem (Achievability

Achievability): Under i.i.d i.i.d. G Gaussi ssian r random sa

• m samp

mpling, with exponentially high probability

Theor

• rem (Con
• nverse

se): The minimax capacity loss per H

Hertz obeys:

— Sharp concentration: exponentially high probability — Un

Universa sality p phenome

• mena n

not

• t sh

show

• wn⋯

— We have only shown the results for i.i.d. Gaussian sampling

— Gaussian sampling is Minimax !

Concluding Remarks

The power of random sampling

• - Near-optimal in an overall sense (minimax)
• - Large random samplers behave in deterministic ways

(sharp concentration + universality)

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

A Non-Asymptotic analysis of random

channels

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/1304.7751, April 2013,