Blind ReconstructionBlind Reconstructionof Sparse Multiband of Sparse Multiband

SignalsSignalsPerformed by:Performed by: Eli Sorin Eli Sorin

Zvika ShiraziZvika Shirazi

Supervisor: Supervisor: Michael Michael YampolskyYampolsky

Mid PresentationMid PresentationWinter 09/10Winter 09/10

Sampling & Reconstruction device

The SystemThe System

x(t)x(t)

Sampling Block

.

.

.

h(t)

Analog Block

h(t)

h(t)

.

.

.

1( )Np t

1( )p t

( )Np t

MathematicalAlgorithm

The Idea In GeneralThe Idea In General

The The sampling blocksampling block constructs a constructs a matrix Amatrix A that operates on the signal: that operates on the signal: y(t)=Ax(t) y(t)=Ax(t)

The The reconstruction blockreconstruction block - - Reconstructs the Reconstructs the support Ssupport S of x out of A of x out of A

and y.and y. Now, Using Now, Using SS, we invert , we invert AA and and

reconstruct Xreconstruct X.. The algorithm was successfully The algorithm was successfully

simulated in Matlab.simulated in Matlab.

Our MissionOur Mission Implement Implement the reconstruction the reconstruction

algorithm by software (C++) algorithm by software (C++) with hardware orientation.with hardware orientation.

Profiling -Profiling -Analyze Algorithm’s Analyze Algorithm’s subblocks:subblocks:bottlenecks, parallelism, bottlenecks, parallelism,

serialization.serialization. logically massive sub-blocks.logically massive sub-blocks. time-consuming sub-blocks, time-consuming sub-blocks,

latency.latency.

How we accomplished it?

Gave up profiler tools produce irrelevant data, overkill lack relevant data great overhead in tuning the tools

Plant diagnostics code easy adjusted to our needs

What did we look for?

Execution time – rough estimation for hardware latency

Arithmetic operations – adders, multipliers, divisors

The Reconstruction The Reconstruction AlgorithmAlgorithm

Q=∑y[n] ∙ y’[n]

Q=V∙V`

I=As ∙As`

Construct frame V Construct frame V for yfor yCostruct Costruct QQ

DecomposDecompose Qe Q

CTF BlockCTF Block

Reconstruct joint Reconstruct joint supportsupport

V=A∙U

MMV MMV sloverslover

U[supp(ui)]

y V

APseudo Pseudo

InvertInvert

Calculate Calculate ZZAs

`zy

2 parts for the Algorithm

CTF+`As` Pseudo-invertion Not real Time – operate only once

Serial

Calculating ZReal-Time

Clocking

Internal system clock vs. Sample clock System clock - limited by the FPGA

device Typical rates:

FPGA clock - 200-250MHz Sample rate – 50 MHz

Sample rate can be reduced by adding more channels.

But….

But…. Trade Offs

More channels More Logic

Reduced sample Rate Smaller Buffer, Loose

Timing

Contruct Frame Q (CFQ)

Q = ∑y[n] ∙ y`[n] Operates @ sample rate Finite number of samples (about

40...) Relatively light block:

Q Factorization Block (QFB)

Q = V ∙ V` Cholesky factorization – Lower &

Upper Requires Positive-defined Q

Add 1e-06 delta to Diagonal The delta forces precision

How big it is? Not so...

MMV Block

Find the sparsest solution to the system V=AU

NP-hard no polynomial algorithm Several sub-optimal algorithms exist. All - iterations-based limited

Parallelization. We’ve selected M-OMP. How Big?

well.. XXL... !!

Pseudo Invert Block

Inverting is expensive but

worthwhile. How big? Medium-Large…

1H Hs s sA A A

Calculate Z

Operates @Real-Time. Only Multipliers and Additions! Theoretically can be Fully Parallel. How big? Extra Small…

†sZ A y

Summary

Summary

Most heavy Blocks: MMV - by farPseudo Invert – silver medal

Optimization FocusMMV – wait until

Optimization Suggestions

MMV Fastest possible clock Largest device Exploit idle arithmetic units

Pseudo-Invert Can partially operate parallel to CTF,

but… Great resource waste.

Another Conclusion…

Another Conclusion…

Arithmetic elements vs. Channels - - Non-

Linear

Low amount of channels – Low marginal cost when adding a channel.

Large amount of channels – Great marginal cost.

Next step….

Fixed Point

Good Good Luck!Luck!