Sub-Nyquist SamplingContinuous to Finite Module

Orthogonal Matching Pursuit Block

• Supervisors: Inna Rivkin & Moshe Mishali• Winter 2009 – Spring 2010

Final Presentation – Part APerformed by : Yoni Smolin

2/8/2010

Sub-Nyquist Digital System

• On every clock cycle, the system solves

• How ?• Expand - computes y from input samples.• CTF - locates non zero elements of z.• DSP - computes their values.

Sampled data

ExpandX3 CTF DSP

ny Supp nz

12

4

Support

y

A

z

sparse

Sampling matrixzsupp

Asupp yzsupp

†SuppA

CTF

• The frame matrix - a basis for recovery:

CTF

Frameconstruction

Support recoveryy frame

matrixSupport

y1y2 y70

…y1H y2

H y70H Q

CTF

• Support recovery – applying OMP for Q:

CTF

Frameconstruction

Support recoveryy frame

matrixSupport

QAUQ

support

OMP – Algorithm (SBR2)

0

2

k i 1 2k

1

Q;i Supp Supp Supp

PSEUDO INVERSE

i Supp Q;i

i

1 initialize: Supp ,

2 loop: do (i'th iteration)

best arg max

Supp Supp best

ˆ

ˆ

while

R Q

a R

U A A A Q

R Q A U

R

2

i-1 FthresholdR

Matching

SolutionApproximation

ResidualUpdate

OMP – Adaptation for Hardware

0

2

k i 1 2k

2

i i-1 F

1 initialize: Supp ,

2 loop: do (i'th iteration)

best arg max

Supp Supp best

while threshold

R Q

a R

R R

Matching

OMP – Adaptation for Hardware

0

2Hk i 1 2k

1 i-1 best 1 i-1 iM.G.S.

NEW COLOLD BASIS

i i-1 i i

1 initialize: Supp ,

2 loop: do (i'th iteration)

best arg max

Supp Supp best

,..., , a ,..., ,

R Q

a R

R R

Hi-1

2

i i-1 Fwhile threshold

R

R R

Modified GramSchmidt

Matching

ResidualUpdate

1 OMP iteration

101 projections

1 M.G.S. iteration

update residual

12

12

2

212

12

12

12

12

OMP – Atomic Operations

Implementation Considerations

• CTF must fit in a single Stratix III FPGA.• Number representation: Fixed-point, Q1.16.• Fully parameterized design: N, L, m.• Guiding principle: Achieve best throughput.

Implying:• A pipelined datapath.• Parallel execution of vector-matrix multiplications.

CTF Block

OMPFrame

A memory

CTF

Merge

Block Diagram - OMP

Block Diagram - Datapath

SUB

2

2 keep

max support

1

stop ?

MAMU

Block Diagram - Datapath

SUB

2

2 keep

max support

1

stop ?

MAMU18bit 21bit

Matrix Multiplier (MAMU)

inputs outputs

1 2 … 12 inputs outputs1

Delay

Multiplication - 2 cycles

Projection - 5 cycles

Hardware consumption

DSP blocks – 64%

. . .

A single column

1

12

+

36 36

36

Inverse Square root

• Implementation alternatives:

• Remark: Input number must be > .

1

megafunction Logic DSP Delay representation accuracy

ALTFP_INV_SQRT 1,000 21 38 cycles at 400MHz

floating point(requires

conversions)─

DW_INV_SQRT 2,292 0 257 nSec fixed point 2-16

ALTSQRT+

LPM_DIVIDE1,256 0 134 nSec fixed point 2-16 (98%

of the time)

14

FSM Controller

OMP iteration - Matching

SUB

2

2 keep

max support

1

stop ?

MAMU

Pipelined calculation = 101 cycles

OMP iteration – Gram Schmidt

SUB

2

2 keep

max support

1

stop ?

MAMU

( i-1 projections ) + normalization

OMP iteration – Update Residual

SUB

2

2 keep

max support

1

stop ?

MAMU

Parllel calculation: stopping cond. & update R

Overflow

• In theory, overflow can occur only for:

→ Solution: Divide R by 2 & recalculate:

• Error detection flags: Overflow: MAMU, , SUB, .

Rsomevector

R/2somevector

2

2 1

Frame OMP

Synthesis Report

* Without debug environment

• Input text files are generated by Matlab.

• Output text files are evaluated in Excel.

Logical Simulation – Test Bench

OMPR memory

A memory

Q.txt

A.txt

supp.txt

Logical Simulation – Example

Simulation output:• Support: → → →

Iteration: 1st 2nd 3rd 4th 5th →

z9 9338 64

f

NYQ

1

2fNYQ

1

2f

Logical Simulation – Results

Type # signals

% exact recovery

mistaken elements

200 92% 1 (7.5%)3 (0.5%)

•Problem : recovery < 100%. Solutions:• Regular mode: CTF can be reinitiated until successful

recovery.• Iterative mode: Support merging can help overcome

mistaken elements.

FMFM

• Swap normal ↔ iterations modes on the fly.

• A must be normalized before system startup.

• Q’s calculation:

• Should be wide to avoid overflow.

→ transfer only 18 relevant bits to OMP.

• Can be accelerated by upsampling y.

• Wide mux (5,184 bits) can pose a latency problem.

• Stopping condition varies with input signal’s energy.

Challenges

Gannt