Non-Uniform sampling and

reconstruction of multi-band signals

M. R. Avendi

Supervisor:

Prof. M. Viberg &

Prof. L. Svensson

M. R. Avendi

June 2010

Agenda

• Multiband Signal Model

• Nonuniform Sampling

• Reconstruction

• Efficient Reconstruction

2

Signal Model & Definitions

signal model:

Spectral support :

3

Spectral Occupancy:

Launda lower bound:

Example

• F={ [0.5,2], [4,5], [8,8.5] } fmax=12

• λ(F)=1.5+1+.5=3

• Ω=0.25

4

Uniform Sampling

Why not?

•As the sparsity increase it will not efficient any more.

•Cost in power, calculation,Storage.

5

Periodic Non-Uniform sampling

• t= (nL +ci)T!!

• x[c1T], x[c2T],…,x[cpT], x[(L+c1)T], x[(L+c2)T],…,x[(L+cp)T],...

6

Sampling parameters

• T= 1/fmax base sample time

• L > 0 period of pattern

• p : number of samples per block, p<L

• C={ c , c , …, c } : sample pattern set

> 0

• C={ c1, c2 , …, cp} : sample pattern set

0 ≤ c1 ≤ c2 ≤ … ≤ cp ≤ L-1

• The average sampling ratio = p/L

7

Example

L=20, p=5

C={0,4,7,12,16}

8

C={2,6,11,15,18}

Implementation view

9

Sampling Formulation

• Categorize p sequences

10

Sampling Formulation

• Express in matrix form

• y is a known p*1 vector

• AC is a sub-DFT matrix , p*L

• s is unkonwn a L*1 vector

p < L ! !

11

Active slots and Spectral set

• Active slots : 1 & 4

• Spectral index set: k={1,4}

• q= number of active slots =|k|=2

12

Reconstruction

• Reduced order model

y(f)=AC s(f) => y(f)=AC(k) z(f)

,,

13

Reconstruction

• y(f)= AC(k) z(f) if p > q =>

• Time domain: x [n]= h[n] * x [n]• Time domain: xhi[n]= h[n] * xi[n]

14

Reconstruction in time

15

Simulation

Ω=0.25 , fmax=5

L=32,p=12 ,

D=1.8!, RMSE=1.9%,

16

Time domain and frequency domain of original and reconstructed signal

Extreme case

In an extreme case each of the L spectral cells may be partly occupied by small

sliver of the spectrum.

Ω=0.2

L=20 , q=20

Efficient Reconstruction

What is the differences between Sampling

Patterns

RMSE=5%

RMSE=105%