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Non-Uniform sampling and
reconstruction of multi-band signals
M. R. Avendi
Supervisor:
Prof. M. Viberg &
Prof. L. Svensson
M. R. Avendi
June 2010
Signal Model & Definitions
signal model:
Spectral support :
3
Spectral Occupancy:
Launda lower bound:
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
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
• 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
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