The Richardson-Lucy Algorithm Based Demodulation Algorithms for ...

Outline 

The Richardson-Lucy Algorithm Based 

Demodulation Algorithms for the Two-dimensional 

Intersymbol Interference Channel 

Zhijun Zhao and Richard E. Blahut 

Department of Electrical and Computer Engineering 

University of Illinois at Urbana-Champaign 

Coordinated Science Laboratory 

1308 West Main Street 

Urbana, IL 61801 

Wednesday, March 23, 2005 

Washington University in St. Louis 


The Richardson-Lucy Algorithm Based Demodulation Algorithm

Outline 

Outline 

1 Introduction 

2 Two-dimensional ISI Channel Model 

3 Wiener Based Demodulation Algorithms 

Wiener Based Demodulation Algorithms 

BER Performance 

4 Richardson-Lucy Based Demodulation Algorithms 

Blind and Nonblind Richardson-Lucy Algorithms 

Richardson-Lucy Algorithm for AWGN Channel 

Richardson-Lucy Based Demodulation Algorithms 

Blind Demodulation with Progressive Thresholding 


5 Comparison of Wiener and Richardson-Lucy Based Algorithms 



Introduction 

Two-dimensional ISI Channel Model 



Comparison of Wiener and Richardson-Lucy Based Algorithms 

Conclusions 

Outline 















Introduction 





Conclusions 

Two-dimensional Data Demodulation Problem 

Viterbi algorithm for one-dimensional channel 

Maximum-likelihood optimal for AWGN ISI channel 

Good for short ISI channel 

Computationally expensive for long ISI channel 

Two-dimensional intersymbol interference channel 

Viterbi-like approach: MVA, computationally expensive 

Image processing approach 

Need an efficient (and optimal) algorithm 

Wiener based demodulation algorithms 

Richardson-Lucy based demodulation algorithms 



Introduction 





Conclusions 

Outline 















Introduction 





Conclusions 

Two-dimensional ISI channel model 

Two-dimensional ISI channel 

v(x, y) = h(x, y) ∗ ∗s(x, y) + n(x, y) (1) 

where (x, y) ∈ Z 2 . 

Fourier domain representation 

V (f x , f y ) = H(f x , f y )S(f x , f y ) + N(f x , f y ) (2) 

where (f x , f y ) ∈ Z 2 . 



Introduction 





Conclusions 

Simulation Setup 

PSF h(x, y) 

h(x, y) = C · e −(x2 +y 2 )/2σ 2 h (3) 

In simulations, σ h = 0.7, 

⎡ 

0.04390 0.12169 

⎤ 

0.04390 

[h] ≈ ⎣ 0.12169 0.33767 0.12169 ⎦ (4) 

0.04390 0.12169 0.04390 

s(x, y) i.i.d., s(x, y) ∈ {−1/2, 1/2}, or s(x, y) ∈ {0, 1}. 

n(x, y) white noise. 



Introduction 





Conclusions 

Original Data Image and Blurred Noisy Image 



Introduction 





Conclusions 

Outline 

















Introduction 





Conclusions 

Wiener Demodulation 



Wiener demodulation algorithm 

Wiener estimation of S(f x , f y ) 

˜S(f x , f y ) = V (f x , f y )G(f x , f y ) (5) 

where G(f x , f y ) is the Wiener filter: 

G(f x , f y ) = 

H ∗ (f x , f y ) 

H(f x , f y )H ∗ (f x , f y ) + Φ n (f x , f y )/Φ s (f x , f y ) 

(6) 

Hard-thresholding 

One-step algorithm 



Introduction 





Conclusions 

Iterative Filtering with Limiting 



Wiener based iterative filtering 

S (r+1) (f x , f y ) = S (r) (f x , f y ) + D (r) (f x , f y )G(f x , f y ) (7) 

where 

D (r) (f x , f y ) = V (f x , f y ) − S (r) (f x , f y )H(f x , f y ) (8) 

Limiting regularization 

⎧ 

⎨ 1/2 if s (q) (x, y) ≥ 1/2 

s (q) (x, y) ⇐ −1/2 if s 

⎩ 

(x, y) ≤ −1/2 

s (q) (x, y) elsewhere 

(9) 




Introduction 





Conclusions 



Iterative Filtering with Progressive Thresholding 

Wiener based iterative filtering 

Progressive thresholding 

⎧ 

⎨ 1/2 if s (q) (x, y) ≥ T (q) 

s (q) (x, y) ⇐ −1/2 if s 

⎩ 

(q) (x, y) < −T (q) 

s (q) (x, y) elsewhere 

(10) 

where T : {0, 1, 2, . . .} → [0, 1/2] is a monotone decreasing 

function. 




Introduction 





Conclusions 

BER Performance Curves 





Introduction 





Conclusions 

Outline 




















Introduction 





Conclusions 






Richardson-Lucy Algorithm – Deconvolution of 

Nonnegative Images 

The Richardson-Lucy Algorithm 

[ 

] 

s (r+1) (x, y) = s (r) v(x, y) 

(x, y) 

s (r) (x, y) ∗ ∗h(x, y) ∗ ∗h(−x, −y) 

Nonnegative everything. 

(11) 



Introduction 





Conclusions 






Blind Richardson-Lucy Algorithm – Blind Deconvolution of 

Nonnegative Images 

Initialize s(x, y) and h(x, y) 

Blind Richardson-Lucy iteration 

Estimate h(x, y) 

[ 

] 

h (r+1) (x, y) = h(r) (x, y) v(x, y) 

∑ 

(x,y) s(x, y) h (r) (x, y) ∗ ∗s(x, y) ∗ ∗s(−x, −y) (12) 

Estimate s(x, y) 

[ 

] 

s (r+1) (x, y) = s (r) v(x, y) 

(x, y) 

s (r) (x, y) ∗ ∗h(x, y) ∗ ∗h(−x, −y) 

Nonnegative everything. 

(13) 



Introduction 





Conclusions 






Richardson-Lucy Algorithm for the AWGN Channel 

Limiting and Biasing 

n 0 (x, y) ∼ Gauss(0, σ 2 n) (14) 

⎧ 

⎨ 0 if n 0 (x, y) ≤ −ν 

n(x, y) = 1 + ν if n 0 (x, y) ≥ ν 

⎩ 

n 0 (x, y) + ν elsewhere 

(15) 

ñ 0 (x, y) = n(x, y) − ν (16) 

In the simulations, 

ν = 4σ n (17) 



Introduction 





Conclusions 

Some Concerns 






The probability of cutting n 0 (x, y) 

P r [ñ 0 (x, y) ≠ n 0 (x, y)] = 1 − erf (4/ √ 2) ≈ 3.67 × 10 −6 (18) 

Where does ν go? 

v(x, y) = h(x, y) ∗ ∗s(x, y) + ñ 0 (x, y) + ν (19) 

Answer: 

v(x, y) = h(x, y) ∗ ∗ [s(x, y) + ν] + ñ 0 (x, y) (20) 



Introduction 





Conclusions 

Richardson-Lucy Algorithm with Limiting 







[ 

] 

s (r+1) (x, y) = s (r) v(x, y) 

(x, y) 

s (r) (x, y) ∗ ∗h(x, y) ∗ ∗h(−x, −y) 

Limiting regularization 

⎧ 

⎨ ν + 1 

s (q) (x, y) ⇐ ν 

⎩ 

s (q) (x, y) 


ŝ(x, y) ⇐ 

if s (q) (x, y) ≥ 1 + ν 

if s (q) (x, y) < ν 

elsewhere 

{ 1 if s (q) (x, y) ≥ 1/2 + ν 

0 if s (q) (x, y) < 1/2 + ν 

(21) 

(22) 

(23) 



Introduction 





Conclusions 






Richardson-Lucy with Progressive Thresholding 


[ 

] 

s (r+1) (x, y) = s (r) v(x, y) 

(x, y) 

s (r) (x, y) ∗ ∗h(x, y) ∗ ∗h(−x, −y) 


⎧ 

⎨ ν + 1 

s (q) (x, y) ⇐ ν 

⎩ 

s (q) (x, y) 

if s (q) (x, y) ≥ 1/2 + ν + T (q) 

if s (q) (x, y) ≤ 1/2 + ν − T (q) 

elsewhere 

where q = r + 1, T : {0, 1, 2, . . .} → [0, 1/2] is a monotone 

decreasing function of iteration. 


(24) 

(25) 



Introduction 





Conclusions 






Blind Richardson-Lucy with Progressive Thresholding 

Initialize s(x, y) and h(x, y) 

Estimate h(x, y) and s(x, y) alternatively 

[ 

] 

h (r+1) (x, y) = h(r) (x, y) v(x, y) 

∑ 

(x,y) s(x, y) h (r) (x, y) ∗ ∗s(x, y) ∗ ∗s(−x, −y) 

(26) 

[ 

] 

s (r+1) (x, y) = s (r) v(x, y) 

(x, y) 

s (r) (x, y) ∗ ∗h(x, y) ∗ ∗h(−x, −y) 


⎧ 

⎨ ν + 1 

s (q) (x, y) ⇐ ν 

⎩ 

s (q) (x, y) 



if s (q) (x, y) ≥ 1/2 + ν + T (q) 

if s (q) (x, y) ≤ 1/2 + ν − T (q) 

elsewhere 

(27) 

(28) 


Introduction 





Conclusions 






BER Curves of the Richardson-Lucy Based Algorithms 



Introduction 





Conclusions 

Outline 















Introduction 





Conclusions 

Comparison of Wiener and Richardson-Lucy Based 

Demodulation Algorithms 



Introduction 





Conclusions 

Conclusions 

Conclusions 

The progressive thresholding applies the binary constraint, and 

gives significant BER improvement. 

The Richardson-Lucy based blind demodulation algorithm with 

progressive thresholding achieves near nonblind performance. 

Richardson-Lucy based algorithms outperform the Wiener 

based algorithms. 

Ongoing and Future Work 

The optimality and convergence of the Richardson-Lucy based 

demodulation algorithms.

The Richardson-Lucy Algorithm Based Demodulation Algorithms for ...

Create successful ePaper yourself

Delete template?

Save as template?