Abstract

In the turbulent atmosphere, wavefront predistortion based on receiver-to-transmitter feedback can significantly improve the performance of optical communication systems that employ sparse aperture transmit and receive spatial diversity. The time evolution of the atmosphere, as wind moves turbulent eddies across the propagation path, can limit any improvement realized by wavefront predistortion with feedback. The improvement is especially limited if the latency is large or the feedback rate is small compared to the time it takes for turbulent eddies to move across the link. In this paper, we develop a physics based channel model that describes the time evolution of atmospheric turbulence. Based on that channel model, we derive theoretical expressions relating latencies—such as feedback latency and channel state estimate latency—and feedback rate to optimal performance. Specifically, we find the theoretical optimal average bit error rate as a function of fundamental parameters such as wind speed, atmospheric coherence length, feedback rate, feedback latency, and channel state estimate latency. Further, we describe a feedback strategy to achieve the optimal bit error rate. We find that the sufficient feedback rate scales linearly with the inverse of the atmospheric coherence time and sub-linearly with the number of transmitters. Under typical turbulence conditions, low-rate feedback, of the order of hundreds of bits per second, with associated latencies of less than milliseconds is sufficient to achieve most of the gain possible from wavefront predistortion.

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References

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  1. V. W. S. Chan, "Free-space optical communications (Invited Paper)," J. Lightwave Technol. 24, (12), 4750‒4762 (2006).
    [CrossRef]
  2. J. H. Shapiro, J. W. Strohehn, ed., "Imaging and optical communication through atmospheric turbulence," Laser Beam Propagation in the Atmosphere, Springer-Verlag, 1978, pp. 172‒183.
  3. J. H. Shapiro, "Normal-mode approach to wave propagation in the turbulent atmosphere," Appl. Opt. 13, (11), 2614‒2619 (1974).
    [CrossRef]
  4. A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront predistortion, and coherent detection," Proc. of the IEEE Conf. on Global Telecommunications, 2009, pp. 1‒8.
  5. E. J. Lee and V. W. S. Chan, "Part 1: Optical communication over the clear turbulent atmospheric channel using diversity," IEEE J. Sel. Areas Commun. 22, (12), 4750‒4762 (2004).
  6. A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront control, and coherent detection," Proc. SPIE 7464, 74640J (2009).
  7. H. S. Lin, Communication model for the turbulent atmosphere [Ph.D. Thesis],, Case Western Reserve Univ., 1973.
  8. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, (6), 1399‒1406 (1972).
    [CrossRef]
  9. G. I. Taylor, "The spectrum of turbulence," Proc. R. Soc. London, Ser. A 164, 476‒490 (1938).
    [CrossRef]
  10. R. L. Mitchell, "Permanence of the log-normal distribution," J. Opt. Soc. Am. 58, 1267‒1272 (1968).
    [CrossRef]
  11. D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge Univ. Press, 2005.

2009 (1)

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront control, and coherent detection," Proc. SPIE 7464, 74640J (2009).

2006 (1)

2004 (1)

E. J. Lee and V. W. S. Chan, "Part 1: Optical communication over the clear turbulent atmospheric channel using diversity," IEEE J. Sel. Areas Commun. 22, (12), 4750‒4762 (2004).

1974 (1)

1972 (1)

1968 (1)

1938 (1)

G. I. Taylor, "The spectrum of turbulence," Proc. R. Soc. London, Ser. A 164, 476‒490 (1938).
[CrossRef]

Chan, V. W. S.

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront control, and coherent detection," Proc. SPIE 7464, 74640J (2009).

V. W. S. Chan, "Free-space optical communications (Invited Paper)," J. Lightwave Technol. 24, (12), 4750‒4762 (2006).
[CrossRef]

E. J. Lee and V. W. S. Chan, "Part 1: Optical communication over the clear turbulent atmospheric channel using diversity," IEEE J. Sel. Areas Commun. 22, (12), 4750‒4762 (2004).

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront predistortion, and coherent detection," Proc. of the IEEE Conf. on Global Telecommunications, 2009, pp. 1‒8.

Lee, E. J.

E. J. Lee and V. W. S. Chan, "Part 1: Optical communication over the clear turbulent atmospheric channel using diversity," IEEE J. Sel. Areas Commun. 22, (12), 4750‒4762 (2004).

Lin, H. S.

H. S. Lin, Communication model for the turbulent atmosphere [Ph.D. Thesis],, Case Western Reserve Univ., 1973.

Mitchell, R. L.

Puryear, A. L.

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront control, and coherent detection," Proc. SPIE 7464, 74640J (2009).

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront predistortion, and coherent detection," Proc. of the IEEE Conf. on Global Telecommunications, 2009, pp. 1‒8.

Shapiro, J. H.

J. H. Shapiro, "Normal-mode approach to wave propagation in the turbulent atmosphere," Appl. Opt. 13, (11), 2614‒2619 (1974).
[CrossRef]

J. H. Shapiro, J. W. Strohehn, ed., "Imaging and optical communication through atmospheric turbulence," Laser Beam Propagation in the Atmosphere, Springer-Verlag, 1978, pp. 172‒183.

Taylor, G. I.

G. I. Taylor, "The spectrum of turbulence," Proc. R. Soc. London, Ser. A 164, 476‒490 (1938).
[CrossRef]

Tse, D.

D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge Univ. Press, 2005.

Viswanath, P.

D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge Univ. Press, 2005.

Yura, H. T.

Appl. Opt. (2)

IEEE J. Sel. Areas Commun. (1)

E. J. Lee and V. W. S. Chan, "Part 1: Optical communication over the clear turbulent atmospheric channel using diversity," IEEE J. Sel. Areas Commun. 22, (12), 4750‒4762 (2004).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

Proc. R. Soc. London, Ser. A (1)

G. I. Taylor, "The spectrum of turbulence," Proc. R. Soc. London, Ser. A 164, 476‒490 (1938).
[CrossRef]

Proc. SPIE (1)

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront control, and coherent detection," Proc. SPIE 7464, 74640J (2009).

Other (4)

H. S. Lin, Communication model for the turbulent atmosphere [Ph.D. Thesis],, Case Western Reserve Univ., 1973.

J. H. Shapiro, J. W. Strohehn, ed., "Imaging and optical communication through atmospheric turbulence," Laser Beam Propagation in the Atmosphere, Springer-Verlag, 1978, pp. 172‒183.

A. L. Puryear and V. W. S. Chan, "Optical communication through the turbulent atmosphere with transmitter and receiver diversity, wavefront predistortion, and coherent detection," Proc. of the IEEE Conf. on Global Telecommunications, 2009, pp. 1‒8.

D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge Univ. Press, 2005.

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Figures (14)

Fig. 1
Fig. 1

Sparse aperture system geometry—a field is transmitted from n t x transmitters in the ρ-plane to n r x receivers in the ρ -plane.

Fig. 2
Fig. 2

Metric space representation of the input/output field ( n t x = 2 ) with the uniform quantization grid overlaid in gray.

Fig. 3
Fig. 3

Metric space representation of the input/output field ( n t x = 2 ) with the vector quantization grid overlaid in gray.

Fig. 4
Fig. 4

(Color online) Asymptotic BER as a function of the normalized rate R u / n t x .

Fig. 5
Fig. 5

Metric space representation of the input/output field ( n t x = 2 ) with the optimal mapping quantization grid overlaid in gray.

Fig. 6
Fig. 6

(Color online) Update average energy versus atmospheric temporal auto-covariance for n t x = n r x .

Fig. 7
Fig. 7

(Color online) Optimal mapping—This figure shows a diagram detailing the optimal algorithm to feed back a full or incremental update to the transmitter.

Fig. 8
Fig. 8

(Color online) Optimal number of transmitters as a function of rate R u .

Fig. 9
Fig. 9

(Color online) Asymptotic BER (exact and bound) as a function of normalized rate R u / n t x . SNR is dimensionless.

Fig. 10
Fig. 10

(Color online) Wavefront predistortion gain versus latency for five cases: (1) no transmitter CSI and perfect receiver CSI, (2) perfect transmitter CSI, perfect receiver CSI, (3) transmitter CSI delayed by τ t x , perfect receiver CSI, (4) no transmitter CSI, receiver CSI delayed by τ r x , and (5) transmitter and receiver CSI delayed by τ r x .

Fig. 11
Fig. 11

(Color online) Average bit error rate versus latency for five cases: (1) no transmitter CSI and perfect receiver CSI, (2) perfect transmitter CSI, perfect receiver CSI, (3) transmitter CSI delayed by τ t x , perfect receiver CSI, (4) no transmitter CSI, receiver CSI delayed by τ r x , and (5) transmitter and receiver CSI delayed by τ r x .

Fig. 12
Fig. 12

(Color online) Wavefront predistortion gain as a function of the update length, R u . The full update is shown as a dashed line while the optimal update, either incremental or full depending on which performs better, is shown as the solid line. For the graph, the number of transmitters is n t x = 10 , the number of receivers is n r x = 10 , the atmospheric coherence time is ρ 0 / v = 1 s, the feedback latency is τ 0 = 0 . 1 s, the rate is 80 bits per second, and the full update length is R u 0 = 10,000 bits.

Fig. 13
Fig. 13

(Color online) Rate required to achieve 99% of the possible gain for various codebook sizes | C | .

Fig. 14
Fig. 14

(Color online) Wavefront predistortion gain, Υ ( T ) , as a function of rate, r, for optimal update length, n t x = n r x = 10 , coherence time, v / ρ 0 = 1 s, and feedback latency, τ 0 = 0 . 1 s.

Equations (73)

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y = SNR n r x H x + w ,
SNR = 2 ( q η / h f ) 2 P l o P laser T b A t x A r x [ ( q 2 η / h f ) P l o + ( q η / h f ) 2 P l o N o + N o c ] ( λ L ) 2 ,
h i j = exp χ ρ i , ρ j + j ϕ ρ i ρ j ,
E x 2 = 1 .
h ρ 1 ρ 1 h ρ 2 , ρ 2 = 1 ( λ L ) 2 exp j π λ L ρ 1 ρ 1 2 ρ 2 ρ 2 2 1 2 D s ρ 1 ρ 2 , ρ 1 ρ 2 ,
D s ( ρ , ρ ) = 2 . 91 k 2 0 L C n 2 ( z ) 1 L | ρ + ρ ( L z ) | 5 / 3 d z ,
R h h ( t ) = ( λ L ) 2 h 0 , 0 h v t , v t = exp 1 2 D s v t , v t .
D s v t , v t = 2 . 91 k 2 0 L C n 2 1 L | v t z + v t ( L z ) | 5 / 3 d z = 2 . 91 k 2 C n 2 L 5 / 3 0 L v t L 5 / 3 d z = 2 . 91 k 2 C n 2 L v t 5 / 3 ,
R h h ( t ) = exp 2 . 91 k 2 C n 2 L 2 v t 5 / 3 = exp 1 2 v t ρ 0 5 / 3 ,
ρ 0 = 1 2 . 91 k 2 C n 2 L 3 / 5 .
t 0 = ρ 0 v .
H c ( t ) = R h h ( t ) H 0 + 1 R h h ( t ) H 1 .
1 n r x H = U Γ V ,
y ̃ i = γ i x ̃ i + w ̃ i ,
BER = min feedback schemes max t BER ,
min R R u BER ,
g ( ) = arg min g ( H c ) : H g ( H c ) R u BER ,
g rate ( H c ) v max ( H c ) ,
g 0 rate ( H c ) = κ ,
E [ Pr ( error ) ]
= Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x + 2 R u n t x ,
n t x = argmax n t x o : n t x o n t x 1 + n t x o n r x 2 1 2 R u n t x o + 2 R u n t x o .
E [ Pr ( error ) ] = min g ( H c ) : H ( g ( H c ) ) R u E [ BER ] = min g ( H c ) : H ( g ( H c ) ) R u E Q 2 SNR ϕ ,
E [ Pr ( error ) ] = min g ( H c ) : H ( g ( H c ) ) R u Q 2 SNR E ϕ .
E [ Pr ( error ) ]
= min g ( H c ) : H ( g ( H c ) ) R u Q 2 SNR E γ min γ max s x , v ( s ) d s ,
E [ Pr ( error ) ] = min g ( H c ) : H ( g ( H c ) ) R u Q 2 SNR 1 n t x n r x 2 1 + n t x n r x 2 s E x , v ( s ) d s .
E [ Pr ( error ) ] = min g ( H c ) : g ( H c ) R u Q 2 SNR 1 + n t x n r x 2 × E x , v 1 + n t x n r x 2 + 1 n t x n r x 2 1 + n t x n r x 2 s E x , v ( s ) d s 1 / 2 .
D = E x v max 2 .
D = 2 R u n t x .
E x 2 = D + E x , v max = 1 .
E x , v max = 1 2 R u n t x .
E [ Pr ( error ) ] = Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x + 1 n t x n r x 2 1 + n t x n r x 2 s E x , v ( s ) d s 1 / 2 .
E [ Pr ( error ) ] = Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x
+ 2 R u n t x 1 n t x n r x 2 1 + n t x n r x 2 s f γ ( s ) d s 1 / 2 ,
E [ Pr ( error ) ]
= Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x + 2 R u n t x .
E [ Pr ( error ) ]
= Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x + 2 R u n t x ,
n t x = argmax n t x o : n t x o n t x 1 + n t x o n r x 2 1 2 R u n t x o + 2 R u n t x o .
n t x max 20 , 2 SNR .
v incremental = v max ( H c ) v t x ,
E [ Pr ( error ) ] = Q 2 SNR 1 + n t x n r x 2 1 Ψ ( τ ) 2 R u n t x + Ψ ( τ ) 2 R u n t x ,
n t x = argmax n t x o : n t x o n t x 1 + n t x o n r x 2 1 Ψ ( τ ) 2 R u n t x o
+ Ψ ( τ ) 2 R u n t x o .
Ψ ( τ ) = E v incremental 2 E v max H c v t x 2 = 2 1 E v max H c v max H t x + D 0 + D 1 = 2 1 E v max H c v max H t x + 2 R u 0 / n t x + Ψ ( τ ) 2 R u / n t x ,
Ψ ( τ ) = 2 1 E v max H c v max H 0 + 2 R u 0 / n t x 1 2 R u / n t x .
min 2 1 R h h ( τ ) + 2 R u 0 / n t x 1 2 R u / n t x , 1 Ψ ( τ ) 1 .
Ψ ( τ ) min 2 1 ( R h h ( τ ) ) 10 + 2 R u 0 / n t x 1 2 R u / n t x , 1 .
t u 1 5 log 2 3 / 5 t 0 0 . 93 t 0 ,
lim R u E [ Pr ( error ) ] = Q 2 SNR 1 + n t x n r x 2 .
lim R u 0 E [ Pr ( error ) ] = Q 2 SNR .
lim R u small E [ Pr ( error ) ] = Q 2 SNR 1 + R u n r x log ( 2 ) .
E [ Pr ( error ) ] Q 2 SNR 1 + n t x n r x 2 1 2 R u n t x + 2 R u n t x .
| ϕ | 2 = v t x H r x H c v t x 2 H r x v t x 2 .
E [ BER ] = Q 2 SNR E [ | ϕ | 2 ] .
Υ = E | ϕ | 2 E | ϕ N F | 2 ,
E | ϕ | 2 = E r H c H c r 2 H c r 2 = 1 .
Υ ( 2 ) = E v max H c H c H c v max H c 2 H c v max H c 2 = E H c v max H c 2 = γ max 2 H c = 1 + n t x n r x 2 ,
Υ ( 3 ) = E v max ( H 0 ) H c H c v max ( H 0 ) 2 H c v max ( H 0 ) 2 = R h h ( τ t x ) γ max 2 ( H 0 ) + ( 1 R h h ( τ t x ) ) = e v τ t x ρ 0 5 / 3 1 + n t x n r x 2 + 1 e v τ t x ρ 0 5 / 3 .
lim τ r x 0 Υ ( 3 ) = Υ ( 2 ) .
lim τ r x Υ ( 3 ) = Υ ( 1 ) .
Υ ( 4 ) = E r H 0 H c r 2 H 0 r 2 = R h h ( τ r x ) .
lim τ r x 0 Υ ( 4 ) = Υ ( 1 ) .
Υ ( 5 ) = E v max ( H 0 ) H 0 H c v max ( H 0 ) 2 H 0 v max ( H 0 ) 2 = R h h ( τ r x ) γ max 2 ( H 0 ) = R h h ( τ r x ) 1 + n t x n r x 2 .
lim τ r x 0 Υ ( 5 ) = Υ ( 2 ) .
Υ ( T ) = E R h h τ 0 + R u r + t H 0 + 1 R h h τ 0 + R u r + t H 1 f v max ( H 0 ) 2 = R h h τ 0 + R u r + t E H 0 f ( v max ( H 0 ) ) 2 + 1 R h h τ 0 + R u r + t E H 1 f ( v max ( H 0 ) ) 2 = R h h τ 0 + R u r + t γ max 2 ( H 0 ) 1 ζ + ζ + 1 R h h τ 0 + R u r + t = e τ 0 + R u / r + t t 0 5 / 3 1 + n t x n r x 2 1 ζ + e τ 0 + R u / r + t t 0 5 / 3 ζ + 1 e τ 0 + R u / r + t t 0 5 / 3 ,
Υ ( T ) = e τ 0 + 2 R u / r t 0 5 / 3 1 + n t x n r x 2 e τ 0 + 2 R u / r t 0 5 / 3 1 + n t x n r x 2 Ψ ( R u / r ) 2 R u / n t x + e τ 0 + 2 R u / r t 0 5 / 3 Ψ ( R u / r ) 2 R u / n t x + 1 e τ 0 + 2 R u / r t 0 5 / 3 .
r s v ρ 0 1 5 log 2 1 + 2 1 / n t x + 2 | C | / n t x 3 5 .
r v ρ 0 1 5 log 2 2 α ( 1 2 1 / n t x ) + 2 | C | / n t x 3 5 .
| C | n t x log 2 1 0 7 2 α 1 2 1 / n t x 25 n t x ,
r v ρ 0 10 n t x α log ( 2 ) 3 / 5 .
Υ ( T ) Υ ( 2 ) e τ 0 t 0 5 / 3 1 + n t x n r x 2 + 1 e τ 0 t 0 5 / 3 .