- Email: [email protected]
Information rates in optical channels
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
INFORMATION RATES IN OPTICAL CHANNELS Cherif BENDJABALLAH
Laboratoire des Signaux et Systdmes du C.N.R.S., Ecole Sup~rieure d'Electricit~ Plateau du Moulon, 91190-Gif-sur-Yvette,France Received 14 November 1975, Revised manuscript received 22 January 1976
Results of numerical computations of the information capacity in a photon communication channels are given and compared for various optical sources of different statistical properties.
In a discrete typical communication system each transmitted symbol x i has a certain probability P(yj/xi) to be detected as a particular symbol yj, because of random interferences, noise, etc., occurring in the channel during the transmission. The rate of transmission of information R through the channel may be defined as the average of I(xi; yi), the mutual information per symbol pairs, where
P(Xi/Y]) I(xi;Yj) = log P(Xi) '
(1)
that is
R = ~ ~. P(Xi, yj)I(x i, yj). i 1
(2)
According to Shannon's definition, the channel capacity C is the maximum value of R, C = max R.
ring the maximum information than an ideal laser, it may be useful to study the corresponding information rates and to compare them from the point of view of their statistical properties. We first consider a communication channel in which the message is transmitted via a fdter with a random transmittance, x (0 ~
(3)
The evaluation of C for some input density is often a difficult problem. However, in certain cases t h e r e exist some methods o f studying the rate of information and computing its maximum. One such case which is of great practical significance is a channel where a single-mode laser is used as the light source associated with a continuously modulated filter. But because of its low output power, communication with such a source is not always feasible. So although light emitted from thermal and other radiations appears to be less interesting for transfer-
1
f dx Pa(x) a(n/x)
R(a) =~--0 0
(4)
Q(n/x) × log f l dz Pa(Z) a(n/z)' C = max R (a),
(5)
a
where Q(n/x) is the probability density (p.d.) of registering n counts in the time interval t when the transmittance is x with the p.d. Pa(x). The parameter a is used to describe a wide range of different forms of 55
Volume 17, number 1
OPTICS COMMUNICATIONS
06
h-
Loser
April 1976
(n 2) - (n) (n) 2
p.d. so that the optimum p.d. will be included and give the maximum rate. The p.d. Pa(x) for various a (0 < a ~< 20) is taken to be
increases from the laser case to the g.m.th, case (1 ~< h ~< 6) [4], the information capacity decreases and in the whole range of (n) is the greatest for the laser case. Let us now deal with another interesting case of transmission o f information which is the case where the photoelectric detector is regarded as a communication channel whose input symbols are the pulsed optical sources considered above and whose output symbols are the registered counts {n} in a given time interval. A rectangular pulse of time of duration 2D is centered at X - X 0, where X 0 is a constant and X is the random variable taking continuously its values in 0 = ( - T , T) with p.d. p(x). For a pulse of coherent radiation, it has been shown that the channel capacity is [21
Pa(X) - 1 - e xa p ( - a ) exp ( - a x ) .
C = max ([1 - p~n)(0)] [H(0) - log 2D]
C
02
~
~
_
~
Th(Ft=4)
~
ThlFt=0)GmL GmTh
1
2
3
4
5
6
3
8
9
10
Fig. ]. Vaziation o f the maximum information rate R M versus mean number of photoelectrons for various light sources in the case o f random transmittance.
The optical signals considered here are: single mode laser light, thermal light (th.) of arbitrarily normalized coherence time (Ft) with exponential time-correlation function, gaussian-modulated laser (g.m.1.) light and gaussian-modulated thermal (g.m.th.) light. The two last sources are considered to have a long coherence time. For all these sources, Q(n/x) is explicitly known (see Appendix). It can be seen, but not easily proved for any p.d. Q(n/x), that there is a maximum, R M, of R(a) for a -~ 1. This maximum may be seen to be very close to the channel capacity [1] as long as a <~ 3 and (n) lies within the interval (0, 10). In fig. 1 we have plotted the variation o f R M versus (n), where {n) is the mean photoelectron count when x = 1. This variation is, as expected, almost logarithmic for the laser light. For the other kinds of radiation quoted above, the variation is more complicated. We notice that for the g.m.th, light which is the most chaotic in this study, the variation of R versus a does not show a pronounced maximum. However, by calculating the derivatives, it is possible to determine with satisfactory accuracy the maximum and to plot R M versus (n), which for (n) > 4 is almost constant. We also remark that although the bunching effect which can be defined as the ratio 56
a
t (6)
with /4(0) = - f p ( x ) log p(x) dx,
(7)
0
where p(n)(k) is given by relation (A1) for x = 1. Eq. (6) can be extended to other p
H(T) - log 2D = - f
dx a exp (-X/Xc)
-T
X log [a exp (-x/xc) ] - log 2D is not too far from its maximum value log (T/D), provided that D ~ T <~x cFig. 2 shows the variation of the channel capacity, computed from eq. (6) with T/D = 10 3, versus (n) for
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
101 -
(2)
H =~=103 05 C,
= 2
001
0.1
10
10
100 Ft
Fig. 3. Variation o f the normalized m a x i m u m i n f o r m a t i o n rate versus normalized counting time interval f o r a thermal
GmTh.
0.5 I
I
I
[
I
10 I
I
1 2 3 /~ 5 6 7 8 9
the different signals studied here. Although, as expected, communication with laser light gives the maximum capacity, we remark that at low level (0 < (n) ~< 1), the second term of (6)
source with exponential time-correlation function in the cases o f random transmittance [curve (1) ] and random pulse position [curve (2)] at low signal level (n) = 2.
normalized such that Cr(°° ) = 1 and Cr(0 ) = 0, versus Pt, the normalized counting time interval, for thermal light with exponential time-correlation function and for the two channel communication models considered here. Comparison of the two curves at the low signal level (n) = 2 for Ft < 1.4 shows that the communication channel as described by eqs. (4) and (5) leads to a better transmission than that described by eqs. (6) and (7). However, for Ft > 1.4 the preceeding conclusion is inversed and we notice that for Ft ~ 40, curve (2) is very close to the limit (C r = 1) while curve (1) is still almost 10% below the limit.
References is, on the contrary, minimum for the laser (see insert fig. 2). For example, it is easy to show from (8) that
Co,~((n)~
0) ~ ¼(n) 2
and C0,th((n) -+ 0)
= lim
(n~-+ 0
l o g ( l + ( n ) ) - - i +(n) (~]~ ~
[1] [2] [3] [4]
R. Jodoin and L. Mandel, J. Opt. Soc. Am. 61 (1971) 191 I. Bar-David, J. Opt. Soc. Am. 63 (1973) 166. G. Bedard, Phys. Rev. 151 (1966) 1038. C. Bendjaballah and F. Perrot, J. Appl. Phys. 44 (1973) 5130.
Appendix
~l(n)2 '
where CO,~ and C0,th are for laser and thermal light, respectively. Notice that eq. (8) becomes important compared to [1 - p(n)(0)] [H(0) - log 2D] when the experimental situation involves low values of T/D. In fig. 3 curves (1) and (2) plotted in semilogarithmic scale show the variation of Cr, the capacity
We briefly summarize here the probability densities o f the various fields considered in this paper. (a) Laser
Q(n/x) = exp (-(n) x) ( (n) x)n /n[
(A1)
(b) Thermal: - long coherence time:
57
Volume 17, number 1
Q(n/x) =
OPTICS COMMUNICATIONS
((n)x)n . {n)x) n+l
(A2)
(1 +
il(z) are the lth order modified spherical Bessel functions o f
Arbitrary coherence time (exponential time-correlation function):
r=0
the first kind. (c) Gaussian-modulated laser field
k-1
k!Q(k/x)= ~
April 1976
t"-~" l~k+r+l(k~.T ¢9( /v/F1.n-r-S-'s=l ()l ~rJ" ~,r,",' (A3)
Q(n/x) -
F(n + ~)(2(n)x) n ~/~r n! (1 +
2On)x)n+1/2
where
F(p) is the factorial function.
(,n)xI't t' l~:l(S) = G(s) e x p ( - l ' t ) \ z ~ ) - ! × "'"
(d) Gaussian-modulated thermal field
×(½Ftil(z(s))+z(s)[l+2~3il
58
l(Z(S)'+z~:~t il-2(z(s) O,
where z = tion.
(A4)
((n)x) -1/2 and Dn(y) is the parabolic cylinder func-
OPTICS COMMUNICATIONS
April 1976
INFORMATION RATES IN OPTICAL CHANNELS Cherif BENDJABALLAH
Laboratoire des Signaux et Systdmes du C.N.R.S., Ecole Sup~rieure d'Electricit~ Plateau du Moulon, 91190-Gif-sur-Yvette,France Received 14 November 1975, Revised manuscript received 22 January 1976
Results of numerical computations of the information capacity in a photon communication channels are given and compared for various optical sources of different statistical properties.
In a discrete typical communication system each transmitted symbol x i has a certain probability P(yj/xi) to be detected as a particular symbol yj, because of random interferences, noise, etc., occurring in the channel during the transmission. The rate of transmission of information R through the channel may be defined as the average of I(xi; yi), the mutual information per symbol pairs, where
P(Xi/Y]) I(xi;Yj) = log P(Xi) '
(1)
that is
R = ~ ~. P(Xi, yj)I(x i, yj). i 1
(2)
According to Shannon's definition, the channel capacity C is the maximum value of R, C = max R.
ring the maximum information than an ideal laser, it may be useful to study the corresponding information rates and to compare them from the point of view of their statistical properties. We first consider a communication channel in which the message is transmitted via a fdter with a random transmittance, x (0 ~
(3)
The evaluation of C for some input density is often a difficult problem. However, in certain cases t h e r e exist some methods o f studying the rate of information and computing its maximum. One such case which is of great practical significance is a channel where a single-mode laser is used as the light source associated with a continuously modulated filter. But because of its low output power, communication with such a source is not always feasible. So although light emitted from thermal and other radiations appears to be less interesting for transfer-
1
f dx Pa(x) a(n/x)
R(a) =~--0 0
(4)
Q(n/x) × log f l dz Pa(Z) a(n/z)' C = max R (a),
(5)
a
where Q(n/x) is the probability density (p.d.) of registering n counts in the time interval t when the transmittance is x with the p.d. Pa(x). The parameter a is used to describe a wide range of different forms of 55
Volume 17, number 1
OPTICS COMMUNICATIONS
06
h-
Loser
April 1976
(n 2) - (n) (n) 2
p.d. so that the optimum p.d. will be included and give the maximum rate. The p.d. Pa(x) for various a (0 < a ~< 20) is taken to be
increases from the laser case to the g.m.th, case (1 ~< h ~< 6) [4], the information capacity decreases and in the whole range of (n) is the greatest for the laser case. Let us now deal with another interesting case of transmission o f information which is the case where the photoelectric detector is regarded as a communication channel whose input symbols are the pulsed optical sources considered above and whose output symbols are the registered counts {n} in a given time interval. A rectangular pulse of time of duration 2D is centered at X - X 0, where X 0 is a constant and X is the random variable taking continuously its values in 0 = ( - T , T) with p.d. p(x). For a pulse of coherent radiation, it has been shown that the channel capacity is [21
Pa(X) - 1 - e xa p ( - a ) exp ( - a x ) .
C = max ([1 - p~n)(0)] [H(0) - log 2D]
C
02
~
~
_
~
Th(Ft=4)
~
ThlFt=0)GmL GmTh
1
2
3
4
5
6
3
8
9
10
Fig. ]. Vaziation o f the maximum information rate R M versus mean number of photoelectrons for various light sources in the case o f random transmittance.
The optical signals considered here are: single mode laser light, thermal light (th.) of arbitrarily normalized coherence time (Ft) with exponential time-correlation function, gaussian-modulated laser (g.m.1.) light and gaussian-modulated thermal (g.m.th.) light. The two last sources are considered to have a long coherence time. For all these sources, Q(n/x) is explicitly known (see Appendix). It can be seen, but not easily proved for any p.d. Q(n/x), that there is a maximum, R M, of R(a) for a -~ 1. This maximum may be seen to be very close to the channel capacity [1] as long as a <~ 3 and (n) lies within the interval (0, 10). In fig. 1 we have plotted the variation o f R M versus (n), where {n) is the mean photoelectron count when x = 1. This variation is, as expected, almost logarithmic for the laser light. For the other kinds of radiation quoted above, the variation is more complicated. We notice that for the g.m.th, light which is the most chaotic in this study, the variation of R versus a does not show a pronounced maximum. However, by calculating the derivatives, it is possible to determine with satisfactory accuracy the maximum and to plot R M versus (n), which for (n) > 4 is almost constant. We also remark that although the bunching effect which can be defined as the ratio 56
a
t (6)
with /4(0) = - f p ( x ) log p(x) dx,
(7)
0
where p(n)(k) is given by relation (A1) for x = 1. Eq. (6) can be extended to other p
H(T) - log 2D = - f
dx a exp (-X/Xc)
-T
X log [a exp (-x/xc) ] - log 2D is not too far from its maximum value log (T/D), provided that D ~ T <~x cFig. 2 shows the variation of the channel capacity, computed from eq. (6) with T/D = 10 3, versus (n) for
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
101 -
(2)
H =~=103 05 C,
001
0.1
10
10
100 Ft
Fig. 3. Variation o f the normalized m a x i m u m i n f o r m a t i o n rate versus normalized counting time interval f o r a thermal
GmTh.
0.5 I
I
I
[
I
10 I
I
1 2 3 /~ 5 6 7 8 9
the different signals studied here. Although, as expected, communication with laser light gives the maximum capacity, we remark that at low level (0 < (n) ~< 1), the second term of (6)
source with exponential time-correlation function in the cases o f random transmittance [curve (1) ] and random pulse position [curve (2)] at low signal level (n) = 2.
normalized such that Cr(°° ) = 1 and Cr(0 ) = 0, versus Pt, the normalized counting time interval, for thermal light with exponential time-correlation function and for the two channel communication models considered here. Comparison of the two curves at the low signal level (n) = 2 for Ft < 1.4 shows that the communication channel as described by eqs. (4) and (5) leads to a better transmission than that described by eqs. (6) and (7). However, for Ft > 1.4 the preceeding conclusion is inversed and we notice that for Ft ~ 40, curve (2) is very close to the limit (C r = 1) while curve (1) is still almost 10% below the limit.
References is, on the contrary, minimum for the laser (see insert fig. 2). For example, it is easy to show from (8) that
Co,~((n)~
0) ~ ¼(n) 2
and C0,th((n) -+ 0)
= lim
(n~-+ 0
l o g ( l + ( n ) ) - - i +(n) (~]~ ~
[1] [2] [3] [4]
R. Jodoin and L. Mandel, J. Opt. Soc. Am. 61 (1971) 191 I. Bar-David, J. Opt. Soc. Am. 63 (1973) 166. G. Bedard, Phys. Rev. 151 (1966) 1038. C. Bendjaballah and F. Perrot, J. Appl. Phys. 44 (1973) 5130.
Appendix
~l(n)2 '
where CO,~ and C0,th are for laser and thermal light, respectively. Notice that eq. (8) becomes important compared to [1 - p(n)(0)] [H(0) - log 2D] when the experimental situation involves low values of T/D. In fig. 3 curves (1) and (2) plotted in semilogarithmic scale show the variation of Cr, the capacity
We briefly summarize here the probability densities o f the various fields considered in this paper. (a) Laser
Q(n/x) = exp (-(n) x) ( (n) x)n /n[
(A1)
(b) Thermal: - long coherence time:
57
Volume 17, number 1
Q(n/x) =
OPTICS COMMUNICATIONS
((n)x)n . {n)x) n+l
(A2)
(1 +
il(z) are the lth order modified spherical Bessel functions o f
Arbitrary coherence time (exponential time-correlation function):
r=0
the first kind. (c) Gaussian-modulated laser field
k-1
k!Q(k/x)= ~
April 1976
t"-~" l~k+r+l(k~.T ¢9( /v/F1.n-r-S-'s=l ()l ~rJ" ~,r,",' (A3)
Q(n/x) -
F(n + ~)(2(n)x) n ~/~r n! (1 +
2On)x)n+1/2
where
F(p) is the factorial function.
(,n)xI't t' l~:l(S) = G(s) e x p ( - l ' t ) \ z ~ ) - ! × "'"
(d) Gaussian-modulated thermal field
×(½Ftil(z(s))+z(s)[l+2~3il
58
l(Z(S)'+z~:~t il-2(z(s) O,
where z = tion.
(A4)
((n)x) -1/2 and Dn(y) is the parabolic cylinder func-