1

Channel Capacity of Optical Fiber Communications

TABLE OF CONTENTS

1 Introduction.................................................................................................3

2 Channel Capacity …………………………… …………...............................4

2.1 Linear Regime with coherent detection………………......................5

2.2 Linear Regime with constant intensity and coherent detection........6

2.3 Linear Regime with direct detection…………...................................7

3 Nonlinear Regime…………………………… …………................................8

4 Dispersion limitations on 2-DPSK and 4-DPSK systems…………............14

5 BER and SER for some modulation schemes…………............................18

5.1 Relationship between Symbol Error Rate and Bit Error Rate…….23

6 OSNR measurements in a WDM system..................................................26

6.1 Receiver Sensitivity Penalty (RSP)………………………………….28

6.2 Eye Opening Penalty (EOP)………………………………………….30

7 References................................................................................................33

2

Introduction

Channel capacity can be calculated theoretically without considering any specific

modulation scheme. The theoretical capacity does not generally tell us which

specific modulation technique we should use in order to achieve the capacity. In

wavelength division multiplexing (WDM) systems the spectral efficiency limit is

obtained from the relation between the capacity per channel and the channel

spacing. The spectral efficiency depends on

The modulation technique used: unconstrained, constant intensity and

binary.

Detection technique: coherent and direct.

Propagation regime: linear (without fiber nonlinearities) and nonlinear.

There are two regimes where the spectral efficiency can be obtained analytically.

The first one does not consider chromatic dispersion or shelf-phase modulation,

cross-phase modulation, four-wave mixing, polarization mode dispersion; it is

called the linear regime. The nonlinear regime considers all these effects.

3

Channel Capacity

The spectral efficiency is expressed as a relation and is written as

f

CC

max (15)

where C is the capacity per channel and f is the maximum bandwidth if a

channel, i.e., the channel spacing.

The capacity is principally determined by the bandwidth and the encoding

technique. The bandwidth in the optical fiber is limited by the intrinsic loss of light

propagating through silica. The maximum bandwidth is considered to be 50 THz

(1.2 m – 1.6 m), because of the spacing of standard frequency grids.

Actually, the optical communication systems have a bandwidth limited to about

15 THz.

Now, the question is how the nonlinearities affect the spectral efficiency of the

fiber? So we proceed to study the principal nonlinearities in the fiber and their

effects on spectral efficiency.

Now, the injected ASE noise at the output of the final amplifier has a power

spectral density given by

hvnGNS spAeq )1( (16)

4

where AN the number of fiber spans, G is the amplifier gain, spn is the

spontaneous emission noise factor of one amplifier, h is Planck’s constant and v

is the optical frequency. The ASE has a total power given by

hvBnGNP spAn )1( (17)

where B is the bandwidth and inP is the input power.

Linear Regime with coherent detection

The classic theory of information developed by Shannon sets limits on the

efficiency of communication channel. The capacity of a communication channel

is the maximal rate at which information can be transferred through the channel

without error. It takes into account noise and limited power and the signal is

modeled as a complex-valued electric field. The capacity can be written as

N

SWC 1log 2 (18)

where W is the spectral bandwidth, n

t

P

P

N

SSNR is the signal power to noise

power ratio. This equation considers a channel with AWGN (additive white

Gaussian noise) with a limited power. The noise (N) is a contribution of ASE

(amplified spontaneous emission) in the system. We assumed that ASE noise is

the principal noise source.

The spectral efficiency limit which we denote maxC can be written as

5

N

S

f

BC 1log 2max (19)

Linear Regime with constant intensity and coherent detection

Some modulations techniques as differential phase shift keying (DPSK) encode

the data using a constant intensity. The receiver recovers the information using

an interferometric detector. The capacity with coherent detection and constant

intensity modulation was developed by J. M. Khan and K. P. Ho. The optimal

transmitted electric field is uniformly distributed on a circle. The capacity at

arbitrary SNR can be written as

0

22 2log)(log)(2 nePdrrfrrfBC (20)

where

n

s

n

s

P

PrI

P

rP

n

eP

rf0

2

2

2

1)(

(21)

At high SNR, the capacity can be a approximated to

10.1log

2

12

n

t

P

PBC (22)

The spectral efficiency is

6

10.1log

2

12max

n

t

P

P

f

BC (23)

This equation has 1.1 b/s/Hz more than half the Shannon limit.

Linear Regime with direct detection

Mecozzi and Shtaif developed the channel capacity in the case of direct

detection considering the high SNR limit. To date, spectral efficiency limits have

not been derived for arbitrary SNR. In this case, the transmitted optical signal is

modeled as a non-negative, real electric field magnitude. The dominant noise is

signal-spontaneous beat noise, which is additive and dependent of the signal.

The capacity is given by

00.1log

2

12

n

t

P

PBC (24)

and the spectral efficiency is given by

0.1log

2

12max

n

t

P

P

f

BC (25)

7

This limit is 1.0 b/s/Hz less than half the Shannon limit.

The figure 52 shows a comparison among the different spectral efficiencies in the

linear regime.

Figure 52: Spectral efficiency limits in a linear regime.

Nonlinear Regime

In typical DWDM systems, the dominant nonlinear impairments arise fro the Kerr

effect. The most important nonlinear effect is the refractive index,

Innn 20 (26)

8

where 0n is the linear refractive index and 2n is a constant.

Spectral efficiency limits in the nonlinear regime have been derived only for

coherent detection

If we consider cross-phase modulation (XPM) as the principal source of

nonlinearities in a dense wavelength division multiplexing (DWDM) system,

where each signal in the fiber perturbs the neighbor channels, then the nonlinear

power for XPM was found to be

effc L

nIn

BDI

22 2

0

(27)

where D is the dispersion, is the distance between two channels, is the

fiber nonlinear coefficient, cn is the number of WDM channels,

Aeff

NL is the

effective length, is the fiber attenuation coefficient.

The maximum spectral efficiency is lower-bounded by

IeP

Ie

f

BC

I

P

n

I

P

LBt

t

2

0

0

1

1log

2

2 (28)

9

This equation requires coherent optical detection, which requires a reference

signal with an identical or very close frequency to the transmitted signal and was

calculated by Mitra and Stark.

As an example, we consider a system with the following parameters:

20cn channels; AN =100 fiber spans;

= 0.2 dB/Km;

Aeff

NL =100 Km;

D = 20 ps/nm/km; Bf 5.1

B = 10 GHz;

= 1 11 KmW , 0.1 11 KmW and 0 (without nonlinearities)

The following shows the figure for these 3 cases

Figure 53: Spectral efficiencies for several values of fiber nonlinear coefficient.

10

The figure shows that the spectral efficiency will decline after the system reach a

maximum capacity given by

t

cP

IBnC 0

2max

2log

3

2 (29)

so, the fiber spectral efficiency is limited by the noise in the optical amplifiers

(ASE) and the nonlinear properties of the fiber. Also the spectral efficiency limit

increases with chromatic dispersion and with channel spacing, because walk-off

decreases the impact of XPM.

The spectral efficiency calculated by Mecozzi in a linear regime using intensity

detection is roughly 1 b/s/Hz less than half that with coherent detection.

C calculated by Mitra and Stark in

a linear regime using coherent

detection

C calculated by Mitra and Stark in

a linear regime using coherent

detection

4 bit 11 Hzs 1 bit 11 Hzs

6 bit 11 Hzs 2 bit 11 Hzs

Many current DWDM systems use on-off keying (OOK) as a binary modulation

technique but the spectral efficiency can not exceed 1 bit/s/Hz ( 0.67 b/s/Hz in a

real system).

Mitra and Stark’s work suggest that the spectral efficiency limits would require a

non-binary encoding technique, such as a multilevel intensity or phase

modulation. It is known that a phase modulation can mitigate the XPM effect.

In conclusion:

11

Systems with constant intensity modulation are less efficient (with respect

to spectral efficiency) that unconstrained modulation.

Systems with direct detection have a significant loss of spectral efficiency.

Systems with coherent detection is the best suited to approximate to the

limits the spectral efficiency.

It was shown that an increase of spectral efficiency to 0.8 bit/s/Hz in 40 Gb/s

based DWDM systems results in increased pulse distortions, because of the

reduced tolerance to implemented narrow-band filtering and larger impact of

multi-channel nonlinearities (particularly XPM). The differences between RZ- and

NRZ based modulation formats vanish in DWDM transmissions, because of the

distortion of RZ pulse shape due to narrow-band filtering needed at the

transmitter side.

It was shown that transmission performance of DWDM systems could profit from

implementation of transmission fibers with a large chromatic dispersion, due to

suppression of multi-channel effects independently of the modulation format in

use. Accordingly, already deployed fibers (e.g. G.652) can be further used in next

generation of DWDM transmission systems.

Furthermore, considering concatenation of identical spans in a DWDM

transmission line, it was observed that XPM-induced impacts superpose

constructively from span to span independently of the implemented dispersion

compensation scheme, resulting in an transmission penalty, which is in high

power regime proportional to number of concatenated spans. This behavior

enables together with already know transmission rules (e.g. Pmax) an efficient

estimation of the maximum transmission performance and maximum

transmission distance in 40 Gb/s DWDM systems.

In an overview of transmission experiments with channel data rates exceeding

2.5Gb/s (Figs. 1.4 and 1.5), two dominant trends can be observed in order to

12

meet the bandwidth demands. First, there is a trend for increasing the overall

system capacity. Summarizing the most important works with channel data rates

of 10, 40 and 160Gb/s regarding the achievable system capacity from the year

1994 up to now.

This tendency can be explained by a reduced complexity in realization and

availability of required 40Gb/s system components e.g. narrow-band filters on the

one hand, and compatibility of 40 Gb/s data rates with existing transmission

infrastructure on the other hand. The increased system capacity in 40Gb/s

experiments is enabled by implementation of some new sophisticated

technologies, e.g. novel modulation formats, new fiber and amplifier types, which

are becoming mature, because of the possibility to realize them using only

electronics or some hybride (electro-optics) solutions, which seems to be

impossible at increased channel data rates e.g. 160Gb/s, due to the fact that 40

Gb/s technology represents the limit for electronics.

Another network evolution trend, which is supported by the need for a better

utilization of the existing system bandwidth, is the enhancement of the system’s

spectral efficiency. The achieved maximum efficiency values in 10Gb/s based

transmissions are not larger than 0.4 bit/s/Hz, which is primary caused by the fact

that narrow-band filters for ultra dense channel spacings (25 GHz or lower) are

not available at the moment. This fact lets the higher channel data rates appear

more convenient for achieving an increased spectral efficiency. Here again, the

40Gb/s based systems indicate the best choice characteristics, which can be

13

explained by the signal generation of 40Gb/s signals (compared to 160Gb/s) and

an easier realization of performance enabling technologies at 40Gb/s, thus

recommending the 40 Gb/s channel data rate as a candidate for the realization of

next generation

WDM and dense WDM (DWDM) optical transmission systems. At the same time,

the implementation of higher channel data rates larger than 10 Gb/s bears issues

e.g. reduced tolerance to fiber dispersion, to polarization mode dispersion (PMD)

and to noise disturbances in the transmission line, whose suppression requires

unconventional technologies and methods for achievement of improved

transmission characteristics, making the high bit rate based transmission

becoming a long term issue with long implementation times.

Dispersion limitations on 2-DPSK and 4-DPSK systems

A Great number of optical communication systems use binary OOK but actually

Metro communications services with high capacity show problems caused by

fiber impairments such as Polarization Mode Dispersion (PMD) and Chromatic

Dispersion (CD) arise the Inter symbol interference (ISI), in other words, produce

a signal distortion at the end of the receiver, degrading the Bit error rate (BER).

It has been proven that multilevel signaling modulation that narrow the optical

spectrum can reduce CD. Also multilevel signaling modulation increase the

symbol duration so more uncompensated CD and PMD can be allowed before

the ISI becomes significant.

Spectral efficiencies than exceed 1 bit/s/Hz are achievable in WDM systems.

However, the use of binary modulation, such as RZ and NRZ, limits the spectral

efficiency to a maximum of 1 b/s/Hz (realistic systems can afford less 1 b/s/Hz).

Systems that require approaching the spectral efficiency limit will require non

binary modulation schemes, such as QPSK (quaternary phase shift keying) and

14

QDPSK (quaternary differential phase shift keying); both of them are multilevel

phase modulation where the information is encoded in optical signals having

constant intensity.

A limitation in these non-binary modulation schemes is the additional OSNR

required to maintain the required BER.

BER and SER for some modulation shemes

The maximum possible data rate R=C is obtained from:

00 N

E

CN

S b (30)

Then the maximum possible data rate is given by

W

C

N

E

W

C b

0

2 1log (31)

Thus the required SNR is obtained from

)12(

12

0

0

WC

b

bWC

C

W

N

E

W

C

N

E

(32)

15

where, C/W is the spectral efficiency in the modulated scheme.

About the question

"How does the bit error rate that you find from your simulations compare with the

Shannon limit?"

Answer: The Shannon limit serve as upper limits on the transmission rate for

reliable communication over a noisy channel. For example, for multilevel PSK

(MPSK), the spectral efficiency increases when M increases, but also at the cost

of required SNR (o

b

N

E), so there is a trade off in every modulation scheme. In

other words, in bandwidth limited systems, spectral efficiency modulation can be

used at the expense of power and in power limited systems; power efficient

modulation can be used at the expense of bandwidth.

If we want to know the power efficiency, then we need to know the BER of every

modulation scheme. Following are the BER (bit error rate) for binary modulation

schemes and SER (symbol error rate) for a multilevel modulation scheme also

the respective bandwidth efficiencies.

We assume that we have a set of M symbols to send:

)(),......(),(s),( 321 tststts m

each of which has a symbol duration of sT

These symbols are formed by modifying the amplitude, frequency or phase of the

carrier. The basic types of modulation are:

1.- Amplitude shift keying – ASK

16

2.- Frequency shift keying – FSK

3,- Phase shift keying – PSK

For a binary modulation we have: BASK, BFSK, BPSK.

Another common modulation scheme includes 4 symbols, so we have: QASK,

QFSK and QPSK.

When we have a system with a number of symbols greater than 4 (M>4) is called

multilevel modulation so we have MASK, MFSK and MPSK.

In a binary system, we have only two symbols so there is no difference between

BER and SER. There is also no difference between Energy symbol sE and

Energy bit bE . The problem happened when we have M (M>2) symbols in the

system.

RELATIONSHIP BETWEEN SYMBOL ERROR RATE AND BIT ERROR RATE

If there are M symbols in the system, then there are 2log M bits encoded en each

symbol. The relation between the number of bits and symbols are:

kM

MK

2

log2

(33)

Now, if we are told that there is an error on the symbol that we receive, then we

know that it could be any of the M-1 symbols that we choose by mistake. Then

the probability of getting a particular symbol when we have an error is:

1M

Pe (34)

17

Now, the question is, how many of these will have n bits in error out of the k

possible bits. The solution is s combination:

)!(!

!

nkn

k

n

k

(35)

thus the probability of receiving a symbol with n bits in error is:

1

M

P

n

ke (36)

The average number of bits in error per symbol is:

k

n

e

M

P

n

kn

1 1 (37)

Now, the probability that a bit is in error is:

k

n

eb

M

P

n

kn

kP

1 1

1 (38)

(it is divided by 1/k because the symbols has k bits)

after some mathematic manipulation, the probability that a bit is in error is:

ek

k

b PP12

2 1

or SERBERk

k

12

2 1

(39)

The BER depends on the energy per symbol Es and the noise power spectral

density ( )oN , so the signal power is given by:

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ssRES (40)

where sR = symbol rate = 0

1

T To is the symbol duration

The signal power in function of energy bit ( bE ) is given by:

bbRES bR bit rate

There are k bits per symbol, then there are k

T0 seconds per bit and therefore

0T

kRb = Rsk (41)

The bandwidth efficiency describes the ability of a modulation scheme to

accommodate data within a limited bandwidth and is given by

B

RB b/s/Hz (42)

The power efficiency (0N

Eb ) is related to the Shannon channel capacity theorem,

where de maximum possible bandwidth efficiency is limited by the noise in the

channel.

)1(log2maxN

S

B

CB (43)

19

The table 1 shows a resume of SER, bandwidth efficiency B and the BER is

obtained from the equation that relates the BER from SER for several modulation

formats.

Modulation

scheme

SER BER B

(b/s/Hz)

Binary

OOK –

ASK

(On-Off

keying)

0N

EQ b

0N

EQ b

2

1

FSK – non

coherent

detection

02

2

1 N

Eb

e

02

2

1 N

Eb

e

1

FSK –

coherent

detection

0N

EQ b

0N

EQ b

2

PSK –

coherent

detection

0

2

N

EQ b

0

2

N

EQ b

0.5

DPSK –

non

coherent

detection

0

2

1 N

Eb

e

0

2

1 N

Eb

e

0.5

M-PAM

0

2 )1(

6(

)1(2

NM

EQ

M

M m

s

)

M

i

s

m

s iE

ME

1

1

00

2

2

2 )1(

log6(

log

)1(2

N

E

NM

MQ

MM

M b

)

2

log2 M

20

QAM

0

2

2

)1(

log3()

11(4

N

E

M

MQ

M

b

2

log2 M

MFSK -

coherent

detection

0

2log()1(

N

MEQM b

3

log2 2

M

M

MFSK –

non

coherent

detection

0

2

2

log

2

1 N

MEb

eM

M

M2log2

MPSK

MN

MEQ b

sinlog2

(20

2

2

log2 M

Transformations can be done to get BER from SER on multilevel signaling.

OSNR measurements in a WDM system

The use of OSNR (Optical signal noise rate) as a signal quality measure is

pointed out that it does not account for signal distortion effects. Today the optical

systems use digital transmission, so the system quality is specified in terms of

BER. Many systems requires longer transmission distances so the systems

require the use of optical amplifiers (OAs). The system requires monitoring of the

optical signal quality throughout the complete transmission route.

The best quality measure is done with the BER, but it is not possible to use

directly in the optical domain. The procedure is to take out part of the signal and

use a demux, detector and finally regenerate this one electrically. All this

requirement is expensive and the advantage of avoiding electro-optical

regeneration in the optical path is partly lost.

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The OSNR does not take into account the distortions on the signal that is caused

by:

Non ideal modulation properties of the laser

The effect of laser phase noise

Fiber dispersion (chromatic and polarization mode dispersion)

Fiber non-linearities

Thus, the OSNR can not give information of errors caused by signal distortions.

Hower, it may still be possible to design WDM systems in such a way that

distortions effects are not significant.

The combination of high data rate and narrow channel spacing limit OSNR as a

measure. To obtain a more definitive measure of signal quality at multi-channel

interface, it is necessary to filter each channel and perform optical eye-diagram

or BER measurements.

The relationship of Q-factor to BER is given by

BER=

22

1 Qerfc (44)

The logarithmic value of Q (in dB) is related to the OSNR is given by

c

dBB

BOSNRQ 0log20 (45)

where 0B is the optical bandwidth of the end device (photo detector) and cB is the

electrical bandwidth of the receiver filter.

22

Some practical cases where dominant source of noise is signal spontaneous

noise, the BER is given by

enBPG

GPQBER

)1(2 (46)

If a series of optical amplifiers are used, the BER is given by:

OSNR

OSNRB

B

QBER e

411

2 0

(47)

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1986.

[2] Green, Paul E., “The Future of Fiber-Optic Computer Networks”, IEEE

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February 1998.

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23

[5] Cantrell, C. D. and Dawn Hollenbeck, “Optical Impairments in Transparent

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[9] Walkin, s. and J. Conradi, “Multilevel Signaling for Increasing the Reach of

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[10] Yonega, K. et al., “Optical Duobinary Transmission System with No

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[11] Wuth T., W. Kaiser, and W. Rosenkranz, “Impact of Self-Phase

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[12] Zaid, M. A., ”Envelope Detection and Correlation of SSB, “Electronics

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