A Performance Analysis of Fiber Optics Communications Systems

Transcription

1 A Performance Analysis of Fiber Optics Communications Systems Jacob Neace, Student and John Darnell, Asst. Professor 1 Abstract This paper presents an analysis of the use and effects of fiber optics as a communications medium. Both benefits and disadvantages are qualified. A modified form of Shannon's theorem is developed. Channel capacity is expressed as a function of bandwidth, and the theoretical limit of channel capacity for shot-noise limited systems is calculated. The limit is calculated for a range of values of system efficiency and incident power. Introduction The advent of the Internet and the World Wide Web, the explosion of local area networks and wide area networks, and the plethora of computer-initiated transactions such as , electronic funds transfer, bar code scanners, point-of-transaction terminals, robotics, distance education, and CD- ROM applications require faster, more reliable, and more secure networks. These requirements necessitate the replacement of copper as a wide-band communications medium. Coaxial cable and twisted wire pairs cannot provide the required capabilities. A medium that provides these capabilities is glass fiber. Fiber optic communications offer the most benefits to the user community and, at the same time, presents the most challenges to network designers. The benefits of fiber optics include: 1. high bandwidth,. rejection of radio frequency interference (RFI) and electromagnetic interference (EMI), 3. increased safety and security, 4. light weight and small size, and 5. increased immunity to corrosion. The high bandwidth, small size, and noise immunity make fiber optics the medium of choice for telephone and cable television companies. Since fiber optic cables are non-current carrying cables, they are safer than copper cables near electrically hazardous places such as power generating plants and sub-stations. Light weight and small size make fiber optics attractive for communications circuits on airplanes and ships. 1 Department of Engineering Technology, 00 STH, Western Kentucky University, 1 Big Red Way, Bowling Green, KY , john.darnell@wku.edu. 1

2 The primary disadvantages of using fiber optics are splicing and the difficulty in installation and maintenance of the connectors that join the fibers. Unlike metallic cables, optical fibers cannot be spliced by soldering or crimping. The fibers must be precisely aligned, and the connectors are relatively expensive. Because of the alignment difficulty, signal losses can be excessive at these junctions. Channel Capacity The channel capacity of a system is widely defined as the maximum number of bits per second that can propagate through the system. Channel capacity is a function of bandwidth and signal-to-noise power and is mathematically expressed as (1) C f log 1+ S N C channel capacity, bits/sec f bandwidth, Hertz S/N ratio of signal-to-noise power Equation (1) is known as Shannon's theorem. Noise is always present on communications systems. The forms of noise that are always present on fiber optic systems are: 1. dark noise,. thermal noise, and 3. shot noise. Dark noise is produced by the reverse current that flows through the receiver's photo-diode. Dark noise is very small compared to either thermal or shot noise and is typically ignored except in the most extreme cases. Thermal noise is produced by randomly moving electrons in the photo-detector's load resistor. Thermal noise can be determined from the equation () N T 4kT f k Boltzmann's constant 1.38 x -3 J/T T temperature, degrees Kelvin.

3 Shot noise is created at the PN junction of the photo-detector diode by incident signal photons from the cable. An equation to determine shot noise is (3) N S ei frs e magnitude of the charge on an electron I average detector current rs dynamic semiconductor resistance If the incident power is very low, the system tends to be thermal-noise limited. However, if the optic power is relatively high, the system tends to be shot-noise limited. Moreover, if the system contains amplifiers, both the input signal and the shot noise are amplified, but thermal noise is unaffected. Therefore, most communications systems are shot-noise limited. Modification of Shannon's Theorem For the shot-noise limited case, the signal-to-noise ratio can be expressed as 1 (4) S N η PI hf f h Plank's constant 6.66 x -34 Js η system efficiency PI incident power to the system. Define β as β ηpi/hf. Channel capacity can then be expressed as β (5) C f log 1+ f Because β is constant with respect to bandwidth, the maximum channel capacity as a function of bandwidth can be analytically determined from calculus by setting the derivative of C with respect to bandwidth equal to zero and solving for f. Using the differentiation identity for logarithms, the chain rule, and changing the base from to, the first derivative of C with respect to bandwidth is (6) dc d ) f + β ) log f X + β ) log d f + β d ( f) f f + β log f + log 3

4 The quotient rule for derivatives produces the final form of dc/d( f). (7) dc log d ) f + β 1 f log β + β ) log This function is a transcendental equation; a closed form expression for the bandwidth that maximizes channel capacity cannot be determined. By setting equation (7) equal to zero and rearranging the terms in the form f + β β (8) log f + β ) the bandwidth that maximizes C can be found by trial and error. Table one lists these values and the resulting channel capacity as β is allowed to range from small to very large values. The second derivative of C with respect to bandwidth is C K K (9) d + d ) f + β ) + β ) where K is a constant. For real values of β and f, the denominator of the positive term will always be greater then the denominator of the negative term. Therefore, the second derivative will always be negative, proving that the solutions to (8) represent maximum values. Results As previously stated, β is allowed to vary from very small values to very large values. This range of β is for academic completeness. TABLE ONE β f(hertz) C(bits/second) x x x x x x x x x 18 4

5 Conclusions The maximum capacity of fiber optic networks as a function of bandwidth is examined. The analysis is based on two assumptions: 1. The system is shot noise limited, and. β, as defined in the paper, is a large number. Both of these assumptions are typical of real systems. High incident power and amplification make shot noise the limiting factor. Planck's constant is the dominant quantity in determining β. The reliability of the calculations is of interest. The solutions of the transcendental equations have a margin of error of +.003% or less. Also, 14 appears to be a limiting value for calculating bandwidth. This limit was reached by calculating log(1+β/ f); as β/ f became very small ( -14 ), both calculators and PCs produced a value of zero for the logarithm. The calculated maximum values of bandwidth appear to be consistent with current physically realizable bandwidths (<300Mbps)on fiber optic systems. The authors concur that the theoretical calculations may vary if more accurate calculators are used. REFERENCES 1. Fiber Optic Communications. Joseph Palais, Fourth Edition, Prentice Hall publisher. page 85. Discovering Computers. Shelly Cashman Series, page An Introduction to Fiber Optics. R. Allen Shotwell, Prentice Hall publisher. 4. Optical Fiber Communications. Gerd Keiser, Second edition, McGraw Hill publisher. 5

6 JACOB NEACE Jacob Neace is a senior engineering technology student at Western Kentucky University. His primary interests are in manufacturing and communication. JOHN H. DARNELL John Darnell received a BEE from Georgia Tech and a MEE from UAB. He is retired from BellSouth, where he worked on network related issues. He has taught at UAB and the University of Alabama, and is currently on the faculty at Western Kentucky University. 6