187 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 3, AUGUST 1990

Estimate of Channel Capacity in Rayleigh Fading Environment

Abstract- The channel capacity of Gaussian noise environment was solved by Shannon in 1949. it provides an upper bound of maximum transmission rate in a given Gaussian noise environment. In this paper the channel capacity in a Rayleigh fading environment has been derived. The result shows that the channel capacity in a Rayleigh fading envi- ronment is always lower than that in a Gaussian noise environment. When operating a digital transmission in a mobile radio environment that has Rayleigh fading statistics, it is very important to know what the degradations are in channel capacity due to Rayleigh fading, and also to what degree the diversity schemes can bring the channel capacity up in a Rayleigh fading environment. The curves are generated to show the degradation of channel capacity in a Rayleigh fading environment and its improvement by diversity schemes.

L

" ll N I. INTRODUCTION

1948, Shannon's Mathematical Theory of Communica- I" tion [ 11, [2] perceived that approaching 1) error-free digital communication on noisy channels and 2) maximum efficiency conversion of analog signal-to-digital form, were dual facets of the same problem. In a Gaussian noise environment the channel capacity of a white bandlimited Gaussian channel can be expressed as [3], [4]

C =Blog , ( I +y) b/s (1) where B is the channel bandwidth and y is the carrier-to-noise ratio as y = C / N , C is the RF carrier power, and N is the Gaussian noise within the channel bandwidth. Equation (1) is called the Channon-Hartley theorem; it is for a continuous channel. First, it tells us the absolute best that the system can provide with given channel parameters, C/N and B . Secondly, with a specified information rate, the power and bandwidth are inversely related to each other. Thirdly, the Shannon-Hartley theorem indicates that a noiseless Gaussian channel has an infinite capacity when CIN approaches infinity. However, the channel capacity does not become infinite when the bandwidth becomes infinite, as seen in (1). This is because the noise power increases with the increase of bandwidth. Let N = NOB where NO is the noise power per Hertz, then ( 1) becomes

lim C = (log, 2)- C = I .U - C E - 0 0 NO NO

Then the upper bound bit error rate of a system with an un-

Manuscript received December 20, 1987; revised December 1, 1988. This paper was presented at the 38th Annual IEEE Vehicular Technology Society Conference, Philadelphia, PA, June 15-17, 1988.

The author is with Pactel Cellular, Inc., 4340 Von Karman Avenue, New- port Beach, CA 92660.

IEEE Log Number 90371 10.

Fig. 1. ( C / N ) in Rayleigh fading environment.

limited bandwidth can be derived with the information of (2) PI.

11. IN A RAYLEIGH FADING ENVIRONMENT [6] The channel capacity in Rayleigh fading has to be calculated

in an average sense. The reason is that the y (=UN varies in time, due to Rayleigh fading as shown in Fig. 1, N is the average noise power over the Gaussian noise. N can also be treated as an average resulting from multiple interference sources which approach a Gaussian-like noise. In a real mobile radio environment, this is the case.

The average channel capacity then, can indicate the average best over the fading environment. It follows the same concept as to obtain the average bit error rate in the Rayleigh fading environment.

Now we would like to find a equation equivalent to (l), but in a Rayleigh fading environment. The carrier-to-noise ratio will no longer be a constant, but a variable following the Rayleigh fading statistics. A maximum value of the channel capacity in this case then can be obtained, but in an average sense.

The probability density function of a Rayleigh variable is [7, p. 3991

(3)

where r is the average power of y, r = (y) = ( C ) / N . We are applying the same technique of obtaining the average bit error rates in a Rayleigh fading environment [7, p. 4071 to find the average channel capacity in the same environment.

0018-9545/90/0800-0187$01 .00 0 1990 IEEE

188 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 39, NO. 3, AUGUST 19'343

Here the average channel capacity is

(4) 1

0 Y (C) = /00 B log, ( 1 + y) . - e-Yir dy.

Equation (4) can be solved [8, p. 5741 as

(C) = -B . log, e . . Ei

where Ei(x) is the exponential-integral function and can be expressed in two different forms [8, p. 9271

00 E;(-x) = E + l n ( x ) + x - ( -Nk

k . k ! k = l

where x > 0, E is Euler constant (E = 0.5772157), and R , is the remainder. substituting (6) into (5) yields

In the case of r > 2, (8) becomes

Equations (1) and (9) are plotted in Fig. 2. The channel capac- ity in a Rayleigh fading environment is reduced 32% at r = 10 dB, and reduced only 11% at r = 25 dB, as expected. A . Infinite Bandwidth Case

the system becomes unlimited, we have to choose (7) in (5) To find the average channel capacity when the bandwidth of

(C) = -B . log, e . [r - r2 + 2! . r3 - 3! . r"' + . . .I. (10) Since J? = ( C ) / N = (C) /NoB, then

lim (C) B-02

Equation (1 1) indicates that when the bandwidth approaches infinity the average channel capacity in the Rayleigh fading environment is the same as shown in (2) for a nonfading en- vironment, i.e., the average channel capacity is finite even though the bandwidth approaches infinity. The average chan- nel capacity is equal to the average (C)/No times a factor of 1.44.

1 -[CHANNEL CAPACITY C OR

LEE: CHANNEL CAPACITY IN RAYLEIGH FADING ENVIRONMENT 189

A . The Extreme Case

(13) as The cumulative probability distribution can be found from

(Y )

P(Y 5 (7)) = 1 P ( Y ) d Y . (15) The extreme case is when M --+ 00; then (15) becomes

and

The probability density function obtained from (15) as M - + m i s

It is indirect proof that (13) becomes a delta function 6(y) when M + 00. Equation (14) then becomes

(6) = B/m log,(l + Y P ( Y = (7)) dr 0

= Blog,( l +y). (19)

Equation (19) is the same as (l), i.e., when M + 02

(C) + C as (7) + y. (20)

We realize that the channel capacity in the Rayleigh fading environment for M = 4 and greater is very close to channel capacity in the Gaussian noise environment.

IV. CONCLUSION Although the average channel capacity is not an absolute

maximum value in the Rayleigh fading environment, it intro- duces other valuable information for the continuous channel system with finite bandwidth. By comparing the actual trans- mission rate with the average channel capacity obtained from Fig. 1, we get a feel for how good the system has been de- signed and how far the actual average channel capacity value will reach. Several points can be summarized as follows.

The channel capacity in a Rayleigh fading environment is in an average sense. The channel capacity in a Rayleigh fading environment is always lower than that of a Gaussian noise environment. The diversity scheme can bring the channel capacity up in a Rayleigh fading environment.

The average channel capacity over Rayleigh fading equals the channel capacity over Gaussian noise when the bandwidth approaches infinite. When the number of diversity branches M approaches infinite, the channel capacity of an M-branch signal in a Rayleigh fading environment approaches the channel capacity in a Gaussian noise environment.

REFERENCES C. E. Shannon and W. Weaver, The Mathematical Theory of Com- munication. Urbana, IL: Univ. Illinois Press, 1949. A. J . Viterbi and J. K. Omura, Principles of Digital Communication and Coding. New York: McGraw-Hill, 1979, ch. 1. H. Taub and D. L. Schilling, Principles of Communication Systems. New york: McGraw-Hill, 1971, p. 421. A. B. Carlson, Communication Systems. New York: McGraw-Hill, 1975, p. 356. R. S. Kennedy, Fading Dispersive Communication Channels. New York: Wiley, 1969, p. 109. W. C. Y. Lee, Mobile Communications Design Fundamentals. New York: Howard W. Sams Co., 1986, ch. 1 . M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. Gradshteyn and Ryzhik, Table of Integrals, Series and Products. W. C. Y. Lee, Mobile Communications Engineerng. McGraw-Hill, 1982, p. 310.

New York:

William C. Y. Lee (M64-SM80-F82) received the B.Sc. degree from the Chinese Naval Academy, Taiwan, and the M.S. and Ph.D. degrees from The Ohio State University, Columbus, in 1954, 1960, and 1963, respectively.

From 1959 to 1963 he was a Research Assistant at the Electroscience Laboratory, The Ohio State University. He was associated with Bell Laborato- ries from 1964 to 1979 where he was concerned with the study of wave propagation and systems, millimeter and optical wave propagation, switching

systems, and satellite communications. He developed a UHF propagation model for use in planning the Bell Systems new Advanced Mobile Phone Service and was a pioneer in mobile radio communication studies. He applied the field component diversity scheme over mobile radio communication links. While working in satellite communications, he discovered a method of calculating the rain rate statistics which would affect the signal attenuation at 10 GHz and above. He successfully designed a 4 x 4 element printed circuit antenna for tryout use. He studied and set a 3-mm-wave link between the Empire State Building and Pan American Building in New York City, experimentally using the newly developed IMPATT diode. He also studied the scanning spot beam concept for satellite communication using the adaptive array scheme. From April 1979 until April 1985 he worked for ITT Defense Communications Division and was involved with advanced programs for wiring military communications systems. He developed several simulation programs for the multipath fading medium and applied them to ground mobile communication systems. In 1982 he was Manager of the Advanced Develop- ment Department, responsible for the pursuit of new technologies for future communication systems. He developed an artificial intelligence application in the networking area and filed a patent application before leaving ITT. He joined PacTel Mobile Companies in 1985, where he is engaged in the improvement of system performance and capacity.