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Performance of DS/CDMA and BFSK Signals in Rayleigh Fading Channels

By

D. Brian Epstein

And

Robert Taylor

06 Dec 99

University of Texas at Dallas

EE6390 Introduction to Wireless Communications

Supervising Professor: Dr. Murat Torlak

ABSTRACT

This paper utilizes the Clark and Gans Fading Model to implement a Rayleigh Fading Simulator using MATLAB. The Rayleigh fading channel is tested using BFSK and DS/CDMA. BFSK is a frequency shift keying type of modulation and DS/CDMA is an example of spread spectrum modulation. Spread spectrum modulation uses a pseudo-noise sequence to spread the signal across a wide bandwidth. Cross-correlating this signal with a locally-generated version of the pseudo-noise sequence de-spreads the signal which restores the modulated message to the same narrow band as the original. The results of this simulation are analyzed using the using bit error rate (BER) to compare their performance. Other performance factors are the number of level crossings and the average fade duration.

INTRODUCTION

Cellular wireless systems experience loss of signal strength due to:

1) Doppler shift in mobile environments, and

2) Scattering due to reflections from natural and manmade obstructions.

R. H. Clarke modeled the mobile channel as a Rayleigh fading channel. Later, M. J. Gans deduced a spectral model from Clarke’s original analysis. John I. Smith simulated the Clarke and Gans model on a computer using the algorithm described below.

DESCRIPTION

In his model [1], R. H. Clarke considers a non-direct line of path between transmitter and receiver in a mobile-radio environment. The signal is reflected and scattered due to obstructions caused by man-made and natural structures. He considers scattered signals of the originally transmitted base station signal to be the only signals received at the mobile end, even though the original signal would be independent of whether it is being transmitted or received by the mobile unit (also referred to as the ‘reciprocal principle’). Clarke also considers the Doppler effect on the wave propagation due to the motion of the mobile unit.

Clarke’s considerations for his model are that, “… at any point, the received field is made up of a number of generally horizontally traveling free-space plane waves whose azimuthal angles of arrival occur at random for different positions of the receiver, and whose phases are completely random such that the phase is rectangularly distributed throughout 0 to 2(. The phase and angle of arrival of each component wave will be assumed to be statistically independent.” Also, he assumes that at any point there are a particular number, N, of carrier waves having the same average amplitude. His model describes the carrier waves arriving at the mobile whose phases are assumed to be Gaussian random variables. The angle of arrival is assumed to be uniformly distributed on the interval (0, 2(]. Because the mobile is in motion with velocity, v, each angle, (, will be associated with the Doppler shift in carrier frequency, i.e.,

(1) [pic]

where,

(2) [pic]

and ( is the carrier wavelength.

According to Clarke, the E-field can be expressed as an in-phase and quadrature component:

(3) [pic]

where,

(4) [pic]

and,

(5) [pic].

Both Tc and Ts are Gaussian random processes. They are uncorrelated zero mean Gaussian random variables with equal variance of:

6) [pic].

The noise spectra is shown in Figure 1.

Figure 1

The E-field is then given by

7) [pic]

where r(t) is Rayleigh distributed, i.e., (Figure 1)

(8) [pic]

Figure 2

For a quarter wavelength antenna and p(() uniform over (0, 2(], the Doppler output spectrum is given by [1] as:

(9) [pic]

where p(() is a fraction of the total incoming power, as shown in Figure 3.

Figure 3

There are two types of fading which are dependant upon the relationship between Coherence Time and Symbol Period.

1) Fast Scale Fading has a high Doppler spread where the coherence time (Tc) is less than the symbol period (Ts). It causes frequency dispersion (time selective fading) since the channel variations are faster than the baseband signal variations. This usually only occurs for very low data rates.

2) Slow Scale Fading has a low Doppler spread where the coherence time (Tc) is greater than the symbol period (Ts). The channel variations are slower than the baseband signal variations.

John I. Smith [2] simulated the model on a computer with the following described algorithm. He used a random number generator to produce two independent Gaussian noise baseband line spectrum with a maximum frequency of fm, the Doppler shifted frequency. There were for positive frequencies. The negative frequency components were constructed by conjugating the positive frequency components. This signal is a purely real Gaussian random process which is used for in-phase and quadrature components, one in each branch of the simulator. The random spectrum is multiplied by the discrete frequency representation [pic]. Smith truncated the edges where [pic]approaches infinity and the slope was extended to the edge.

The IFFT is then taken in each branch and the quadrature branch is created by shifting the phase –90( with the Hilbert transform. The absolute value is then squared. The square root is taken to produce the channel response. The diagram showing the overall method is shown in Figure 4.

Figure 4

Any signal may now be tested in this channel simply by convolution with the channel response, i.e.,

(10) [pic]

where s(t) is the signal through the channel, ( is the Rayleigh random variable produced by the above steps, and ( is the random phase of incidence.

RESULTS

The Rayleigh fading channel was simulated in Matlab by writing an m-file function which was then called for both DS/CDMA and BFSK signal simulation. DS/CDMA was written into an m-file which used BPSK modulation with a 32 bit pseudo-random code for spreading. An n-file was also written for BFSK modulation and a Matlab pre-written function was used.

Figure 5

Figure 5 shows the data and demodulated signal for DS/CDMA. Some errors can be observed in the demodulated waveform. After demodulation, the following was obtained for DS/CDMA:

- BER = 0.0234 = 2.3%

- Number of Errors = 3

- Data Stream = 128 Bits

- Velocity of Mobile = 100 kph (62 mph)

- fm = 83.3 Hz

- fc = 900 MHz

Figure 6 shows the data and demodulated signal (ydemod) for BFSK. And the following was obtained for BFSK:

- BER = 0%

- Number of Errors = 0

- Data Stream = 128 Bits

- Velocity of Mobile = 100 kph (62 mph)

- fm = 83.3 Hz

- fc = 900 MHz

Figure 6

Figure 7

The number of level crossings and average fade duration is plotted versus ( for the given fm of 83.3 Hz. As can be seen in Figure 7, the maximum number of level crossings occurs at ( = 0.707.

CONCLUSION

It has been shown, through the use of the Matlab simulations, that Signals in the Rayleigh fading environment suffer strength loss due to Doppler shift and incidence of arrival which causes cancellation at the receiver, and, that it is clear from the tests that BFSK mitigates these effects whereas DS/CDMA resulted in some errors.

Matlab simulation proved to be very useful for the task and reusable code was the result. For future work, this simulation needs to be extended to a two-ray fading channel to include multipath delay.

REFERENCES

1. R. H. Clarke, “A Statistical Theory of Mobile-Radio Reception,” The Bell Systems Technical Journal, vol. 47, no. 6, pp. 957-1000, July-August 1968

2. John I. Smith, “A Computer Generated Multipath Fading Simulation for Mobil Radio”, IEEE Transactions on Vehicular Technology, vol. 24, no. 3, pp. 39-40, August 1975

3. Theodore S. Rappaport, Wireless Communications: Principles and Practice, Prentice Hall PTR, ISBN 0-13-375536-3, July 1999

4. William C. Y. Lee, Mobile Communications Design Fundamentals, John Wiley & Sons, Inc., ISBN 0-471-57446-5, 1993

5. R. J. Holbeche, Land Mobile Radio Systems, Peter Peregrinus Ltd., ISBN 0-86341-04909, 1985

6. Khaled Ben Letaief, Khurram Muhammad, and John S. Sadowsky, “Fast Simulation of DS/CDMA With and Without Coding in Multipath Fading Channels, IEEE Journal on Selected Areas In Communications, vol. 15, no.4, pp. 626-639, May 1997

7. Louay M. A. Jalloul and Jack M. Holtzman, “Multipath Fading Effects on Wide-band DS/CMDA Signals: Analysis, Simulation, and Measurements”, IEEE Transactions on Vehicular Technology, vol. 43, no. 3, pp. 801-807, August 1994

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-fm 0 fm

[pic]

fm

0

Baseband Doppler Filter

Baseband Gaussian Noise Source

Independent

[pic]

[pic]

[pic]

[pic]

Baseband Doppler Filter

Baseband Gaussian Noise Source

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