DATA
Chapter 11 Simulating Channel Distortion in the Software Radio
11.2 Specifying Ullcorrelated Rayleigh Fading
A communications channel subject to the effects of Rayleigh fading will experience distortion in the amplitude and phase of a transmitted signal. This is due to multi-path effects. However, to simplify the simulation of such a distortion, only the amplitude response will be considered.
The following equation describes the actual Rayleigh distribution:
Chapler 11 Simulating Channel Distortion in the Software Radio
... " ... " ... Equation 11.1
Equation 11.1 generates the graphical distribution seen in Figure 11.2 Graph of Rayleigh Fading Distribution
"
.. .. "
'"
t-••
0• •
Q •
~'
"
'"
•~
0
.. ,
~ •
"
' . .. , " , ,
" ,,
" ,• ., ,
Figure 11.2 - Rayleigh Fading Distribution
Two common statistical distributions will be used to generate the Rayleigh fading coefficients- the uniform distribution and the normal (Gaussian) distribution. Before an examination of the generation of Rayleigh random variables can be undertaken, it is necessary to be able to generate Gaussian random variables from a uniform distribution. As seen in the system overview, the DSP system is used to generate the Rayleigh fading coefficients. It is therefore necessary to generate the Gaussian distributed samples on the DSP card, which is programmed in ANSI C. Unfortunately, ANSI C does not define a standard function for returning samples from a normal distribution. It does, however, have a standard uniform number generator that is can be used to create the Gaussian random deviate.
11.2. 1 Ge ne ratin g G allss ian Random Sampl es
There are many techniques available for transforming a uniform random sample space to a normal random sample space. However, the developer has to be constantly aware of the target implementation platform. This is because the choice of generation technique usually involves a trade-off between the speed of generation and the accuracy of the generated samples. For this
95
Chapter 11 Simulating Channel Distortion in the Software Radio
first generation technique, speed is not considered, while a premium is placed on the final accuracy. This is because the samples are being generated in the DSP, where a plethora of standard ANSI C maths functions allow for accurate mathematical manipulation of variables. However, the generation of a single sample within the time frame of a transmit interrupt on the SC-BUS is not possible, since the DSP is not quick enough. Instead of this approach, a sufficiently large number of samples from the Kayleigh random space are stored during an initialisation period. These are then retrieved sequentially during processing, and sent to the software radio.
The Gaussioll Distributioll
[A. Ralston, 1967], in his work on mathematical methods as implemented on digital computers, (20), states that the unit nOnllal (Gaussian) probability distribution is defined as:
... F.quation 11.2
• Ru is a normalised uniform random variable
• RN is a normalised normal random variable
In this case, there is no close-form solution for RN and any teChnique must rely on an approximate generating equation. To this end, and with accuracy being foremost, there is a means of generating a Gaussian random variable given two uniformly distributed random variables:
... Equation 11.3
• Ru and RII+1 are the two uniformly distributed random variables
• RN is the normal random deviate
To test the success of this approach, MATLAB was used to generate a sample space of 20,000 unifonnly distributed random variable samples, making use of the standard randO function. These fonned the 10,000 pairs required to generate the 10,000 samples for the Gaussian approximation. using the above equation. The following graph shows the results of the test, revealing the familiar bell shape associated with a normal distribution.
Chapter II Simulating Channel Distortion in the Software Radio
••
Normalised Gaussian l'robalJility Distribution Generated from a Uniform Distribution
Di5lribulion v.tu,
Figure 11.3 - Gaussiun Deviate rrom Unirorm Distribution
,
Once these samples from the Gaussian distribution have been generated. they can then be used in the next stage, which is the generation of the Rayleigh random variables
11.2.2 Generating Vncorrelated Rayleigh Random Variables
The generation of accurate uncorrelated Rayleigh Random variable samples requires the manipulation of 2 Gaussian random variable samples. Further, the desired mean of the distribution needs to be specified before calculation can proceed. The variance of the distribution is actually a function of the mean, and so specitying the mean automatically detennines the variance. The equation, which is used in the generation of the Rayleigh samples, makes use of two normal distribution samples:
... Equ3tion 11.4
• N I and N2 are two Gaussian random variables
• MR is the desired mean for the Rayleigh random variable
• AR is the Rayleigh random variable, in this case, an Amplitude coefficient Further
97
Chapter 11 Simulating Channel Distortion in the Software Radio
... Equation II.S
and
F¥
Ma
= R, f (
...•... Equation 11.6
In this generation technique, the standard deviation is a function of the desired mean. For simulation purposes, the mean should ideally be set to the path loss attenuation. However, for simulation purposes, the mean will be set to I, indicating that, on average, the received signal will not be attcnuated. Simulation of this generation technique produces the following graph:
~
"
0
•
= .,.
• -
...
"
.~•
•
E-
z 0
.. ••
"
• • •
• • •
••
• • •
.,
• • •
H.csulting Normalised Rayleigh Probability Distribution
'. • • • " " •
Distribulion Sample Value
Figure 11.4 - Haylcigh Dcviate from Gaussian Distribution
In Figure 11.4, the uncorrelated Rayleigh samples were taken from a space consisting of 10000 points, with a mean of I. The measured mean of the given sample space is 1.0074, showing that the generator produces not only the correct Rayleigh distribution graph, but also the correct mean.
Chapter I I Simulating Channel Distortion in the Software Radio