APPENDIX 1 NEYMAN PEARSON CRITERIA

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Ashlee Maxwell
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1 54 APPENDIX NEYMAN PEARSON CRITERIA The design approaches for detectors directly follow the theory of hypothesis testing. The primary approaches to hypothesis testing problem are the classical approach based on the Neyman Pearson criteria and the Baysean approach (Kay 998). The choice of an appropriate approach depends on the knowledge about the probability of occurrence of the various hypotheses. The design of detectors is usually based on the theory of binary hypothesis testing where we must choose between two hypotheses H (null hypothesis) and H (alternative hypothesis). Detector compares the observed datum value with the threshold and decides on the underlying hypothesis. With this scheme, the detectors can make two types of errors namely Type I error and Type II error. If the detector decide H but H is true, then it make a Type I error. On the other hand, if the detector decides H but H is true, it makes a Type II error. These two errors are unavoidable to some etent but may be traded off against each other by changing the threshold value. It is not possible to reduce both error probabilities simultaneously. A typical approach in designing an optimal detector is to hold one error probability fied while minimizing the other.

2 55 P ( H i : H j ) denotes the probability of deciding H i when H j is true. P H : H ) is Type I error probability and is referred to as the probability of ( false alarm denoted by P in engineering terminology. P ( H : H) is type II error probability and is referred to as probability of miss detection. P ( H : H ) is just P H : H ) and is called the probability of detection denoted by P D. ( In Neyman- Pearson approach to hypotheses testing PD is maimized subject to the constraint P H : H ) P (. The Neyman Pearson theorem tells how to choose the threshold if we are given pdf of underlying hypothesis and probability of false alarm P. decide H if Theorem ( Neyman Pearson): To maimize P D for a given P, L( ) P( : H ) P( : H ) th Where the threshold th is found from P : L( ) th P ( : H ) d The function L () is the test statistics which is compared with the threshold for making decision. For a typical scenario, Figure A. show the possible errors and Figure A. illustrate the trading off errors by adjusting threshold.

3 56 Figure A. Possible hypothesis testing errors and their probabilities Figure A. Trading off errors by adjusting threshold

4 57 APPENDIX SUMMARY OF STATISTICAL DISTRIBUTIONS In wireless communications, Nakagami distribution, Weibull distribution, Rician distribution, and Rayleigh distribution are used to model scattered signals that reach a receiver by multiple paths. Depending on the density of the scatter, the signal will display different fading characteristics. Rayleigh and Nakagami distributions are used to model dense scatters, while Rician distributions model fading with a stronger line-of-sight. Nakagami distributions can be reduced to Rayleigh distributions, but give more control over the etent of the fading. The Rayleigh distribution is a special case of the Weibull distribution. If A and B are the parameters of the Weibull distribution, then the Rayleigh distribution with parameter b is equivalent to the Weibull distribution with parameters A b and B =. A. WEIBULL DISTRIBUTION Weibull fading, named after Waloddi Weibull, is a simple statistical model of fading used in wireless communications and based on the Weibull distribution. Empirical studies have shown it to be an effective model in both indoor and outdoor environments.

5 58 The probability density function of a Weibull random variable is: f ( ;, k) k k e k where k is the shape parameter and is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched eponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the eponential distribution (k = ) and the Rayleigh distribution (k = ). A. RICE DISTRIBUTION Rician fading occurs when one of the paths, typically a line of sight signal, is much stronger than the others. In Rician fading, the amplitude gain is characterized by a Rician distribution. A Rician fading channel is described by two parameters: K and. K is the ratio between the power in the direct path and the power in the other, scattered, paths. is the total power from both paths ( ), and acts as a scaling factor to the distribution. K and K The received signal amplitude is Rice distributed with parameters variable with rice distribution is:. The probability density function of random ( K ) ( K ) ( K ) f ( ) ep K I K( K ) Where, I (.) is the th order modified Bessel function of the first kind.

6 59 A.3 RAYLEIGH DISTRIBUTION Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the receiver. If there is no dominant component to the scatter, then such a process will have zero mean and phase evenly distributed between and radians. The envelope of the channel response will be Rayleigh distributed. distribution is: Probability density function of the random variable r with rayleigh R r r p ( r ) e, r where E( R ) Rayleigh fading can be a useful model in heavily built-up city centres where there is no line of sight between the transmitter and receiver and many buildings and other objects attenuate, reflect, refract, and diffract the signal. In tropospheric and ionospheric signal propagation the many particles in the atmospheric layers act as scatterers and this kind of environment may also approimate Rayleigh fading. A.4 NAKAGAMI DISTRIBUTION The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter and a second parameter controlling spread.

7 6 Its probability density function (pdf) is f ( ) ( ;, ) ep The parameters and are E ( X Var( X ) ) and E( X ) The Nakagami distribution is related to the gamma distribution. In particular, given a random variable Y ~ Gamma( k, ), it is possible to obtain a random variable X ~ Nakagami(, ), by setting k,, and taking the square root of Y : X Y

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