Fairleigh Dickinson University Gildart Haase School Of Compu

Fairleigh Dickinson Universitygildart Haase School Of Computer Science

Fairleigh Dickinson Universitygildart Haase School Of Computer Science

We consider a Rayleigh fading signal. Suppose that a single branch receiver has a 10% chance of being 5 dB below a certain SNR threshold γo. a) Determine the mean of the Rayleigh fading signal as reference to the threshold mentioned in the above, i.e., γo. b) What is the probability that the receiver signal will be 8 dB below γo if b1) a single branch receiver b2) a two branch selection diversity receiver b3) a three branch selection diversity receiver, or b4) a four branch selection diversity receiver is used, respectively.

Paper For Above instruction

The problem outlined involves understanding the statistical behavior of Rayleigh fading signals and their implications in wireless communication systems, particularly in the context of diversity reception techniques aimed at mitigating fading effects.

Understanding Rayleigh Fading and Its Parameters

Rayleigh fading models the statistical variations in signal amplitude caused by multipath propagation, which results in rapid fluctuations in received signal strength. The probability density function (PDF) of the Rayleigh distribution characterizes these fluctuations, and its properties are crucial in analyzing system performance.

Part 1: Mean of the Rayleigh Fading Signal Relative to a Threshold

The problem states that there is a 10% probability that the signal falls below 5 dB relative to a given SNR threshold γo. Mathematically, this can be expressed as:

P(SNR < γo - 5 dB) = 0.10

Converting the dB difference to linear scale:

Alpha = 10^(-5/10) = 10^(-0.5) ≈ 0.3162

Using the Rayleigh fading model, the cumulative distribution function (CDF) for the amplitude |h| is:

F_{|h|}(x) = 1 - e^{ -x^2 / (2\sigma^2) }

where σ is the scale parameter of the Rayleigh distribution. The SNR is proportional to |h|², so the probability that the SNR is below a threshold can be related to the amplitude's distribution.

Given the probability that the SNR is below 5 dB, the corresponding amplitude threshold is:

|h|_{threshold} = √(γo * Alpha)

To find the mean of |h|, recognizing that for a Rayleigh distribution, the mean is:

E[|h|] = σ√(π/2)

and given the constraints, the scale parameter σ can be derived from the pdf and the probability condition. Considering the link between |h| and the probability, the mean amplitude can be expressed relative to the threshold value, leading to an analysis that shows the expected value in relation to the given probability and dB margins.

Eventually, assuming the given probability corresponds with the upper tail of the Rayleigh distribution, the mean of |h| can be determined numerically or through analytical integration; typically, through calculations involving the incomplete gamma function or standard tables, the mean amplitude relative to the threshold γo is approximately 0.886σ (since the mean of the Rayleigh distribution is σ√(π/2)).

Part 2: Probability of Signal Being 8 dB Below Threshold at Different Receiver Configurations

The question asks for the probability that the signal is 8 dB below the threshold γo, considering various diversity receiver configurations. This involves calculating the probability that the combined received SNR or amplitude falls below a certain level, conditioned on multiple branches.

Converting 8 dB below γo to linear scale:

Beta = 10^{ -8/10 } = 10^{ -0.8 } ≈ 0.1585

The probability calculations depend on the combined channel amplitude distribution across diversity branches. For a single branch, it directly pertains to the Rayleigh amplitude distribution:

  • Single branch receiver (b1): The probability that |h|² is less than Beta*γo can be obtained directly from the Rayleigh distribution’s CDF.
  • Two, three, and four branch receive diversity (b2, b3, b4): These employ selection diversity, where the maximum of multiple independent Rayleigh fading amplitudes is considered. The distribution of the maximum of N independent Rayleigh fading amplitudes involves the order statistics, leading to more robust performance.

The CDF of the maximum of N independent Rayleigh random variables with the same distribution is:

F_{max}(x) = [F_{|h|}(x)]^{N}

which indicates that the probability that the maximum amplitude is less than x is the product of individual probabilities. Therefore, the probability that the combined signal is below the specified threshold involves calculating:

P( max |h_i|^2 < Threshold ) = [ F_{|h|}(√(Beta * γo)) ]^{N}

Applying this to each case:

  • For two branches (b2): N=2
  • Three branches (b3): N=3
  • Four branches (b4): N=4

Finally, substituting the known values and computing the probabilities yields an increasing likelihood as the number of diversity branches increases, demonstrating the effectiveness of selection diversity in combating fading.

Conclusion

This analysis underscores the significance of diversity techniques in wireless communications. By exploiting multiple antennas or branches, systems can substantially enhance signal reliability and reduce outage probability, especially in environments characterized by Rayleigh fading. Understanding the statistical properties of wireless channels, including their mean amplitudes and probability distributions, is fundamental for designing robust communication systems capable of maintaining high performance despite channel impairments.

References

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