ENEL 529



ENEL 529

Lab 1 Report,

Characteristics and Simulation of Rayleigh Fading Channel: Part I

Submitted by:

Maggie Zhang (241693)

Kwok Cheng (224784)

Date: October 31, 2005

Objectives

A non-line-of-sight (NLOS) received wireless signal could be simulated using two approaches: vector sum of multiple arrived signals, or inverse discrete Fourier transform of two independently filtered Gaussian signals.

This laboratory attempts to model a radio frequency (RF) received signal using the first approach using MathWorks Matlab. With this simulation, a better understanding of the relationship between the number of source paths and Rayleigh fading channel modeling, as well as the impact of mobile station (MS) mobility on the received RF signal is pursued.

Implementation

A. Theoretical Approach

Part I - simulated receiving signal by a stationary MS

The simulation assumes a receiver receive a signal from N independent paths. That is,

[pic].

Recall that sinusoid signals can be express as

[pic]

where [pic] and [pic]

Then, the envelope of the received signal after demodulation is given by

[pic]

Part II - simulated receiving signal by a non-stationary MS

Assuming MS is moving in a speed of V, the Doppler shifting frequency fd is given by

[pic]

Therefore, the envelope of receive signal becomes

[pic]

where [pic] and [pic]

Then, the envelope of the received signal after demodulation is given by

[pic]

B. Laboratory Approach

In this laboratory, all simulations were done in Matlab. The amplitudes of signals were simulated by generating random Weibull-distributed values on each single signal path amplitude ai and applying a random uniformly-distributed path phase φi. Built-in function weibrnd(α,β) with α=3 and β=1 is used on gain and built-in function unifrnd(0, 2π) is used on phase. Function demod(signal, fc, fs, ‘qam’) is used to get inphase and quadrature components, respectively, I(t) and Q(t) from receive signal s(t) so that received signal envelop R(t) could be calculated. User-defined function SimRayleighFading() calculates one time sample of receive signal. Details please see attached SimRayleighFading.m file.

Parameters used in this laboratory are as follows:

- carrier frequency, fc = 900 MHz

- sample frequency, fs = 4fc

- number of sample point in each approach, M = 2000

- In Part II, MS moving speed V = 90km/hr.

Equations used in calculation and analyses are as follows:

- The root-mean-square (rms) value of envelope [pic]

- Mean of the simulated envelope [pic]

- Rayleigh parameter [pic]

- Theoretical Rayleigh signal envelope pdf (probability density function) [pic]

- Theoretical Rayleigh signal power pdf [pic]

- average chi-square statistic [pic]

Part I - simulated receiving signal by a stationary MS

Test 1 - Conduct a Chi-square test of the simulated envelope

1. Select number of bins K = 10 for grouping the simulated envelope

2. Select signal number of signal path N = 5

3. Call function SimRayleighFading() to get a sample signal s and its envelope R

4. Call function hist() using K and R as parameter and then get the empirical number of envelope samples mk that fall in the kth interval

5. Calculate vector X=[min(R) : ((max(R)- min(R))/K) : max(R)]

6. Mean of the simulated envelope E[R] and deviation σ

7. Get Rayleigh CDF(cumulative distribution function) FR = raylcdf(X,σ)

8. Theoretical probability pk that envelopes would fall in the kth interval will be pk=FR(k+1)-FR(k)

9. Calculate the average chi-square statistic X1

10. Repeat step 2 to 9 forty-nine times to generate a total of 50 samples of X1

11. Calculate the mean of X1, [pic]

12. Check whether E[X1] is less then X2 which is the threshold chi-square statistic corresponding to K=10 bins at 95th percentile (X2 = 15.51)

13. If E[X1] > X2, the test fails, go back to step 2 with N= N+1 (N increased by 1)

14. If E[X1] < X2, the test is passed, and we get an N value

Test 2 - Compare theoretical Rayleigh pdf and histogram on envelop data

1. Select N from the output of Test 1

2. Call function SimRayleighFading() to get a sample signal s and its envelope R

3. Plot received rf signal in volts vs. time in nanoseconds

4. Calculate rms value of envelope Rrms

5. Plot normalized received signal envelope in dB unit RdB=20log10(R/Rrms)

6. Sort envelope R as Rs

7. Calculate mean of the simulated envelope E[R] and deviation σ

8. Calculate theoretical Rayleigh signal envelope pdf fR(Rm) and normalize by its maximum value

9. Normalize envelope sorted envelope Rs

10. Plot the normalize Rayleight envelop pdf vs. normalized sorted envelope Rs

11. Select number of bins K = 20 for grouping the simulated envelope

12. Compute the histogram of envelope by calling function hist() using K and R as parameter

13. Normalize the histogram of envelope by its maximum value

14. Plot the histogram vs. normalizes bin interval [0: 1/K : (1 – 1/K)], and then superimpose this plotting into the figure in step 7. (Please see the attachment for the diagram)

15. Calculate [pic]

16. Calculate signal power P=R2

17. Calculate theoretical Rayleigh signal power pdf fP(Pm) and normalizes by its maximum value

18. Plot the normalize Rayleight power pdf vs. normalized sorted power P

19. Calculate [pic]

All codes are available as attachments.

Part II - simulated receiving signal by a non-stationary MS

Procedures are the same as in part I, using equations from Part II of Theoretical Approach of the Implementation section. In other words, simulation is similar to part I with the inclusion of a Doppler frequency shift term fd max in the received signal and the signal’s quadrature and in-phase components.

Analysis

Part I

Perform steps of Part I test 1 to find the minimum N which passes the Chi-square test. Result was plotted in Figure 1 shown below. From the figure, we find N = 27. From the figure, we can see the trend of the Chi-Square statistic X1 decrease as N increases. That is because when the received signal is collected from more paths, the effect of noise on the received RF signal becomes less. There are some variances on the main trend in the diagram. This is due to random fluctuations of path amplitude and phase. Flat tail of the diagram shows that we cannot improve signal quality by increasing path N while number of paths reaches to a specific value.

With N = 27, we perform the second test, and a series of diagrams was obtained. In figure 2, a sample of received signal collected with 27 paths is shown, along with signal envelope and normalized signal envelope. We can see that the signal received is pretty random with the effect from noise.

Using the signal shown in figure 2, statistical calculations on envelope distribution and signal power distribution were performed. Compare this empirical data with the theoretical Rayleigh distribution, shown in figure 3.

From figure 3, we can conclude that when a signal passes the Chi-square statistics test, the statistic of experiment data on envelope distribution matches the theoretical Rayleigh distribution and signal power matches exponential distribution.

The calculated [pic] = 1.9429 and [pic]= 1.0364

[pic]

Part II

Similar analyses and calculations were performed as in part I. Plotting the Chi-square statistic against the number of paths:

From this graph, we can deduce the number of paths/signal sources should be around 18 to 22 in order to pass the Chi-square statistic test. Choosing N=21, we get the following graphs:

From figure 7, the empirical pdf curve is subjectively similar to the theoretical Rayleigh pdf, thus passing the envelope data fit test.

Conclusion

Comparing figure 3 with figure 7, a peak Rayleigh pdf value is reached with a smaller value of normalized received signal for the case of the receiver with mobility. For the case of the RF signal and envelope with mobility, the fluctuations are more frequent and, at the same time, the amplitude of the envelope tends to be smaller during those fluctuations, thus contributing to a NLOS signal more difficult to receive.

This laboratory proved to be a good refresher for the use of Matlab and many communication concepts including demodulation and in-phase and quadrature components.

The simulations were relatively easy to set up. In general, predicted results are the same as actual results. Such as the resemblance of the empirical Rayleigh pdf to the theoretical pdf is directly proportional to the number of signal paths. However, due to the random nature of the input (received RF signal amplitude and phase etc.), “surges” of values appear in the signal envelope thus the Chi-square statistic values. Because of this, results of analyses were often out of the expectation of the group; especially when the number of trials is increased, the “spike values” appeared more often.

This laboratory exercise, in the group’s opinion, is a good extension to the course material covered in lectures. More emphasis could be placed on the relationship between lecture concepts. For example, how would the plotted graphs differ with/without the existence of Doppler shift or how would the signals look like when various types of fading occurs.

Reference

[1] A. Fapojuwo, ENEL 529 Lab 1 manual. Calgary: 2005.

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Figure 4 Plot of expected Chi-square statistic vs. number of source signal paths

Figure 6 Received RF signal before demodulation

Figure 5 Signal envelope after demodulation

Figure 7 Empirical and theoretical Rayleigh probability distribution function

Figure 8 Plot of normalized received signal envelope vs. time

Figure 1 Average Chi-Square Statistic vs. Number of Paths (Stationary MS)

Figure 2 Received Signal and Signal Envelops (Stationary MS)

Figure 3 Empirical vs. Theoretical Rayleigh distribution (Stationary MS)

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