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This Matlab exercise requires you to calculate the symbol error probability PM for M-PSK where M=2,4,8, or 16

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This Matlab exercise requires you to calculate the symbol error probability PM for M-PSK where M=2,4,8, or 16. For each value of M given above I would like you to plot PM versus Eb=N0. As you know, we have an exact formula for PM, an upper bound, and a lower bound. We can also calculate PM using Monte Carlo simulation. So, for each value of M, you should do the following: ² Calculate PM versus Eb=N0 using the exact formula: PM = 1 ¡ 2 Z 1 0 e¡(r0¡ p Es)2=N0 p ¼N0 ÃZ r0 tan (¼=M) 0 e¡r2 1=N0 p ¼N0 dr1 ! dr0 : ² Calculate the upper, and lower bounds for each M: Q Ãr 2Eb N0 (log2M) sin2 ³ ¼ M ´! < PM < 2Q Ãr 2Eb N0 (log2M) sin2 ³ ¼ M ´! ² Calculate PM using Monte Carlo simulations with baseband representation of the signals and the noise: 1. Fix the number of signals in the PSK constellation (M) 2. Select the desired value of Eb=N0 (without loss of generality you can assume that N0 = 1) 3. Select the transmitted signal index m from the set f1; 2; : : : ;Mg with equal probability 4. Calculate Es = Eb log2M and generate the two components of the received signal as r1 = p Es cos(2¼(m ¡ 1)=M) + n1 r2 = p Es sin(2¼(m ¡ 1)=M) + n2 ; where n1 and n2 are generated as independent Gaussian random variables with mean 0 and variance N0=2 = 1=2. Since the mth signal is transmitted, we make an error when (r1; r2) falls outside the mth decision region. 5. After we generate (r1; r2), we test if it falls in the right decision region. If so, we count no error. If not, an error has occurred. 6. Go back to step 3 until 100 errors occur. 7. For the current value of Eb=N0, estimate PM by taking the ratio 100 divided by the number of iterations required to get those 100 errors. Clearly, we could chose a number other than 100, and you might choose to change it to see how it changes your plot. So, for each M, and each Eb=N0, the program goes through the above steps. Then, for that M, you plot the PM versus Eb=N0. You should get something very close to the exact expression for PM. Present these results in a series of plots. The plots should illustrate the following: ² How tight are the upper and lower bounds for each M? 1 ² How close is the Monte Carlo results to the exact expression? In answering these questions, please be quantitative, and use the conventional approach of ¯xing an appropriate BER and comparing the di®erence in Eb=N0 in dBs, when comparing the curves.