I’m working on a mathlab multi-part question and need guidance to help me understand better.
1 INTRODUCTION This project investigates a very common problem in communication of digital information, the variation of the bit error rate (or probability of bit error) as a function of changing the signal-to-noise ratio at the detector input. It is complicated, but I’ve tried to break out the steps as clearly as possible. This is a real, and immensely important, application of maximum likelihood detection techniques. Take your time. Do not let it go to the last minutes. In revision 5, the changes are in red, so you can find them quickly. There is an additional task in 2.5. I’ve made these changes based on my experience in crafting a solution to Project 4, as well as comments or questions from students. 1.1 Due Date Project 4 is due on Wednesday May 5, at 11:59 PM. Project 5 is due on Monday, May 17 at 11:59 PM. 1.2 Background Binary Amplitude Shift Keying (BASK) is a technique of sending binary information by sending one of two signals levels, with one level assigned to be a binary 1 and one level assigned to be a binary 0. One common mapping is to send a level of when the binary information is a 1 and to send at when the binary information is a 0. For this project we will assume a BASK system with the input being a binary random variable and The random variable is a simple mapping and the random variable corresponds to the symbol bit in a string of symbols, generated by the mapping of the input bit . For the various values for different values are independent and identically distributed, with . (1.1) As in Project 2, this sequence of i.i.d. random values is corrupted by additive white Gaussian noise, where the pdf of the noise is −A +A B ∈{0,1} Pr⎡B = 0 ⎣ ⎤ ⎦ = p0 , Pr⎡B = 1 ⎣ ⎤ ⎦ = 1− p0. M b = 0 ! m = −A, b = 1! m = A Mk k-th Bk Bk and Mk k fB (b) = p0 b = 0 1− p0 b = 1 ⎧ ⎨ ⎪ ⎩ ⎪ f M (m) = p0 m = +A 1− p0 m = −A ⎧ ⎨ ⎪ ⎩ ⎪2 . (1.2) Our model for our received signal is , (1.3) and we can assume that the random variables are independent. Hint: This is the same model used in Project 3, except we will be varying the noise variance. Hint: Notice that we are NOT using M and N as the number of rows and number of columns in your R matrix, as we have done before! Please be careful and don’t confuse them! We are tasked to design a Maximum A Priori (MAP) detection method that will examine the value received random variable, at the time instant , and estimate the value of the corresponding input binary digit, where the is the binary digit used to create the BASK value , and is the estimate at the output of the MAP detector. For this simple model, the Signal to Noise ratio, , is defined as , (1.4) and is often expressed in decibels, . Throughout your implementation of this Project, please assume that the value of , so that and Figure 1 shows the bits, messages, noise, received value, estimates and errors. The pmf for the input bits is not changed for any particular set of experiments. The entire process is repeated for various values of , which specify the variance by means of (1.4) fN (n) ~ N(0,σ2 ) = 1 2πσ2 e−n2 /(2σ2 ) , − ∞ < n < ∞ R = M + N M,N r k k ˆ bk bk ∈{0,1} mk ˆ bk γ γ = A2 σ2 γ dB = 10log10 (γ ) A = 1 γ = 1 σ2 γ dB = −10log10 (σ2 ) = −20log10 (σ ) γ dB3 Figure 1 Overall System Model for Project 4 (revised 4/19/2021) 2 PROJECT TASKS Perform the following tasks, document your results and submit them in written form in accordance with the instructions in Section 3, below. You may use this document as a format. 2.1 Design the MAP detector Design a MAP detector by the following steps. 1) Write expressions for the conditional pdfs, and . 2) Transform those conditional pdfs into the joint pdfs and by application of Bayes’ Rule. The answer will explicitly contain . 3) Using these joint pdfs, set up the ratio (think through what this means!). You will see that this ratio contains exponentials, the Gaussian constant, and a ratio of terms explicitly containing When , the numerator is larger than the denominator and thus the input is more likely. 4) Carefully take the natural log of the ratio and set the result equal to zero, because (Note that there is no derivative necessary here; we’re not formally maximizing, we’re just comparing the two pdfs.) Solve for , the value that makes the equation true. Hint: The MATLAB routine log takes the natural logarithm. log10 is logarithm to the base 10. BASK + MAP (or ML) Count Errors fB (b) = p0 b = 0 1− p0 b = 1 ⎧ ⎨ ⎪ ⎩ ⎪ b1,b2 ,…,bk ,…bn m1,m2 ,…,mk ,…mn r 1,r 2 ,…,r k ,…r n ˆ b1, ˆ b2 ,…, ˆ bk ,…ˆ b b1 n ,b2 ,…,bk ,…bn e1,e2 ,…,ek ,…en ∈{0,1} XOR pBX = Nerror Ntrial = Nerror n n1,n2 ,…,nk ,…nn iid fN (n) = N(0,σ2 ) b = 0 ! m = + A b = 1! m = − A fR|M =A(r | M = A) f r|M =− A(r | M = −A) fR,M (r, A) fR,M (r,−A) p0 L = fRM (n,m | b = 0) fRM (n,m | b = 1) p0. L > 1 b = 0 log(1) = 0. r = τ MAP Λ = log(L) = 04 Your MAP detector uses this voltage threshold, , to decide if each individual received value (in volts) should be interpreted as a binary 1 or a binary 0. That is (1.5) Explain how your value of does or does not depend on For a noise variance equivalent to a fixed value of (remember the assumption that ), plot the values of your MAP threshold, , as a function of over the range . Discuss why your MAP threshold curve makes sense in terms of the interpretation of Why does the threshold move in one direction or the other based on ? 2.2 Investigate the MAP detector For a value of and a value of , generate a large number of i.i.d. samples of the random variable and plot the histogram corresponding to the pdf . Using this value of , and the equivalent of , plot the analytical value of on the same figure. Indicate on your plot where the MAP threshold would be for this set of parameters. Look at the relative amplitudes of the “humps” of the Gaussian histograms and think about what they mean. 2.3 Evaluate the ML Detector 2.3.1 Find the ML threshold Modify your MAP detector to be a maximum likelihood (ML) detector by assuming that a binary 1 and a binary 0 are equally likely, and determine the ML threshold, . Explain why this value makes sense. 2.3.2 Derive the probability of error. Then, using the value of your ML threshold, analytically compute the theoretical . Using the following steps 1) Write the conditional pdf when the input bit is 2) Using the threshold in (1.5), write the limits of the region where the ML decision selects a 1 instead of a 0. 3) Integrate the conditional pdf over that region to find the conditional probability of error given that the input was a 0. Hint: The Q function will be of use here, consider your change of variables from an early homework. 4) Write the conditional pdf when the input bit is τ MAP r k ˆ bk = 1 r k < τ MAP 0 r k ≥ τ MAP ⎧ ⎨ ⎪ ⎩ ⎪ τ MAP p0. γ dB = 10 dB A = 1 τ MAP p0 0.01≤ p0 ≤ 0.99 p0. p0 p0 = 0.6 γ dB = 10 dB R fR (r) p0 σ2 fR (r) τ ML pBT b = 0. b = 1.5 5) Using the threshold in (1.5), write the limits of the region where the ML decision selects a 0 instead of a 1. 6) Integrate the conditional pdf over that region to find the conditional probability of error given that the input was a 1. 7) Use the Principle of Total Probability to write an expression for the unconditional probability of error. This is . For full credit, combine the terms wherever possible. Hint: Remember the Q function. MATLAB knows about erfc, but not Q. 2.3.3 Simulate the ML detector. Evaluate the bit error performance of the ML detector as a function of the signal-to-noise ratio by varying from 1 dB to 8 dB in steps of 0.5 dB, and then from 8.5 dB to 13 dB in steps of 0.25 dB. At each value of , generate a very large number of trials of the random variable , apply your ML decoder, and compare with the estimate . Hint: errors = mod(bk-bkhat,2) Count all of the errors at each value of , and compute your experimental probability of bit error, (1.6) at each value of , where is the number of errors counted and is the number of trials. Save the values of , for use in Section 2.5. Plot the simulated results on the y-axis against on the x-axis. On the same figure plot a smooth curve of using your theoretical value. Discuss any differences between your simulated values and your analytical curve. Hint: At , there should be about 1 error in every 100,000 trials, so you may need several hundred thousand trials for the higher values of . Hint: To plot the analytical values, you can use a much finer step size for to get a smooth curve, because you are just doing the computation, not a simulation at each point. You might also want to plot the simulated values using only symbols to make it clear the exact coordinates of the simulation points. Hint: Your curves should have a negative slope: as increases, decreases. Hint: Consider the MATLAB function semiology for the plots. pBT γ dB γ dB R bk ˆ bk γ dB pBX = Nerror Ntrials γ dB Nerror Ntrials (γ dB , pB ) log10 ( pBX ) γ dB log10 ( pBT ) γ dB = 12.6 dB γ dB γ dB γ dB pB6 2.4 Evaluate the MAP Detector 2.4.1 Derive the probability of error using the MAP threshold Following the same steps as 2.3.2 write an expression for the probability of error for the MAP threshold. Your answer should explicitly contain . 2.4.2 Simulate the MAP detector Following the same steps as 2.3.3, simulate the bit error performance of the MAP detector. Remember to use when establishing your random binary bits, . Plot the simulated results on the y-axis against on the x-axis. On the same figure plot a smooth curve of using your theoretical value. Discuss any differences between your simulated values and your analytical curve. 2.5 Compare the MAP and ML Detector performance Finally, plot the analytical for ML detector (from 2.3) and the analytical for MAP detector (from 2.4), on the same x-axis, that is, plot two curves on the same figure. Discuss any differences. Then, on a separate figure, plot the ratio (1.7) on the (linear) y-axis against the value of on the x-axis. How does this ratio compare with a value of 1, which would indicate that the probability of error was the same for MAP and ML? Why does the ratio (compared to 1) make sense? Hint: What does tell us? 3 INSTRUCTIONS FOR PROJECT REPORT 3.1 Report Format The project report shall be in the same form as this document, with an introduction, simulation and discussion section, and a “what I learned” section. Each section shall contain the content identified in Section 2 and the appropriate Section 3 subsection below. The report shall be in Times New Roman 11 point font. MATLAB pictures shall be pasted in-line in the report (this is a useful skill to know!); shall be numbered consecutively; shall be appropriately titled; the axes shall be appropriately labeled; the curves shall be appropriately identified by an appropriate legend. 3.2 Section 2 Content Section 2 of the report shall be titled “Simulation and Discussion” and shall contain the simulation plots and a discussion of each plot. The discussion shall address the points identified in Section 2, and any other interesting p0 p0 bk log10 ( pBX ) γ dB log10 ( pBT ) log10 pBT log10 pBT γ dB ρ = pBT for p0 = 0.6 (MAP) pBT for p0 = 0.5 (ML) γ DB p0 ≠ 0.57 observations that occur to you. Remember, I know this stuff: you don’t. So take a look at the plots and tell me what you see and what it means to you. 3.3 Section 3 Content Section 3 of the report shall be titled “What I learned” and shall contain a summary of what information you observed, what insights you gained, etc. Section 3 shall also contain a subsection critiquing the project and suggesting improvements that I could institute for next spring. Finally, Section 3 shall contain an estimate of how much time you spent on the project, including reading, research, programming, writing, and final preparation. 3.4 Questions I will accept questions regarding the project through the Ask the Professor discussion forum through 6 PM on Tuesday April 13, 2021. Please plan to check the Ask the Professor discussion forum frequently to learn of clarifications and hints (if I give any!). In my opinion, Project 3 is actually easier than Project 2, so I’m less likely to give direct assistance. 3.5 Project Grading The project will be graded in the following way: 75% of the project score shall depend on the technical, theoretical, and graphical presentations of the tasks set out in Section 2 of this document. 25% of the project score shall be based on an evaluation of the technical writing against the Rubric on Technical Writing, posted on Blackboard, including grammar, clarity, organization, etc. For the purpose of this document, you can assume that the intended audience consists of your CMPE320 classmates
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