- May 19, 2020

1 ENG332 Semester Project During the design of the new electrical engineering course at UTAS during 2017, UTAS decided to terminate the old course unit kne333 and replace it with a new unit eng332. The new unit is not based on in-semester tests as was the old kne333 unit. Instead a comprehensive project will be used to assess the in-semester formative component of the student learning. This project will be assessed based on three reports, each dealing with one of the three parts of the unit. Linear systems and signals is a fundamental unit taken by all electrical engineers all over the world, but especially important for all electrical engineers, including power engineers. The unit deals with the properties of systems in general, it does not matter if it is power, electrical machines or electronics. The language we use to analyse and predict the behaviour of systems is mathematics. The unit assumes you are proficient in linear algebra, analysis, numerical methods, differential equations and calculus. If you are not proficient in these topics, you will need to work on it ASAP, else you will be disadvantaged in this unit. The reports must be comprehensive and professional, and each report can be written either individ- ually, or in a group of two maximum. The choice is yours. If you work in a group, then each member of the group receives an equal mark. Assessment criteria for the reports will be placed on MYLO. I recommend you to make use of LATEX to write your reports. Even though its harder to learn LATEX than word, it provides a much superior product that will make your report appear professional. Probably using overleaf is the easiest way to start using LATEX – see https://www.overleaf.com. LATEX is free, and even works on windoze (see https://miktex.org/about). There are many excellent resources that deals with linear systems and signals, some of the books on linear systems are classics and have been around for decades. These include the books by Papoulis, Oppenheim, Lathi, Kailath, and Roberts, to name but a few. These books are comprehensive, but make a difficult read for the newcomer. If you are up to it, you are welcome to invest by buying some of these books. There are also very good videos on linear systems, for example the MIT open courseware by Oppenheim is very good.1 The books I recommend for the unit are [1, 2]. However for these projects you may need to consult other resources also, there are a lot of material available on the internet. The unit has three parts, and there is a project for each part. 1. Part 1: Time domain analysis. This part of the unit deals with direct time domain processing, which is almost universal now in industry, due to the availability of powerful microprocessors and DSP chips or ASIC’s. We will deal with the continuous time domain as that is the basis of systems theory, but then also with the sampled (digital) domain, as that is how computers process systems. 2. Part 2: Spectral domain analysis. Often linear operations based on differential equations tend to simplify if we transform it under suitable transformations. Examples include Laplace, Fourier and Z transformations. This part of the unit deals with these transformations, both in the continuous and the discrete spectral domain. The treatment of these topics are “state of the art” and includes the DFT and FFT. 3. Part 3: Stochastic signals analysis. Due to renewable energy now becoming important to comple- ment dispatch-able energy, as well as complications due to random noise that are always present in 1I would recommend that only the very capable and able students attempt reading Kailath, as he tends to consider systems from a pure mathematics point of view that is not very clear to most engineers. Roberts is quite readable and quite entertaining to read, but tends to be verbose (you will have to be patient). 1 data obtained from sensors, we have to have tools that deal with this issue effectively. The tool we deploy is stochastic analysis. This part of the unit is also “state of the art”, and is quite modern. 2 Project Part I: Time Domain Analysis: Due Sunday night, week 4, 22h00 on MYLO Weeks 1,2,3 will present lectures and a weekly tutorial/feedback session on this part of the unit. Then during week 4 there are no lectures and students are provided an opportunity to work on the project, and to submit the report (Sunday night of week 4, online on MYLO). Week 4 will also present a “exam preparation drill” Tut week, where we help you prepare for exam conditions through directed tutorials. As clearly, doing the project wont help you become exam ready. I highly recommend you attend and participate in these sessions. Its very important you plan and work ahead, as in my opinion it is not possible for the average student to start and complete the Part I project during week 4. Planning ahead, and time management is very important. Also note that based on UTAS policy (explained in the unit outline) if you are late in submitting the report, you will loose marks, and the later you submit the more marks you loose. The only way to prevent loosing marks if you are late, is if you are sick during week 4, and you can submit a doctors note to me. Part I will require two reports to get up to 100%. The first part described in section 2.2 is report 1 and counts 75% of the Part I overall mark. Then section 2.3 describes report 2, for the other 25% of the Part I mark. Part I counts 13 of the internal mark. 2.1 Project report layout for Part I For your report, you must include an introduction that contains a historical survey of the relevant material, and an introduction/explanation of the theory you used. Then you can have a section on the technical parts (presented below), and then a conclusions part where you present your conclusions. 2.2 Technical content for Part I Below the investigative project Refer to the system shown in Figure 1. This system is linear, and the L L 1 2 C R C1 2 v (t)i v (t)o R1 2 3 R Figure 1 A linear system. input voltage is given by vi (t) = u(t) − u(t − τ) τ cos(ωct). (1) u(t) designate the Heaviside unit step function. The system output is designated as vo(t). 2 1. Set up a system of linear and ordinary differential equations that models the system. Using a finite difference approximation to the differential operators you can solve the system on your computer. You cannot use MATLAB build in functions. You must use your own functions that you write yourself. Choose your own values for the system components and ω, and choose a very small τ. Small means that further reduction does not change the output – thus your output approximates the impulse response. Also you need to have τ T , where T is a period of the cosine. Explain why this input approximates a Dirac delta. Make sure your normalise the area, so that if you change τ your input energy does not change. You need to carefully choose an appropriate sampling time for you computer simulation, it must resolve the input pulse. Some info on how to solve a DE using finite differences you can find in [3] (also see chapter 30 on transfer functions) and many other online resources (see MIT open- course videos). With the input now resembling a Dirac delta, the output must resemble the system impulse response. Explain why this is the case and plot all your results. 2. Now choose τ ≈ 10T . Depending on the values you chose for the parts of the system, the duration of the transient will vary. This will determine if the output achieves an equilibrium state (pure phase shifted cosinusoid). You need to explain all these aspects in your report. Show pictures of the system output as a function of time. This must clearly show the transient part, and a part where the output is moving towards equilibrium (but possibly not attaining it), so choose appropriate values of the system components and frequency to make this possible. For example see Figure 30.5 in [3] as well as section 30.2 in [3]. 3. Now choose τ T . Measure the amplitude of the output during equilibrium, and repeat over a range of frequencies. Then determine the transfer function magnitude over this range of frequencies – the transfer function magnitude is the amplitude of the output of the system cosinusoid, divided by the amplitude of the input cosinusoid, and denoted ‖H (ω)‖. See chapter 30 in [3] for more information. Plot 20 log ‖H (ω)‖ versus ω. Use a log scale (see Figure 30.5 in [3]), and comment and interpret what you observe. Explain behaviour at low and high frequencies. Its important you show that you have insight into what is going on in the system. For example, why did we have to make τ very large before we could define the transfer function? Importantly check your result based on phasor analysis (H (ω) = Yout (ω)Yin (ω) ) you learned in Y1. 4. Now you must make use of the impulse response you obtained above. Using the convolution integral, compute the output with the input as defined in case 2 above. You can make use of a numerical approximation to the convolution integral, and show that you obtain the same output for case 2 above (using the same input). In other words, you must verify numerically the convolution theorem. You must program the functions yourself, you cannot use build-in MATLAB integration functions. Document all results with descriptions of what you did, and figures of results. 5. Design a discrete equivalent of the system shown in Figure 1. It is up to you to decide how to do this, but there are examples in [1] on how this is done – see for example Figure 5.4 and section 5.5. The idea is that given the sample time is known, then the discrete system impulse response must approximate the impulse response of the analogue (continuous time) system. Show that this is the case. But remember, there is a scaling issue when you compare the discrete and continuous impulse response. 6. Choose a mid range value of τ, say τ ≈ 10T . Now sample the input, and show that the discrete convolution sum produces the correct output over discrete time – do this by comparing it to direct recursion of your discrete system with the same input. You can also compare to the continuous time results. 3 2.3 Investigative project for Part I The project described above will get a student who does an outstanding job up to a maximum of 75% for Part I. The other 25% for Part I will be based on an investigative project, described here. For this project, I will place noisy data on MYLO. The data is a linear chirp cos(ωcn + αn2 + φ) that has been corrupted by additive random noise. Your job is to compute ωc, α, φ. You can decide how you want to do this computation. Your mark will be determined by how close you are to the values I used to make the data, and the quality of your report. 3 Project Part II: Spectral Domain Analysis: Due Sunday night, week 8, 22h00 on MYLO 3.1 Project report layout for Part II There is a considerable amount of history underpinning the spectral domain and transformation of the convolution integral. This includes the development of the Laplace and Fourier transformations, as well as the Z transformation. You need to present a summary of this history, and your own perspective on how these breakthrough theories influences your insight and point of view regarding system analysis and behaviour. For your report, you must include an introduction to the theory you will use. Then you can have a section on the technical parts (presented below), and then finally a conclusions part where you present your conclusions. 3.2 Technical content of Part II 1. Perform an s-plane analysis of the system shown in Figure 1. Determine the system impulse response using the Laplace transformation, and you are allowed to use any resource, including mathematica or Wolfram|Alpha if you want to. You could also do the work by hand if you wanted to. Compare your impulse response to the results from Part I. 2. Assume that the circuit is at rest, and τ ≈ 10T . Obtain the system response in the time domain through Laplace transformation. Compare your results to that obtained from Part I. 3. Obtain the system transfer function H (ω) = Yout (ω)Yin (ω) based on the Fourier transformation – you are allowed to use a line cut through the s-domain provided you are able to explain why that produces the Fourier transformation. Compare your results with the results you obtained in Part I. 4. Based on the discrete system model from Part I, make use of the Z transformation to determine the transfer function, do this by following the theory of section 10.11 in [1]. Study the effect of time sampling time as shown in Figure 10.7 in [1]. Explain how the Z transformation (when a cut on the unit circle of the Z plane is used) provides the basis of the phasor you studied in first year. What I am looking for here is insight as to how the Z transformation generalises the phasor theory of Steinmetz. Back up all your writings and explanations with mathematics and figures. 4 4 Project Part III: Stochastic signals and systems: Due Sunday night, week 12, 22h00 on MYLO 4.1 Project report layout for Part III The report layout for this part is quite free and I am not going to provide guidelines. I leave it up to you. But as always historic perspectives are important, as are detail and results of course. 4.2 Technical content for Part III For Part III you can choose any one topic of the list below, and submit your report on it. The first topic (call-out game) I consider the project with the best chance for a positive outcome. The others are harder. 4.2.1 The call-out game Imagine you have a fair dice and a crooked dice (that tends to land more often on a high number) in your hand, and both these dice look identical from the outside. Also you have the ability to toss either dice without your friend (who is observing the dice outcome) being able to notice you switching a dice before tossing. Now let us imagine you toss the fair dice N times (where N is a very large number), then after a random number of additional tosses of the fair dice, you start tossing the crooked dice instead. Thus you have to toss the fair dice at least N times. Also once you start tossing the crooked dice, you cannot revert back to the fair dice. Thus you and your friend know the rules: 1) You will toss a fair dice at least N times. 2) You will start tossing a crooked dice after a random number of tosses beyond N. 3) You will not revert back to the fair dice once you started tossing the crooked dice. 4) You will keep tossing the crooked dice until your friend correctly calls you out for tossing a crooked dice. You also stop when your friend calls you out while you are tossing the fair dice (so-called false alarm). The game objective and points: 1) Your friend gains M points if he/she calls you out when you are tossing the crooked dice. The game is over, and a new game can commence. 2) For each toss of the crooked dice where your friend does not call you out, he/she looses P points. You don’t tell your friend when this happens. Typically P M, so the loss per toss is small, but over time of course loss will start to accumulate if your friend delays calling out too long. 3) If your friend calls you out (for using a crooked dice), but you are still using the fair dice, he/she looses Q points. The game is over, and a new game can commence. This is known as a false alarm or false positive in decision theory. Typically Q P, so causing a false alarm is a bad idea, as its costly. 4) Your objective is to end up with as many points as possible after you and your friend played a fixed number of games. Obviously you want to win. 1. The objective here is to program this game in MATLAB (or whatever language you prefer) and you need to design/create a decision device on software in your computer, which will tell you when to call-out. You can select N,M,P,Q as you want. Study the performance of your decision device as N,M,P,Q is varied. You can make the crooked dice as crooked as you want to in software. Obviously the more crooked it is (less uniform PDF), the easier the call-out job is. You may use MATLAB’s build in random number generators. 5 2. Obviously the objective is to call-out as soon as is possible, but an over-eager call-out algorithm will incur a penalty for false alarms. But waiting too long also is costly eventually. In theory, the longer the decision device waits to call-out, the more reliable the call-out (the higher the probability for positive detection). 3. There are many call-out strategies in the literature. This problem has been studied by many people. You can use your own ideas, or use methods from the literature. Just cite any ideas from the literature you use. So now we come to performance metrics. How do we measure the performance of your decision de- vice? The idea is to play many games on your computer with fixed N,M,P,Q. Then you collect statistics : A) probability of correct detection B) mean number of crooked tosses until detection occurred C) probability of false alarm. A few things to consider: 1. Note that you have to compare your statistics to another group/student who may have used a different method than you did. Its futile to compare your statistics to another student’s statistics if he/she used the same method, and they will be the same if you play enough games. 2. Present all the theory of how your decision device works, and the comparative statistics (results) in your report. 3. You do not have to include computer code in your report. 4. Here are some optional things to consider for those of you seeking an HD. Is there some optimal strategy for this game, that will produce the best possible performance as measured by the statis- tics? That is, a decision device that when used to play a large number of games against other decision devices, will always either win or draw? Depending on how ambitious you are, see for example [4]. 5. How many games must you play before your statistics are reliable? You need to explain how you decided this! This type of strategy to test your algorithm, that is playing many random games and estimating the statistics, is known as a Monte-Carlo analysis. Monte-Carlo analysis was invented during the war (1946) to solve problems so complex that other solutions do not exist [5]. Background information to help you A bit of history and background. There is an entire field of mathematics devoted to these sorts of games involving risk and payoff, known as game theory. Game theory was invented by the mathematician John Von Neumann and the economist Oscar Morgenstein. An important part of game theory is known as decision theory in mathematics, and it has a lot of application in practice – it has a nice wiki page devoted to it, just google it. For example its used in production lines to control quality, for data mining, quantitive finance and online preventative maintenance. Your group could play competitive games with other students/groups over the network. All you have to do is set N,M,P,Q and then play a large number of games, and see who ends up with the most points. You don’t have to report this in your report, but it will be interesting to hear who in the class came up with the best performing decision device. Years ago I had students play each other in a prac, and the mark they received for the project was directly proportional to their total number of points scored in playing games over the network. But many students didn’t like competing against other students, so I decided not to do that again. 6 4.2.2 Ergodic theorem Study and discuss the historic development and applications of the ergodic theorem. This includes its use in the kinetic theory of gases, as well as electrical/electronics engineering. 4.2.3 Wiener process Study and discuss a Wiener process, and provide practical examples of such a process. 4.2.4 Stock exchange Study the statistical properties of stock exchange prices and price changes. For example Mandelbrot (from fractal fame) showed that commodity price changes on the NY stock exchange do not follow a normal distribution. Show that stock futures prices are not predictable. 4.2.5 Nash equilibrium Study and discuss the concept of a Nash equilibrium in economics and game theory. References [1] https://us.artechhouse.com/Linear-Systems-and-Signals-A-Primer-P1988.aspx [2] https://www.accessengineeringlibrary.com/content/book/9780071829465 [3] https://us.artechhouse.com/Electric-Circuits-A-Primer-P1947.aspx [4] https://projecteuclid.org/euclid.aos/1024691466 [5] https://en.wikipedia.org/wiki/Monte_Carlo_method 7