WHEW. WOOOOOOOOOOWIEE. As this unusual yet unique semester comes to an end, I wanted to look back at the things I've learned as a sophomore. It's insane to think that Junior Year is approaching, and I think the best way to commemorate this school year is to go through the lessons/the things I learned about, especially because of how the method of learning has changed because of these challenging times. Also, to the class of 2020, CONGRADS!!! I wish you all the best for your future, and I know you are going to rock it! So I learned lots of things from all my classes (Of course!!), but I think my favorite class by far was the Systems and Signals class. The concepts I learned in my signals class were so applicable to real life scenarios! For instance, when you send a message to another device in your WiFi, an electronic signal transmits the bits of info (data) to the router which transmits It to the receiver! Signals are everywhere! So, these were some of the key points I felt were pretty 'special' to me this semester: 1. Frequency Domain vs. Time DomainUnderstanding the two different types of domains in signals and systems is important and very interesting! The time domain graph shows how a particular signal (waveform, sinusoid) changes over time. "Frequency" is essentially the number of times a particular instance of a signal occurs. So, the frequency time domain graph shows how much of the signal is in a frequency band over a range of frequencies. 2. Lesson on Fourier transformThis topic was very intriguing, and I wanted to dedicate to it in a separate post (coming soon!!). 3. Thinking "outside" of the box ProblemsThere were many problems in my classes that were challenging to solve, but one particular problem that comes to my mind was in my Digital Logic Design class. We were asked to design a grading system, using only a certain number logical components (a 2-by-1 MUX, 2x4 Decoder, and 2 Comparators). Since there was a restriction as to how many digital logical components you can use, I couldn't just use as many components as I wanted, wherever I want. I needed to understand the function of every component and place them in the appropriate place. It was a tough one, since I had to essentially "experiment" with every possible combination placement of these components. I only had two comparators, and I had to decide whether the input number was a grade A, B, C or D. It was a tricky one, and I finally had to see the answer because my brain was literally about to explode lol. But as soon as I saw the solution, I finally had my AHA moment and realized that I could use the MUX as a component to restrict the possible range of our input. Thinking outside the box is all about understanding the function and using it where it is the most applicable (especially if you don't have that many resources). 4. Real Life Scenario ConnectionsReal Life Scenarios. Super important, especially in my two EE classes. Digital Logic Design is all about the digital components that exist in our computers today, and we essentially looked into the several smaller components that make up these important parts. It's just insane to learn about what an SRAM or DRAM (new posts, maybe?!) looks like, since we have been hearing these words forever but I never really understood what they meant (or how they looked liked) UNTIL NOW!! (no flex) 5. ReflectionsI'd like to end it off by saying, its been a great and CRAZY year. I know it sounds cliché, but sophomore year was a pretty valuable year to me. I actually learned A LOT, and not just about academics/electrical engineering/computer science, but about life. How important it is to be a be a good citizen, stay home, help others, and help yourself of course. How life can give you the most unexpected turns, and how you should just keep on going, and keep on moving. I hope you enjoyed your year as a freshman/sophomore/junior/senior, and once again thank you for reading! I wish you the best, and GOOD LUCK!
Thank you, Aarushi Ramesh :)
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A reasonable probability is the only certainty. - E.W. Howe The best way to describe the term "Probability" is that its a combination of very interesting concepts with 500 things to keep in mind about (ok maybe not 500 but a LOT of formulas lol). Yeah, I feel like Probability has always been the math thats "weird" and different--not in the bad way though! I think once you learn a lot about its applications in the real world, it gets very interesting and exciting! If you're interested in the probability concepts of sets, unions and intersections, I got you covered! Its on this link! If you want you learn more about Probability models and functions, this is for you! (hopefully lol) probability models & PMFsSo whenever we are considering the probability of a certain event to occur, we are also considering the total number of possible outcomes. How can we lay out the possible outcomes for a particular situation (ex. flipping a fair coin n times, or finding the number of k successes in t trials) with certain values associated with it, like for example the number '1' to represent the probability of heads to appear, and the number '0' to represent the probability of tails to appear?? (yeah that was the longest question in the world lol). In order to answer this question, we need to learn about the Probability Mass Function, aka PMF. The PMF basically maps out certain values to the probabilities (outcomes) of an experiment or situation. But what are those 'certain values'? These are called random variables, or discrete random variables. A Random Variable is a variable that assigns a value to a certain outcome of an experiment. We use a capital letter to represent the random variable of an experiment. So for example, lets say the experiment was to test 10 different circuits and see if they work (a success, denoted as s) or they need improvement (an error, denoted as e). Every observation in this case is a sequence of 10 different letters (s or e). So the sample space or the total number of possible outcomes (sequences) is 1024 (2^10). The random variable in this case, lets say K, can be the number of successful circuits in a sequence. So for a certain outcome, sssssseeee, the random variable K = # of successes = 6. Since the sequence is a set of 10 letters, the range of K has to be from 0-10. K is a discrete random variable, since the range of K can be listed. (even if it was infinitely long). goal scoring problemOk lets look at another example with pmf and random variable (RV) stuff. Suppose you are playing soccer and have two free kicks. A free kick can lead to two possibilities: a score (in the goal, denoted as S) or no score (N). What is the PMF of the random variable G, the number of free kicks scored? We can make a table to highlight the probability of the scoring a certain outcomes and its relationship with the random variable G! Since we are assuming that each outcome is equally likely, the probability of getting a goal in the first try and not getting a goal in the second try is just = 1/2*1/2. This is the same for every other outcome.
Since the random variable G has three possible values {0, 1, 2}, the probabilities of these three possible values are: P[X=0] = 1/4, P[X=1] = 1/4 + 1/4 = 1/2, P[X=2] = 1/4. One way of representing the PMF is by a plot or graph: Note: The PMF/probability is represented with a P and then a subscript of the random variable, and then a lower case version of the random variable in the brackets, because the notation says that this is "the probability of a random variable G equaling some value g (which is in the set of outcomes)". So g is an actual number, which is in the range of G, the random variable. binary symmetric channelOk this is a trickier problem, and what I really liked about this one was that it is has such a great connection with electrical engineering. So, we are sending a 1000 bits into some data channel, and the probability there is a bit error (the bit was the wrong bit received, for example 1 is a 0 or 0 is a 1) = P[error] = 0.02. So 1-P[error] = success. So the question is, what is the probability there are 10 errors? Ok so first, we need to think about how many possible combinations are there if there are 10 errors out of 1000 bits sent into the channel. In order to figure out the total number of possibilities we do 1000 choose 10 (the combinations formula). We would then use the binomial distribution (since there are only 2 possible values, 1 or 0 to find the PMF: If we plug in P for 0.02, we get about P[10 errors] ~~ 0.0055, which is a relatively low probability!
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