Complex numbers: One of the main mathematical concepts when it comes to describing physical systems, such as signals, and circuit analysis! Let's delve deep into what complex numbers are and more about its applications. What makes a number "complex"? And what is the "real" definition of a "real" number? (ha!) real vs. imaginary numbersReal numbers consist of integers, whole numbers, positive or negative, rational numbers and irrational numbers (or a combination of those!!). Imaginary Numbers on the other hand, consist of numbers with a property where it's square is negative. An imaginary number, i, represents the square root of -1, where i^2 = -1. NOTE: Sometimes imaginary numbers are denoted with a j. a complex numberA complex number is a number that's made up of a real part, and an imaginary part (that's what makes it 'complicated'). A complex number (usually denoted as z), is written in the form: a + bi, where a represents the real part of the complex number, and b represents the imaginary part of the complex number z. Note that a and b are real numbers, but since b is being multiplied by an imaginary number i, b is the imaginary part, or Im(z). A really interesting part about a complex number is that it can be represented graphically, just like how we plot coordinates in a Cartesian plane. (Like (2,4) on the Cartesian plane for example). The difference is we plot complex numbers in the Complex Plane, where the vertical axis is the imaginary axis (where imaginary numbers are plotted - i, 2i, 3i..), and the horizontal axis is the real axis (where real numbers are plotted - 1, 2, 3..). To plot a complex number a + bi, you would essentially move a steps horizontally (horizontal real axis) from the origin, and then b steps vertically or perpendicularly (vertical imaginary axis). The the magnitude of z, represents the distance from the origin [(0, 0)] to the complex number z's point. The distance is calculated by sqrt((difference in x axis)^2 + (difference in y axis)^2). So it's calculated by adding the squares of a and b and taking the square root of the sum. (Just like how you would calculate the distance in the cartesian plane or in a map, for example). For any real number, a one -dimensional number line would satisfy to plot the number. But for a complex number, we would require a 2 dimensional plane to plot it's imaginary part and real part. The really interesting part of this complex number representation is the angle, or the phase. To calculate the angle, we can use the distance between the origin and the point, r! We can say that a, the horizontal portion is equal to the cos(of the phase angle) * r. (Since cosine is adjacent over hypotonuse) and b the vertical portion is equal to sin(of the phase angle) * r. The tan of (the phase) is then equal to b/a! Since the r's cancel out when dividing. So you can then find the phase of the complex number (also called argument of z) by finding inverse tan of b/a. euler's formulaThe Euler's Law relates the these cosines and sines we see above with a complex exponential using a revolutionary formula! The Euler's formula states that: So essentially the component to the left of the equation is a complex exponential function, e raised to the power of (imaginary number * a phase/angle x). This is equal to the cos(x) + imaginary number i * sin(x). What makes this formula so intriguing is that there is a way to represent sines and cosines using complex exponentials, which can be very useful in a lot of applications. applicationsThere are numerous applications of complex numbers and complex exponentials! One of the applications is in Signals and Systems/Processing, where you can express periodic signals as a bunch of sines and cosines (sinusoidal functions in Fourier Analysis). These sinosoids can also be expressed as complex exponentials, since in LTI systems (Linear and Time Invariant systems), complex exponential inputs can make calculations much more efficient/easier! (These are also called eigenunctions of LTI systems!) Thank you!
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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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