Reflection: What have you learned about the behavior of sinusoidal functions and the effect of changing the parameters A, f. What affects the symmetry? What generalizations can you make?
All sinusoidal functions have the same general shape; however, they are not all identical. The amplitude (A) can be described as the maximum distance between the horizontal axis and the vertical position of the signal; moreover, it is the average of the difference between the maximum and minimum points of the signal ((Max-Min)/ 2). Changing this parameter will effect the maximum and minimum peaks of the signal along the vertical axis in relation to the horizontal axis. On the other hand, the fundamental frequency (f) illustrates a full cycle of the sinusoid function; moreover, changing this parameter effects how quickly/frequently the signal completes a full cycle or the number of times the signal can be observed to repeat itself. The fundamental frequency is also inversely related to the period (T0 (period) = 1 dived by f), it is described as the distance between two peaks of the sinusoid or the distance it takes for it to repeat. For basic sine and cosine functions the period is 2pi, the length of a complete cycle of the sinusoid. Lastly, the angular frequency is equal to 2pi times the fundamental frequency (w0 (angular frequency) = 2pi * f). Putting this all together, creates the equation of a continuous-time sinusoidal signal ( f(t) = Acos(w0t + theta)), where theta would describe a phase shift of the sinusoid and the sine function could also be applied in the same manner. In this Flash Lab, theta was not observed meaning no phase shifts were in any of the signals. Additionally, signals can be identified as odd or even. An even signal is any signal f such that f(t) = f(-t); even signals can be easily identified as they are symmetric around the vertical axis. An odd signal, on the other hand, is a signal f such that f(t) = -f(-t), meaning the signal has been rotated by 180 degrees about the origin. Any signal can be written as a combination of an even and an odd signal. The cosine function is an even signal, cos(-x) = cos(x). The sine function is an odd signal, sin(-x) = -sin(x). A signal f(t) is periodic if there exists a positive constant T0 such that, f(t + T0) = f(t), Vt. The smallest value of T0 which satisfies such relation is the period of the signal and a periodic signal remains unchanged when time-shifted or for integer multiples of the period. If a signal is not periodic it can be aperiodic or nonperiodic, such signals changes constantly without exhibiting a pattern or cycle that repeats over time. It has been proved by a Fourier transform that any aperiodic signal can be decompose into an infinite number of periodic signals. The first part of this lab examined periodic signals and their "vital signs" (A, f, w0, T0, odd/even/neither) where changing the vital signs lead to the effects described earlier in this submission. Adding all the equations together to form the sinusoidal function "z" illustrated that any periodic signal x(t) with fundamental frequency w0 can be represented by a linear sum of the basis functions. In the second part of the lab, it was observed that sine and cosine are the same function, just offset from each other by pi/2. It also illustrated the Fourier series expansion for a square wave being the sum of odd harmonics. Changing sin with cos on all the formulas changed how the graph looked for all graphs except for the Gibbs effect and the 3D representation. The first sine graph was a normal sine function when switched to cosine it showed the normal cosine function. The second graph with where the formula was sine with the addition of the third harmonic showed the same sine graph but with the peak inverted to create another curve. The third graph illustrated the same effect but with more inverted waves at the three peaks. Switching all these to cosine has the inverse effect where the peaks remained the same and additional waves were added along the sides of the peaks in the same fashion. Changing the alternating signs with respect to sine in the formulas on lines 4 and 6 changed the length of the peaks and in respect to cosine created graphs which were a single line.