Question

Problem 2: Consider the following periodic signals x(t), a square wave, and yt), a saw tooth 2T The pulses width of x(t) т, wave. Both have the same amplitude A and the same frequency - equal T. The duty-cycle of x(t) is defined as d- T. -A From tables of Fourier Series ofvarious periodic signals, the following formulas are given for your convenience x(= Ad+2Adnacos at+2Ad sna cos 2at+2Adl sun 3xdcos3at яd 2лd Зяd 24 (sin a 1 sin 2asin3ajain 4s +..) у)- t-- The hamonics of a periodic signal don't necessarly become smaller as the hamonic index k increases. In y(t) it happens (for the absolute value the harmonics). In x(t) the harmonics may come in bundles, and the bundles, on the over all, become smaller and smaller in their absolute values. If we approximate each of the signals by adding N (finitely many) harmonics, let us neglect N+1 and higher hamonics that are less than 5% of the first harmonic (in y(t)). In x(t) let's neglect of the first bundle peak. bundles of harmonics in which the bundle maximm is smaller than 2.1. How many harmonics N are needed to approximate y(t)? Plot on the same scale the original saw tooth function and the one approximated by N harmonics. Take A and oo of your choice and using MATLAB plot 2-3 periods 2.2. Repeat 2.1 for x(t), assuming that d 0.5 2.3. Repeat 2.2, assuming that d=0.2. Is it true that the smaller d, the larger is N (needed for the 5% approximation)?