Question

could someone please help me solve the marked question question? (3-19)

x (r) 147 FIGURE P3-18 3-19./The phenomenon non of "beats" when two slightly out of tune band instruments play the same nomi- nal note at the same time may be seen by considering the signal Use trigonometric identies to rewrite this in the form That is, find explicit expressions for A (t) and e0 in terms of Δ Plot the waveform for the case x(1)-cos(ay) + cos[(a0 + Δω)11 of ay 2n: 1000 hertz and Δα/2 π 5 hertz. Discuss how this would sound to the ear. Section 3-7 an. Pot the two-sided amplitude and phase spectra for the ful ave tifed sice ve rectif 2of Table 3-1). Convert the two-sided plots to single-sided plots. forms 4 and S of compare amplitude spectra for the triangular and square waves (wave handwidth for a given fidelity of reproduction?

Answer 1

x(t) \ = cos (w_o t) + cos [(w_0 - delta w)t] = A(t) cos [w_0 t + theta (t)] using identity cos (C) + cos (D) = 2 cos (C + D/2) cos (C - D/2) Here C = w_0 t D = (w_0 + delta w)t C + D/2 = (2 w_0 + delta w)t/2 = (w_0 - delta w/2)t C - D/2 = (w_0 - w_0 - delta w) t/2 = -delta wt/2 Now putting value of C + D/2 and C + D/2 cos (w_0 t) + cos (w_0 - w_0 - delta w) t = 2 [(w_0 + delta w/2) t] . cos (- delta w t/2) Now cos (- theta) = cos theta So, = 2 cos (delta w t/2) cos [(w_0 + (- delta w t/2) t] Here A(t) = 2 cos (delta t/2) theta(t) = (delta w t/2)