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(a) (3 marks) Prove cos 3x = 4 cos x – 3 cos x . (b) (17 points) You are asked to design a DSB-SC modulator to generate a modulated signal km(t) cos wct, where m(t) is a signal bandlimited to B Hz (See Figure 2). Figure 3 shows a DSB-SC modulator available. The bandpass filter is tuned to wc. The carrier generator available generates not cos wet but cos' wet. Mw) A -21B 0 27B w Figure 2: Frequency spectrum of m(t) for Problem 3(b). m (t) km(t)cost Bandpass Filter శారా cos300 (Carrier) Figure 3: DSB-SC modulator Problem 3(b). (i) (5 marks) Explain whether you would be able to generate the desired signal using only this modulator. If so, what is the value of k as shown in point c in Figure 3? (ii) (4 points) Determine the signal spectra at points b and c given in Figure 3, and indicate the frequency bands occupied by these spectra. (iii) (2 points) What is the minimum usable value of wc? (iv) (4 points) Would this scheme work if the carrier generator output were cosa wet? Explain. (v) (2 points) Would this scheme work if the carrier generator output were cos” wet for any integer n > 2?