# Signals and systems39

###### What is the fundamental period of the below signal? You need to find the fundamental frequency of e and e 5 separately. The fundamental period of [n] is the least common multiple of the se two numbers.
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###### Define signal y(t) = %∞ k=−∞ x(0.5t − 10k), where x(t) = e−2t t ≥ 1 0 t < 1 (a) Determine the constant a such that the signal x(−2t +a) is borderline anticausal. (b) Is the signal y(t) periodic? If so, determine the period Ty. If not, explain why y(t) is not periodic.
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###### 1.35: Given x1(t) = cos (t), x2(t) = sin(πt), and x3(t) = x1(t)+x2(t). (a) Determine the fundamental periods T1 and T2 of signals x1(t) and x2(t). (b) Show that x3(t) is not periodic, which requires T3 =k1T1 =k2T2 for some integers k1 and k2. (c) Determine the powers Px1 , Px2, and Px3 of signals x1(t), x2(t), and x3(t).
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###### Consider the signal x(t) shown in Fig. P1.1-10. Outside the interval shown, x(t) is zero. Determine the signal energy E[x(t)].
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###### In Fig. P1.2-3, express signals x1(t), x2(t), x3(t), x4(t), and x5(t) in terms of signal x(t) and its time-shifted, time-scaled, or time-reversed versions
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###### Exa) Calculate the output 𝒚𝒌of LTID system with the following input, 𝒙𝒌, and impulse response, 𝒉𝒌 𝑥𝑛=(0.4)^𝑛*𝑢[𝑛] ℎ𝑛=(0.8)^𝑛*𝑢[𝑛]
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###### Determine the DTFT of the following signal 𝑥𝑛=4(1/2)^𝑛*𝑢(𝑛)
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###### 6 Determine the power and the rms value for each of the following signals: (a) 5+10 cos(100t +π/3) (b) 10 cos(100t +π/3)+16 sin(150t +π/5) (c) (10+2 sin 3t) cos 10t (d) 10 cos 5t cos 10t (e) 10 sin 5t cos 10t (f) ejαt cosω0t
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###### Figure P1.1-7 shows a periodic 50% duty cycle dc-offset sawtooth wave x(t) with peak amplitude A. Determine the energy and power of x(t).
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###### If the inductor voltage vL(t) is taken as the output, show that the RLC circuit in Fig. 1.34 has an input–output equation of (D2 +3D+2)vL(t) = D2x(t).
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###### If the capacitor voltage vC(t) is taken as the output, show that the RLC circuit in Fig. 1.34 has an input–output equation of (D2 +3D+2)vC(t) = 2x(t)
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###### Show that if ω1 = ω2, the power of x(t) = C1 cos(ω1t +θ1)+C2 cos(ω2t +θ2) is [C1 2 +C2 2 + 2C1C2 cos(θ1 −θ2)]/2, which is not equal to the Ex. 1.2b result of (C1 2 +C2 2)/2.
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