Question #226175

Classify the following signal as energy signal or power signal. Find the

normalized energy or normalized power.

(i) sin 2t + 3 cos 4t

Expert's answer

E=\int_{-\infin}^\infin\|{x(t)}\|^2dt\\ x(t)=\sin{2t}+3\cos{4t}\\ E=\int_{-\infin}^\infin({ \sin{2t}+3\cos{4t}})^2dt\\ E=\int_{-\infin}^\infin({ \sin^2{2t})dt+\int_{-\infin}^\infin( 9\cos^2{4t}})dt+\int_{-\infin}^\infin(6\sin{2t}\cos{4t})dt\\ E=\infin

Signal has infinite energy so it's not an energy signal

x(t)=\sin{2t}+3\cos{4t}\\ x_1(t)=\sin{2t}\\ \omega_1=2\\ T_1=\dfrac{2\pi}{\omega_1}\\ T_1=\dfrac{2\pi}{2}=\pi\\ x_2(t)=3\cos{4t}\\ \omega_2=4\\ T_2=\dfrac{2\pi}{\omega_2}\\ T_2=\dfrac{2\pi}{4}=\dfrac{\pi}{2}\\

T_o=T_1=2T_2=1\\

P=\dfrac{1}{T_o}\int_0^{T_o}\|{x(t)}\|^2dt\\ P=\dfrac{1}{1}\int_0^{1}\|{\sin{2t}+3\cos{4t}}\|^2dt\\ P=\int_0^{1}\sin^2{2t}dt+9\int_0^{1}\cos^2{2t}dt+6\int_0^{1}\sin{2t}\cos{3t}dt=5-\sin4=4.93J

signal is a power signal

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