GATE 2023 EE – Question 49
The period of the discrete-time signal $x[n]$ described by the equation below is $N=$ ______________ (Round off to the nearest integer).
$$x[n]=1+3\sin\left(\frac{15\pi}8n+\frac{3\pi}4\right)-5\sin\left(\frac\pi3n-\frac\pi4\right)$$
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Correct answer: 48
Explanation
$\dfrac{15\pi}{8}=2\pi\cdot\dfrac{15}{16}$, so the second term has period 16. $\dfrac\pi3=2\pi\cdot\dfrac16$, so the third term has period 6. The signal repeats when both repeat: $\mathrm{lcm}(16,6)=48$.