# Given a Fourier transform pair x(t) ↔ X(jω) = $\frac{2 sin⁡ω}{ω}$, where, x(t) = 1 for |t|<1 and 0, otherwise. Then the Fourier transform of y(t) having the shape of a triangular waveform from t=-2 to t=2 and maximum peak value=2 is ___________

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Given a Fourier transform pair x(t) ↔ X(jω) = $\frac{2 sin⁡ω}{ω}$, where, x(t) = 1 for |t|<1 and 0, otherwise. Then the Fourier transform of y(t) having the shape of a triangular waveform from t=-2 to t=2 and maximum peak value=2 is ___________

(a) $\frac{4 sin^2 ω}{ω^2}$

(b) $\frac{2 sin^2 ω}{πω^2}$

(c) $\frac{8π sin^2 ω}{ω^2}$

(d) $\frac{8 sin^2 ω}{ω^2}$

This question was posed to me in examination.

This intriguing question originated from Characterization and Nature of Systems topic in chapter Laplace Transform and System Design of Signals and Systems

by (42.1k points)
Right choice is (a) $\frac{4 sin^2 ω}{ω^2}$

To elaborate: We know that, y (t) = x (t)*x (t)

Or, Y (jω) = (X (jω))^2 = $\frac{4 sin^2 ω}{ω^2}$.

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