Question

If $x(t) = \sum_{n=-\infty}^{\infty} \cos(t)\delta(t - n)$ has Fourier transform $X(\omega)$, then what is the value of $\int_{-\infty}^{\infty} X(\omega)d\omega$? (a) $2\pi$ (b) $\pi$ (c) $4\pi$ (d) $\pi/2$ (e) $\pi/4$ (f) $2\pi^2$ (g) $4\pi^2$ (h) $2/\pi$ (i) 2 (j) 4 (k) 0 (l) None of these

          If $x(t) = \sum_{n=-\infty}^{\infty} \cos(t)\delta(t - n)$ has Fourier transform $X(\omega)$, then what is the value of $\int_{-\infty}^{\infty} X(\omega)d\omega$? 
(a) $2\pi$
(b) $\pi$
(c) $4\pi$
(d) $\pi/2$
(e) $\pi/4$
(f) $2\pi^2$
(g) $4\pi^2$
(h) $2/\pi$
(i) 2
(j) 4
(k) 0
(l) None of these

If x(t) = ∑n=-∞^∞cos(t)δ(t - n) has Fourier transform X(ω), then what is the value of ∫-∞^∞ X(ω)dω?
(a) 2π
(b) π
(c) 4π
(d) π/2
(e) π/4
(f) 2π^2
(g) 4π^2
(h) 2/π
(i) 2
(j) 4
(k) 0
(l) None of these

Added by Victor Manuel W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

01/12/2023

Step 1

Step 1: The Fourier transform of $x(t)$ is given by: $$X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt$$ Substituting the given expression for $x(t)$, we get: $$X(\omega) = \int_{-\infty}^{\infty} \sum_{n=-\infty}^{\infty} cos(t)\delta(t-n)e^{-j\omega t} Show more…

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If $x(t) = \sum_{n=-\infty}^{\infty} cos(t)\delta(t-n)$ has Fourier transform $X(\omega)$, then what is the value of $\int_{-\pi}^{\pi} X(\omega)d\omega$? (a) $2\pi$ (b) $\pi$ (c) $4\pi$ (d) $\pi/2$ (e) $\pi/4$ (f) $2\pi^2$ (g) $4\pi^2$ (h) $2/\pi$ (i) $2$ (j) $4$ (k) $0$ (l) None of these