Multiple choice

The Fourier transform of a signal h(t) is H (j$\omega$) = (2 cos$\omega$) (sin2$\omega$)/$\omega$. The value of h(0) is

  1. ¼

  2. ½

  3. 1

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We need h(0) from H(jω). Using inverse Fourier: h(t) = (1/2π)∫ H(jω)e^(jωt) dω. At t=0: h(0) = (1/2π)∫ H(jω) dω. H(jω) = 2cos(ω)sin(2ω)/ω = [sin(3ω) + sin(ω)]/ω using identity 2sin(A)cos(B) = sin(A+B)+sin(A-B). The integral ∫[sin(ax)/x]dx from -∞ to ∞ = π for a>0. So h(0) = (1/2π)[∫ sin(3ω)/ω dω + ∫ sin(ω)/ω dω] = (1/2π)[π + π] = 1.

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