The left-hand column below shows four (1-4) continuous-time signals. The Fourier transform of each signal appears in the right-hand column in mixed-up order. Match the signal to its Fourier transform.
Added by Robert D.
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Step 1
Signal #1 is a constant signal with a value of 1.5. The Fourier transform of a constant signal is an impulse (Dirac delta function) at frequency 0. So, the Fourier transform of signal #1 is the one with an impulse at 0, which is B. Show more…
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Fourier Transform Pairs Table 1 e^(-at)u(t) 1/(a + jw) a > 0 2 e^(at)u(-t) 1/(a - jw) a > 0 3 e^(-a|t|) 2a/(a^2 + w^2) a > 0 4 te^(-at)u(t) 1/(a + jw)^2 a > 0 5 t^ne^(-at)u(t) n!/(a + jw)^(n+1) a > 0 6 delta(t) 1 7 1 2pi delta(w) 8 e^(jw0t) 2pi delta(w - w0) 9 cos w0t pi[delta(w - w0) + delta(w + w0)] 10 sin w0t jpi[delta(w + w0) - delta(w - w0)] 11 u(t) pi delta(w) + 1/jw 12 sgn t 2/jw 13 cos w0t u(t) pi/2[delta(w - w0) + delta(w + w0)] + jw/(w0^2 - w^2) 14 sin w0t u(t) pi/2j[delta(w - w0) - delta(w + w0)] + w0/(w0^2 - w^2) 15 e^(-at) sin w0t u(t) w0/((a + jw)^2 + w0^2) a > 0 16 e^(-at) cos w0t u(t) (a + jw)/((a + jw)^2 + w0^2) a > 0 17 rect(t/tau) tau sinc(wt/2) 18 W/pi sinc(Wt) rect(w/2W) 19 delta(t/tau) tau/2 sinc^2(wt/4) 20 W/2pi sinc^2(Wt/2) delta(w/2W) 21 sum_{n=-infinity}^{infinity} delta(t - nT) w0 sum_{n=-infinity}^{infinity} delta(w - nw0) w0 = 2pi/T 22 e^(-t^2/2sigma^2) sigma sqrt(2pi)e^(-sigma^2w^2/2)
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Complete Table of Fourier Transform Pairs Function f(t) Fourier Transform, F(Δ) Definition of Fourier Transform Definition of Inverse Fourier Transform f(t) F(Δ) = ∫ f(t)e^{-jωt} dt f(t - t_0) F(ω)e^{-jωt_0} f(αt) (1/|α|)F(ω/α) f'(t) F(t) sgn(t) cos(ω_0t) sin(ω_0t) e^{jω_0t}
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