Question

Given the Fourier Transform pairs below, find the Fourier transform for each signal.

e⁻ᵇᵗu(t)↔ 1 / jω + b , where b > 0
δ(t-c)↔ e⁻ʲʷᶜ, where c is any real number


X₁(t) = e⁻³ᵗ eʲ⁴ᵗu(t)
(Hint: use Fourier Transform Properties. Please indicate which properties you used.

Answer 1

The Fourier transform of the given signals are 1/(jω + b), e^(-jωc), and 1 / ((jω + 3)(j(ω - 4))) respectively.

Step-by-step explanation:

To find the Fourier transform for each signal, we can use the Fourier Transform properties. For the first signal, e^(-bt)u(t), the Fourier transform is 1/(jω + b), where b > 0. For the second signal, δ(t-c), the Fourier transform is e^(-jωc), where c is any real number. For the third signal, X₁(t) = e^(-3t)e^(4t)u(t), we can use the time-shifting property of the Fourier transform. The Fourier transform of e^(-3t) is 1 / (jω + 3) and the Fourier transform of e^(4t)u(t) is 1 / (j(ω - 4)). Multiplying these two transforms together, we get the Fourier transform of X₁(t) as 1 / ((jω + 3)(j(ω - 4))).