Question

Compute the continuous Fourier Transform for (a) x(t)=u(t)−u(t−4); (b) e −∣t∣ . [Hint: for the absolute value, break it up into positive and negative values to compute the integral].

Answer 1

To compute the continuous Fourier Transform for x(t) = u(t) - u(t-4), use the formula F(w) = ∫-∞∞ x(t)e-jwtdt. To compute the continuous Fourier Transform for x(t) = e^-|t|, break it up into positive and negative values and integrate separately.

Step-by-step explanation:

(a) To compute the continuous Fourier Transform for x(t) = u(t) - u(t-4), we use the formula:

F(w) = ∫-∞∞ x(t)e-jwtdt

Plugging in the given expression for x(t) into the formula, we get:

F(w) = ∫-∞∞ (u(t) - u(t-4))e-jwtdt

(b) To compute the continuous Fourier Transform for x(t) = e^-|t|, we need to break it up into positive and negative values:

F(w) = ∫-∞0 e^te-jwtdt + ∫0∞ e^-te-jwtdt