Question

Find the Fourier transform for each of the following signals, using the Fourier integral:

x(t) = 2[u(t) - u(t - 4)]

Answer 1

To find the Fourier transform of x(t) = 2[u(t) - u(t - 4)], we integrate it with respect to time in the limits from 0 to 4, and simplify the resulting expression to obtain X(f) = (1/jπf)(1 - e^{-j8πf}).

Step-by-step explanation:

To find the Fourier transform of the signal x(t) = 2[u(t) - u(t - 4)], where u(t) is the unit step function, we use the definition of the Fourier transform. The Fourier transform, X(f), of a signal x(t) is given by:

X(f) = ∫ x(t)e^{-j2πft}dt

Since x(t) is non-zero between 0 and 4, we can rewrite the integral as:

X(f) = 2 ∫_0^4 e^{-j2πft}dt

Now, we can integrate this expression:

X(f) = 2 [-rac{1}{j2πf}e^{-j2πft}]_0^4

Evaluating the limits gives us:

X(f) = rac{2}{j2πf} (1 - e^{-j2πf × 4})

Finally, this can be simplified to get the Fourier transform of our signal:

X(f) = rac{1}{jπf} (1 - e^{-j8πf})

Where j is the imaginary unit.

Answer 2

tGPT 3.5



You

Given e⁻|ᵗ| F 2/ω²+1, Find the Fourier transform of the following: ⇔ d/dt e⁻|ᵗ

The Fourier transform of the derivative of a function is given by multiplying the Fourier transform of the original function by the imaginary unit multiplied by the angular frequency. In this case:

�{�/��(�−∣�∣)}=���{�−∣�∣}F{d/dt(e−∣t∣)}=jωF{e−∣t