StudyGPT Just Leveled Up – Say Hello to Knowee! 🚀
Knowee
Questions
Features
Study Tools

Evaluate the Fourier Transform of following continuous time signal:[i] x(t) = u(-t) * eA(-at) [ii] x(t) = 6 (t)

Question

🧐 Not the exact question you are looking for?Go ask a question

Solution

To evaluate the Fourier Transform of the given continuous-time signals, we will follow the steps outlined.

1. Break Down the Problem

We have two signals to analyze:

  1. x(t)=u(t)eat x(t) = u(-t) e^{-at}
  2. x(t)=6δ(t) x(t) = 6 \delta(t) (assuming 6(t) 6(t) is a delta function)

2. Relevant Concepts

The Fourier Transform X(f) X(f) of a continuous-time signal x(t) x(t) is defined as:

X(f)=x(t)ej2πftdt X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt

We will apply this definition to both signals.

3. Analysis and Detail

For the first signal x(t)=u(t)eat x(t) = u(-t) e^{-at} :

  • The unit step function u(t) u(-t) is equal to 1 for t<0 t < 0 and 0 for t0 t \geq 0 . Thus, the expression simplifies to: x(t)=eatfor t<0 x(t) = e^{-at} \quad \text{for } t < 0
  • The Fourier Transform becomes: X(f)=0eatej2πftdt X(f) = \int_{-\infty}^{0} e^{-at} e^{-j2\pi ft} dt

We can simplify this integral: X(f)=0e(a+j2πf)tdt X(f) = \int_{-\infty}^{0} e^{-(a + j2\pi f)t} dt Evaluating the integral results in: X(f)=[e(a+j2πf)t(a+j2πf)]0=1a+j2πf X(f) = \left[ \frac{e^{-(a + j2\pi f)t}}{-(a + j2\pi f)} \right]_{-\infty}^{0} = \frac{1}{a + j2\pi f}

For the second signal x(t)=6δ(t) x(t) = 6 \delta(t) :

  • The Fourier Transform of the delta function is given by: X(f)=6δ(t)ej2πftdt X(f) = 6 \cdot \int_{-\infty}^{\infty} \delta(t) e^{-j2\pi ft} dt Using the sifting property of the delta function: X(f)=6 X(f) = 6

4. Verify and Summarize

We have evaluated the Fourier Transforms for both signals:

  1. For x(t)=u(t)eat x(t) = u(-t) e^{-at} : X(f)=1a+j2πf X(f) = \frac{1}{a + j2\pi f}

  2. For x(t)=6δ(t) x(t) = 6 \delta(t) : X(f)=6 X(f) = 6

Final Answer

  • For x(t)=u(t)eat x(t) = u(-t) e^{-at} : X(f)=1a+j2πf X(f) = \frac{1}{a + j2\pi f}
  • For x(t)=6δ(t) x(t) = 6 \delta(t) : X(f)=6 X(f) = 6

This problem has been solved

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.