Question

Colculate the FT of ( e⁻²⁽ᵗ⁻¹⁾ u(t-1) . Sketch & label the mapnitude of each FT

Answer 1

Let's find the Fourier transform of the given function \( x(t) = e^{-2(t-1)}u(t-1) \), where \( u(t) \) is the unit step function. First, express \( x(t) \) in a piecewise form:

\[ x(t) = \begin{cases}

e^{-2(t-1)} & \text{if } t > 1 \\

0 & \text{otherwise}

\end{cases}

\]

Now, the Fourier transform \( X(f) \) of \( x(t) \) is given by:

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

Performing the integration:

\[ X(f) = \int_{1}^{\infty} e^{-2(t-1)} e^{-j2\pi ft} dt \]

Solving this integral yields the Fourier transform. The magnitude plot of the Fourier transform will depend on the specific values obtained.

The key features to observe when sketching the magnitude plot include any exponential decay and how it affects the overall shape of the function.

If you provide specific numerical values or ask for a particular part of the plot, I can offer more detailed information.

Answer 2

The Fourier Transform of e^{-2(t-1)} u(t-1) is found by applying the time-shifting property to the transform of e^{-2t}u(t) which yields e^{-j\omega}/(2+j\omega). The magnitude decays symmetrically as the frequency increases, which can be sketched and labeled accordingly.

Step-by-step explanation:

The question involves calculating the Fourier Transform (FT) of a given function, e^{-2(t-1)} u(t-1), where u(t-1) is the unit step function that is activated at t=1. To find the Fourier Transform, we can use the property of time-shifting and the known transform of e^{-at}u(t). In this case, a=2 and the time shift is 1 unit. We compute the FT as:

1. Recognize that e^{-2(t-1)} u(t-1) can be expressed as e^{-2t}u(t) shifted to the right by 1 unit.
2. Find the Fourier Transform of e^{-2t}u(t), which is \frac{1}{2+j\omega}.
3. Apply the time-shifting property: e^{-j\omega}\times\frac{1}{2+j\omega} is the FT of the original function.

The magnitude of this Fourier Transform is \frac{1}{\sqrt{4+\omega^2}}, and it decays as \omega increases.

To sketch, plot the magnitude against frequency \omega, starting from zero and decaying as \omega moves away from zero in both the positive and negative directions, creating a symmetric curve around the vertical axis. Label the curve with its expression for magnitude.