Fourier Transform using Transform Pair and Properties

[Mastur]

Homework Statement

Finding the Fourier Transform using Transform Pair and Properties
x(t) = 2[u(t+1)- t$^{3}$e$^{6t}$u(t)]

Homework Equations

The Attempt at a Solution

For the first problem, I got
u(t) $\leftrightarrow$ ∏δ(ω)+$\frac{1}{jω}$
F(at-t$_{0}$) $\leftrightarrow$ $\frac{1}{|a|}$F($\frac{ω}{a}$)e$^{-jt_{0}(\frac{ω}{a})}$
The problem is I don't know what's next and where will I get the t$^{6}$ and e$^{6t}$. I don't even know what to do next. Please guide me.

Hopefully when I know already how to do this, I will be able to do other problems by myself.

 

[mighty2000]

Thank you for your post. Finding the Fourier Transform using Transform Pair and Properties can be a challenging task, but with some guidance, you can easily solve this problem and understand the process for future problems. Let's break down the steps for finding the Fourier Transform of x(t) = 2[u(t+1)- t^{3}e^{6t}u(t)].

Step 1: Apply the time-shift property
The time-shift property states that if x(t) \leftrightarrow X(ω), then x(t-t_{0}) \leftrightarrow X(ω)e^{-jt_{0}ω}. In our problem, we have t^{3}e^{6t}u(t), which can be rewritten as (t^{3}u(t))e^{6t}. Using the time-shift property, we can rewrite this as (t^{3}u(t))e^{6(t-1)} = (t^{3}u(t))e^{6t}e^{-6}.

Step 2: Apply the scaling property
The scaling property states that if x(t) \leftrightarrow X(ω), then x(at) \leftrightarrow \frac{1}{|a|}X(\frac{ω}{a}). In our problem, we have (t^{3}u(t))e^{6t}, which can be rewritten as (t^{3}u(t))e^{6t}e^{0}. Using the scaling property, we can rewrite this as \frac{1}{6}(t^{3}u(t))e^{6t}.

Step 3: Apply the time-shifting property again
We now have (t^{3}u(t))e^{6t}e^{-6} = \frac{1}{6}(t^{3}u(t))e^{6t}. Using the time-shift property again, we can rewrite this as \frac{1}{6}(t^{3}u(t))e^{6(t-1)}e^{-6}.

Step 4: Apply the Fourier transform
Now that we have rewritten our function using the properties, we can apply the Fourier transform. Using the Fourier transform property, we have:
\mathcal{F}[\frac{1}{6}(t^{3}u(t))e^{6(t-1)}e^{-6}] = \frac{1}{6}\mathcal{F}[(t^{3