I am reading a book by Zoran Gajic titled Linear Dynamic Systems and Signals.
There were two problems in chapter 3 that I was curious about.
The first question asks for the Fourier Transform of a function $x^2(t-5)$, given the Fourier Transform of $x(t) = \mathrm{X}(jw)$. Is my approach correct?
$$\mathrm{X}(jw) = \frac{1}{jw}$$
Then, from the tables, we know:
$$ x(t) = \frac{1}{2} \text{sgn} (t)$$
Then, $$ x^2(t) = \begin{cases} \dfrac{1}{4}, t \neq 0 \\ \\ 0, t = 0 \end{cases}$$
The Fourier Transform of two different functions $f(t)$ and $g(t)$ are the same if they are both bounded and differ at a finite number of points. Thus, the Fourier Transform of $x^2(t)$ is the same as the Fourier Transform of $g(t) = \frac{1}{4}$.
$$ \mathfrak{F}\left(\frac{1}{4}\right) = \frac{\pi}{2} \delta(w)$$
Then, applying the time-shifting property,
$$ \mathfrak{F}\left(x^2(t-5)\right) = e^{-jw5} \frac{\pi}{2} \delta(w) = \frac{\pi}{2} \delta(w)$$
My second question is, what is the Inverse Fourier Transform (or how do I approach the Inverse Fourier Transform) of the following?
$$ \mathrm{Y}(jw) = \frac{jw}{5 + jw}$$
Specifically, it is the $jw$ term in the numerator that is making the problem difficult for me. I know the inverse of the denominator, and I am aware of the Fourier Transform Property that convolution in the time domain corresponds to multiplication in the frequency domain.
Thank you for your assistance.
Best Answer
Given that $$x(t)\Longleftrightarrow X(j\omega)$$ the given signal can be written as $$x^2(t-5) = x^2(t)\star \delta(t-5)$$ where $\star$ denotes convolution and $\delta(t)$ is Dirac delta function. Thus, by using multiplication in time-domain, convolution in time-domain and time-shifting properties we easily find that $$x^2(t-5)\Longleftrightarrow \dfrac{1}{2\pi}\left[X(j\omega)\star X(j\omega)\right]e^{-j5\omega}$$