Fourier transform of Rect and sinc functions

Question

How to transform rect function $\Pi(t/2)$ to $8 \text{sinc}(4t)$

Is there a way without using the inverse formula?

Answer 1

First of all, let me say that your question was not clear. Eventhough, I will proceed computing the Fourier transform of $x(t) = \Pi (t/2)$, which is, I guess, what you are asking for. Note that $x(t)$ can be expressed as

$$ x(t) = \begin{cases} 1 \quad \text{if |x|<1} \\ 0 \quad \text{otherwise} \end{cases}. $$ Let us take $x(t)$ and simply apply the general Fourier transform definition and see what happens. I will be using the ordinary frequency $f$.

\begin{equation} \begin{split} \mathcal{F}\{x(t)\} &= \int_{-\infty}^{\infty} x(t) ~e^{-j2 \pi f t} dt = \int_{-1}^{1}1 \cdot e^{-j2 \pi f t} dt = \int_{-1}^{1} ~e^{-j2 \pi f t} dt= \bigg[\frac{e^{-j2 \pi f t}}{-j2 \pi f}\bigg]_{-1}^1 = -\frac{1}{ j2 \pi f}\bigg[{e^{-j2 \pi f t}}\bigg]_{-1}^1 \\ &= \frac{1}{j2 \pi f} (e^{j2 \pi f}-e^{-j2 \pi f}), \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(1)} \end{split} \end{equation}

where $j = \sqrt{-1}$. By definition, we have that $\sin(x) = \frac{e^{jx}-e^{-jx}}{2j}$, and hence we can reformulate (1) as

\begin{equation} \mathcal{F}\{x(t)\} = \frac{\sin(2\pi f)}{\pi f} = \frac{2 \sin(2\pi f)}{2 \pi f} \end{equation}

By definition, we know that $\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$, and hence we finally obtain that

\begin{equation} \mathcal{F}\{x(t)\} = 2\text{sinc}(2f) \end{equation}

Answer 2

Fourier Transform is used for digital signal processing. The standard form of F.T. is F(ω)=∫f(t)e^(-iωt) dt for -∞≤t≤∞. For example, find the F.T. of a rectangle function, rect(t), for rect(t)={1 if -1/2<t<1/2, 0 otherwise}: The product f(t)rect(t) can be understood as the signal turned on at t=-1/2 and turned off at t=1/2. So, F(ω)=∫(1)e^(-iωt) dt. Integrate the integral from -1/2 to 1/2 and the F.T. for f(t) is F(ω)=sinc(ω/2). The sinc function is written as sinc(x) and has two forms: normalized and unnormalized. Digital signal processing usually uses the normalized form.