FT of periodic signals; rect pulse and constant

Fourier transform of periodic signals

A periodic signal has infinite energy, so its FT is found through its Fourier series. If $x(t)$ has period $T_0$ and $\omega_0 = 2\pi/T_0$:

$$x(t) = \sum_{k=-\infty}^{\infty} a_k\,e^{jk\omega_0 t}$$

Using the pair $e^{jk\omega_0 t} \leftrightarrow 2\pi\,\delta(\omega - k\omega_0)$ (because the inverse FT of $2\pi\delta(\omega-k\omega_0)$ is $e^{jk\omega_0 t}$) and linearity:

$$X(j\omega) = \sum_{k=-\infty}^{\infty} 2\pi\,a_k\,\delta(\omega - k\omega_0)$$

So the FT of a periodic signal is a train of impulses at the harmonics $k\omega_0$, with strengths $2\pi a_k$. Example: $\cos\omega_0 t \leftrightarrow \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$.

FT of a rectangular pulse

Let $x(t) = 1$ for $|t| < T_1$ and $0$ otherwise (width $2T_1$).

$$\begin{aligned} X(j\omega) &= \int_{-T_1}^{T_1} e^{-j\omega t}dt = \frac{e^{j\omega T_1} - e^{-j\omega T_1}}{j\omega} \\ &= \frac{2\sin(\omega T_1)}{\omega} = 2T_1\,\text{sinc}\left(\frac{\omega T_1}{\pi}\right) \end{aligned}$$

Explanation: The spectrum is a real sinc function with peak $2T_1$ (the pulse area) at $\omega = 0$ and zeros at $\omega = \pm\pi/T_1, \pm 2\pi/T_1, \dots$. A time-limited pulse has an infinitely wide spectrum. Most of the energy lies in the main lobe $|\omega| < \pi/T_1$, and a narrower pulse gives a wider main lobe (inverse time–bandwidth relation).

 x(t)                    X(jw)
 1 +-----+               2T1 ^
   |     |                  .|.
---+--+--+---> t           . | .
 -T1  0  T1          .  . .  |  . .  .
                   ----'-----+-----'-----> w
                        -pi/T1  pi/T1

FT of a constant amplitude $A$

$x(t) = A$ for all $t$ is not absolutely integrable. Take $X(j\omega) = 2\pi A\,\delta(\omega)$; its inverse is $\frac{1}{2\pi}\int 2\pi A\,\delta(\omega)e^{j\omega t}d\omega = A$. Therefore

$$A \;\longleftrightarrow\; 2\pi A\,\delta(\omega)$$

Explanation: A constant has no variation, so all its content is at $\omega = 0$: a single impulse. This also follows as the limit of the rectangular pulse as $T_1 \to \infty$: the sinc becomes taller ($2T_1A$) and narrower, approaching $2\pi A\,\delta(\omega)$. Together, the two results show the inverse relation between time and frequency: a signal that is infinitely wide in time is infinitely narrow in frequency, and a narrow pulse has a wide spectrum.