11.7. Derivation of Fourier Transform of Common Functions

Let
$f(t)= \begin{cases} 1 & |t| \leq T \ 0 & \text{otherwise} \end{cases}$
This is a rectangular pulse of width T.

Fourier Transform:
$F(\omega)= \int_{-T/2}^{T/2} e^{-i\omega t} dt = \left[\frac{e^{-i\omega t}}{-i\omega}\right]_{-T/2}^{T/2}$
$= \frac{2\sin(\omega T/2)}{\omega} \quad \Rightarrow \quad F(\omega) = T \cdot \text{sinc}\left(\frac{\omega T}{2\pi}\right)$
where sinc(x) = \frac{\sin(\pi x)}{\pi x}.
This transform is fundamental in signal analysis.

The Fourier Transform of a rectangular pulse is derived by integrating the exponential function multiplied by our pulse function over its defined limits. The rectangular pulse is defined as having a value of 1 within a certain length T and 0 otherwise. The integration provides us with a sine function, leading to the relation with the sinc function, which is common in signal processing as it describes the shape of the Fourier Transform in relation to a fundamental frequency.

Imagine turning on a light in a dark room just for a moment and then turning it off again—this is like a rectangular pulse. The light being on represents the pulse (value of 1), while it being off represents when no light is there (value of 0). The Fourier Transform helps us analyze how this 'moment of light' translates into frequency components, similar to how music is made up of different notes.

Let
$f(t)= e^{-a t}u(t), \quad a > 0, \quad u(t) \text{ is the unit step function.}$
$F(\omega)= \int_{0}^{\infty} e^{-a t} e^{-i\omega t} dt = \int_{0}^{\infty} e^{-(a+i\omega)t} dt$
Calculating this integral gives:
$= \left[\frac{e^{-(a+i\omega)t}}{-(a+i\omega)}\right]_{0}^{\infty} \quad = \frac{1}{a+i\omega}.$

To derive the Fourier Transform of the exponential decay function, we multiply our function by a complex exponential and integrate from 0 to infinity. The unit step function, u(t), ensures that the function is zero for t < 0. The integral simplifies to a rational expression, revealing how decay translates into frequency components. The presence of the parameter 'a' gives us insight into the rate of decay and affects the positioning of the result in the frequency domain.

Consider how a candle burns down over time. At first, it has a strong glow, and as time passes, the glow diminishes—this is similar to how an exponential decay function behaves. The Fourier Transform allows us to see not just the brightness of the candle but also how that changes over time in terms of frequency, providing a deeper understanding of the underlying process.

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