Fourier series enable us to represent a periodic function as sum of sinusoids and to obtain the frequency spectrum from the series. The Fourier transform allows us to extend the concept of a frequency spectrum to non-periodic functions.
The Fourier transform is an integral transform like the Laplace transform. The Fourier transform is very useful in communications systems and digital signal processing, in situations where the Laplace transform does not apply.
The Fourier transform is an integral transformation of \(f(t)\) from the time domain to the frequency domain.
\(\boxed{F(\omega) = \mathcal{F}\left[f(t)\right] = \int_{-\infty}^{\infty}f(t)\,e^{\displaystyle -j\omega t}\,dt}\)
where \(\mathcal{F}\) is the Fourier transform operator.
In general, \(F(\omega)\) is a complex function; its magnitude is called the amplitude spectrum, while its phase is called the phase spectrum. Thus \(F(\omega)\) is the spectrum.
The inverse Fourier transform can be obtained by
\(\boxed{f(t) = \mathcal{F}^{-1}\left[F(\omega)\right] = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\,e^{\displaystyle\,j\omega t}\,d\omega}\)
The Fourier transform \(F(\omega)\) exists when the Fourier integral converges. A sufficient but not necessary condition that \(f(t)\) has a Fourier transform is that it be completely integrable in the sense that
\(\displaystyle \int_{-\infty}^{\infty}\left|f(t)\,e^{\displaystyle -j\omega t}\right|\,dt < \infty\)
But \(\displaystyle \left|e^{\,j\omega t}\right| = 1\) for any value of \(t\),
\(\boxed{\int_{-\infty}^{\infty}\left|f(t)\right|\,dt < \infty}\)
To avoid the complex algebra that explicitly appears in the Fourier transform, it is sometimes expedient to temporarily replace \(j\omega\) with \(s\) and then replace \(s\) with \(j\omega\) at the end.
Properties of Fourier Transform
| Property | \(\displaystyle f(t) = \mathcal{F}^{-1}\left[F(\omega)\right]\) | \(\displaystyle F(\omega) = \mathcal{F}\left[f(t)\right]\) |
|---|---|---|
| Linearity | \(\displaystyle a_1 f_1(t) + a_2 f_2(t)\) | \(\displaystyle a_1 F_1(\omega) + a_2 F_2(\omega)\) |
| Scaling | \(\displaystyle f(at)\) | \(\displaystyle \frac{1}{\lvert a \rvert}F\left(\frac{\omega}{a}\right)\) |
| Time shift | \(\displaystyle f(t - a)\) | \(\displaystyle e^{\displaystyle -j\omega a}F(\omega)\) |
| Frequency shift | \(\displaystyle e^{\displaystyle j\omega_{0}t}f(t)\) | \(\displaystyle F(\omega - \omega_0)\) |
| Modulation | \(\displaystyle \cos(\omega_{0}t)\,f(t)\) | \(\displaystyle \frac{1}{2}\left[F(\omega + \omega_0) + F(\omega - \omega_0)\right]\) |
| Time differentiation | \(\displaystyle f^{(n)}(t)\) | \(\displaystyle (j\omega)^{n}F(\omega)\) |
| Time integration | \(\displaystyle \int_{-\infty}^{t}f(\tau)\,d\tau\) | \(\displaystyle \frac{F(\omega)}{j\omega} + \pi F(0)\,\delta(\omega)\) |
| Frequency differentiation | \(\displaystyle t^{n}f(t)\) | \(\displaystyle (j)^{n}F^{(n)}(\omega)\) |
| Reversal | \(\displaystyle f(-t)\) | \(\displaystyle F(-\omega) = F^{\ast}(\omega)\) |
| Duality | \(\displaystyle F(t)\) | \(\displaystyle 2\pi f(-\omega)\) |
| Convolution in \(t\) | \(\displaystyle f_1(t) \ast f_2(t)\) | \(\displaystyle F_1(\omega)\,F_2(\omega)\) |
| Convolution in \(\omega\) | \(\displaystyle f_1(t)\,f_2(t)\) | \(\displaystyle \frac{1}{2\pi}F_1(\omega)\ast F_2(\omega)\) |
Fourier Transform Pairs
| Function | \(\displaystyle f(t) = \mathcal{F}^{-1}\left[F(\omega)\right]\) | \(\displaystyle F(\omega) = \mathcal{F}\left[f(t)\right]\) |
|---|---|---|
| unit impulse | \(\displaystyle \delta(t)\) | \(\displaystyle 1\) |
| \(\displaystyle 1\) | \(\displaystyle 2\pi\,\delta(\omega)\) | |
| delayed unit impulse | \(\displaystyle \delta(t - t_0)\) | \(\displaystyle e^{\displaystyle -j\omega t_0}\) |
| unit step | \(\displaystyle u(t)\) | \(\displaystyle \pi\,\delta(\omega) + \frac{1}{j\omega}\) |
| \(\displaystyle u(t + \tau) - u(t - \tau)\) | \(\displaystyle \frac{2\sin(\omega\tau)}{\omega}\) | |
| \(\displaystyle \lvert t \rvert\) | \(\displaystyle -\frac{2}{\omega^2}\) | |
| signum | \(\displaystyle \mathrm{sgn}(t)\) | \(\displaystyle \frac{2}{j\omega}\) |
| \(\displaystyle e^{-at}u(t)\) | \(\displaystyle \frac{1}{a + j\omega}\) | |
| \(\displaystyle e^{at}u(-t)\) | \(\displaystyle \frac{1}{a - j\omega}\) | |
| \(\displaystyle t^{n}e^{-at}u(t)\) | \(\displaystyle \frac{n!}{(a + j\omega)^{n + 1}}\) | |
| \(\displaystyle e^{-a\lvert t\rvert}\) | \(\displaystyle \frac{2a}{a^2 + \omega^2}\) | |
| \(\displaystyle e^{j\omega_{0}t}\) | \(\displaystyle 2\pi\,\delta(\omega - \omega_0)\) | |
| \(\displaystyle \sin(\omega_{0}t)\) | \(\displaystyle j\pi\left[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)\right]\) | |
| \(\displaystyle \cos(\omega_{0}t)\) | \(\displaystyle \pi\left[\delta(\omega + \omega_0) + \delta(\omega - \omega_0)\right]\) | |
| \(\displaystyle \sin(\omega_{0}t)\,u(t)\) | \(\displaystyle \frac{j\pi}{2}\left[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)\right] + \frac{\omega^2}{\left.\omega_{0}\right.^2 - \omega^2}\) | |
| \(\displaystyle \cos(\omega_{0}t)\,u(t)\) | \(\displaystyle \frac{\pi}{2}\left[\delta(\omega + \omega_0) + \delta(\omega - \omega_0)\right] + \frac{j\omega}{\left.\omega_{0}\right.^2 - \omega^2}\) | |
| \(\displaystyle e^{-at}\sin(\omega_{0}t)\,u(t)\) | \(\displaystyle \frac{\omega_0}{(a + j\omega)^2 + \left.\omega_{0}\right.^2}\) | |
| \(\displaystyle e^{-at}\cos(\omega_{0}t)\,u(t)\) | \(\displaystyle \frac{a + j\omega}{(a + j\omega)^2 + \left.\omega_{0}\right.^2}\) |