5.3 The Fourier transform

We are finally ready to define the Fourier transform of a signal \(x(t)\):

Definition 16 (Fourier transform)

The Fourier transform of a signal \(x(t)\) is

\[X(\omega) = \int_{-\infty}^{\infty} x(t)\, e^{-j\omega t}\, dt.\]

It maps an angular frequency \(\omega\) to a single complex number, \(X(\omega) : \mathbb{R} \to \mathbb{C}\).

At last, a function of frequency. Frequency is the input, and the output is a single complex number summarizing how much of that frequency is present in \(x(t)\) (and at what phase).

Note

Calculus is not a focus of this book. The Fourier transform does contain an integral, but you will not be asked to work through tricky integration here. Later, we will derive a discrete version of the transform that replaces the integral with a finite sum, turning it into a concrete computational tool rather than a mathematical one.

This definition is the direct payoff of the previous section. Look closely at the integrand: the term \(e^{-j\omega t}\) is a phasor at frequency \(\omega\), rotating clockwise (the minus sign reverses the direction). The transform multiplies the sound \(x(t)\) by this phasor, rotating the signal in the complex plane, and then integrates the result over all time. That multiply-by-a-phasor step is the central operating principle of the Fourier transform, and it is why we spent so long building up the complex sinusoid. We will develop the intuition for why this isolates the amount of frequency \(\omega\) shortly. First, let us rewrite the transform in a more concrete form.

Although \(X(\omega)\) is complex, we can split it into two real-valued integrals using Euler’s formula. Since \(e^{-j\omega t} = \cos(\omega t) - j\sin(\omega t)\),

\[ X(\omega) = \int_{-\infty}^{\infty} x(t)\big[\cos(\omega t) - j\sin(\omega t)\big]\, dt = R(\omega) + j\, I(\omega), \]

where we name the real and imaginary parts \(R(\omega) \triangleq \Re\big(X(\omega)\big)\) and \(I(\omega) \triangleq \Im\big(X(\omega)\big)\):

\[ R(\omega) \triangleq \Re\big(X(\omega)\big) = \int_{-\infty}^{\infty} x(t)\cos(\omega t)\, dt, \qquad I(\omega) \triangleq \Im\big(X(\omega)\big) = -\int_{-\infty}^{\infty} x(t)\sin(\omega t)\, dt. \]

That is the full definition. It probably still feels mysterious, which is completely expected. We will spend the rest of the chapter unpacking what it means and why it works.