Fourier series
Fourier series are mathematical series that can represent a periodic signal as a sum of sinusoids at different amplitudes and frequencies. They are useful for solving for the response of a system to periodic inputs. However, they are probably most important conceptually: they are our gateway to thinking of signals in the frequency domain—that is, as functions of frequency (not time). To represent a function as a Fourier series is to analyze it as a sum of sinusoids at different frequencies1 \(\omega_n\) and amplitudes \(a_n\). Its frequency spectrum is the functional representation of amplitudes \(a_n\) versus frequency \(\omega_n\).
Let’s begin with the definition.
Definition
The Fourier analysis of a periodic function \(y(t)\) is, for \(n\in \mathbb{N}_0\), period \(T\), and angular frequency \(\omega_n = 2\pi n/T\), \[\begin{aligned} a_0 &= \frac{2}{T} \int_T y(t) dt \\ a_n &= \frac{2}{T} \int_T y(t) \cos(\omega_n t) dt \\ b_n &= \frac{2}{T} \int_T y(t) \sin(\omega_n t) dt. \end{aligned}\] The Fourier synthesis of a periodic function \(y(t)\) with analysis components \(a_n\) and \(b_n\) corresponding to \(\omega_n\) is \[\begin{aligned} y(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(\omega_n t) + b_n \sin(\omega_n t). \end{aligned}\]
Let’s consider the complex form of the Fourier series, which is equivalent to . It may be helpful to review Euler’s formula(s)—see section C.4.
Definition
The Fourier analysis of a periodic function \(y(t)\) is, for \(n\in \mathbb{N}_0\), period \(T\), and angular frequency \(\omega_n = 2\pi n/T\), \[\begin{aligned} c_n &= \frac{1}{T} \int_T y(t) e^{-j \omega_n t} dt\quad\text{for }n \in \mathbb{Z} \end{aligned}\] The Fourier synthesis of a periodic function \(y(t)\) with analysis components \(c_n\) corresponding to \(\omega_n\) is \[\begin{aligned} y(t) = \sum_{n=-\infty}^\infty c_n e^{j \omega_n t}. \end{aligned}\]
We call the integer \(n\) a harmonic and the frequency associated with it, \[\begin{aligned} \omega_n = 2\pi n/T, \end{aligned}\] the harmonic frequency. There is a special name for the first harmonic (\(n=1\)): the fundamental frequency. It is called this because all other frequency components are integer multiples of it.
It is also possible to convert between the two representations above.
Definition
The complex Fourier analysis of a periodic function \(y(t)\) is, for \(n\in \mathbb{Z}\) and \(a_n\) and \(b_n\) as defined above, \[c_n = \begin{cases} \frac{1}{2} \left( a_{n} - j b_{n} \right) & n \ge 0 \\ \frac{1}{2} \left( a_{|n|} + j b_{|n|} \right) & n < 0. \end{cases}\] The sinusoidal Fourier analysis of a periodic function \(y(t)\) is, for \(n\in \mathbb{N}_0\) and \(c_n\) as defined above, \[\begin{aligned} a_n &= c_{n} + c_{-n}\text{ and} \\ b_n &= j\left(c_{n} - c_{-n}\right). \end{aligned}\]
The harmonic amplitude \(C_n\) is, for \(n\in \mathbb{N}_0\), \[\begin{aligned} C_n &= \sqrt{a_n^2 + b_n^2} \\ &= 2 \sqrt{ c_{n} c_{-n} }. \label{eq:harmonic_amplitude_complex} \end{aligned}\] A magnitude line spectrum is a graph of the harmonic amplitudes as a function of the harmonic frequencies. The harmonic phase is \[\begin{align} \theta_n &= -\arctantwo(b_n,a_n) \tag{see \cref{eq:two_to_one}} \\ &= \arctantwo(\Im(c_n),\Re(c_n)). \label{eq:harmonic_phase_complex} \end{align}\]
The illustration of fig. ¿fig:fourier_time_freq? shows how sinusoidal components sum to represent a square wave. A line spectrum is also shown.
Let us compute the associated spectral components in the following example.
Compute the first five harmonic amplitudes that represent the line spectrum for a square wave in the figure above.
Assume a square wave with amplitude 1. Compute an: Compute bn: Therefore,
It’s important to note that the symbol \(\omega_n\), in this context, is not the natural frequency, but a frequency indexed by integer \(n\).↩︎
Online Resources for Section 6.1
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