2.5. Fourier Series#

For a continuous-time periodic signal \(s(t)\) of period \(T\), i.e., \(s(t+T) = s(t)\) for all \(t \in \mathbb{R}\):

  • Synthesis:

    (2.9)#\[\begin{equation} s(t) = \sum_{n=-\infty}^{\infty} a_n e^{j\frac{2\pi n t}{T}} \end{equation}\]

  • Analysis:

    (2.10)#\[\begin{equation} a_n = \frac{1}{T} \int_{-T/2}^{T/2} s(t) e^{-j\frac{2\pi n t}{T}} dt, ~~~n \in \mathbb{Z} \end{equation}\]

The complex numbers \(\{a_n\}_{n \in \mathbb{Z}}\) are called the Fourier series (FS) coefficients of \(s(t)\).

  • Considering the “change of variable” \(t = -\frac{\hat\omega T}{2\pi}\), \(s( -\frac{\hat\omega T}{2\pi})\) is periodic in \(\hat\omega\) with period \(2\pi\). Treating \(s( -\frac{\hat\omega T}{2\pi})\) as \(X(e^{j\hat\omega})\) and \(a_n\) as \(x[n]\), the FS synthesis and analysis formulas, (2.9) and (2.10), become the forward and inverse DTFT formulas, (2.1) and (2.2), respectively. As a result, all the convergence results for DTFT apply to FS under this translation.

  • In particular, the Riesz-Fisher theorem tells us that for each (up to an equivalence class) periodic \(s(t) \in L^2\left[-\frac{T}{2}, \frac{T}{2}\right]\) (i.e., \(s(t)\) has finite power), there is a unique FS coefficient sequence \(\{a_n\}_{n \in \mathbb{Z}} \in \ell^2\) such that both FS synthesis and analysis formulas, (2.9) and (2.10), hold with the convergence in the infinite series on the RHS of the synthesis formula interpreted as

    \[\begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{T} \int_{-T/2}^{T/2} \left| s(t) - \sum_{n=-N}^{N} a_n e^{j\frac{2\pi n t}{T}} \right|^2 dt = 0. \end{equation*}\]

    The power of \(s(t)\) is given by \(\displaystyle \frac{1}{T} \int_{-T/2}^{T/2} |s(t)|^2 dt = \sum_{n=-\infty}^{\infty} |a_n|^2\).

  • The extension of DTFT for infinite-energy signals also translates to a similar extension of FS for FS coefficient sequences \(\{a_n\}_{n \in \mathbb{Z}} \notin \ell^2\).

    Tip

    In fact, we may use the DTFT tables to determine the FS coefficients of some common periodic signals by the change of variable \(\hat\omega = -\frac{2\pi t}{T}\) and the association \(a_n \leftarrow x[n]\) and \(x\left( -\frac{2\pi t}{T} \right) \leftarrow X(e^{j\hat\omega})\).

  • Example: From the example in Section 2.4.1 and linearity, we have \(\frac{1}{T} \stackrel{\text{DTFT}}{\longleftrightarrow} \frac{2\pi}{T} \delta (e^{j\hat{\omega}})\). By the above change of variable, the FS coefficients of the periodic (period \(=T\)) signal \(\frac{2\pi}{T} \delta \left(e^{-j \frac{2\pi t}{T}}\right) = \frac{2\pi}{T} \delta \left(e^{j \frac{2\pi t}{T}} \right)\) (recall \(\delta (e^{j\hat{\omega}})\) is an even function) are \(a_n = \frac{1}{T}\) for all \(n \in \mathbb{Z}\). Hence, the FS synthesis formula (2.9) becomes

    (2.11)#\[\begin{equation} \frac{2\pi}{T} \delta \left(e^{j \frac{2\pi t}{T}} \right) = \frac{1}{T} \sum_{n=-\infty}^{\infty} e^{j\frac{2\pi n t}{T}} \end{equation}\]

    which is often referred to as the Poisson sum formula.

    Tip

    Note that the Poisson sum formula (2.11) above is simply a mnemonic for the limiting process of the mapping between the window signal \(w_{\lambda}[n]\) and its DTFT \(W_{\lambda}(e^{j\hat\omega})\), i.e., \(\displaystyle W_{\lambda}\left(e^{-j \frac{2\pi t}{T}} \right) = \sum_{n=-\infty}^{\infty} \lambda^{|n|} e^{j\frac{2\pi n t}{T}}\), as \(\lambda \rightarrow 1\).