MIT 6.007 Signals and Systems Course Notes 3

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Lecture 7: Continuous-Time Fourier Series

Recall in Lec 4

We want to decompose input signals into a linear combination of basic signals, i.e.

\[x(t) = \sum_{n=0}^{\infty} a_n \phi_n(t)\]

, such that if \(\psi_n(t)\) is the response of \(\phi_n(t)\), then the response of \(x(t)\) can be computed as:

\[y(t) = \sum_{n=0}^{\infty} a_n \psi_n(t)\]

due to linearity.

Specifically, for time-invariant systems, we chose delayed unit impulses as basic signals, leading to the concept of convolution.

In this lecture, we will use complex exponentials with unity magnitude as basic signals, leading to the concept of Fourier Analysis.

Fourier series vs Fourier transform

Fourier series is used for periodic signals, which is what we will discuss in this lecture.

Fourier transform is used for non-periodic signals. Its basic idea is to treat the signal as periodic with period \(T\), and then apply Fourier series followed by letting the period \(T\) goes to infinity.

Fourier series

How a periodic signal with period \(T_0\) can be represented in the form of a linear combination of complex exponentials?

• Fourier synthesis equation

  \(x(t) = \sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t}\) , where \(\omega_0 = \frac{2\pi}{T_0}\) being the fundamental frequency.

  To determine \(a_k\), we need:

• Fourier analysis equation

  \[a_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-jk\omega_0 t} dt\]

  That can be easily verified using the fact that:

  \[\int_{T_0} e^{jk\omega_0 t} dt = T_0 \text{ if } k = 0 \text{ else } 0\]

The advantage of using complex exponentials

Complex exponentials is the eigenfunction of the LTI system. Its eigenvalues can be computed by directly computing its response using convolution integral.

Convergence of Fourier Series

If \(x(t)\) is square integrable, the energy of the error between the partial sum of \(2N+1\) terms and the Fourier Series converges to zero as \(N\) goes to infinity.

If \(x(t)\) satisfies Dirichlet conditions, the energy converges to zero as \(N\) goes to infinity except at discontinuities.