Fourier Series introduction

Aaron Curtis- good TA

Approximating and representing signals

While $x$ might be complicated, maybe we can express $x$ either approximately or exactly as a weighted sum of many simple basis (building block signals)

• This is the idea of Fourier analysis

Fourier series of even periodic signal

Assume that $x$ is real-valued, even, and periodic with fundamental period $T_0$

$$ \phi_0(t) = \frac{1}{2}, \quad \phi_1(t) = \cos(\omega_0 t), \quad \phi_2(t) = \cos(2\omega_0 t), \ldots $$

• We thereform try to approximate $x$ as:

  $$ \hat{x}(t) = \frac{a_0}{2} + \sum_{\ell=1}^{\infty} a_\ell \cos(\ell \omega_0 t), \quad a_0, a_1, \ldots \text{ are real constants} $$

  

• To find that coefficient $a_0$, you integrate $x(t)$ over one period, and due to the orthogonality property of cosine, the integrals of the cosine terms go to 0:

  $$ a_0 = \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t) \, dt $$

• To find the coefficients $a_k$ for $k≥1$, multiply both sides of the Fourier series by $\cos (k \omega _0 t)$ and integrate over one period

  $$ a_k = \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t) \cos(k\omega_0 t) \, dt, \quad \text{for } k \geq 1 $$

  • All terms where $l≠k$

• Orthogonality property: when the $\cos (k \omega _0 t)$ and $\cos (l \omega _0 t)$ is integrated over one period, the result is zero if $k≠l$ and $\frac{T_0}{2}$ if $k=l$

  $$ \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos(k\omega_0 t) \cos(\ell\omega_0 t) \, dt = \begin{cases} 0 & \text{if } k \neq \ell \\1 & \text{if } k = \ell \end{cases} $$

  • Cannot be decomposed into a linearly dependent combination of other cosines

• Calculation of $a_k$ for $k≥1$, intergating the product of the signal $x(t)$ and the cosine term $\cos (k \omega _0 t)$ over one period

  $$ \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t) \cos(k\omega_0 t) \, dt = \sum_{\ell=1}^{\infty} a_\ell \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos(k\omega_0 t) \cos(\ell\omega_0 t) \, dt = a_k $$

Theorem 3.1: If $x$ is a real-values, even, and periodic CT signal with fundamental period $T_0$, then we can represent $x$ via the cosine Fourier series:

• Cosine Fourier series representation for a real-valued, even, and periodic CT signal

  $$ x(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} a_k \cos(k\omega_0 t), \quad \omega_0 = \frac{2\pi}{T_0} $$

• Calculation of coefficients $a_k$

  $$ a_k = \frac{2}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t) \cos(k\omega_0 t) \, dt, \quad k \in \{0, 1, 2, \ldots\}. $$

  • Note that $\omega _0$ is the fundamental angular frequency, related to the fundamental period $T_0$ by $\omega _0 = \frac{2 \pi}{T_0}$
  • The coefficients $a_0, a_1, \cdots$ are real constants that scale the amplitude of the corresponding cosine basis functions in the series
  • The integral can be taken over any interval of length $T_0$ due to the periodicity of the signal

Fourier series of a square wave

For $T_0 >0$ and $0< \tau ≤ T_0$, consider:

$$ x_{\text{fin}}(t) = \begin{cases}1 & \text{if } -\frac{T_0}{2} \leq t \leq \frac{T_0}{2} \\0 & \text{otherwise}\end{cases} $$