Editing Convolution theorem (section)

=== Periodic convolution (Fourier series coefficients) ===

Consider <math>P</math>-periodic functions <math>u_{_P}</math> &nbsp; and &nbsp; <math>v_{_P},</math> which can be expressed as [[periodic summation]]s:

:<math>u_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} u(x-mP)</math> &nbsp; and &nbsp; <math>v_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} v(x-mP).</math>

In practice the non-zero portion of components <math>u</math> and <math>v</math> are often limited to duration <math>P,</math> but nothing in the theorem requires that.

The [[Fourier series]] coefficients are:

:<math>\begin{align}
   U[k] &\triangleq \mathcal{F}\{u_{_P}\}[k] = \frac{1}{P} \int_P u_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}; \quad \quad \scriptstyle \text{integration over any interval of length } P\\
   V[k] &\triangleq \mathcal{F}\{v_{_P}\}[k] = \frac{1}{P} \int_P v_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}
 \end{align}</math>

where <math>\mathcal{F}</math> denotes the '''Fourier series integral'''.

* The product:  <math>u_{_P}(x)\cdot v_{_P}(x)</math> is also <math>P</math>-periodic, and its Fourier series coefficients are given by the [[Convolution#Discrete convolution|discrete convolution]] of the <math>U</math> and <math>V</math> sequences:

:<math>\mathcal{F}\{u_{_P}\cdot v_{_P}\}[k] = \{U*V\}[k].</math>

*The convolution:

:<math>\begin{align}
\{u_{_P} * v\}(x)\ &\triangleq \int_{-\infty}^{\infty} u_{_P}(x-\tau)\cdot v(\tau)\ d\tau\\
&\equiv \int_P u_{_P}(x-\tau)\cdot v_{_P}(\tau)\ d\tau; \quad \quad \scriptstyle \text{integration over any interval of length } P
\end{align}</math>

is also <math>P</math>-periodic, and is called a '''[[periodic convolution]]'''.

{{Collapse top|title=Derivation of periodic convolution}}
:<math>\begin{align}
\int_{-\infty}^\infty u_{_P}(x - \tau) \cdot v(\tau)\,d\tau
  &= \sum_{k=-\infty}^\infty \left[\int_{x_o+kP}^{x_o+(k+1)P} u_{_P}(x - \tau) \cdot v(\tau)\ d\tau\right] \quad x_0 \text{ is an arbitrary parameter}\\
  &=\sum_{k=-\infty}^\infty \left[\int_{x_o}^{x_o+P} \underbrace{u_{_P}(x - \tau-kP)}_{u_{_P}(x - \tau), \text{ by periodicity}} \cdot v(\tau + kP)\ d\tau\right] \quad \text{substituting } \tau \rightarrow \tau+kP\\
  &=\int_{x_o}^{x_o+P} u_{_P}(x - \tau) \cdot \underbrace{\left[\sum_{k=-\infty}^\infty v(\tau + kP)\right]}_{\triangleq \ v_{_P}(\tau)}\ d\tau
\end{align}</math>
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The corresponding convolution theorem is''':'''

{{Equation box 1
|indent=|cellpadding=0|border=0|background colour=white
|equation={{NumBlk|:|
<math>\mathcal{F}\{u_{_P} * v\}[k] =\ P\cdot U[k]\ V[k].</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}} }} }}

<!--{{math proof|title=Derivation of Eq.2| proof = -->
{{Collapse top|title=Derivation of Eq.2}}
:<math>\begin{align}
\mathcal{F}\{u_{_P} * v\}[k] &\triangleq \frac{1}{P} \int_P \left(\int_P u_{_P}(\tau)\cdot v_{_P}(x-\tau)\ d\tau\right) e^{-i 2\pi k x/P} \, dx\\
&= \int_P u_{_P}(\tau)\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k x/P} dx\right) \, d\tau\\
&= \int_P u_{_P}(\tau)\ e^{-i 2\pi k \tau/P}
\underbrace{\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k (x-\tau)/P} dx\right)}_{V[k], \quad \text{due to periodicity}} \, d\tau\\
&=\underbrace{\left(\int_P\ u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} d\tau\right)}_{P\cdot U[k]}\ V[k].
\end{align}</math>
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}}