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      <text>23.6.4.1 Parseval’s theorem</text>
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      <text><![CDATA[<h4>Parseval’s theorem</h4>

This is essentially a mathematical theorem but has, as we shall see, an important
engineering interpretation particularly in electrical engineering. Parseval's theorem states that if \(f(t)\) is a periodic function with period \(T\) and if \(c_n\) \( (n=0,\pm 1,\pm 2,\ldots)\) denotes the complex Fourier coefficients of \(f(t)\), then
\[ \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} f^2(t)\, dt=\sum_{n=-\infty}^\infty
|c_n|^2.\]
In words the theorem states that the mean square value of the signal \(f(t)\) over one period
equals the sum of the squared magnitudes of all the complex Fourier coefficients.
<p></p>
<b>Proof of Parseval’s theorem</b>
<p></p>
Assume \(f(t)\) has a complex Fourier series of the usual form:
\[ f(t)=\sum_{n=-\infty}^{\infty} c_n e^{ in\omega_0t}\qquad
\left(\omega_0=\frac{2\pi}{T}\right) \]
where
\[  c_n=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)e^{- in\omega_0t}dt \]
Then
\[ f^2(t)=f(t)f(t)=f(t)\sum c_n e^{ in\omega_0t}=\sum c_n f(t) e^{in\omega_0t} \]
Hence
\[
\begin{eqnarray*}
\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} f^2(t)\,dt&amp;=&amp;
\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}\sum c_n f(t) e^{ in\omega_0t}dt\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\
\\
&amp;=&amp;\frac{1}{T}\sum c_n \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t) e^{ in\omega_0t}dt\\
\\
&amp;=&amp;\sum c_n {c^*_n}\\
\\
&amp;=&amp;\sum_{n=-\infty}^\infty |c_n|^2
\end{eqnarray*}
\]
which completes the proof.]]></text>
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