This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.
Parseval's theorem
Continuous Time Fourier Series preserves signal energy
i.e.:
$$\begin{array}{ccc}\hfill \underset{0}{\overset{T}{\int}}{\left|f\left(t\right)\right|}^{2}dt& =T\sum _{n=-\infty}^{\infty}{\left|{C}_{n}\right|}^{2}\hfill & \hfill \phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\text{unnormalized}\phantom{\rule{4.pt}{0ex}}\text{basis}\phantom{\rule{4.pt}{0ex}}{e}^{j\frac{2\pi}{T}nt}\\ \hfill \underset{0}{\overset{T}{\int}}{\left|f\left(t\right)\right|}^{2}dt& =\sum _{n=-\infty}^{\infty}{\left|{C}_{n}^{\text{'}}\right|}^{2}\hfill & \hfill \phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\text{unnormalized}\phantom{\rule{4.pt}{0ex}}\text{basis}\phantom{\rule{4.pt}{0ex}}\frac{{e}^{j\frac{2\pi}{T}nt}}{\sqrt{T}}\\ \hfill \underset{{L}^{2}[0,T)energy}{\underbrace{{\left|\right|f\left|\right|}_{2}^{2}}}& =\underset{{l}^{2}\left(Z\right)energy}{\underbrace{\left|\right|{C}_{n}^{\text{'}}{\left|\right|}_{2}^{2}}}\hfill & \end{array}$$
Prove: plancherel theorem
$$\begin{array}{cc}\hfill \text{Given}\phantom{\rule{4.pt}{0ex}}f\left(t\right)& \stackrel{CTFS}{\to}{c}_{n}\hfill \\ \hfill g\left(t\right)& \stackrel{CTFS}{\to}{d}_{n}\hfill \\ \hfill \text{Then}\phantom{\rule{4.pt}{0ex}}\underset{0}{\overset{T}{\int}}f\left(t\right){g}^{*}\left(t\right)dt& =T\sum _{n=-\infty}^{\infty}{c}_{n}{d}_{n}^{*}\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\text{unnormalized}\phantom{\rule{4.pt}{0ex}}\text{basis}\phantom{\rule{4.pt}{0ex}}{e}^{j\frac{2\pi}{T}nt}\hfill \\ \hfill \underset{0}{\overset{T}{\int}}f\left(t\right){g}^{*}\left(t\right)dt& =\sum _{n=-\infty}^{\infty}{c}_{n}^{\text{'}}{\left({d}_{n}^{\text{'}}\right)}^{*}\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\text{normalized}\phantom{\rule{4.pt}{0ex}}\text{basis}\phantom{\rule{4.pt}{0ex}}\frac{{e}^{j\frac{2\pi}{T}nt}}{\sqrt{T}}\hfill \\ \hfill {\u3008f,g\u3009}_{{L}_{2}(0,T]}& ={\u3008c,d\u3009}_{{l}_{2}\left(\mathbb{Z}\right)}\hfill \end{array}$$
Periodic signals power
$$\begin{array}{cc}\hfill \text{Energy}\phantom{\rule{4.pt}{0ex}}& ={\left|\right|f\left|\right|}^{2}=\underset{-\infty}{\overset{\infty}{\int}}{\left|f\left(t\right)\right|}^{2}dt=\infty \hfill \\ \hfill \text{Power}\phantom{\rule{4.pt}{0ex}}& =\underset{T\to \infty}{lim}\frac{\text{Energy}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}[0,T)}{T}\hfill \\ & =\underset{T\to \infty}{lim}\frac{T{\sum}_{n}{\left|{c}_{n}\right|}^{2}}{T}\hfill \\ & =\sum _{n\in \mathbb{Z}}{\left|{c}_{n}\right|}^{2}\phantom{\rule{4.pt}{0ex}}\text{(unnormalized}\phantom{\rule{4.pt}{0ex}}\text{FS)}\hfill \end{array}$$
Fourier series of square pulse iii -- compute the energy
$$\begin{array}{cc}\hfill f\left(t\right)& =\sum _{n=-\infty}^{\infty}{c}_{n}{e}^{j\frac{2\pi}{T}nt}\stackrel{\mathbb{FS}}{\to}{c}_{n}=\frac{1}{2}\frac{sin\frac{\pi}{2}n}{\frac{\pi}{2}n}\hfill \\ \hfill \text{energy}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{time}\phantom{\rule{4.pt}{0ex}}\text{domain:}\phantom{\rule{4.pt}{0ex}}& {\left|\right|f\left|\right|}_{2}^{2}=\underset{0}{\overset{T}{\int}}{\left|f\left(t\right)\right|}^{2}dt=\frac{T}{2}\hfill \\ \hfill \text{apply}\phantom{\rule{4.pt}{0ex}}\text{Parseval's}\phantom{\rule{4.pt}{0ex}}\text{Theorem:}\phantom{\rule{4.pt}{0ex}}& T\sum _{n}{\left|{c}_{n}\right|}^{2}\hfill \\ & =\frac{T}{4}\sum _{n}{\left(\frac{sin\frac{\pi}{2}n}{\frac{\pi}{2}n}\right)}^{2}\hfill \\ & =\frac{T}{4}\frac{4}{{\pi}^{2}}\sum _{n}\frac{{\left(sin,\frac{\pi}{2}n\right)}^{2}}{{n}^{2}}\hfill \\ & =\frac{T}{{\pi}^{2}}\left[\frac{{\pi}^{2}}{4},+,\underset{\frac{{\pi}^{2}}{4}}{\underbrace{{\sum}_{n\phantom{\rule{4.pt}{0ex}}\text{odd}}\frac{1}{{n}^{2}}}}\right]\hfill \\ & =\frac{T}{2}\square \hfill \end{array}$$
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Plancharel theorem
Plancharel theorem
The inner product of two vectors/signals is the same as
the
$\ell ^{2}$ inner product of their expansion coefficients.
Let
$\{{b}_{i}\}$ be an orthonormal basis for a Hilbert Space
$H$ .
$x\in H$ ,
$y\in H$
$$x=\sum {\alpha}_{i}{b}_{i}$$
$$y=\sum {\beta}_{i}{b}_{i}$$ then
$${x\cdot y}_{H}=\sum {\alpha}_{i}\overline{{\beta}_{i}}$$
Applying the Fourier Series, we can go from
$f(t)$ to
$\{{c}_{n}\}$ and
$g(t)$ to
$\{{d}_{n}\}$
$$\int_{0}^{T} f(t)\overline{g(t)}\,d t=\sum_{n=()} $$ ∞
∞
c
n
d
n inner product in time-domain = inner product of Fourier
coefficients.
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$$x=\sum {\alpha}_{i}{b}_{i}$$
$$y=\sum {\beta}_{j}{b}_{j}$$
$${x\cdot y}_{H}=\sum {\alpha}_{i}{b}_{i}\cdot \sum {\beta}_{j}{b}_{j}=\sum {\alpha}_{i}({b}_{i}\cdot \sum {\beta}_{j}{b}_{j})=\sum {\alpha}_{i}\sum \overline{{\beta}_{j}}({b}_{i}\cdot {b}_{j})=\sum {\alpha}_{i}\overline{{\beta}_{i}}$$ by using
inner product rules
${b}_{i}\cdot {b}_{j}=0$ when
$i\neq j$ and
${b}_{i}\cdot {b}_{j}=1$ when
$i=j$
If Hilbert space H has a ONB, then inner products are
equivalent to inner products in
$\ell ^{2}$ .
All H with ONB are somehow equivalent to
$\ell ^{2}$ .
Point of interest square-summable sequences
are important
Plancharels theorem demonstration
Interact (when online) with a Mathematica CDF demonstrating Plancharels Theorem visually. To Download, right-click and save target as .cdf.
Parseval's theorem: a different approach
Parseval's theorem
Energy of a signal = sum of squares of its expansion
coefficients
Let
$x\in H$ ,
$\{{b}_{i}\}$ ONB
$$x=\sum {\alpha}_{i}{b}_{i}$$ Then
$$(, x)^{2}=\sum \left|{\alpha}_{i}\right|^{2}$$
Directly from Plancharel
$$(, x)^{2}={x\cdot x}_{H}=\sum {\alpha}_{i}\overline{{\alpha}_{i}}=\sum \left|{\alpha}_{i}\right|^{2}$$
Fourier Series
$\frac{1}{\sqrt{T}}e^{i{w}_{0}nt}$
$$f(t)=\frac{1}{\sqrt{T}}\sum {c}_{n}\frac{1}{\sqrt{T}}e^{i{w}_{0}nt}$$
$$\int_{0}^{T} \left|f(t)\right|^{2}\,d t=\sum_{n=()} $$ ∞
∞
c
n
2
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