An introduction to the general properties of the Fourier series
Introduction
In this module we will discuss the basic properties of the Continuous-Time Fourier Series. We will begin by refreshing your memory of our basic
Fourier series equations:
Let
$\mathcal{F}(\xb7)$ denote the transformation from
$f(t)$ to the Fourier coefficients
$$\mathcal{F}(f(t))=\forall n, n\in \mathbb{Z}\colon {c}_{n}$$$\mathcal{F}(\xb7)$ maps complex valued functions to sequences of
complex numbers .
Linearity
$\mathcal{F}(\xb7)$ is a
linear transformation .
If
$\mathcal{F}(f(t))={c}_{n}$ and
$\mathcal{F}(g(t))={d}_{n}$ .
Then
$$\forall \alpha , \alpha \in \mathbb{C}\colon \mathcal{F}(\alpha f(t))=\alpha {c}_{n}$$ and
$$\mathcal{F}(f(t)+g(t))={c}_{n}+{d}_{n}$$
Easy. Just linearity of integral.
$\mathcal{F}(f(t)+g(t))=\forall n, n\in \mathbb{Z}\colon \int_{0}^{T} (f(t)+g(t))e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega}_{0}nt)}\,d t+\frac{1}{T}\int_{0}^{T} g(t)e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon {c}_{n}+{d}_{n}={c}_{n}+{d}_{n}$
$\mathcal{F}(f(t-{t}_{0}))=e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}$ if
${c}_{n}=\left|{c}_{n}\right|e^{i\angle ({c}_{n})}$ ,
then
$$\left|e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}\right|=\left|e^{-(i{\omega}_{0}n{t}_{0})}\right|\left|{c}_{n}\right|=\left|{c}_{n}\right|$$$$\angle (e^{-(i{\omega}_{0}{t}_{0}n)})=\angle ({c}_{n})-{\omega}_{0}{t}_{0}n$$
$\mathcal{F}(f(t-{t}_{0}))=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t-{t}_{0})e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(t-{t}_{0})e^{-(i{\omega}_{0}n(t-{t}_{0}))}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(\stackrel{~}{t}())e^{-(i{\omega}_{0}n\stackrel{~}{t})}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon e^{-(i{\omega}_{0}n\stackrel{~}{t})}{c}_{n}$
A differentiator
attenuates the low
frequencies in
$f(t)$ and
accentuates the high frequencies. It
removes general trends and accentuates areas of sharpvariation.
A common way to mathematically measure the smoothness of a
function
$f(t)$ is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of thesignal, specifically how fast they
decay as
$n\to $∞ .If
$\mathcal{F}(f(t))={c}_{n}$ and
$\left|{c}_{n}\right|$ has the form
$\frac{1}{n^{k}}$ ,
then
$\mathcal{F}(\frac{d^{m}f(t)}{dt^{m}})=(in{\omega}_{0})^{m}{c}_{n}$ and has the form
$\frac{n^{m}}{n^{k}}$ .So for the
${m}^{\mathrm{th}}$ derivative to have finite energy, we need
$$\sum \left|\frac{n^{m}}{n^{k}}\right|^{2}$$∞ thus
$\frac{n^{m}}{n^{k}}$ decays
faster than
$\frac{1}{n}$ which implies that
$$2k-2m> 1$$ or
$$k> \frac{2m+1}{2}$$ Thus the decay rate of the Fourier series dictates
smoothness.
Fourier differentiation demonstration
Integration in the fourier domain
If
$\mathcal{F}(f(t))={c}_{n}$
then
$\mathcal{F}(\int_{()} \,d \tau )$∞tfτ1ω0ncn
If
${c}_{0}\neq 0$ , this expression doesn't make sense.
Integration accentuates low frequencies and attenuates high
frequencies. Integrators bring out the
general
trends in signals and suppress short term variation
(which is noise in many cases). Integrators are
much nicer than differentiators.
Fourier integration demonstration
Signal multiplication and convolution
Given a signal
$f(t)$ with Fourier coefficients
${c}_{n}$ and a signal
$g(t)$ with Fourier coefficients
${d}_{n}$ ,
we can define a new signal,
$y(t)$ ,
where
$y(t)=f(t)g(t)$ .
We find that the Fourier Series representation of
$y(t)$ ,
${e}_{n}$ ,
is such that
${e}_{n}=\sum_{k=()} $∞∞ckdn-k .
This is to say that signal multiplication in the time domainis equivalent to
signal convolution in the frequency domain, and vice-versa: signal multiplication in the frequency domain is equivalent to signal convolution in the time domain.The proof of this is as follows
Like other Fourier transforms, the CTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.
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