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.
Properties of the ctfs
Property
Signal
CTFS
Linearity
$ax\left(t\right)+by\left(t\right)$
$aX\left(f\right)+bY\left(f\right)$
Time Shifting
$x(t-\tau )$
$X\left(f\right){e}^{-j2\pi f\tau /T}$
Time Modulation
$x\left(t\right){e}^{j2\pi f\tau /T}$
$X(f-k)$
Multiplication
$x\left(t\right)y\left(t\right)$
$X\left(f\right)*Y\left(f\right)$
Continuous Convolution
$x\left(t\right)*y\left(t\right)$
$X\left(f\right)Y\left(f\right)$
Questions & Answers
can someone help me with some logarithmic and exponential equations.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.