# 6.3 Properties of the ctfs

 Page 1 / 1
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:

$f(t)=\sum_{n=()}$ c n ω 0 n t
${c}_{n}=\frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega }_{0}nt)}\,d t$
Let $ℱ(·)$ denote the transformation from $f(t)$ to the Fourier coefficients $ℱ(f(t))=\forall n, n\in \mathbb{Z}\colon {c}_{n}$ $ℱ(·)$ maps complex valued functions to sequences of complex numbers .

## Linearity

$ℱ(·)$ is a linear transformation .

If $ℱ(f(t))={c}_{n}$ and $ℱ(g(t))={d}_{n}$ . Then $\forall \alpha , \alpha \in \mathbb{C}\colon ℱ(\alpha f(t))=\alpha {c}_{n}$ and $ℱ(f(t)+g(t))={c}_{n}+{d}_{n}$

Easy. Just linearity of integral.

$ℱ(f(t)+g(t))=\forall n, n\in \mathbb{Z}\colon \int_{0}^{T} (f(t)+g(t))e^{-(i{\omega }_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega }_{0}nt)}\,d t+\frac{1}{T}\int_{0}^{T} g(t)e^{-(i{\omega }_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon {c}_{n}+{d}_{n}={c}_{n}+{d}_{n}$

## Shifting

Shifting in time equals a phase shift of Fourier coefficients

$ℱ(f(t-{t}_{0}))=e^{-(i{\omega }_{0}n{t}_{0})}{c}_{n}$ if ${c}_{n}=\left|{c}_{n}\right|e^{i\angle ({c}_{n})}$ , then $\left|e^{-(i{\omega }_{0}n{t}_{0})}{c}_{n}\right|=\left|e^{-(i{\omega }_{0}n{t}_{0})}\right|\left|{c}_{n}\right|=\left|{c}_{n}\right|$ $\angle (e^{-(i{\omega }_{0}{t}_{0}n)})=\angle ({c}_{n})-{\omega }_{0}{t}_{0}n$

$ℱ(f(t-{t}_{0}))=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t-{t}_{0})e^{-(i{\omega }_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(t-{t}_{0})e^{-(i{\omega }_{0}n(t-{t}_{0}))}e^{-(i{\omega }_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(\stackrel{~}{t}())e^{-(i{\omega }_{0}n\stackrel{~}{t})}e^{-(i{\omega }_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon e^{-(i{\omega }_{0}n\stackrel{~}{t})}{c}_{n}$

## Parseval's relation

$\int_{0}^{T} \left|f(t)\right|^{2}\,d t=T\sum_{n=()}$ c n 2
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.
Parseval tells us that the Fourier series maps $L(\left[0 , T\right])^{2}$ to $l(\mathbb{Z})^{2}$ .

For $f(t)$ to have "finite energy," what do the ${c}_{n}$ do as $n\to$ ?

$\left|{c}_{n}\right|^{2}$ for $f(t)$ to have finite energy.

If $\forall n, \left|n\right|> 0\colon {c}_{n}=\frac{1}{n}$ , is $f\in L(\left[0 , T\right])^{2}$ ?

Yes, because $\left|{c}_{n}\right|^{2}=\frac{1}{n^{2}}$ , which is summable.

Now, if $\forall n, \left|n\right|> 0\colon {c}_{n}=\frac{1}{\sqrt{n}}$ , is $f\in L(\left[0 , T\right])^{2}$ ?

No, because $\left|{c}_{n}\right|^{2}=\frac{1}{n}$ , which is not summable.

The rate of decay of the Fourier series determines if $f(t)$ has finite energy .

## Even signals

• $f\left(t\right)=f\left(-t\right)$
• $\parallel {c}_{n}\parallel =\parallel {c}_{-n}\parallel$
• ${c}_{n}=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(-t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(-t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)\left[exp,\left(ı{\omega }_{0}nt\right),\phantom{\rule{0.166667em}{0ex}},d,t,+,exp,\left(-ı{\omega }_{0}nt\right)\right]\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)2cos\left({\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$

## Odd signals

• $f\left(t\right)=\mathrm{-f}\left(\mathrm{-t}\right)$
• ${c}_{n}={c}_{-n}$ *
• ${c}_{n}=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt-\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(-t\right)exp\left(ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=-\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)\left[exp,\left(ı{\omega }_{0}nt\right),\phantom{\rule{0.166667em}{0ex}},d,t,-,exp,\left(-ı{\omega }_{0}nt\right)\right]\phantom{\rule{0.166667em}{0ex}}dt$
• $=-\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)2ısin\left({\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$

## Real signals

• $f\left(t\right)=f$ * $\left(t\right)$
• ${c}_{n}={c}_{-n}$ *
• ${c}_{n}=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{\frac{T}{2}}\phantom{\rule{-0.166667em}{0ex}}f\left(-t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(-t\right)exp\left(-ı{\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)\left[exp,\left(ı{\omega }_{0}nt\right),\phantom{\rule{0.166667em}{0ex}},d,t,+,exp,\left(-ı{\omega }_{0}nt\right)\right]\phantom{\rule{0.166667em}{0ex}}dt$
• $=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)2cos\left({\omega }_{0}nt\right)\phantom{\rule{0.166667em}{0ex}}dt$

## Differentiation in fourier domain

$(ℱ(f(t))={c}_{n})\implies (ℱ(\frac{d f(t)}{d t}})=in{\omega }_{0}{c}_{n})$

Since

$f(t)=\sum_{n=()}$ c n ω 0 n t
then
$\frac{d f(t)}{d t}}=\sum_{n=()}$ c n t ω 0 n t n c n ω 0 n ω 0 n t
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 $ℱ(f(t))={c}_{n}$ and $\left|{c}_{n}\right|$ has the form $\frac{1}{n^{k}}$ , then $ℱ(\frac{d^{m}f(t)}{dt^{m}})=(in{\omega }_{0})^{m}{c}_{n}$ 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.

## Integration in the fourier domain

If

$ℱ(f(t))={c}_{n}$
then
$ℱ(\int_{()} \,d \tau )$ t f τ 1 ω 0 n c n
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.

## 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 ${e}_{n}=\sum_{k=()}$ c k d n - 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

${e}_{n}=\frac{1}{T}\int_{0}^{T} f(t)g(t)e^{-(i{\omega }_{0}nt)}\,d t=\frac{1}{T}\int_{0}^{T} \sum_{k=()} \,d t$ c k ω 0 k t g t ω 0 n t k c k 1 T t 0 T g t ω 0 n k t k c k d n - k
for more details, see the section on Signal convolution and the CTFS

## Conclusion

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.

 Property Signal CTFS Linearity $ax\left(t\right)+by\left(t\right)$ $aX\left(f\right)+bY\left(f\right)$ Time Shifting $x\left(t-\tau \right)$ $X\left(f\right){e}^{-j2\pi f\tau /T}$ Time Modulation $x\left(t\right){e}^{j2\pi f\tau /T}$ $X\left(f-k\right)$ Multiplication $x\left(t\right)y\left(t\right)$ $X\left(f\right)*Y\left(f\right)$ Continuous Convolution $x\left(t\right)*y\left(t\right)$ $X\left(f\right)Y\left(f\right)$

is the law of independent assortment same as the law of dominance?
explain the process of digestion in animal
discuss the ueses of human cell
what is hormone
they are liquid substance which are essential for different body functions
Ed
the meaning of warm classroom
How is an image formed at the retina
what is gland
what's cells?
hi santino Cells is a basic unit of life
Yohannes
the structural and functional units of life is known as cell
Rathod
good night all friends
Rathod
u too
Papaye
y do plants have a single respiratory system?
Papaye
hey
Chinwe
Chinwe
what is a meiosis
what is a mitosis
Mercy
meiosis is a cell division that produces 4 daughter cell that doesn't look alike
Damilola
mitosis is a cell division that produces two daugther cell that looks alik
Damilola
what is call division
mitosis
Damilola
what are the effects of transpiration
Are extraterrestrials mostly sited on land or in the sea also?
what's sperm
what is gaseous exchange
what is respiration
Lubna
hi lubna respiration is the breakdown of food with or without the use of oxygen to produce energy, carbon dioxide and water and it takes place in cells
caleb
fart😥😂
Golden
what is the structure of a cell
cell membrane, nucleus and cytoplasm lie upon the two
Jamiu
is a cell membrame
KIHEMBO
types of diffusion
KIHEMBO
two types of transpiration
KIHEMBO
Passive and active transport
Dillenia
nidhi
Nidhi
Got questions? Join the online conversation and get instant answers!