Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Properties of the CTFS

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Signals and Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection: "Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Properties of the CTFS

Module by: Justin Romberg, Benjamin Fite. E-mail the authors

Summary: 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:

ft= n = c n ei ω 0 nt f t n c n ω 0 n t
(1)
c n =1T0Tfte(i ω 0 nt)d t c n 1 T t 0 T f t ω 0 n t
(2)
Let · · denote the transformation from ft f t to the Fourier coefficients ft= n ,nZ: c n f t n n c n · · maps complex valued functions to sequences of complex numbers.

Linearity

· · is a linear transformation.

Theorem 1

If ft= c n f t c n and gt= d n g t d n . Then α ,αC:αft=α c n α α α f t α c n and ft+gt= c n + d n f t g t c n d n

Proof

Easy. Just linearity of integral.

ft+gt= n ,nZ:0T(ft+gt)e(i ω 0 nt)d t = n ,nZ:1T0Tfte(i ω 0 nt)d t +1T0Tgte(i ω 0 nt)d t = n ,nZ: c n + d n = c n + d n f t g t n n t 0 T f t g t ω 0 n t n n 1 T t 0 T f t ω 0 n t 1 T t 0 T g t ω 0 n t n n c n d n c n d n
(3)

Shifting

Shifting in time equals a phase shift of Fourier coefficients

Theorem 2

ft t 0 =e(i ω 0 n t 0 ) c n f t t 0 ω 0 n t 0 c n if c n =| c n |ei c n c n c n c n , then |e(i ω 0 n t 0 ) c n |=|e(i ω 0 n t 0 )|| c n |=| c n | ω 0 n t 0 c n ω 0 n t 0 c n c n e(i ω 0 t 0 n)= c n ω 0 t 0 n ω 0 t 0 n c n ω 0 t 0 n

Proof

ft t 0 = n ,nZ:1T0Tft t 0 e(i ω 0 nt)d t = n ,nZ:1T t 0 T t 0 ft t 0 e(i ω 0 n(t t 0 ))e(i ω 0 n t 0 )d t = n ,nZ:1T t 0 T t 0 f t ~ e(i ω 0 n t ~ )e(i ω 0 n t 0 )d t = n ,nZ:e(i ω 0 n t ~ ) c n f t t 0 n n 1 T t 0 T f t t 0 ω 0 n t n n 1 T t t 0 T t 0 f t t 0 ω 0 n t t 0 ω 0 n t 0 n n 1 T t t 0 T t 0 f t ~ ω 0 n t ~ ω 0 n t 0 n n ω 0 n t ~ c n
(4)

Parseval's Relation

0T|ft|2d t =T n =| c n |2 t 0 T f t 2 T n c n 2
(5)
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.

Note:

Parseval tells us that the Fourier series maps L2 0 T L 0 T 2 to l2Z l 2 .

Figure 1
Figure 1 (pars.png)

Exercise 1

For ft f t to have "finite energy," what do the c n c n do as n n ?

Solution

| c n |2< c n 2 for ft f t to have finite energy.

Exercise 2

If n ,|n|>0: c n =1n n n 0 c n 1 n , is fL2 0 T f L 0 T 2 ?

Solution

Yes, because | c n |2=1n2 c n 2 1 n 2 , which is summable.

Exercise 3

Now, if n ,|n|>0: c n =1n n n 0 c n 1 n , is fL2 0 T f L 0 T 2 ?

Solution

No, because | c n |2=1n c n 2 1 n , which is not summable.

The rate of decay of the Fourier series determines if ft f t has finite energy.

Parsevals Theorem Demonstration

Figure 2: Interact (when online) with a Mathematica CDF demonstrating Parsevals Theorem. To download, right click and save file as .cdf.
ParsevalsDemo

Symmetry Properties

Rule 1: Even Signals

Even Signals

  • f(t)=f(-t)f(t)=f(-t)
  • cn=c-ncn=c-n

Proof

  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t

Rule 2: Odd Signals

Odd Signals

  • f ( t ) = -f ( -t ) f ( t ) = -f ( -t )
  • cn=c-ncn=c-n*

Proof

  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t - 1 T T 2 T f ( - t ) exp ( ı ω 0 n t ) d t = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t - 1 T T 2 T f ( - t ) exp ( ı ω 0 n t ) d t
  • = - 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t - exp ( - ı ω 0 n t ) d t = - 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t - exp ( - ı ω 0 n t ) d t
  • = - 1 T 0 T f ( t ) 2 ı sin ( ω 0 n t ) d t = - 1 T 0 T f ( t ) 2 ı sin ( ω 0 n t ) d t

Rule 3: Real Signals

Real Signals

  • f(t)=ff(t)=f*(t)(t)
  • cn=c-ncn=c-n*

Proof

  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t

Differentiation in Fourier Domain

(ft= c n )(dftd t =in ω 0 c n ) f t c n t f t n ω 0 c n
(6)

Since

ft= n = c n ei ω 0 nt f t n c n ω 0 n t
(7)
then
dd t ft= n = c n dei ω 0 ntd t = n = c n i ω 0 nei ω 0 nt t f t n c n t ω 0 n t n c n ω 0 n ω 0 n t
(8)
A differentiator attenuates the low frequencies in ft f t and accentuates the high frequencies. It removes general trends and accentuates areas of sharp variation.

Note:

A common way to mathematically measure the smoothness of a function ft f t is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of the signal, specifically how fast they decay as n n . If ft= c n f t c n and | c n | c n has the form 1nk 1 n k , then d m ftd t m =in ω 0 m c n t m f t n ω 0 m c n and has the form nmnk n m n k . So for the m th m th derivative to have finite energy, we need n |nmnk|2< n n m n k 2 thus nmnk n m n k decays faster than 1n 1 n which implies that 2k2m>1 2 k 2 m 1 or k>2m+12 k 2 m 1 2 Thus the decay rate of the Fourier series dictates smoothness.

Fourier Differentiation Demonstration

Figure 3: Interact (when online) with a Mathematica CDF demonstrating Differentiation in the Fourier Domain. To download, right click and save file as .cdf.
FourierDiffDemo

Integration in the Fourier Domain

If

ft= c n f t c n
(9)
then
tfτd τ =1i ω 0 n c n τ t f τ 1 ω 0 n c n
(10)

Note:

If c 0 0 c 0 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

Figure 4: Interact (when online) with a Mathematica CDF demonstrating Integration in the Fourier Domain. To download, right click and save file as .cdf.
fourierIntDemo

Signal Multiplication and Convolution

Given a signal ft f t with Fourier coefficients c n c n and a signal gt g t with Fourier coefficients d n d n , we can define a new signal, yt y t , where yt=ftgt y t f t g t . We find that the Fourier Series representation of yt y t , e n e n , is such that e n = k = c k d n - k e n k c k d n - k . This is to say that signal multiplication in the time domain is 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 =1T0Tftgte(i ω 0 nt)d t =1T0T k = c k ei ω 0 ktgte(i ω 0 nt)d t = k = c k (1T0Tgte(i ω 0 (nk)t)d t )= k = c k d n - k e n 1 T t 0 T f t g t ω 0 n t 1 T t 0 T k 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
(11)
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.

Table 1: Properties of the CTFS
Property Signal CTFS
Linearity a x ( t ) + b y ( t ) a x ( t ) + b y ( t ) a X ( f ) + b Y ( f ) a X ( f ) + b Y ( f )
Time Shifting x ( t - τ ) x ( t - τ ) X ( f ) e - j 2 π f τ / T X ( f ) e - j 2 π f τ / T
Time Modulation x ( t ) e j 2 π f τ / T x ( t ) e j 2 π f τ / T X ( f - k ) X ( f - k )
Multiplication x ( t ) y ( t ) x ( t ) y ( t ) X ( f ) * Y ( f ) X ( f ) * Y ( f )
Continuous Convolution x ( t ) * y ( t ) x ( t ) * y ( t ) X ( f ) Y ( f ) X ( f ) Y ( f )

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks