<< Chapter < Page Chapter >> Page >
Describes continuous time systems.

Introduction

As you already now know, a continuous time system operates on a continuous time signal input and produces a continuous time signal output. There are numerous examples of useful continuous time systems in signal processing as they essentially describe the world around us. The class of continuous time systems that are both linear and time invariant, known as continuous time LTI systems, is of particular interest as the properties of linearity and time invariance together allow the use of some of the most important and powerful tools in signal processing.

Continuous time systems

Linearity and time invariance

A system H is said to be linear if it satisfies two important conditions. The first, additivity, states for every pair of signals x , y that H ( x + y ) = H ( x ) + H ( y ) . The second, homogeneity of degree one, states for every signal x and scalar a we have H ( a x ) = a H ( x ) . It is clear that these conditions can be combined together into a single condition for linearity. Thus, a system is said to be linear if for every signals x , y and scalars a , b we have that

H ( a x + b y ) = a H ( x ) + b H ( y ) .

Linearity is a particularly important property of systems as it allows us to leverage the powerful tools of linear algebra, such as bases, eigenvectors, and eigenvalues, in their study.

A system H is said to be time invariant if a time shift of an input produces the corresponding shifted output. In other, more precise words, the system H commutes with the time shift operator S T for every T R . That is,

S T H = H S T .

Time invariance is desirable because it eases computation while mirroring our intuition that, all else equal, physical systems should react the same to identical inputs at different times.

When a system exhibits both of these important properties it allows for a more straigtforward analysis than would otherwise be possible. As will be explained and proven in subsequent modules, computation of the system output for a given input becomes a simple matter of convolving the input with the system's impulse response signal. Also proven later, the fact that complex exponential are eigenvectors of linear time invariant systems will enable the use of frequency domain tools such as the various Fouier transforms and associated transfer functions, to describe the behavior of linear time invariant systems.

Consider the system H in which

H ( f ( t ) ) = 2 f ( t )

for all signals f . Given any two signals f , g and scalars a , b

H ( a f ( t ) + b g ( t ) ) ) = 2 ( a f ( t ) + b g ( t ) ) = a 2 f ( t ) + b 2 g ( t ) = a H ( f ( t ) ) + b H ( g ( t ) )

for all real t . Thus, H is a linear system. For all real T and signals f ,

S T ( H ( f ( t ) ) ) = S T ( 2 f ( t ) ) = 2 f ( t - T ) = H ( f ( t - T ) ) = H ( S T ( f ( t ) ) )

for all real t . Thus, H is a time invariant system. Therefore, H is a linear time invariant system.

Got questions? Get instant answers now!

Differential equation representation

It is often useful to to describe systems using equations involving the rate of change in some quantity. For continuous time systems, such equations are called differential equations. One important class of differential equations is the set of linear constant coefficient ordinary differential equations, which are described in more detail in subsequent modules.

Consider the series RLC circuit shown in [link] . This system can be modeled using differential equations. We can use the voltage equations for each circuit element and Kirchoff's voltage law to write a second order linear constant coefficient differential equation describing the charge on the capacitor.

The voltage across the battery is simply V . The voltage across the capacitor is 1 C q . The voltage across the resistor is R d q d t . Finally, the voltage across the inductor is L d 2 q d t 2 . Therefore, by Kirchoff's voltage law, it follows that

L d 2 q d t 2 + R d q d t + 1 C q = V .
A series RLC circuit.
Got questions? Get instant answers now!

Continuous time systems summary

Many useful continuous time systems will be encountered in a study of signals and systems. This course is most interested in those that demonstrate both the linearity property and the time invariance property, which together enable the use of some of the most powerful tools of signal processing. It is often useful to describe them in terms of rates of change through linear constant coefficient ordinary differential equations.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask