<< Chapter < Page Chapter >> Page >
Discussion of linear time invariant systems.


Linearity and time invariance are two system properties that greatly simplify the study of systems that exhibit them. In our study of signals and systems, we will be especially interested in systems that demonstrate both of these properties, which together allow the use of some of the most powerful tools of signal processing.

Linear time invariant systems

Linear systems

If a system is linear, this means that when an input to a given system is scaled by a value, the output of the system is scaled by the same amount.

Linear scaling

In [link] above, an input x to the linear system L gives the output y . If x is scaled by a value α and passed through this same system, as in [link] , the output will also be scaled by α .

A linear system also obeys the principle of superposition. This means that if two inputs are added together and passed through a linear system,the output will be the sum of the individual inputs' outputs.

Superposition principle

If [link] is true, then the principle of superposition says that [link] is true as well. This holds for linear systems.

That is, if [link] is true, then [link] is also true for a linear system. The scaling property mentioned above still holds in conjunction with the superposition principle. Therefore,if the inputs x and y are scaled by factors α and β, respectively, then the sum of these scaled inputs will give the sum of the individual scaled outputs:

Superposition principle with linear scaling

Given [link] for a linear system, [link] holds as well.

Consider the system H 1 in which

H 1 ( f ( t ) ) = t f ( t )

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

H 1 ( a f ( t ) + b g ( t ) ) ) = t ( a f ( t ) + b g ( t ) ) = a t f ( t ) + b t g ( t ) = a H 1 ( f ( t ) ) + b H 1 ( g ( t ) )

for all real t . Thus, H 1 is a linear system.

Got questions? Get instant answers now!

Consider the system H 2 in which

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

for all signals f . Because

H 2 ( 2 t ) = 4 t 2 2 t 2 = 2 H 2 ( t )

for nonzero t , H 2 is not a linear system.

Got questions? Get instant answers now!

Time invariant systems

A time-invariant system has the property that a certain input will always give the same output (up to timing), without regard to when the input was applied to the system.

Time-invariant systems

[link] shows an input at time t while [link] shows the same input t 0 seconds later. In a time-invariant system both outputs would be identical exceptthat the one in [link] would be delayed by t 0 .

In this figure, x t and x t t 0 are passed through the system TI. Because the system TI is time-invariant, the inputs x t and x t t 0 produce the same output. The only difference is that the output due to x t t 0 is shifted by a time t 0 .

Whether a system is time-invariant or time-varying can be seen in the differential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations . A constant coefficient differential (or difference) equation means that the parameters of thesystem are not changing over time and an input now will give the same result as the same input later.

Consider the system H 1 in which

H 1 ( f ( t ) ) = t f ( t )

for all signals f . Because

S T ( H 1 ( f ( t ) ) ) = S T ( t f ( t ) ) = ( t - T ) f ( t - T ) t f ( t - T ) = H 1 ( f ( t - T ) ) = H 1 ( S T ( f ( t ) ) )

for nonzero T , H 1 is not a time invariant system.

Got questions? Get instant answers now!

Consider the system H 2 in which

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

for all signals f . For all real T and signals f ,

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

for all real t . Thus, H 2 is a time invariant system.

Got questions? Get instant answers now!

Linear time invariant systems

Certain systems are both linear and time-invariant, and are thus referred to as LTI systems.

Linear time-invariant systems

This is a combination of the two cases above. Since the input to [link] is a scaled, time-shifted version of the input in [link] , so is the output.

As LTI systems are a subset of linear systems, they obey the principle of superposition. In the figure below, we see the effect of applying time-invarianceto the superposition definition in the linear systems section above.

Superposition in linear time-invariant systems

The principle of superposition applied to LTI systems

Lti systems in series

If two or more LTI systems are in series with each other, their order can be interchanged without affecting the overall output of the system.Systems in series are also called cascaded systems.

Cascaded lti systems

The order of cascaded LTI systems can be interchanged without changing the overall effect.

Lti systems in parallel

If two or more LTI systems are in parallel with one another, an equivalent system is one that is defined as the sum of these individual systems.

Parallel lti systems

Parallel systems can be condensed into the sum of systems.

Consider the system H 3 in which

H 3 ( f ( t ) ) = 2 f ( t )

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

H 3 ( a f ( t ) + b g ( t ) ) ) = 2 ( a f ( t ) + b g ( t ) ) = a 2 f ( t ) + b 2 g ( t ) = a H 3 ( f ( t ) ) + b H 3 ( g ( t ) )

for all real t . Thus, H 3 is a linear system. For all real T and signals f ,

S T ( H 3 ( f ( t ) ) ) = S T ( 2 f ( t ) ) = 2 f ( t - T ) = H 3 ( f ( t - T ) ) = H 3 ( S T ( f ( t ) ) )

for all real t . Thus, H 3 is a time invariant system. Therefore, H 3 is a linear time invariant system.

Got questions? Get instant answers now!

As has been previously shown, each of the following systems are not linear or not time invariant.

H 1 ( f ( t ) ) = t f ( t )
H 2 ( f ( t ) ) = ( f ( t ) ) 2

Thus, they are not linear time invariant systems.

Got questions? Get instant answers now!

Linear time invariant demonstration

Interact(when online) with the Mathematica CDF above demonstrating Linear Time Invariant systems. To download, right click and save file as .cdf.

Lti systems summary

Two very important and useful properties of systems have just been described in detail. The first of these, linearity, allows us the knowledge that a sum of input signals produces an output signal that is the summed original output signals and that a scaled input signal produces an output signal scaled from the original output signal. The second of these, time invariance, ensures that time shifts commute with application of the system. In other words, the output signal for a time shifted input is the same as the output signal for the original input signal, except for an identical shift in time. Systems that demonstrate both linearity and time invariance, which are given the acronym LTI systems, are particularly simple to study as these properties allow us to leverage some of the most powerful tools in signal processing.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.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?