For linear time invariant system, we only need to know the impulse response h(t) of the system (or equivalently frequency response H(omega)) in order to predict the output of the system in response to any input. This is done by convoluting the input with the impulse response. So a linear time invariant system is a lot easier to analyze. 

This is not true for nonlinear or time variant system. 

Time invariance is absence of t in the formula, isn't it? The formula can be non linear: f = sin(x^2)+ln(y), or linear f = 5x + y.

i think this chapter can help you

i wish to you best

http://www.site.uottawa.ca/~jpyao/courses/ELG3120_files/ch2.pdf

Mr. Dmitri thank you but your answer regards mathematics not in DSP

thank you Miss. SARAH for your help

Dear Mr. Oday

If the operator T in satisfies the following two conditions, then T is called a linear operator and the system represented by a linear operator T is called a linear system:

condition 1: Additivity

Given that Tx1 = y1 and Tx2 = y2, then

T{x1 +x2} =y1 +y2

for any signals x1 and x2.

Condition2: Homogeneity (or Scaling)

T{αx} = αy

for any signals x and any scalar a.

Any system that does not satisfy above 2 conditions is classified as a nonlinear system. The above Equations can be combined into a single condition as:

T{α1x1 +α2x2} =α1y1 +α2Y2

where α1, and α2, are arbitrary scalars. The last eq. is known as the superposition

property. Examples of linear systems are

y(t) = R x(t)

Examples of nonlinear systems are :

y =x^2  

y = cos x

A system is called time-invariant if a time shift (delay or advance) in the input signal causes the same time shift in the output signal. Thus, for a continuous-time system, the system is time-invariant if

T{ x(t - t') } = y{ t - t' }

for any real value of (t'). For a discrete-time system, the system is time-invariant (or shift-invariant ) if :

T{ x[n-k] }= y[n-k]

If the system is linear and also time-invariant, then it is called a linear time-invariant (LTI) system.

For linear time invariant system, we only need to know the impulse response h(t) of the system (or equivalently frequency response H(omega)) in order to predict the output of the system in response to any input. This is done by convoluting the input with the impulse response. So a linear time invariant system is a lot easier to analyze. 

This is not true for nonlinear or time variant system. 

thank you Mr. Mohammed that what i need exactly for the difference between them but what is the advantage of them in practical work?

thank you Mr. ken

you are right

Another advantage is that you can analyze the output of your system decomposing the input. For example if you have Z as input and you are able to decompose Z such that Z= a + b+ c .....  you can study how the system responds to a to b to c and so on and then sum up those responses. Usually the response of the system to a, b, c ... is much easier 

Advantage? Many :)

You can analyse the system without the time factor or taking memory into account! The  output of such a system depends only on the current values of the input rather than what had happened sometime back!

thank you Mr. prasad

Mr.Oday I totally agree with miss Wei Hu .

LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant system to an arbitrary input signal. Trajectories of these systems are commonly measured and tracked as they move through time (e.g., an acoustic waveform), but in applications like image processing and field theory, the LTI systems also have trajectories in spatial dimensions. Thus, these systems are also called linear translation-invariant to give the theory the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.