Chapter 2linear time invariant systems

continuous time systemsPrepared by

Dr. Taha MAhdy

What is a system?

A system is a mathematical model that represents the transformation of some input signal x(t) into an output signal y(t).

Systems

• In digital domain, the system may be a digital filter

Systems

• For our purposes, we aren't going to worry about the particulars of how the system is implemented.

• It will be a black box and our only concern will be the mathematical properties of the system.

Systems

• Mathematically, we represent the system by a transformation or operator that we will denote by T.

• Then we can write the action of a system on an input signal as

• Y(t) = T{x(t)}

• A system is known as a continuous time system if x = x(t) and y = y(t) are continuous time signals.

System properties: Memoryless Systems

• If the output y(t) of a given system depends only on the input x(t) at the same time, then the system is called memoryless.

• A simple example of a memoryless system is an output that depends on a constant multiple of the input y(t) = ax(t)

• where a is some real constant.

Memory system

• A system with memory is one where the output depends on the values of the

• input at previous times. An example of such a system is one where we add up

Example

Causal and Noncausal Systems

• If a system output y(t) depends only on the input at present or earlier times, we say that the system is causal. Another way to say this is that the output does not anticipate future values of the input.

• All memoryless systems are causal.

Non causal system

Linear Systems

Example

• Check the linearity (soln. a,c linear)

(soln. a,c linear)

Time (Shift) Invariance

System Stability

Example

cosh

The Unit Impulse Function

Properties of unit impulse

More properties

Very important

The Unit Step Function

The Unit Step Function

The Unit Step Function

The Unit Step Function

Square pulse using unit step

Relationship between unit impulse and unit step

Assignment

• Solve the problems of the chapter