1. EE 309 Signal and Linear System Analysis
Linear Time-Invariant Systems
Lecture 5
EE 309 Signal and Linear System Analysis

2. Summary of Last Lecture
Total Energy
Average Power
Sum of power Time
Energy / Time
Lecture 5
EE 309 Signal and Linear System Analysis

3. Linear Time-Invariant Systems
A system produces an output signal in response to stimulus from an input signal.
What is a Linear Time-Invariant (LTI) System?
A Linear system produces an output that is a linear function of the input
A Time-Invariant system produces an output that is NOT explicitly dependent on time
A time-shifted input results in an only time-shifted output!!
Input
Output
Lecture 5
EE 309 Signal and Linear System Analysis

4. Linear Time-Invariant Systems Linear Systems
A function f(…) is said to be linear if:
Scaling Property: 𝑓(𝑎𝑥)=𝑎 𝑓(𝑥)
Additivity Property: 𝑓( 𝑥 1 + 𝑥 2 )=𝑓( 𝑥 1 )+𝑓( 𝑥 2
Is the function 𝑦(𝑥)=m 𝑥+𝑐 a linear function?
Assume, m and c to be constants.
Lecture 5
EE 309 Signal and Linear System Analysis

5. Linear Time-Invariant Systems Linear Systems
Scaling Property:
Consider the following:
Lecture 5
EE 309 Signal and Linear System Analysis

6. Linear Time-Invariant Systems Linear Systems
It the system scalable?
Given the relationship between the input (𝑥(𝑡)) and output (𝑦(𝑡)):
Is the system scalable?
Input
Output
System
If, x = c x and y = c y,
Is the eqn unchanged?
Lecture 5
EE 309 Signal and Linear System Analysis

7. Linear Time-Invariant Systems Linear Systems
Additivity Property:
Also known as the superposition property!!
An Example: Determine the current I in resistor R2
Lecture 5
EE 309 Signal and Linear System Analysis

8. Linear Time-Invariant Systems Linear Systems
A system is time-invariant if a time shift in the input results only in the same time shift in the output
Lecture 5
EE 309 Signal and Linear System Analysis

9. Linear Time-Invariant Systems Linear Systems
MATLAB Code
Lecture_05_LTI_systems.m
Some Examples:
System
Linear?
Time Invariant?
Scaling?
Additive?
Yes
Yes
Yes
Yes
Yes No No No
Yes
Yes
Yes No No No
Yes
Lecture 5
EE 309 Signal and Linear System Analysis

10. Linear Time-Invariant Systems Linear Systems
Graphical review of
System
y(𝑡)
𝑥(𝑡)
Is this a TI system?
System
y 𝑡−1 ?
𝑥(𝑡−1)
Lecture 5
EE 309 Signal and Linear System Analysis

11. EE 309 Signal and Linear System Analysis
Next Lecture
Impulse Response of LTI Systems
Reading Assignment: Chap. 2.2
Lecture 5
EE 309 Signal and Linear System Analysis