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

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

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

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

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

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

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

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

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

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

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