1. EE 309 Signal and Linear System Analysis
Impulse Response of LTI Systems
Lecture 6
EE 309 Signal and Linear System Analysis

2. Summary of Last Lecture
Linear Time-Invariant System:
Linear: Scalable & additive (i.e., superposition)
Time-Invariant : System response remains unaffected by absolute time
for any 𝑡 0
Lecture 6
EE 309 Signal and Linear System Analysis

3. Impulse Response of LTI Systems
The impulse response:
Output of a system in response to an impulse input
The step response:
Output of a system in response to an step input
Knowing the impulse response of a system will allow the determination of the system’s response to any other input
Relationship
between Impulse and step?
Lecture 6
EE 309 Signal and Linear System Analysis

4. Impulse Response of LTI Systems
An Example:
Compute the impulse response of an RC circuit
The input is a voltage source (𝑥(𝑡)), and
The output is the voltage across the capacitor (𝑦(𝑡))
Assume that the initial voltage across the capacitor is zero
How would you test to see if this is a LTI system?
Scaling test and
Additive test
Lecture 6
EE 309 Signal and Linear System Analysis

5. Impulse Response of LTI Systems
The loop voltage eqn:
Voltage across the capacitor:
The input/output eqn:
Integrating factor?
Aside:
The i/p: x(t) = (t) (i.e., an impulse)
The o/p: y(t) = h(t) (i.e., the impulse resp.)
Lecture 6
EE 309 Signal and Linear System Analysis

6. Impulse Response of LTI Systems
Impulse response (let 𝜏 𝑐 =𝑅𝐶):
Step Response (let 𝜏 𝑐 =𝑅𝐶):
Relationship betw. ?
Knowing the step and impulse response we can use superposition to determine the
Response to any i/p that can be expressed as a combination of step/impulse inputs
Lecture 6
EE 309 Signal and Linear System Analysis

7. Impulse Response of LTI Systems
TI Property
Superposition
Scalable
Lecture 6
EE 309 Signal and Linear System Analysis

8. Impulse Response of LTI Systems
Impulse response from step response
Ramp response from step response
Lecture 6
EE 309 Signal and Linear System Analysis

9. EE 309 Signal and Linear System Analysis
Next Lecture
Convolution
Reading Assignment: Chap. 2.3
Lecture 6
EE 309 Signal and Linear System Analysis