Signals and Systems

1. Lecture17
Properties of LTI Systems

2. Properties of LTI Systems
Invertibility (Cont.)
2. A system 𝑦[𝑛] has an impulse response ℎ 𝑛 = 𝑢[𝑛]. Is this system
invertible?
• An accumulator or a summer (𝑦 𝑛 = 𝑘=−∞
𝑛
𝑥[𝑘]) is a system that
computes the running sum of all the values of the input up to the
present time.
• The inverse system of an accumulator or a summer can be obtained
using the first difference operation (i.e., 𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 − 1]).
(Example 2.11, 2.12, Oppenheim)

3. Properties of LTI Systems
Invertibility (Cont.)
2. A system 𝑦[𝑛] has an impulse response ℎ 𝑛 = 𝑢[𝑛]. Is this system
invertible?
• An accumulator or a summer (𝑦 𝑛 = 𝑘=−∞
𝑛
𝑥[𝑘]) is a system that
computes the running sum of all the values of the input up to the
present time.
• The inverse system of an accumulator or a summer can be obtained
using the first difference operation (i.e., 𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 − 1]).
(Example 2.11, 2.12, Oppenheim)

4. Properties of LTI Systems
Causality: When impulse response ℎ[𝑛] is a causal signal, the corresponding LTI system is called a causal system
DT: ℎ 𝑘 = 0 𝑓𝑜𝑟 𝑘 < 0
y 𝑛 =
𝑘=−∞
∞
𝑥 𝑘 ℎ 𝑛 − 𝑘 =
𝑘=−∞
∞
ℎ 𝑘 𝑥 𝑛 − 𝑘
y 𝑛 =
𝑘=−∞
∞
ℎ 𝑘 𝑥 𝑛 − 𝑘
y 𝑛 =
𝑘=0
∞
ℎ 𝑘 𝑥[𝑛 − 𝑘]
CT: ℎ 𝜏 = 0 𝑓𝑜𝑟𝜏 < 0
𝑦 𝑡 =
−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏
𝑦 𝑡 =
0
∞
ℎ 𝜏 𝑥 𝑡 − 𝜏 𝑑𝜏

5. Properties of LTI Systems
Stability

6. Properties of LTI Systems
Stability
DT: 𝑘=−∞
∞
ℎ[𝑘] < ∞
CT: 𝑡=−∞
∞
ℎ(𝜏) 𝑑𝜏 < ∞

7. A systm is the following impulse response. Is the
this system BIBO stable, causal and memoryless

8. Properties of LTI Systems
Stability – Examples:
i. Pure Time Shift System
DT: 𝑛=−∞
∞
ℎ[𝑛] = 𝑛=−∞
∞
𝛿[𝑛 − 𝑛0] = 1
CT: −∞
∞
ℎ(𝜏) 𝑑𝜏 = −∞
∞
𝛿(𝜏 − 𝑡0) 𝑑𝜏 = 1
ii. Accumulator
DT: 𝑛=−∞
∞ 𝑢[𝑛] = 𝑛=0
∞
𝑢[𝑛] = ∞
CT: −∞
∞
𝑢(𝜏) 𝑑𝜏 = 0
∞
𝑑𝜏 = ∞
(Example 2.13, Oppenheim)

9. Properties of LTI Systems
Unit Step Response
The response of the system when the input is a unit step signal i.e.,
DT: 𝑠 𝑛 = 𝑢 𝑛 ∗ ℎ 𝑛
• Step response of a DT LTI system is the running sum of its impulse response.
𝑠 𝑛 =
𝑘=−∞
𝑛
ℎ[𝑘]
• Impulse response of a DT LTI system is the first difference of its step response.
ℎ 𝑛 = 𝑠 𝑛 − 𝑠 𝑛 − 1
CT:
𝑠(𝑡) = 𝑢(𝑡) ∗ ℎ(𝑡)
• Step response of a CT LTI system is the running integral of its impulse response.
𝑠 𝑡 = −∞
𝑡
ℎ(𝜏) 𝑑𝜏
• Impulse response of a CT LTI system is the first derivative of the unit step response.
ℎ 𝑡 =
𝑑𝑠 𝑡
𝑑𝑡
= 𝑠′(𝑡)