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Properties of LSI Impulse Response

Module by: Richard Baraniuk

Summary: An overview of the properties of LSI impulse responses.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Figure 1
Figure 1 (fig1.png)

Causality

An LSI system is causal if the current output is a function of only current and past input values.

Figure 2
Figure 2 (fig2.png)
What is the constraint on h h?

Causality of LTI Systems

When, in addition to LTI, is a system causal?

Definition 1: Causal
y n o y n o depends on xn x n only for n≤ n o n n o .
Definition 2: Causality
hn=0 h n 0 for all h<0 h 0
Figure 3
Figure 3 (fig3.png)

Exercise 1: Causal or Non-Causal?

Figure 4
Figure 4 (fig4.png)

Solution

No

Exercise 2

Figure 5
Figure 5 (fig5.png)

Solution

Yes

Exercise 3

Figure 6
Figure 6 (fig6.png)

Solution

Yes

Exercise 4

Figure 7
Figure 7 (fig7.png)

Solution

No

Exercise 5

Figure 8
Figure 8 (fig8.png)

Solution

No

FIR - Finite Impulse Response

An LSI system is FIR if the current output is the function of only a finite number of input values.

Figure 9
Figure 9 (fig9.png)
Otherwise it is called IIR - infinite impulse response.

What are the constraints on h h to be FIR?

Impluse Response Duration

Figure 10
Figure 10 (fig10.png)
yn=∑k=-∞∞xkhn−k y n k - x k h n k

Type 1: Finite Impulse Response (FIR)

-∞< N 1 N 1 and N 2 <-∞ N 2

Figure 11
Figure 11 (fig11.png)
→ sum in convolution is finite → practical.

Type 2: Infinite Impluse Response (IIR)

Either N 1 N 1 or N 2 N 2 or both =∞

Figure 12
Figure 12 (fig12.png)
→ sum in convolution is infinite → impractical.

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