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Properties of LTI Systems

Module by: Richard Baraniuk

Summary: Introduction to the properties of LTI systems.

All LTI systems are characterized by the convolution Figure 1

yn=∑k=-∞∞xkhn-k y n k x k h n k (1)
Figure 1
fig10.png
  • The impulse response hn h n is a complete characterization of the properties of a specific LTI system.
  • For LTI systems, we often use this diagram.
Figure 2
fig9.png
Let's look at some of these properties and how they are reflected in hn h n .

Convolution is Commutative

This means:

xn*hn=hn*xn x n h n h n x n (2)
or Figure 3.
Figure 3
fig8.png

Proof

The proof is a simple change of variables:

yn=xn*hn=∑k=-∞∞xkhn-k=∑m=∞-∞xn-mhm=∑m=-∞∞hmxn-m=hn*xn y n x n h n k x k h n k m x n m h m m h m x n m h n x n (3)
where m=n-k m n k , which means k=n-m k n m .

Cascade Connecion of LTI Systems

Compute Figure 4.

Figure 4
fig7.png

Two Methods

  1. Brute Force and Massive Ignorance: Use yn=vn* h 2 n y n v n h 2 n and substitute in vn=xn* h 1 n v n x n h 1 n . Try this at home.
  2. Elegant and Quick: Just find the overall impulse response, Figure 5, which implies Figure 6.
Figure 5: h 1 n* h 2 n h 1 n h 2 n is the overal impulse response.
fig6.png
Figure 6
fig5.png

Parallel Connection of LTI Systems

Same elegant reasoning gives Figure 7.

Figure 7
fig4.png

Example 1: Moving Window Filter

Figure 8
fig3.png

What is the impulse response of Figure 8?

Is it an FIR or IIR? Figure 8?

What is yn y n of Figure 8?

Example 2: Pure Recursive Filter

Figure 9
fig2.png

What is the impulse response of Figure 9?

Is it an FIR or IIR ? Figure 9?

What is yn y n of Figure 9?

Proof That the Filter Structure Computes the Correct Output

Find its impulse response (Figure 10) and show that it is hh.

nn Input δ 0 n δ 0 n
0 1 0 0 0 … 0 0
1 0 1 0 0 … 0 0
2 0 0 1 0 … 0 0
3 0 0 0 1 … 0 0
4 0 0 0 0 … 1 0
⋮ ⋮ ⋮ ⋮ ⋮ … ⋮ ⋮
Figure 10
fig1.png
n n hn h n
0 h0 h 0
1 h1 h 1
2 h2 h 2
3 h3 h 3
4 h4 h 4
⋮ ⋮
where the hn h n 's make up h h, which is the impulse response we desire.

Example 3: Combination Moving Window/Recursive Filter

Figure 11.

Figure 11
figlast.png

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