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Impulse Response Properties of an LTI System

Module by: Richard Baraniuk

Summary: An introduction to the impulse response properties of an LTI system.

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LTI is linear and time-invariant: Figure 1, Figure 2, Figure 3.

Figure 1
Figure 1 (fig1.png)
Figure 2
Time Invariance
Time Invariance (fig2.png)
Figure 3
Linearity
Linearity (fig3.png)

Bottom Line

For an LTI system, it is easy to compute output due to the sum of shifted and scaled impulses.

Amazing Fact:

By the sifting property of impulses, any signal can be decomposed in terms of an infinite sum (integral) of shifted, scaled impulses.

Sifting

ft=∫-∞∞fτδt−τdτ f t τ f τ δ t τ (1)
where the integral is an "infinite sum" (Figure 4).
Figure 4: Peaks up where t=τ t τ .
Figure 4 (fig4.png)
This can be derived from the properties of δ δ, but it also has a nice physical interpretation.

Interpret f(t)

Equation 1.

  1. Recall δt=limΔ→0 δ t Δ 0 (Figure 5).
  2. Figure 6.
  3. Sum over -∞≤n≤∞ n (Figure 7).
  4. Take limΔ→0∑n=-∞∞fnΔ δ Δ t−nΔΔ=∫-∞∞fτδt−τdτ Δ 0 n f n Δ δ Δ t n Δ Δ τ f τ δ t τ .
Figure 5: Call this δ Δ t δ Δ t .
Figure 5 (fig5.png)
Figure 6: The height is fnΔ f n Δ .
Figure 6 (fig6.png)
Figure 7
Figure 7 (fig7.png)

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