ECE 210: Analog Signal Processing

Deriving the Impulse Response \(h(t)\) from the Step Response \(y_s(t)\)

System Selection

System Context (Circuit Domain)

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Why do these relate? Because the input \(u(t)\) is the integral of \(\delta(t)\), the step response \(y_s(t)\) is the integral of the impulse response \(h(t)\). Therefore, \(h(t) = \frac{d}{dt}y_s(t)\).

The Core Concept

\( h(t) = \underbrace{y'_{cont}(t)}_{\text{Continuous Derivative}} + \sum \underbrace{\left[y_s(t_k^+) - y_s(t_k^-)\right]}_{\text{Jump Size}} \delta(t - t_k) \)

1. Unit-Step Response \(y_s(t)\)

2. Continuous Derivative \(y'_{cont}(t)\)

3. True Impulse Response \(h(t)\)

4. Comparison: Missing the Jumps?

Instructor Notes & Pedagogical Design

How to use this applet:

Use the dropdown to select different LTI systems. Scrub your mouse across any plot to view instantaneous values synced across all domains. This tool visually proves that standard continuous differentiation is insufficient for finding \(h(t)\) if the step response contains jump discontinuities.

Misconceptions Addressed:

  • Missing the Impulse: Students routinely differentiate \(e^{-t}u(t)\) as \(-e^{-t}u(t)\), forgetting the chain rule on \(u(t)\). Card 4 makes the missing \(\delta(t)\) graphically obvious.
  • Physical connection: Tooltips connect abstract \(y_s(t)\) expressions to real circuit voltages (e.g., across a capacitor vs. a resistor).
  • Time-invariance: The delayed pulse preset shows that shifting the input directly shifts the output and the resulting impulses.