Unit 6 of 13

Laplace Transforms and Causal Response

  • 6 lessons
  • Videos 31 to 36

The Laplace transform turns a constant-coefficient initial-value problem into algebra. This unit defines it, inverts it and uses it for switched inputs, convolution, impulses and transfer functions.

The one-sided Laplace transform exists for piecewise continuous functions of exponential order, on a half-plane of convergence, and partial fractions invert rational Laplace transforms. The Laplace transform of a derivative carries the initial values into the algebra. Step functions describe switched inputs, and the second shifting theorem needs them rewritten in the delayed variable. Convolution gives the zero-state response, and the Dirac delta is a distribution that makes the velocity jump. Transfer-function poles describe input-output behavior, not the stability of every internal state.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

The Laplace Transform and Its Inverse

  1. The Laplace Transform

    How the one-sided Laplace transform weights a signal by a decaying exponential, why piecewise continuity and exponential order guarantee it on a region of convergence, and how linearity builds new pairs.

    Video 31

  2. Inverse Laplace Transforms and Partial Fractions

    How to recover a signal from a rational Laplace transform with partial fractions, giving every power of a repeated pole its own term and each irreducible quadratic factor a linear numerator over a completed square.

    Video 32

Initial-Value Problems and Switched Inputs

  1. Laplace Transforms of Derivatives and Initial-Value Problems

    How integration by parts gives the Laplace transform of a derivative, how the initial-value terms enter an algebraic equation for the Laplace transform, and how this solves linear initial-value problems.

    Video 33

  2. Step Functions, Time Shifts, and Periodic Inputs

    How the Heaviside step function switches an input on, how to write piecewise inputs with steps, why the second shifting theorem needs a true time shift, and how to find the Laplace transform of a periodic function.

    Video 34

Convolution, Impulses and Transfer Functions

  1. Convolution and Linear-System Response

    How causal convolution adds up past inputs, why the convolution theorem turns it into a product of Laplace transforms, how the impulse-response kernel gives the zero-state response, and where Volterra integral equations come from.

    Video 35

  2. Impulses, Transfer Functions, and Poles

    How the Dirac delta distribution models a unit impulse, why an impulse makes the velocity jump, and how a transfer function and its poles describe the zero-state response and bounded-input bounded-output stability.

    Video 36