Signals & Systems #19 | LTI Differential & Difference Equations
Автор: EngineeringENG - AcEdumy
Загружено: 2026-03-03
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In this lesson, we learn how LTI (Linear Time-Invariant) systems are described by differential equations (continuous-time) and difference equations (discrete-time).
For CT systems, we solve the differential equation in two steps:
1. *Homogeneous solution* y_h — from the characteristic equation and its roots
2. *Particular solution* y_p — using a trial function matched to the input type
Then we work through a complete CT example: ÿ + 5ẏ + 6y = 2x + ẋ with x(t) = e^{-t}, finding y_h, y_p, and the full general solution.
For DT systems, the difference equation rearranges into a *recursive formula* that computes each output sample from current and past inputs plus past outputs. We compute y[0], y[1], y[2] step-by-step for the system y[n] + (1/4)y[n-1] = x[n] + 2x[n-2] with x[n] = u[n].
Topics covered:
General form of CT differential equations for LTI systems
Characteristic equation and homogeneous solution y_h
Trial function table for particular solution y_p
Full CT example: 2nd-order ODE step-by-step
General form of DT difference equations
Recursive computation from rearranged difference equation
Full DT example: table of y[0], y[1], y[2]
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