Exponentially Damped Sinusoidal Signals

x(t) = A(e^(-αt))sin(ωt + Φ)

• Exponentially Damped Sinusoid (Continuous-Time)

• A sinusoidal signal multiplied by a decaying exponential

The oscillation amplitude decreases exponentially and approaches zero as t approaches ∞

Behavior of Damped Sinusoids

Parallel LRC Circuit Differential Equation

Satisfies the voltage across the capacitor

Solution of LRC Circuit

Natural Angular Frequency (Damped Case)

Discrete-Time Exponentially Damped Sinusoid

0 < |r| < 1

• Decay Condition for Discrete-Time Damped Sinusoid

• The signal decays if the r is in the following range

Discrete-Time Unit-Step Function

Continuous-Time Unit-Step Function

t = 0, u(0)

The unit-step function is discontinuous at _____; _____ is undefined

Models the sudden application of a DC source or switch closure.

Physical Interpretation of Unit-Step

Rectangular Pulse Signal

A finite-duration signal with constant amplitude.

Mathematical Definition of Rectangular Pulse

Rectangular Pulse Using Step Functions