13.5 : Basic Continuous Time Signals

Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.

The unit step function, denoted u(t), is zero for negative time values and one for positive time values, exhibiting a discontinuity at t=0. This function often represents abrupt changes, such as the step voltage introduced when turning a car's ignition key. The derivative of the unit step function is the unit impulse function, denoted δ(t). The unit impulse function is zero everywhere except at t=0, where it is undefined. It is a short-duration pulse with a unit area, signifying an applied or resulting shock.

Integrating a function with the impulse function yields the value of the function at the impulse point, a characteristic known as sampling. Mathematically, this is expressed as,

Integrating the unit step function results in the unit ramp function, denoted r(t). The unit ramp function is zero for negative time values and increases linearly for positive time values, representing a function that changes steadily over time. These basic continuous-time signals are fundamental in signal processing and system analysis due to their unique properties and applications.