Question

How is the continuous time impulse function defined in terms of the step function?

• A. \( u(t) = \frac{d}{dt} \delta(t) \)
• B. \( u(t) = \delta(t) \)
• C. \( \delta(t) = \frac{d}{dt} u(t) \)
• D. \( \delta(t) = u(2t) \)

Hint:

Continuous Impulse and Step. The Dirac delta function \(\delta(t)\) is the derivative of the Heaviside step function \(u(t)\): \(\delta(t) = du(t)/dt\). The step function is the integral of the delta function.

Answer 1

The Correct Option is C

The continuous-time unit step function, \(u(t)\), is defined as: $$ u(t) = \begin{cases} 1 & \text{if } t>0
0 & \text{if } t<0 \end{cases} $$ (Value at t=0 can vary, often taken as 0
5 or undefined)
The continuous-time unit impulse function (Dirac delta function), \(\delta(t)\), is defined by its properties, primarily the sifting property: \(\int_{-\infty}^{\infty} f(t)\delta(t-t_0) dt = f(t_0)\)
It can be thought of as the limit of a pulse with unit area as its width goes to zero and height goes to infinity
Mathematically, the Dirac delta function is the derivative of the Heaviside step function \(u(t)\) in the sense of distributions: $$ \delta(t) = \frac{d}{dt} u(t) $$ Conversely, the unit step function is the integral of the unit impulse function: $$ u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau $$ Therefore, the correct relationship is \(\delta(t) = \frac{d}{dt} u(t)\)