Unit step function symbol

Question

I've been studying communications and i see that the book used the symbol $u_{-1}(t)$ for unit step function , why ?

Answer 1

The standard notation for the unit step function is $u(t)$. I've seen the notation $u_k(t)$, with integer $k$ in Oppenheim's Signals and Systems, where $k$ is the order of the derivative of the Dirac delta impulse. I.e., $u_1(t)=\delta'(t)$, etc. Using the same logic, negative values of $k$ indicate integrals of the Dirac delta impulse. So $u_{-2}(t)$ is the unit ramp: $u_{-2}(t)=tu(t)$.

In Signals and Systems, Oppenheim uses $u_k(t)$ mainly for $k\ge 1$ and $k\le -2$, because he prefers to use the standard symbols $\delta(t)$ and $u(t)$ for the Dirac impulse and the unit step, respectively.

But using the logic explained above, $u_{-1}(t)$ is the unit step function, being the integral of the Dirac impulse. The Dirac impulse itself would then be $u_0(t)$. I guess that that was the intention of the author of the book you refer to.