1

u(t)

The given time signal is:

$$u(t) = -3\sigma(t+4) + 6\sigma(t) - 3\sigma(t-4)$$

$\sigma$ - unit step function

The same signal can be describes with the following mathematical relation between $u(t)$ and $u_1(t)$:

$$u(t) = u_1(t-t_0) - u_1(t+t_0)$$

Now I have to determine the signal $u_1(t)$.

First attempt

$$ \begin{align} u(t) &= u(t)\\ -3\sigma(t+4)+6\sigma(t)-3\sigma(t-4) &= u_1(t-t_0)-u_1(t+t_0)\\ \color{red}{-3\sigma(t+4)+3\sigma(t)}\color{blue}{-3\sigma(t-4)+3\sigma(t)} &= \color{red}{u_1(t-t_0)}\color{blue}{-u_1(t+t_0)} \end{align} $$

At this point I see two problems:

  1. Ignoring the parameters, the signs does not look correctly. Expecting sign change between blue and red on the left side.
  2. Not ignoring the parameters, could the signs be right and should I assume $t_0 = 4$?

Second attempt

Canceled because of a good answer.

Marc
  • 113

1 Answers1

1

You need to see that by writing

$$u(t)=u_1(t-t_0)-u_1(t+t_0)$$

you simply decompose the signal $u(t)$ in two equally shaped signals with different signs and different time shifts. By looking at the signal $u(t)$ you see immediately that you can construct it by using a rectangular signal with height $3$ and width $4$, once shifting it to the left with a negative sign, and adding another such signal shifted to the right. Since both shifts need to be the same (with different signs), the signal $u_1(t)$ must be centered around $t=0$, i.e. it must be symmetric. So $u_1(t)$ is given by

$$u_1(t)=3[\sigma(t+2)-\sigma(t-2)]$$

which is a rectangle with height $3$ in the interval $[-2,2]$.

With a shift $t_0=2$ you get

$$u(t)=3[\sigma(t)-\sigma(t-4)-\sigma(t+4)+\sigma(t)]=-3\sigma(t-4)+6\sigma(t)-3\sigma(t+4)$$

which is indeed equal to the given $u(t)$.

Matt L.
  • 10,636