What is the Energy of a Signal with Rectangular Components?

[banfina]

Homework Statement

Given two constants, A and B, what is the energy of the following signal?

$$f(t) = A*rect(t) + B*rect(t-0.5)$$

Homework Equations

$$E_f = \int_{-\infty}^{\infty} |f(t)|^2$$

The Attempt at a Solution

$$E_f = \int_{-\infty}^{\infty} [A*rect(t) + B*rect(t-0.5)]^2 dt$$
$$= \int_{-\infty}^{\infty} [A^2*rect^2(t) + 2AB*rect(t)rect(t-0.5) + B^2rect^2(t-0.5)] dt$$
$$= A^2\int_{-\infty}^{\infty} rect^2(t) dt + 2AB\int_{-\infty}^{\infty} rect(t)rect(t-0.5) dt + B^2\int_{-\infty}^{\infty} rect^2(t-0.5) dt$$
$$= A^2 + 2AB + B^2$$
$$= (A + B)^2$$

This seems wrong to me somehow; I guess my real question is does $$\int_{-\infty}^{\infty} rect^2(\frac{t}{\tau}) = \tau$$?

 

[HallsofIvy]

What do you mean by "rect(x)"?

 

[banfina]

What do you mean by "rect(x)"?

$$rect(t) = \left\{ \begin{array}{11} 0 & \mbox{if } |t| > \frac{1}{2} \\ \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\ 1 & \mbox{if } |t| < \frac{1}{2} \end{array} \right.$$
http://en.wikipedia.org/wiki/Rectangular_function