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\{<br /> \begin{array}{11}<br /> 0 &amp; \mbox{if } |t| &gt; \frac{1}{2} \\<br /> \frac{1}{2} &amp; \mbox{if } |t| = \frac{1}{2} \\<br /> 1 &amp; \mbox{if } |t| &lt; \frac{1}{2}<br /> \end{array}<br /> \right.
http://en.wikipedia.org/wiki/Rectangular_function