# How is the energy of $x_1\cdot x_2$ related to the energies of $x_1$ and $x_2$?

Let's say first signal x1 = [1 2 3 4], second signal x2 = [0.08 0.77 0.77 0.08] (Hamming window), third signal x3 = x1.*x2 = [0.08 1.54 2.31 0.32].

Energy of x1: E1=30, energy of x2: E2=1.1986, energy of x3: E3=7.8165.

What is energy relation between x1,x2 and x3?

• This reads a bit like homework... Jun 17 '16 at 17:45

Knowing the energies of $x_1$ and $x_2$ is not sufficient for determining the energy of $x_3=x_1x_2$. What you can do is determine an upper bound for the energy of $x_3$ given the energies of $x_1$ and $x_2$ and their maximum values:

$$E_3=\sum_{k}\big|x_1[k]x_2[k]\big|^2\le\begin{cases}\max_k\big|x_1[k]\big|^2\sum_k\big|x_2[k]\big|^2=\max_k\big|x_1[k]\big|^2E_2\\\max_k\big|x_2[k]\big|^2\sum_k\big|x_1[k]\big|^2=\max_k\big|x_2[k]\big|^2E_1 \end{cases}\tag{1}$$

From $(1)$ an upper bound for $E_3$ is

$$E_3\le\min\left\{\max_k\big|x_1[k]\big|^2E_2,\max_k\big|x_2[k]\big|^2E_1\right\}\tag{2}$$

Unfortunately, this bound is generally not very tight. For your example you get

$$E_3\le\min\{19.178,17.787\}=17.787$$

but the actual value is $E_3=7.8165$.

• As an example of why knowing the energies of $x_1$ and $x_2$ is insufficient, consider the case where $x_1 = [0, 1, 0, 1, ...]$ and $x_2 = [1, 0, 1, 0, ...]$. They each have equal energies, but the energy of their elementwise product is zero. Jun 17 '16 at 17:43
• @JasonR : Was just about to add an example like that! Good stuff! :-)
– Peter K.
Jun 17 '16 at 18:21