# Product of power signal and energy signal

I tend to believe that if we multiply an energy signal with a power signal the product is an energy signal.

My reasoning is that the energy signal will have finite bandwidth and consequenty the product will have finite bandwith and will be an energy signal.

Can anybody confirm that I am right? Or am I missing something?

• What about a signal whose spectrum is infinite but with finite area? A Gaussian, for instance. – MBaz Sep 22 '17 at 16:13
• I see your point. But again, a gaussian multiplied by a power signal is an energy signal, isn't it? – gdaras Sep 22 '17 at 16:38
• What about the product of a signal with Gaussian spectrum and finite area, times a power signal whose spectrum is 1/Gaussian? (or any other spectrum which grows as $f$ grows). – MBaz Sep 22 '17 at 16:47
• (BTW, i had never thought of this question and I don't know the answer, I'm just throwing questions out there in case it helps you) – MBaz Sep 22 '17 at 16:48

If you are not looking for a rigorous proof, then the following would suffice for a particular class of power signals which has the property that their magnitude is bounded; i.e, $|x_p(t)| \leq K$ for all $t$.
Numerous examples exist for such power signals like sinusoidal waves, complex exponentials, all kinds of periodic continuous-time signals which are defined via convergent Fourier series sums... Then for such a power signal $x_p(t)$ and a given energy signal $x_e(t)$ you can show that their product $y(t) = x_p(t) x_e(t)$ will be an energy signal as follows; given
$$E_x = \int_{-\infty}^{\infty} |x_e(t)|^2 dt$$
and $|x_p(t)| \leq K$ for all $t$ then the energy of y(t) is
$$E_y = \int_{-\infty}^{\infty} |y(t)|^2 dt = \int_{-\infty}^{\infty} |x_p(t) x_e(t)|^2 dt \leq \int_{-\infty}^{\infty} K^2 |x_e(t)|^2 dt = K^2 E_x$$
Hence $y(t)$ will also be an energy signal.