Q:

sum of 2 signals

I have $2$ signals: $x(t)$ and $y(t)$. They have the same bandwidth $B_x=B_y$. But the bounds for the bandwidth are not the same. For example, $B_x=f_{max}-f_{min}$ and $B_y=g_{max}-g_{min}$ with $f_{max},f_{min}>g_{max}$.
This is my question:
Can we sum these $2$ signals ($z(t)=x(t)+y(t)$) even if the bounds of the bands are not the same? I would like an answer with a precise justification.

A:

Yes, of course, you can sum them. The bandwidth of the resulting signal is simply the min/max of the individual signals. If we assume
$$z(t)=x(t)+y(t)$$
Then then bandwidth of $z(t)$ will simply be $[min(f_{min},g_{min}),max(f_{max},g_{max})]$, so in general the you will have $B_z > B_x$
Keep in mind that for any real valued signal $x(t)$ the spectrum has complex conjugate symmetry, i.e. $f_{min}=-f_{max}$. The whole idea of having two different signals with the same bandwidth but different frequency limits can only happen if the time domain signals are complex.

