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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.

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    $\begingroup$ Yes, you can sum them. In order to give a "precise justification", it'd be useful to know why do you think they cannot be added. $\endgroup$ – MBaz Oct 30 at 19:49
  • $\begingroup$ In fact, I took a special case with just $2$ signals. In practice, I have a family of signals that I want to sum $2$ to $2$ for trying to create an orthogonalization process. $\endgroup$ – user45939 Oct 30 at 20:15
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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.

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