# 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.

• 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. – MBaz Oct 30 '19 at 19:49
• 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. – user45939 Oct 30 '19 at 20:15

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.