Superposing Signals

If a signal $x[n]$ has a peak or RMS amplitude $A$ (in some fixed window), then the scaled signal $k \cdot a[n]$ (where $k \ge 0$) has amplitude $kA$. The RMS power of the scaled signal changes by a factor of $k^2$. The situation gets more complicated when two different signals are added together; just knowing the amplitudes of the two does not suffice to know the amplitude of the sum. The two amplitude measures do at least obey triangle inequalities; for any two signals $x[n]$ and $y[n]$,

$${A_{\mathrm{peak}}} \{x[n]\} + {A_{\mathrm{peak}}} \{y[n]\} \ge {A_{\mathrm{peak}}} \{x[n]+y[n]\}$$

$${A_{\mathrm{RMS}}} \{x[n]\} + {A_{\mathrm{RMS}}} \{y[n]\} \ge {A_{\mathrm{RMS}}} \{x[n]+y[n]\}$$

If we fix a window from $M$ to $N+M-1$ as usual, we can write out the mean power of the sum of two signals:

$$P \{x[n] + y[n]\} = P \{x[n]\} + P \{y[n]\} + 2 \cdot {\mathrm{COR}} \{ x[n] , y[n] \}$$

where we have introduced the correlation of two signals:

Correlation

$${\mathrm{COR}} \{ x[n] , y[n] \} = { {x[M]y[M] + \cdots + x[M+N-1]y[M+N-1]} \over N }$$

The correlation may be positive, zero, or negative. Over a sufficiently large window, the correlation of two Sinusoids with different frequencies is negligible. In general, for two uncorrelated signals, the power of the sum is the sum of the powers:

$$P \{x[n] + y[n]\} = P \{x[n]\} + P \{y[n]\} , \hspace{0.1in} \mathrm{whenever} \ {\mathrm{COR}} \{ x[n] , y[n] \} = 0$$

Put in terms of amplitude, this becomes:

$${{\left ( {A_{\mathrm{RMS}}} \{x[n]+y[n]\} \right ) } ^ 2} ... ...2} + {{\left ( {A_{\mathrm{RMS}}} \{y[n]\} \right ) } ^ 2} .$$

This is the familiar Pythagorean relation. So uncorrelated signals can be thought of as vectors at right angles to each other; positively correlated ones as having an acute angle between them, and negatively correlated as having an obtuse angle between them.

For example, if two uncorrelated signals both have RMS amplitude $a$, the sum will have RMS amplitude ${\sqrt 2} a$. On the other hand if the two signals happen to be equal--the most correlated possible--the sum will have amplitude $2a$, which is the maximum allowed by the triangle inequality.