audio signal arithmetic

BACKGROUND: I have been messing around with audio signals in Audacity and realized that by inverting a signal and adding it to the same signal I get no signal. Thus (S + invS) is really (S-S). (S+S) = 2S and S increased by 6db is also 2S (checked by subtracting an inverted S+S signal and got no signal).

QUESTION: What would be S*S, 1/S, and sqrt(S)?

-

When working with audio signals, you must look at the signals as phasors rather than plain numbers. As such, the signals will be represented as complex numbers. For example, a given sinusoidal signal $Se^{j\phi}$ has an amplitude of S, and a phase angle of $\phi$. Inverting the signal means increasing its phase angle by 180 degrees, or $\pi$ radians. So when adding the two signals together what you are doing is represented by the equation (omitting conjugate pairs):

$$S_{sum}=Se^{j\phi}+Se^{j(\phi+\pi)}$$ $$S_{sum}=Se^{j\phi}+Se^{j\phi}e^{j\pi}$$ $$e^{j\pi}=-1$$ $$S_{sum}=Se^{j\phi}-Se^{j\phi}=0$$

In terms of the intensity going up by 6dB when the intensity is doubled, this can be calculated by using the equation: $$10log(I_2/I_1)dB$$

Intensity is directly proportional to the square of the amplitude, so doubling the amplitude will result in the equation: $$10log(4S^2/S^2)dB=6dB$$

Doing the other calculations you mentioned is a simple matter of trigonometric arithmetic. For example, in the case of squaring the sinusoidal signal, the result is a sinusoid of twice the frequency and half the phase. This is demonstrated by the identity: $$cos^2(x)=\frac{1}{2}(cos(2x)+1)$$

-
A really good way to experiment more in depth with signal arithmetic would be to use MATLAB to make a signal, and play it using the "sound" function. – Ataraxia Mar 2 '13 at 6:33
Thank you. It looks like I will need to learn MATLAB and mess around with this as I don't see how I could implement these in Audacity. – Michelle Mar 2 '13 at 14:17
@Michelle no problem. Is there anything you're still unclear on? – Ataraxia Mar 2 '13 at 15:34
Yeah, I haven't had trig yet (next year). But I looked up some identities and read some about sine and cosine. So cos is really just sin + pi/2. I guess I am unclear why you used cos^2x instead of sin^2x. I also haven't figured out how to type the math symbols here. Even with these confusions though I now see the importance of viewing the signals as sine waves. Which leads to the importance of figuring out how the trig math works out. – Michelle Mar 2 '13 at 19:48
Downloading MATLAB now, so maybe I'll be messing with it tonight or tomorrow. thnx – Michelle Mar 2 '13 at 19:56