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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)? |
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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: $$cos^2(x)=\frac{1}{2}(cos(2x)+1)$$ |
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