Thursday, January 12, 2012

Mixing Two Tones (Take 2)

The previous solution for mixing two tones was only an approximation. Using a result from trigonometry one can show that the sum of two cosine functions of equal magnitude is proportional to the produce of the cosines of half the sum and half the difference of the two original frequencies.

The beats correspond to the peaks of the absolute value of the amplitude function. The frequency of the oscillations is average of the reference frequency ω0 and the second frequency ω. The result is closer to what one would expect.

Allowing negative values for the amplitude results in a second oval in the phase diagram which is the mirror image of the one for positive amplitudes.

The previous solution appears to be missing a small term of the same frequency of the single cosine function since the amplitude is repeatedly distorted. The error results from assuming that the solution only affects the amplitude and phase of a cosine function. When the coefficients of a sine and a cosine function are constants one can easily calculate the phase. Apparently this doesn't work properly when the coefficients are also function of time.

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