AM Theory for Tones with Unequal AmplitudeAM Theory for Tones with Unequal AmplitudeIn his article 'Beating Sinusoids and Pitch Changes'[1] Lloyd A. Jeffress describes the IF deviations seen in the previous section. This technical note is largely concerned with the perception of these 'pitch changes' however he goes on to define the following equation for the combination of sine waves:
y(t) = (a+b)cos(2*PI*n*t)cos(2*PI*m*t) + (a-b)sin(2*PI*n*t)cos(2*PI*m*t)
where m = (f1 + f2)/2, n = (f1 - f2)/2, and a and b are the amplitudes of frequencies f1 and f2.
This equation descibes the combination of unequal sine waves as before, with m being the frequency of the AM waveform (at least at peak envelope) and n the period of the envelope beating. If we apply this equation to the results found in the last two experiments of ' Single Channel AM Experiments ' we find that the theoretical values back up those found during the last two experiments.
However, if the equation can describe how two tones can be combined we should be able to apply the inverse and calculate the consituent tones of a modulated waveform. A useful method for extracting the variables from the AM signal for component separation would be to examine the most obvious cues, i.e. the IF deviations or 'spikes' at the envelope troughs (see previous section). Jeffress describes the deviations as an equation of frequency change D(delta)f as follows:
Df = n((a^2 -b^2)/(a^2 + b^2 +2ab*cos(4*PI*n*t)))
Now we have an equation for the deviation of IF the next step is to find specialised equations for when IF deviation is at its maximum (the spike) and minumum. By substituting k = a/b, and finding the minimum (when cos(4*PI*n*t) is 1 ) and maximum (when cos(4*PI*n*t) is -1 ) Df we find that:
Df(min) = n(k-1)/(k+1)
Df(max) = n(k+1)/(k-1)
Since we know that Df(min) and Df(max) occurs at the envelope peak (pe) and troughs (tr), and converting Df the IF minus the centre frequency (cf) of the AM signal. Substituting this into the equations above we get the following:
pe - cf = n(k-1)/(k+1) - minimum deviation [1]
tr - cf = n(k+1)/(k-1) - maximum deviation [2]
Now there is an equation for cf in terms of either the peak or trough IF, since we can measure these values, as well as n ( by measuring the period between IF peaks or troughs), all that is required is a value for k. Taking the equation for minimum deviation [1] we can find k in terms of pe,cf, and n. This results in two possible equations for k as follows:
2 2 2 1/2
4 n - 2 (4 n + pe - 2 pe tr + tr )
k := {1/2 ---------------------------------------, [3]
- pe + tr
2 2 2 1/2
4 n + 2 (4 n + pe - 2 pe tr + tr )
1/2 ---------------------------------------} [4]
- pe + tr
Equation [3] is > 0 if a > b Equation [4] is > 0 if a < b Therefore equation 1 will give correct k if a < b, and equation 2 will give the answer when a > b if trough and peak values are assigned correctly. If not the opposite will be true. When k has been found there are two possible equations for finding cf, either of which may be used. We can now use these equations with [1] and [2] to find the cf:
n (k - 1) n (k + 1)
cf := {- ---------- + pe, - ---------- + tr} [5]
k + 1 k - 1
Using these equations it is possible to find the AM centre frequency by simply measuring the peak and trough IF values and the 'beating' period of the AM. Once cf has been found then the two component frequencies of the AM signal lie at cf + n and cf -n.
In order to test this theory a program has been written which will take the output of the gammatone filterbank (giving envelope, IF, etc.) to detect peak's and troughs in IF in order to find values for pe,tr, and the period n. This avoids having to analyse the envelope for peak/troughs (since they always occur in the same positions). Unfortunately, depending on which tone is dominant, higher or lower frequency, the IF either peak or trough when the envelope reaches a minimum. Therefore without knowing which tone is dominant (impossible) or examining the envelope for peaks and troughs as well as IF there is no way of matching the IF peaks/troughs with pe or tr. So using this method there will always be two estimations for cf, one for each estimation of k. Fortunately these estimations are widely different, therefore it is an easy matter to select the correct value since it will always be the one lying closest to the filter's CF (since the other estimation would lie so far outside the bandwidth of the filter as to be undetectable).
Using this estimator on clean two tone AM signals found a very low error ratio of 0.1% in the estimation of the component tone's frequency (caused in the most part by sampling quantisation error). However, when noise was added to the signal the error percentage rose sharply, this is due to the ease at which the envelope trough IF (tr) can be corrupted, since it is based on the analysis of a relatively low amplitude signal (the AM trough).
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