Amplitude Modulation equation help

[dmorr]

Hey everyone,

I am trying to understand the phenomena behind the technique of using amplitude modulation in sound synthesis. When you have two frequencies, for example, w1=200hz and w2=300hz, and you use the amplitude of one (w1) to modulate the amplitude of the other (w2), it creates 3 signals, one of which is the original carrier frequency, plus 2 others which are called the "sum" and "difference".

you can read more about it here:

http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm

Anyways, to get the point here... the equation to explain this looks like this:


A1= a1cos(w1t)) + 1/2 [a2cos(w1+w2)t] + 1/2[a2cos(w1-w2)t]

so "1" is the carrier and "2" is the modulator frequency

A= instantaneous amplitude
a= maximum amplitude possible (just using an arbitrary unit here)
w= frequency
t= time

It's probably a basic trigonometry equation, but I haven't done trig since high school, and it's all a blur. What my goal is is to compare two sets of outcomes of data going through this equation. I just don't know how to do the math, so if someone could help me plug in these pairs of data into the equation, that would be awesome.

I want to compare the outcome of this carrier and modulator pair with another.

1. FIRST PAIR>> carrier: a1 = 100, w1= 100hz, t=3 seconds (also arbitrary)
modulator: a2 = 50, w2= 1hz, t =3

2. SECOND PAIR>> carrier: a1 = 100, w1= 100hz, t=3 seconds (also arbitrary)
modulator: a2 = 50, w2= 80hz, t =3

I would like to see what the instantaneous amplitude of the carriers would be in each cases, ideally it would be better if someone could just explain to me how to do the math, and then I could do a bunch of calculation myself.

Thank you so much if someone is able to help me.

dylan

 

[haruspex]

Why don't you drop these formulae into a spreadsheet and graph the result?
Btw, Eqn 4 at the link you provided is wrong. It's OK in Eqn 5.

 

[dmorr]

Why don't you drop these formulae into a spreadsheet and graph the result?

Because I don't know how to solve trigonometry problems. So even though I have the data and the equation, I am not able to plug in numbers into it and find the instantaneous amplitude of the carrier.

Since you were able to recognize that the equation in the link was wrong haruspex would you be able to put one of my pairs of data into the equation and show your work so I can do the rest on a spreadsheet graph myself?

cheers,

dmorr

 

[haruspex]

I don't understand why you think you have to solve any trig.
Create cells for the constants, a1, aq2, w1, w2.
Create a column for time, the first being 0, the rest being "above cell + .01".
Put the formula in the next column, and away you go.

Btw, you don't need to use eqn 5. You can use the original eqn:
(a1 + a2.cos(w2.t)).cos(w1.t)
The point of eqn 5 is to reveal certain properties.

 

[dmorr]

sweet, i will do that. thanks!