From: aephraim@physics.Berkeley.EDU (Aephraim M. Steinberg)
Newsgroups: sci.physics
Subject: Re: Simple Sin Wave form Question
Date: 7 Nov 1992 20:11:57 GMT
Organization: University of California, Berkeley
NNTP-Posting-Host: physics.berkeley.edu

In article <10292312.16173.15208@kcbbs.gen.nz> Steve_Grant@kcbbs.gen.nz (Steve Grant) writes:
>Hello,
>     As some of you may or may not know when a number is pressed on 
>the key pad of a telephone two tones are emitted, I wish to copy the 
>resulting sound to make a autodialer but- I am told by the people in 
>telecom land that the resultant frequencie is NOT the average. I know 
>that SIN (x + y) = SIN(x)COS(y) + COS(x)SIN(y) but when i plot this 
>on my plotting program for say the frequencies of say 1209, 657 it just 
>evaluates the numbers and draws a straight line parallel to the x axis, 
>is the y value of this line the resultant freq? ie:
>
>SIN (1209 + 657) = SIN(1209)COS(657) + COS(1209)SIN(657)

You should not be using 1209 and 657 as the arguments but 1209t and
657t, first of all, but perhaps you understood that.

Secondly, the sine of a sum is not the sum of the sines.  If one wave
is sin(1209t) and the other is sin(657t) the overall waveform is
sin(1209t) + sin(657t), not just sin(1209t+657t) = sin (1866t).


>
>I know that I should be able to figure this one out by myself but cannot 
>and are getting very frustrated. Any help in the form of dummy proof 
>explinations would be very much appreciated.
>
>Al Grant
>
>PS the question is: A) is the freq. the average?
>                    B) whats the equation?

The carrier frequency is indeed the average.  In the case where the two
superposed tones are not too different, it makes sense to talk about one
carrier frequency, modulated relatively slowly by a "beat frequency."
The beat frequency is the difference between the two.

That is, if you play a 440Hz tone and a 442Hz tone at the same time,
it will sound like a 441Hz tone which gets louder and softer sinusoidally
at a frequency of 2Hz.  There is not just "one frequency," however.
For that matter, if you play an A-440, but only let the note last for
half a second, it actually "contains" a whole range of tones from approximately
438 Hz to 442Hz.  This is because the "envelope," the attack and decay
of the note, can be seen as a more complicated form of "beat notes."

If you have something which only produces one tone at a time, you can
still make something sound like a dialer as long as you can vary the
loudness of your tone at the beat frequency.  If you really needed two
tones at 1209 Hz and 657 Hz, you would have to be able to control the
intensity at about 552 Hz-- i.e., to get a relatively nice waveform,
you'd have to readjust the loudness at least a few times every millisecond.

The equation is

Cos (A t) + Cos (B t) = 2 cos[(A+B)t/2] cos[(A-B)t/2]

The carrier oscillates at (A+B)/2, the average of A and B. 
The envelope oscillates at (A-B)/2, but the intensity goes as the square
of the amplitude, so we hear this as a modulation at a frequency (A-B).

If you're not scared of complex numbers, the derivation is much easier
by using them rather than 3 or 4 trig identities in a row:

exp (iAt) + exp (iBt)  =
		       exp[i(A+B)t/2] {exp[i(A-B)t/2] + exp[i(B-A)t/2]}
                    =  exp[i(A+B)t/2] {2 cos[(A-B)t/2]}


Hope I've confused the issue sufficiently!


-- 
Aephraim M. Steinberg                    | "WHY must I treat the measuring
UCB Physics                              | device classically??  What will
aephraim@physics.berkeley.edu            | happen to me if I don't??"
                                         |       -- Eugene Wigner