Louis said:

I have seen reference to both "adding" (mixing) two waveforms AND

"multiplying" them.

Can anyone please define what is meant by the later term?

For example, how does one multipy a sinewave with a second one of the

same frequency and amplitude but with a phase difference. And what

does the result look like.

What I am about to suggest might not seem relevant to your question,

but it is.

As you fiddle with the mathematics of functions, it is very helpul to

stop thinking of them as being continuous. There is no continuum -

there is only the discrete, and limits toward the continuum. Instead,

think of discrete points along the x-axis, with discrete f(x) values

for those points, and anytime you do math, use the discrete values, for

both positions on x and y, then imagine the inter-point spaces getting

smaller and smaller as the points converge. It seems silly, but it

helped me to begin retaining what i learned.

For example, take your last question above. Think of the two waveforms

y1(x) and y2(x) being multiplied together, and think what the output

y3(x)=y1(x)*y2(x) will look like. It is helpful to think of y1 and y2

as having the points on their "slanted slope" multiplied. Now imagine

a y1(x) being multiplied by a y2(x) value where both their slanted

slopes hover above x-axis. The result will be some number y3(x). Now

imagine y3(x+dx) = y1(x + dx)*y2(x+dx), where dx is a small, positive,

incremntal value. y1 might have a steep slope, while y2 might have a

not-so-steep slope at x. Whatever the slopes are, each jump of dx,

y3(x+dx) will change its _value_ from y3(x) at a _rate_ depending upun

the _sum_ of the rates of change of y1 and y2. You should think about

this for a bit until you are convinced. Both y1 and y2 are already

trying to change, regardless, but if you multiply them together, the

change becomes more pronounced in their product, whether their signs

are equal or opposite. How pronounced? Well, naturally, according to

the combined rates of change of each. (You can see the Chain Rule from

calculus in this last statement, not just from two signals being

multiplied, but the entire space of functional operators). This view

allows intuitive appreciation of the math of modulation and Euler's

formula. When multiplication of two waves are involved, the

frequencies add, both in a spectrogram and in the exponents of the

product wave.

Finally, if you think about it this way, you get one other benefit.

When electrical engineers speak of modulation, filtering, etc...the

waves are often idealistically pure sine waves with impulses in the

frequency domain. In fact, the waves often have a continuum of

frequencies and look weird in the time domain. If you focus on each

point along the curve instead of the curve itself, it becomes clear

that the frequency of a signal is actually its instantaneous rate of

change of phase. Note in the preceeding sentence, the word

"instantenous" is implicitly required to be inserted before the word

"frequency", as you cannot speak of a waveform having a "frequency" if

there are multiple frequencies in it. If it has only one frequency,

then the instantaneous rate of change of phase is constant. A signal

that is the product of two other signals gets its instantaneous rate of

change of phase from the addition of the instantaneous rates of change

of those two signals. When you work with phase detection, signal

recovery, VCO's, PLL's, it is helpful to abandon the notion of

frequency altogether, at least momentarily, and instead start thinking

about instantaneous rates of change of phase:

There was a post about adding two wave forms whose frequencies where

close, and multiplying them, which might explain better the difference

between modulation and adding tones for detection by the human ear:

http://groups.google.com/group/sci....=le+chaud+lapin&rnum=6&hl=en#b46c813a11a450dd
-Le Chaud Lapin-