Multiplying a complicated frequency

[Averagesupernova]

There is very little about summing, and frequency mixing that I don't understand. I know spectrally what the outputs look like relative to one another. Go ahead and continue, maybe we will get a description out of the OP as to what they want more specifically. I just wanted to point out that summing, is most certainly different than frequency mixing which generates new frequencies.
-
Maybe I misinterpreted you studiot and am sorry if that has happened. Summing signals together with a linear amplifier such as the one in post #12 never, ever generates new frequencies. I have interpretted you to have said that summing does in fact generate new frequencies in this thread studiot and if I have been wrong I am very sorry.

 

[Studiot]

There is nothing linear about a sinusoid of itself. The function f(x)=sin(x) is non linear.

There is nothing linear or non linear about the statement

sin(A) + sin(B)

any more than there is about the statement 5+7

Both are simply numbers.

A sinusoid function is a member of the class of infinitely differentiable continuous functions.

As such it may be added to other members of this class by the rules of linear algebra edit: when there is an integral (integer) relationship between A and B.
I have shown that the product of two sinusoids is equivalent to the sum of two sinusoids twice already in this thread.
This is standard stuff that may be found in most textbooks on the subject.

 

[Averagesupernova]

Ummmmmm, not sure what to say. Seems like you are avoiding comment on what I have posted. I don't know what your comment about nothing linear about a sinusoid has to do with anything. A sinusoid is a sinusoid, nothing else nothing more. Did my mention of a linear amplifier prompt you to comment about sinusoids not being linear? A linear amplifier has nothing to do with a sinusoid. All it means is that incoming signals (such as but not limited to sine waves) are amplified by a factor of X. It does NOT mean that incoming signals (such as but not limited to sine waves) are multiplied by a factor of X as well as multiply with each other.

 

[jim hardy]

now I'm getting confused - two old rules conflict

This shows that the sum of two sine waves is the product of a sine wave and a cosine wave..

Conversely the product of a cosine wave and a sine wave is equal to the sum of two sine waves

true enough at any instant from our high school trig identities. i looked them up.
yet i also know if you add two Fourier polynomials you get a different result than when you multiply them.

and one must multiply to modulate.

what gives?
if A = jw(a)t
and B = jw(b)t

does identity still hold?

i know it's just one of those mental wrong turns I've taken someplace.
Ahh the joys of aging...

 

[Studiot]

Seems like you are avoiding comment on what I have posted

?

The only avoidance I can see is that I have twice posted a standard formulae showing that the product of two sinusoids may also be represented as the sum of two (different) sinusoids and vice versa.

so if you have sin (a) +sin(b) you must have sin(c) times cos(d) where c and d are different from a and b

 

[Studiot]

Hi, Jim.

The answer to this conundrum is in post#20.

When we (amplitude) modulate the maths is given as in post20.
This results in three terms, not two, in the resulting expression.
Two of these terms are the relevant transformation between the product and sum of two sinusoids.
The third is to do with the relative amplitudes of the two sinusoids.

(Note my trig formula assumes this is unity in both directions)


This is saying that there is a difference between the three black boxes labelled modulator, multiplier and adder.

If you feed two sinusoids sin(ω₁t) and sin(ω₂t)

into an adder you will get an identical waveform to that feeding sin(ω₃t) and sin(ω₄t) into a multiplier.

But you will not get the same result if you feed

Asin(ω₁t) and Bsin(ω₂t) into the adder and

Asin(ω₃t) and Bsin(ω₄t) into a multiplier

does this help?

 

[sophiecentaur]

I have said this before and I'll say it again. Addition is a linear operation and multiplication is a non-linear operation.

It's horses for courses but we still don't seem to know what course we are on. afaics, the OP talks in terms of an input signal that consists of a plurality of frequency components but does not specify exactly what he wants to do with it and what the output signal needs to be.

I am always uneasy with threads in which the OP takes such a back seat and everyone else talks at cross purposes and gets irate.

I think we should have a self-imposed rule that threads which are not kept alive by the OP should be allowed to die after one page.

Now - YOU put the phone down... no YOU put it down ... no YOU put it down first

 

[jim hardy]

""Addition is a linear operation and multiplication is a non-linear operation.
""

with that i am comfortable.

next step for me is find the examples of both using Fourier polynomials. i have them somelace...



Thanks Studiot - that a condition is involved, your unity, is a relief.
""(Note my trig formula assumes this is unity in both directions)""

phone down.

old jim

 

[Averagesupernova]

My phone is still up, I am just choosing not to speak at the moment. If you all can catch my drift. ;)

 

[Studiot]

\begin{array}{l}<br /> 3{x^2} + 4{y^2} = 16 \\ <br /> 2{x^2} - 3{y^2} = 5 \\ <br /> \end{array}

Here is a pair of simultaneous equations, both containing addition.

This is a non linear system, with more than one solution for x and y.

To obtain these we make make a transformation to a linear system

\begin{array}{l}<br /> X = {x^2} \\ <br /> Y = {y^2} \\ <br /> 3X + 4Y = 16 \\ <br /> 2X - 3Y = 5 \\ <br /> \end{array}

Which has one unique solution for X and Y.

Now tell me that addition is always linear.