Multiplying a complicated frequency

[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]

$$\begin{array}{l} 3{x^2} + 4{y^2} = 16 \\ 2{x^2} - 3{y^2} = 5 \\ \end{array}$$

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

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

Which has one unique solution for X and Y.

Now tell me that addition is always linear.