 ### Network L

#### Louis Sternberg

Jan 1, 1970
0
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.

Thank you,

Louis S.

G

#### Genome

Jan 1, 1970
0
Louis Sternberg 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.

Thank you,

Louis S.

Don't know.... if I was seriously bovvered about your question then I'd type
something like....

sinwt+sinwt

and

sinwt*sinwt

into Google and see what happened.

Then I would come back here and explain it to you.

NOT

DNA

P

#### Phil Allison

Jan 1, 1970
0
"Louis Sternberg"
I have seen reference to both "adding" (mixing) two waveforms AND
"multiplying" them.

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

** Exactly what it says - the instantaneous values of two independent
waveforms ( X and Y ) are electronically multiplied to get the product XY
at the output. The output takes regard of the sign of X and Y when needed.

For example, how does one multiply a sinewave with a second one of the
same frequency and amplitude but with a phase difference. And what
does the result look like.

** You can draw two such waves on 5mm grid paper and do it yourself.

To make it easy, assign unity amplitude to each wave.

.......... Phil

J

#### J.A. Legris

Jan 1, 1970
0
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.

Thank you,

Louis S.

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html

"Product-to-Sum Formulas"

For example, sin(u)*sin(v) = 1/2 [(cos(u-v) - cos(u+v)]

Suppose your angular frequency is w (i.e. 2pi*f), the phase difference
is p, and time is t.

Then u= wt and v = wt+p.

sin(wt)*sin(wt+p) = 1/2 [cos (wt - (wt + p)) - cos (wt + (wt + p))]
= 1/2 [cos (-p) - cos(2wt + p)]

The first term in the square brackets is a constant that depends only
on the phase difference. The second term is another wave at twice the
original frequency shifted in time by the same phase difference. The
resulting wave has half the amplitude of the original and a D.C. shift
depending on the phase difference. In the special case where p=0 (i.e.
a squared sine wave) you get a cos wave at twice the frequency that
oscillates between 0 and 1.

R

#### Rich Grise

Jan 1, 1970
0
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.

Without doing a bunch of arithmetic, the simple answer is, for each
point on the waveform (for each instant), multiply the value of one
by the value of the other, and plot the answer.

This is the difference between "mixing" and "modulation", BTW.

Cheers!
Rich

E

#### Eeyore

Jan 1, 1970
0
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

With a multiplier.

of the
same frequency and amplitude but with a phase difference. And what
does the result look like.

There wouldn't be much point with one of the same frequency !

I'm sure you can do the math though.

Graham

J

#### J.A. Legris

Jan 1, 1970
0
Eeyore said:
With a multiplier.

There wouldn't be much point with one of the same frequency !

I'm sure you can do the math though.

And if you did the math you might see that multiplying a sine by
another at the same frequency can be used for extracting a
phase-modulated signal. It can also be used for synchronous detection
of AM. And for frequency doubling. Etc. Etc.

A

#### Andrew Holme

Jan 1, 1970
0
Louis Sternberg 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.

sin(A) . sin (B) = [cos (A-B) - cos(A+B)] / 2

sin(wt + phi) . sin(wt) = [cos(phi) - cos(2wt + phi)] / 2

i.e. a sine wave of twice the frequency, and the half the amplitude, with a
DC offset.

J

#### John Larkin

Jan 1, 1970
0
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.

Thank you,

Louis S.

The term "mix" is ambiguous. Audio people use it to mean add, and RF
people usually use it to mean multiply.

Classic trigonmetric identities, in the backs of handbooks, show what
happens when you multiply waveforms of the same or different
frequencies, and various phases. Bottom line,

if you multiply different frequency sine waves, the output is the sum
and difference frequencies

if you multiply two sines of the same frequency, you get 2F plus a DC
term that depends on the phase angle of the inputs.

John

J

#### John Larkin

Jan 1, 1970
0
With a multiplier.

There wouldn't be much point with one of the same frequency !

Actually, it can be very handy, as in synchronous demodulation,
lock-in techniques, PLLs, frequency doublers, all sorts of things.

John

L

#### Le Chaud Lapin

Jan 1, 1970
0
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:

-Le Chaud Lapin-

D

#### Don Bowey

Jan 1, 1970
0
With a multiplier.

There wouldn't be much point with one of the same frequency !

Sure there is. It's called a Direct Conversion receiver.

Replies
4
Views
79
Replies
5
Views
901
Replies
48
Views
5K
Replies
10
Views
1K
Replies
1
Views
1K