Multiplying a complicated frequency

[samuelcupham]

Hi,

Suppose I have a signal that is the sum of sin waves of varying frequencies. That is, the signal S(x) = Sin(ax) + Sin(bx) + ... where {a,b,...} are integers.

Is there some kind of circuit or other mechanism that could multiply the frequency of the signal by some scalar λ? In general, I do not know what the signal is, or what the constants {a,b,...} are.

I've read about circuits that multiply the frequency of a signal, but everything I saw only dealt with multiplying the frequency of a simple signals of the form a*Sin(b*x) with a,b real numbers. I cannot tell whether these types of circuits generalize to more complex signals with unknown frequency.

I do not have a background in electrical engineering, so forgive me if my terminology is wrong. Let me know if my question is unclear. Thanks everyone.

Charles

 

[Studiot]

Hello Charles (edit that was confusing), welcome to Physics Forums.

How much trigonometry have you studied?
You cannot have a single frequency signal that the sum of two different sine waves

$$\sin (a) + \sin (b) = 2\sin \frac{1}{2}\left( {a + b} \right)\cos \frac{1}{2}\left( {a - b} \right)$$

As you can see from this formula if you only have two different sine waves you get two new frequencies introduced. If you have many sine waves you get many frequencies.

So your signal does not have a single frequency as soon as you introduce another sine wave.

This phenomenon is called beat generation or intermodulation.

 

[Averagesupernova]

You cannot have a single frequency signal that the sum of two different sine waves

Studiot, you may want to rephrase that, or maybe I am not reading you correctly. Summing two signals will not generate new signals.

 

[samuelcupham]

But a function f(x) = Sin(a) + Sin(b) is still periodic, correct?

In more proper terminology, I suppose this means that I want to double the period between beats.

I'm looking for a circuit device that, given input f(x), produces output g(x) such that g(x) = f(λx) for some integer λ.

Thanks for your help,
Charles

 

[skeptic2]

Is there some kind of circuit or other mechanism that could multiply the frequency of the signal by some scalar λ? In general, I do not know what the signal is, or what the constants {a,b,...} are.

In more proper terminology, I suppose this means that I want to double the period between beats.

In post #1, first quotation, you say you want to multiply the frequency by an integer but in the second quote, post #4, you say you want to double the period. Which do you want to do?

One way to multiply or divide your signal is to record your signal in static RAM and play it back at a different speed. There are two ways of doing that.

1. To double the frequency you could double the clock rate at which the signal is played back from RAM.

2. To multiply the frequency by N, N being an integer, just output every Nth byte from the memory.

 

[samuelcupham]

Sorry I was inconsistent. I want to either multiply the frequency, period or the wavelength. It doesn't matter - I'm trying to "mark" a certain signal by multiplying one of these characteristics by λ.

 

[fleem]

You can convert it to the frequency domain, upsample or downsample it accordingly, and then convert back to time domain. If you just want to do it to an existing file of time domain data, then most audio editors (audacity, cooledit, goldwave, etc.) can do it for you.

 

[Studiot]

Studiot, you may want to rephrase that, or maybe I am not reading you correctly. Summing two signals will not generate new signals.

Just to clarify here is the result of adding sin(x) and sin(1.1x) :ie a=1, b=1.1

The red trace is the sum of the sine waves.

The blue trace is the single sine wave sin(ax) with a=1 for reference

The green trace is the new frequency (beat) frequency that appears at 0.1 units and is cos((b-a)x). I have shown this as a cosine wave as this is in sync with the red wave as shown by my earlier formula.
Charles is correct that you get a repetitive wavetrain at this frequency.

Charles, if you would like to clarify exactly what you are trying to achieve we can help more.
Please let us know if you need a hardware or software solution and can the signals be sampled (digitised) for processing, either in hardware or software?

 

[samuelcupham]

Thanks for the diagram Studiot.

I've attached a diagram of what I want the device to do. In the diagram I've used general notation f(x), but I intend f(x) to be some function that is the sum of sine waves. That is

$f(x) = sin(c_1 x) + sin (c_2 x) + ...$

I am looking primarily for a hardware solution.

If I can, I'd like to avoid digitizing the signal: I'm trying to construct an analog calculator of sorts, so I'm wary of digital components. That said, if the only solution to this problem is a digital one, I welcome any advice to that end.

 

[Studiot]

Are you trying to build a sort of analog computer?

This can be an interesting project and was done for real (the only available way) in the past first with mechanical and later with electronic units.

I will see what references I can dig up.

However why can you not represent your signal by voltages? This makes scaling much easier.

 

[samuelcupham]

Yes I am thinking along the lines of an analog computer. Any good analog computing references would be much appreciated.

What exactly do you mean by representing my signal as voltages? I'm not very knowledgeable about electrical engineering or circuit design so I apologize if that is an overly basic question. If there is a way to represent the signal other than as an alternating current, I am open to that sort of idea.

As you've probably noticed, my ideas are still in the early stages - none of the specifics are 100% rigid so long as I can output a signal with an arbitrary integer scaled frequency/wavelength/period.

 

[Studiot]

Here is a rough sketch showing what I mean.

Let us say we have a voltage V volts equal to 10x.
We can very simply divide this down either by a switch as shown in discrete steps or continuously by a potentiometer.

Thus we can generate a voltage = nx for any n.

An analog circuit block that takes an input voltage v and outputs the sine of this voltage =sin(v) is a readily obtainable/ constructable unit.

If you take a number of these for n₁, n₂, n₃ etc you can easily combine them to obtain the sum of these voltages which is therefore the sum you require.

This is the basis of an analog computer, many other functions besides the sine are availbale this way.

Traditional analog computers used fixed voltages, which is equivalent to looking up the sine of a certain number in tables.
If you require scanning you can substitute a ramp voltage at the input so that you can obtain an output voltage which tracks the variation of x at the input and produces an output which is the sum you require over the range of x input.

 

[elibj123]

Any non-linear device (see, diode) outputs all the harmonies of an input sinusoidal signal. This is due to the fact that the expression
$$(sin(\omega t))^{n}$$

contains (among other frequencies), the harmony

$$sin(n \omega t)$$

given that you know the input frequency\ies and the desired harmonies n*w are spaced enough, you can cascade band-pass filters at the output of a non-linear device, and thus at the output of the entire system you will have only the desired harmonies.

Studiot, what kind of analog devices are there that compue a sine function of the input, never heard of these. And how do they help at generating a sine with the input voltage as a frequency?

 

[elibj123]

Hello Charles (edit that was confusing), welcome to Physics Forums.

How much trigonometry have you studied?
You cannot have a single frequency signal that the sum of two different sine waves

$$\sin (a) + \sin (b) = 2\sin \frac{1}{2}\left( {a + b} \right)\cos \frac{1}{2}\left( {a - b} \right)$$

As you can see from this formula if you only have two different sine waves you get two new frequencies introduced. If you have many sine waves you get many frequencies.

So your signal does not have a single frequency as soon as you introduce another sine wave.

This phenomenon is called beat generation or intermodulation.

Your equation is only a nice way to introduce an envelope with frequency f1-f2 and the oscillator (of high frequency f1+f2) that is bound by it. It does not mean the signal contains more than the two original frequencies f1 & f2, you still get two pure spectral lines only at f1 & f2.

But, when you multiply two sinusoids of different frequencies f1 and f2, only then you get the sum of two other frequencies (sum and difference), and looking at the spectrum you get spectral lines at f1+f2 & f1-f2.

 

[Studiot]

And how do they help at generating a sine with the input voltage as a frequency?

I didn't say they did.

I said that if you represent x by a voltage you can easily obtain another voltage given by

f(x) = sin(n₁x)+sin(n₂x)+sin(n₃x)+..+

where the n_i are coefficients.

From what we have been told, I think Charles would be better served working with voltage to create his required functions.

 

[jim hardy]

i had a friend once who took a signal,

clipped it with diodes to produce harmonics
then applied analog filter to detect harmonic of interest.

he was trying to measure frequency of a nominal 60hz signal in less than one cycle, for purpose of detecting impending grid disturbance..

it seemed to work on his test bench but we got distracted and i never knew final result.

it would seem fft might be an approach for you.

 

[samuelcupham]

Does a diode / other non-linear device generate harmonics regardless of the input signal or does it only work for "nice" signals, i.e. sin(x), cos(x), ... ?

 

[samuelcupham]

So suppose I had a periodic signal f(x) = sin(ax) + sin(bx) + ... Suppose I want to generate a signal g(x) with k times the frequency of f(x).

I don't know what {a,b,...} are but I know that they are elements of some finite set of coefficients C.

Suppose I build a bank of LC circuits tuned sin(cx) for all c in C. If I run the signal through this bank, it seems like it would "separate" f(x) into its constituent sine waves. Then, for each "separated" sine wave, I could use another LC circuit tuned to the k'th harmonic of that sine wave. Then I could recombine all the outputs to yield g(x).

Does this make sense? Would I have to put a diode between the "separating" LC circuit and the "k'th harmonic" LC circuit to generate all the harmonics of a given sine wave?

 

[Averagesupernova]

Studiot your diagram in post#8 shows me more than 2 signals on the red trace. It is an AM DSB signal, carrier not suppressed. You cannot achieve that without multiplication, and summing is obviously not multiplication. I think that is what elibj123 is saying and where I was heading in my previous post.

 

[Studiot]

Averagesupernova & elibji23.

Studiot your diagram in post#8 shows me more than 2 signals on the red trace. It is an AM DSB signal, carrier not suppressed. You cannot achieve that without multiplication, and summing is obviously not multiplication. I think that is what elibj123 is saying and where I was heading in my previous post.

Not quite

First look carefully at the equation in post#2.

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

ie they are the same thing.

So if you have one, of necessity you have the other.

My traces are a printout of directly adding two sine waves.


************************

Now in amplitude modulation we do indeed approach from the point of view of multiplying two sine waves. That is we take a carrier

A_csin(ω_ct)

and replace A_c by A_ssin(ω_st)

where c and s represent carrier and signal respectively.

When we transform the product of these two sine waves we use the formula

$$\sin (A)\sin (B) = \frac{1}{2}\{ \cos (A - B) - \cos (A + B)\}$$

which is different from mine, although the resultant waveform is very similar since they both use sinusoids.

 

[Windadct]

Hello Sam...

I do not think you are really providing enough info to accurately answer the question - it seems to me you are looking to multiply the frequency of an UNKNOWN input - in real time ( without storing the signal ) - correct?

If so Just my 2 c - here, I would say this is impossible to MULTIPLY ( increase) the frequency because you have to predict the future - and to DIVIDE (slow down) you need to STORE information.

Consider starting with an input of simple sine ( pure single frequency component) - period x - as long as you know a single cycle stays pure - you can complete a new signal in 1/2 of the time - but if you wish to double the frequency ( half the period) - you have completed a full cycle before the input signal has completed a cycle - should this signal change in the second half - your new output is complete, but the input has changed.

On the dividing frequency signal - a similar situation - if an input takes 10 Seconds, your output will take 20 seconds. You need to store the "info" about the input as you complete the output.

Physically - this is similar to Doppler effect - HOWEVER there HAS to be a time difference between the transmitter / sender ( input) and the Receiver (output ) - so it is not real time and this time difference in a physical system will come to zero at some point. In this case the medium that the signal is in is "Storing" the information - and it is them being received faster than it was transmitted and there is a delay.

 

[jim hardy]

thanks studiot for that clear example. shows nicely where AM sidebands come from.


since you guys' math is so superior to mine, may i ask a dumb question ---

Fourier says one can make a square wave(or any other wave) by summing sinewaves.

is there another math transform that'd give a series of square waves that, when summed, would produce a sinewave?
seems that'd be a nice synthesizer tool.

 

[Studiot]

is there another math transform that'd give a series of square waves that, when summed, would produce a sinewave?

Kinda depends upon your meaning of square waves.
I don't think you can assemble a sine wave from a series of different square waves, all of 50% duty cycle.
However if you allow very tight rectangular pulses (needle pulses or delta functions) you could do it. This is equivalent to digital synthesis of a sine wave, which can be very accurate these days.

Studiot, what kind of analog devices are there that compue a sine function of the input, never heard of these. And how do they help at generating a sine with the input voltage as a frequency?

Sinusoids arise naturally as the solution to certain simple second order differential equations. Another word for solution is 'integrate' and one of the fundamental blocks in an analog computer is an integrator. These can be used to produce sine waves.

As to the use, Charles is asking for an electrical representation of sin(n₁t)+sin(n₂t)+..+
I am recommending the use of voltage not frequency to represent n₁t and n₂t. This can easily be achieved either manually or by using a voltage ramp.

Charles, do you wish to pursue this?

 

[Averagesupernova]

I stated in my previous thread that the signal is a DSB AM with carrier. Upon closer inspection I don't believe it is. It is difficult to tell but it is suppressed carrier. However, I'm still not sure I am on the same track as you studiot. What you call the green trace which is .1 difference between the two inputs. You are correct that this does in fact represent the difference between the two input frequencies unless I am misunderstanding you. BUT, it isn't really there. We perceive it is, but it isn't. We can hear a 'beat' due to the log response of our hearing. If we use a summing circuit as in post #12 and feed it with 100 Khz and 100.1 Khz sine waves we will have an output that looks like the red trace in post #8 and the 'beat' as it has been referred to will be 100 hertz similar to the green trace in post. BUT, it is not actually there. Feed this output into a spectrum analyzer and it will not be there. Filter for it and it will not be there. There will be NO 100 hertz signal in this output.

 

[sophiecentaur]

We seem to have some confusion in this thread between the linear addition (sum) of signals and the effect of non-linear mixing (modulation / multiplication etc). Are we 'adding frequencies' or 'adding signals'?

Producing a frequency offset on a narrow band signal is fairly straightforward if you up-convert and then down-convert. This will 'add' the same frequency offset to all components of the signal and is how 'superhet' receivers manage to use just one i.f. filter for a whole range of received signal frequencies.

 

[Studiot]

We seem to have some confusion in this thread between the linear addition (sum) of signals and the effect of non-linear mixing (modulation / multiplication etc). Are we 'adding frequencies' or 'adding signals'?

Producing a frequency offset on a narrow band signal is fairly straightforward if you up-convert and then down-convert. This will 'add' the same frequency offset to all components of the signal and is how 'superhet' receivers manage to use just one i.f. filter for a whole range of received signal frequencies.

I look forward to your more detailed explanation.

 

[sophiecentaur]

I should look at superhet on Wiki and a number of other google hits. It involves a few diagrams so it's probably best to find out that way.
Many receivers use 'double conversion' to give the benefit of a filter that is easier to make at low frequencies but also the benefit of a wide frequency range and good 'image rejection' for the receiver (RF spectrum analysers often have a ridiculously high first IF frequency but it works).

The basic principle is ever so common.

 

[Studiot]

Whilst I am not quite sure what the OP wants to do I am sure he is not interested with frequency changing as in superhets or other up/down converters.

I am concerned that these diversions are frightening him off PF however.

Averagesupernova,

I don't know if you are misunderstanding my comments or your original class on amplitude modulation.

When you create an amplitude modulated wave from two signals with frequencies say G and H ( G>> H) the wave you create consists of three frequencies viz G, (G+H) and (G-H). It does not contain H.
In my book, this means two new frequencies that were not present in the original.
For every additional modulating signal you add two further sum and difference frequencies.
This can easily be shown on a spectrum analyser.

Since the original question, repeated several times, asks about many modulating frequencies it follows that there resultant waveform will contain many frequencies.

All I am asking is which of these frequencies is the OP referring to when he says he want to double, treble etc the frequency of the 'sine' wave and pointing out that some of these will not be at any of the original frequencies.
I further observe that this is a very complicated and difficult way to go about what I am guessing (apparently correctly) is the objective of the OP and that voltage multiplication would serve the purpose much better.

I would happily discuss the finer details of AM in another thread, but that is off topic in this one.

go well

 

[sophiecentaur]

You may be right but, apart from saying that a diode will mix any number of signals and produce harmonics and intermodulation products - (in principle) - what more is there to say without a much more detailed question.

It could all be down to terminology but, if he wants to produce 'multiples' of an original signal, it is no use just going for 'harmonics' because each multiple of the original will also have proportionally stretched sidebands etc etc, If you want a replica of a signal at a higher (integer multiple or not) carrier frequency, you need to 'Mix'.
The word "harmonic" is bandied about on many of these threads and it is only a fraction of the story.

Reading the second post in detail, I see I am right. You can't get what he wants by just generating 'harmonics'. That is, unless his bank of narrow band filters are good enough to eliminate any of the intermodulation products that are bound to occur if the original signal is blasted through a non-linearity at high enough level to produce the wanted 'harmonics'. In between all of those, there will be dozens of products of the same order as the harmonics which could bang next to the wanted frequencies, depending on the original spectrum.
In fact, exactly the required result would be very complicated to produce unless you started with a bank of oscillators and 'beat them all up individually'. The filtering would be relatively easy in that particular implementation.
It may help to know the source of this original signal, how it's generated and if he really has stated the problem accurately.

 

[Studiot]

This is the closest I have got

I'm trying to construct an analog calculator of sorts,

see posts 8 and 9

 

[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