Understanding Beat Frequency in Sinusoidal Waves: Exploring the Envelope Concept

[Firefox123]

So far every source I have read define "beat" frequency as the difference between two frequencies or f₂-f₁. The common example is in the audio world where we have two tones very close in value, say 100 MHz and 101 MHz. In this case the beat frequency would simply be 1MHz.

These same sources pull out one of the trig terms in the result for adding two sine waves and refer to the beat frequency as the "envelope" of the sum of the two waves.

This envelop or beat frequency is easy to see with two sinusoid waves close in frequency, but when I have two waves far apart in frequency...say 100 MHz and 1 MHz the "envelope" visually looks like the lower frequency of 1 MHz although the math says it should be at 99 MHz since it's the difference frequency.

So what is the issue here? Am I misunderstanding the definition of beat frequency or am I simply tracing out the envelope incorrectly?

Would the "beat" frequency or "envelops" be 99 MHz if I had two waves adding together at 100 MHz and 1 MHz?

 

[Naty1]

Here is the same and an illustrative diagram...

http://en.wikipedia.org/wiki/Beat_(acoustics )

 

[Averagesupernova]

Something related to this (modulation and mixing) has been discussed at great length here on PF. Do a search. I think one thread was entitled radio numbers or something. You say that technically the difference frequency should be the envelope. Why do you say that? Can you tell me what a 1 Mhz signal would look like with a 99 Mhz envelope? The 'envelolpe' is actually a perception. It isn't real.

 

[Firefox123]

Here is the same and an illustrative diagram...

http://en.wikipedia.org/wiki/Beat_(acoustics )

I have already read the wikipedia page and more...please do not reply with a link unless it adds something to the discussion or answers the question.

Thanks.

 

[Firefox123]

Something related to this (modulation and mixing) has been discussed at great length here on PF. Do a search. I think one thread was entitled radio numbers or something. You say that technically the difference frequency should be the envelope. Why do you say that? Can you tell me what a 1 Mhz signal would look like with a 99 Mhz envelope? The 'envelolpe' is actually a perception. It isn't real.

Mixing and modulation are quite different then what I am referring to though...this is not "mixing" in the sense of using a non linear device to multiply two signals together and getting a frequency shift...

I say it should be the "envelope" because that is what the sources I am reading are telling me...and keep in mind this is the difference frequency when we are adding the signals together...not multiplying them.

I do not know what a 1 MHz signal with a 99 MHz envelope would look like...but that isn't the situation...what you would have in my example is the sum of a 100 MHz signal and a 1 MHz signal that would have an "envelope" of 99 MHz...assuming the sources I am reading are correct. What that envelope would look like...I'm not really sure...it seems a bit weird, which is why I'm asking the question.

I agree that, in a way, the envelope isn't "real"...although you can detect an AM signal by using the "envelope"...all you need is a peak detector.

 

[0xDEADBEEF]

When the beat frequency is larger then the lower of the two frequencies the picture changes qualitatively. You get a line squiggling fast around a sinusoidal base line, not "waists and bellies" I don't think you even call it a beating.

 

[Averagesupernova]

Mixing and modulation BOTH create sum and difference frequencies. Tell me how that is different from what you are referring to. I suspect that what you claim to refer to does not 'shift' the frequency. It creates the two original frequencys as well as the sum and difference frequencies.

 

[vk6kro]

If you send two sinewaves with different frequencies through an amplifier (assuming it is a perfect amplifier with no IM distortion), you do not get sum and difference products. You get the original frequencies.
They may influence each other's loudness and we perceive this as "beats" but no new frequencies are generated.

[PLAIN]http://dl.dropbox.com/u/4222062/linear%20mixing.PNG

This is what you get. The graph shows the voltage across the 1 K resistor. Both inputs are 100 volts peak.

Each input affects the position of the other in the output at any moment, but there is no 100 KHz or 98KHz output. (this is not shown, but a magnified version of the horizontal scale just shows the 99 KHz sinewave ).

Incidentally, if you mix 100 KHz and 99 KHz, you get a 1 KHz "modulation" effect, but you wouldn't hear this because both input frequencies are above the human hearing range.
If you half wave rectified this, then you would hear it because there is an actual audio frequency sinewave present (with a DC component). You could feed this to a speaker and listen to it.

 

[Firefox123]

When the beat frequency is larger then the lower of the two frequencies the picture changes qualitatively. You get a line squiggling fast around a sinusoidal base line, not "waists and bellies" I don't think you even call it a beating.

Let me make sure I understand what you mean here...

If I have one frequency of 99 MHz and one frequency of 1 MHz then the beat frequency would be 98 MHz, which is much larger (faster) then the smaller (1 MHz) of the two original frequencies...

So you are saying that for such a case the picture changes qualitatively so much that the phrase "beat frequency" isn't even applicable.

Is that what you are saying?

 

[Firefox123]

Mixing and modulation BOTH create sum and difference frequencies. Tell me how that is different from what you are referring to. I suspect that what you claim to refer to does not 'shift' the frequency. It creates the two original frequencys as well as the sum and difference frequencies.

Mixing and modulation both create sum and difference frequencies because modulation usually incudes mixing as part of the process...

Using RF terminology, when you multiply or "mix" two signals with frequencies f₁ and f₂ you get the sum and difference frequencies...you can see this mathematically simply by using trigonometric identities...in practise this is done by using non-linear circuit devices like diodes or transistors...

What I was referring to was ADDING or SUMMING...this DOES NOT create the sum and difference frequencies. All you see on a spectrum analyzer in this case are the two original frequencies. So two signals with f₁ and f₂ when added together in time simply give a composite signal where the voltage levels add for each moment in time. On a spectrum analyzer the frequencies you would see in this case would be f₁ and f₂, but that's it.

The sources I have read are claiming that the ENVELOPE of the composite signal created by adding the two signals together would yield a frequency equal to the difference between f₁ and f₂.

Keep in mind I am ADDING these signals not MULTIPLYING them...the two cases are different both mathematically and physically.

 

[Averagesupernova]

I'm not sure if that is what he is saying or not but any time you take two frequencies and mix them together (mathematically multiply) you will create 2 new frequencies which are the sum and difference frequencies of the two originals. I'm not sure technically which one is called the beat frequency or if they both are. The way it looks on a scope is simply a result of the crests and troughs of mulitple signals periodically canceling each other out in the time domain. Whether you can see it or recognize it is irrelevant. I would estimate off the top of my head that if you have a frequency that is about a fifth of the frequency of another and they are mixed you will be able to recognize the 'waistes and bellies' of the resultant waveform on a scope. If the frequencies get much closer than this you might not be able to recognize it on the scope but the same thing is still happening which is the creation of sum and difference frequecies.
-
Edit: You beat my post.

 

[Firefox123]

If you send two sinewaves with different frequencies through an amplifier (assuming it is a perfect amplifier with no IM distortion), you do not get sum and difference products. You get the original frequencies.
They may influence each other's loudness and we perceive this as "beats" but no new frequencies are generated.

Okay...no disagreement there...

[PLAIN]http://dl.dropbox.com/u/4222062/linear%20mixing.PNG

This is what you get. The graph shows the voltage across the 1 K resistor. Both inputs are 100 volts peak.

Nice picture...I was basically doing the same thing, but in Excel...

Each input affects the position of the other in the output at any moment, but there is no 100 KHz or 98KHz output. (this is not shown, but a magnified version of the horizontal scale just shows the 99 KHz sinewave ).

Right...I agree that there is no 100 KHz or 98 KHz output. A Spectrum Analyzer will simply show the two original frequencies...

But the sources I am reading are claiming that the "frequency" of the ENVELOPE of the composite signal will be the difference of the two frequencies...this they label the "beat" frequency. Is this correct? Every source I have found seems to have this definition...if I use two signals of say 99 KHz and 98 KHz I can sort of visually "see" this 1 KHz "envelope", yet for other examples I have tried and for the picture you just referred to...there doesn't seem to be an easily seen "envelope". Are the sources I am reading defining terms incorrectly or am I misunderstanding what beat frequency is?

Incidentally, if you mix 100 KHz and 99 KHz, you get a 1 KHz "modulation" effect, but you wouldn't hear this because both input frequencies are above the human hearing range.
If you half wave rectified this, then you would hear it because there is an actual audio frequency sinewave present (with a DC component). You could feed this to a speaker and listen to it.

When you say "mixing" in this case do you mean adding the two signals together as you have in your example or are you referring to "mixing" in the sense of multiplying the two signals together?

 

[Averagesupernova]

Mixing and modulation both create sum and difference frequencies because modulation usually incudes mixing as part of the process...

Using RF terminology, when you multiply or "mix" two signals with frequencies f₁ and f₂ you get the sum and difference frequencies...you can see this mathematically simply by using trigonometric identities...in practise this is done by using non-linear circuit devices like diodes or transistors...

What I was referring to was ADDING or SUMMING...this DOES NOT create the sum and difference frequencies. All you see on a spectrum analyzer in this case is the two original frequencies. So two signals with f₁ and f₂ when added together in time simply give a composite signal where the voltage levels add for each moment in time. On a spectrum analyzer the frequencies you would see in this case would be f₁ and f₂, but that's it.

The sources I have read are claiming that the ENVELOPE of the composite signal created by adding the two signals together would yield a frequency equal to the difference between f₁ and f₂.

Keep in mind I am ADDING these signals not MULTIPLYING them...the two cases are different both mathematically and physically.

I would say those sources are wrong.

 

[Firefox123]

I would say those sources are wrong.

Perhaps...this does sometimes occur...but that would imply that the majority of sources from what I have read are incorrect on this issue.

If that is the case...then what is the definition of "beat frequency" and how is it related to the "envelope" of adding two signals together?

 

[Averagesupernova]

Are you sure you aren't misinterpreting your sources?

 

[Firefox123]

Are you sure you aren't misinterpreting your sources?

At this point yes...I'm fairly sure. They all seem to define "beat" frequency as the difference in the two frequencies and most of them relate this to the "envelope"...in fact they say it is the envelope.

Do a search in some EE or physics textbooks or do a google search on beat frequency...maybe you will find something I am missing or misreading...

 

[Averagesupernova]

I've seen more stuff on modulation/mixing/heterodyning vs. summing in textbooks than I care to discuss and they all pretty much agree. I've never seen any textbook make a claim that a summing amplifier creates new frequencies. The only place I've seen this confusion is on internet message boards such as this. In this thread in fact I was assuming you were not referring to simple voltage adding (summing amplifier) since nowhere in the first post is it mentioned. In other threads a few posters have had some serious confusion about modulation/mixing/heterodyning vs. simple voltage addition as well.
-
The only way I can see that a beat frequency is formed when summing two signals is when both frequencies reach your ear the nonlinearity of the human ear causes you to hear a beat. Or, possibly nonlinearities of a loudspeaker could cause a perceived beat frequency. VK6KRO implied something like this in post #8.

 

[0xDEADBEEF]

Let me make sure I understand what you mean here...

If I have one frequency of 99 MHz and one frequency of 1 MHz then the beat frequency would be 98 MHz, which is much larger (faster) then the smaller (1 MHz) of the two original frequencies...

So you are saying that for such a case the picture changes qualitatively so much that the phrase "beat frequency" isn't even applicable.

Is that what you are saying?

Right. As you see in the pictures that others have posted, you get the 1MHz wave with a broad wiggle around it, and not the waists and bellies typical for a beating amplitude.

 

[Firefox123]

I've seen more stuff on modulation/mixing/heterodyning vs. summing in textbooks than I care to discuss and they all pretty much agree.

Thats nice...so have I. Unfortunately none of those are what I am talking about.

I've never seen any textbook make a claim that a summing amplifier creates new frequencies.

And you won't...because it doesn't create new frequencies. An "envelope" isn't a new frequency...you won't see it on a Spectrum Analyzer...you need some kind of a detector to "pull" the envelope off.

The only place I've seen this confusion is on internet message boards such as this.

Believe me...I'm not confused in the least...I have looked at the math and it makes sense...my only issue was more of a visual question with respect to the envelope.

In this thread in fact I was assuming you were not referring to simple voltage adding (summing amplifier) since nowhere in the first post is it mentioned.

Not only is it clear from my first post, but it is clear from a few other posts of mine trying to explain this to you...maybe you should look up "beat frequency".

In other threads a few posters have had some serious confusion about modulation/mixing/heterodyning vs. simple voltage addition as well.

Thats nice...I am not one of them.

The only way I can see that a beat frequency is formed when summing two signals is when both frequencies reach your ear the nonlinearity of the human ear causes you to hear a beat. Or, possibly nonlinearities of a loudspeaker could cause a perceived beat frequency. VK6KRO implied something like this in post #8.

The specifics of the human ear DO have an impact on hearing a beat frequency...but not on the existence or formation of it. The ear is a detector...that is all.

 

[Firefox123]

Right. As you see in the pictures that others have posted, you get the 1MHz wave with a broad wiggle around it, and not the waists and bellies typical for a beating amplitude.

Gotcha...

That answer makes sense.

 

[Averagesupernova]

I don't know what you want anymore but I'm going to assume you are satisfied. If you've seen so much info on mixing vs. summing I can't understand how you could have had a question in the first place. The only question I would have is what you didn't understand and why.

 

[Firefox123]

I don't know what you want anymore but I'm going to assume you are satisfied. If you've seen so much info on mixing vs. summing I can't understand how you could have had a question in the first place. The only question I would have is what you didn't understand and why.

Okay...so here was the original problem I had...

So I'm reading about beat frequency and I see a few examples using say 98 MHz and 99 MHz or 98 KHz and 99 KHz...when you look at the combined waveform made by adding the two frequencies you can visually see the beat frequency as a 1MHz or 1KHz "envelope" depending on which example you choose...

But when I tried it with two frequencies that had a much larger difference...say 98 MHz and 1 MHz I couldn't really "see" an obvious envelope...so I just wanted to make sure I wasn't misunderstanding the concept.

I think the answer is along the lines of what DeadBeef posted...the picture will change so much that it will be very difficult to see a clear "envelope" like in the other examples.

 

[Averagesupernova]

Gotchya.