Which is what I've been saying. You're the one who has been claiming that AM modulation is the linear addition of the sidebands.]

I see, so you are saying sum and difference is a multiplication process? I don't think so. Perhaps you need to look up their definitions. The sum and difference is centered around the RF carrier. This is not multiplicative.

Which is a nonlinear process.

True enough, and your point being?

Which why it is semantics to say that one description is wrong and the other is right.

Not really, but you do seem to be arguing for the sake of it.

The act of heterodyne is by definition non-linear, however, the spectral image is identical. An important consideration.

Any other non-linearities will distort the spectral image, which will distort the information, the audio.

 

One of the objections folk are raising is that they can't reconcile the notions of simple sideband plus carrier additions with the requirement for multiplicative operations for sideband generation in the AM signal.

Bill noted his calculator experiment in which he produces an AM signal by the additive process. This works fine regardless of what people say. What was not noted is the means by which the individual parts could be generated in the first instance. The carrier generation part is simple. The problem arises with generating one or both of the sidebands to add to the carrier. I would suggest a multiplicative process is required to generate the latter.

The two concepts are actually used together in practice right under our noses - it's called Frequency Division Multiplexing [FDM].

NB: It would be interesting to know if autodidact is any the wiser.

 

Perhaps, I thought that the Sum and the Difference with heterodyne was understood.


Modulation

Sum: Carrier Frequency (in Hz) + Audio Frequency (in Hz) = Upper Sideband Frequency (in Hz)
Difference: Carrier Frequency (in Hz) - Audio Frequency (in Hz) = Lower Sideband Frequency (in Hz)


Demodulation

Sum: Carrier Frequency (in Hz) + Upper Sideband Frequency (in Hz) = Higher Frequency (in Hz, but this is rejected and not used.)
Difference: Carrier Frequency (in Hz) - Upper Sideband Frequency (in Hz) = Audio Frequency (in Hz)

Sum: Carrier Frequency (in Hz) + Lower Sideband Frequency (in Hz) = Higher Frequency (in Hz, but this is rejected and not used.)
Difference: Carrier Frequency (in Hz) - Lower Sideband Frequency (in Hz) = Audio Frequency (in Hz)


In this case the sign doesn't indicate phase. Or if it does the phase of the lower side band is also inverted in the modulation process, I'm not sure.

In any case it is not a multiplicative process. The new frequencies are created as part of the heterodyne function.

A side thought I was going to add is that precision heterodyne is non-linear, but in a very specific way. It is possible to have other types of non-linearities that will ultimately distort the results. Done right, the transmitted information (audio in this case) is reproduced exactly.

Quite a bit of research since RF was discovered is making precision heterodyne circuits. For simple one transistor type transmitters this is more the exception than the rule, but the principle of close enough works. In many of these circuits you get a mix of AM, FM and other.

If I do convert this into an article you guys have showed where I have to be precise in my definitions, as assumptions can creep in where I didn't expect them. Thanks.

 

I see, so you are saying sum and difference is a multiplication process? I don't think so. Perhaps you need to look up their definitions. The sum and difference is centered around the RF carrier. This is not multiplicative.

But, as I've asked before, you are summing a 1.000MHz signal and a 1.001MHz signal and saying that this is the same as modulating a 1MHz carrier with a 1kHz signal. Leaving aside the issue of the lower sideband, the question remains: How did you take a 1kHz signal and come up with the 1.001MHz signal needed to add to the 1.000MHz signal? You can't get there without using a nonlinear process.

True enough, and your point being?

The point being that you said, "A statement was made that AM is not linear, in an ideal system I would dispute that too. "

Please describe an ideal linear system that can take a 1kHz signal and modulate it onto a 1MHz carrier to produce signals at 999kHz, 1000kHz, and 1001kHz (or any subset that includes at least one of the sidebands).

 

<snip>

The act of heterodyne is by definition non-linear, however, the spectral image is identical. An important consideration.

Any other non-linearities will distort the spectral image, which will distort the information, the audio.

Perhaps, I thought that the Sum and the Difference with heterodyne was understood.

<snip>

A side thought I was going to add is that precision heterodyne is non-linear, but in a very specific way. It is possible to have other types of non-linearities that will ultimately distort the results. Done right, the transmitted information (audio in this case) is reproduced exactly.

<snip>

I think I covered that, in several place, in many ways, in almost every response. What part are you arguing with?

How many times do I have to repeat the same things?

Is it the concept of what heterodyne, or the sum/difference frequencies you are having problem with?

I admit to some aggravation, but if there is a real misunderstanding I would like to cover it. I just don't see where the objection is.

It could be you are not understanding I was demonstrating how to create an exact duplicate of a SSB signal, and how this relates to AM.

Was it made with AM, no, but you could not tell the difference at the receiver. If you can not tell the difference, then it is an AM signal, though not made with traditional means. I was demonstrating how a sideband works with an AM signal, and why a sideband is the modulation, not the carrier varying its amplitude. Using test equipment, be it an oscilloscope or a spectrum analyzer, you could not tell them apart.

 

I think I covered that, in several place, in many ways. What part are you arguing with?

I'm mot sure if this is directed at me or not. Assuming it was, let's take a step back and be sure that we are at least working from a common intersection.

Do you still dispute that, in an ideal system, AM is nonlinear?

If no, then you have changed your position and I didn't catch that and have been harping on a point that has been overtaken by events.

If yes, however, then I dispute your dispute and claim that AM is inherently nonlinear. My basis for that claim is that you can't generate the sideband frequencies from the baseband frequencies without a nonlinear process since linear processes cannot produce frequencies that aren't already there. All I ask for to prove me wrong (and for which I will be quite grateful) is a description of a linear system, no matter how ideal, that can accomplish this.

 

WBahn asks a good question with respect to the origin of the sidebands in the first place.

I (for one) have no objection to the notion that one can create the AM signal from a linear addition of the component signals. In principle one can extract the components of an AM signal and re-combine them by linear addition to form the original AM signal.

I also agree that one could not differentiate the means by which the construction of a received AM signal was undertaken at the modulation stage back at the transmitter.

It's a simple case to construct an AM [SSBWC] signal by the linear addition of a sinusoidal "carrier" and a single frequency sinusoidal "sideband" of appropriate amplitude. Imagine a black box with input modulating signal which is an arbitrary audio speech pattern. The two outputs from the black box are upper and lower sidebands of a 1MHz carrier. How were the outputs generated by linear methods? I think that's the question for which we seek an answer.

 

<snip>
Do you still dispute that, in an ideal system, AM is nonlinear?
<snip>

I have not changed my position, modulation is not linear, but the side bands are an exact match spectrally to the audio frequencies, to me this is a form of linear. Can you point to where I said heterodyne was linear? I can point to early on where I said it requires a nonlinear response to create the effect. Being nonlinear is not enough though, it requires a very specific form of nonlinear to be pure, otherwise you wind up with massive harmonics in bad places.

I will repeat, where is the specific argument? You started out saying the RF carrier varied in amplitude, this is not the case.

I showed where the apparent amplitude showed on an oscilloscope is not in fact variation of the carrier, but a result of the sideband interfering with the carrier.

You then said a sideband next to a carrier is not AM if it was not a result of modulation. Given it is in fact indistinguishable from an AM signal I would state this is a false argument. You have tried to dismiss this as semantics, when in fact it is not.

Did I get the arguments wrong?

I am not arguing the way AM is taught, but I am saying it is an analogy. That is why the OP started the thread. The real facts of AM are much more complex, which is why the analogy is a good thing for beginners. This particular analogy is much better than most, it actually generates other concepts that seem valid on the surface, but also have much deeper explanations.

Nature is full of things that the intuitive explanations are incorrect. This is just one of them. Electronics in particular has several cases where we teach students analogies to get them started, but teach the correct explanations later, sometimes much later.

Don't get me started on FM, I don't have the math tools to cover it. I have seen the sidebands on spectrum analyzers though, it isn't just the carrier frequency moving back and forth. But it is another good analogy, one that will get most students well on the road.

 

My understanding is that the problem with not having a carrier signal to sync to, the receiver is extremely difficult to tune since a mismatch of a few tens of hertz can make the signal annoying or even unintelligible by the time you start getting toward 100Hz.

Back in the day when I was a "sparky" we depended on very accurate and stable master oscillator standards when using SSB-SC for data/voice communications so manual tuning was almost never necessary. A depot service adjusted all the standards to a primary standard so all stations were matched within a few hertz at 10mhz and each receiver/transmitter used the local standard to drive a internal frequency synthesizer.

Example equipment:
http://www.navy-radio.com/rcvrs/r1051.htm
http://www.navy-radio.com/xmtrs/urt23-31.JPG

We had 100hz resolution to 30Mhz in the HF band and used SSB for just about everything. The main limitation to intelligibility and data errors was not equipment related but was caused by propagation factors that distorted the signals phase and frequency response within the audio channel channel. The types of very good synchronous detectors normally seen in SSB receivers can help AM reception fade but it's normally only seen on the top-end radios today.

http://www.youtube.com/watch?v=i5GejGwCA7Y

SSB signal tuning with 1970 era analog receiver.
http://www.youtube.com/watch?v=ANsosY7bkcs

 

I have not changed my position, modulation is not linear

Then what did you mean by "A statement was made that AM is not linear, in an ideal system I would dispute that too. "?

What is it that you would be disputing?

Can you point to where I said heterodyne was linear?

No. But I can point to where you said that AM is not linear in an ideal system. Since you acknowledge that heterodyning is nonlinear, that can only mean that you are talking about an ideal system that achieves AM modulation without using any type of heterodyning (or any other nonlinear process). All I am asking for is a description of that system.

I will repeat, where is the specific argument? You started out saying the RF carrier varied in amplitude, this is not the case.

And that is a matter of semantics.

If you have a waveform that is described by:

x(t) = A(t)sin(wt)

Then how is it wrong to say that this is a sinusoid that has a time-varying amplitude?

What would the equation for a sinusoid that had a time-varying amplitude look like?

This is in no way saying that the spectrum of this signal has no sidebands. The only signal that can have no sidebands is a pure sinusoid.

You then said a sideband next to a carrier is not AM if it was not a result of modulation. Given it is in fact indistinguishable from an AM signal I would state this is a false argument.

The purpose of modulation is to impress information onto a carrier in some way, correct? That means you start with the information you want to convey and, through some sequence of steps, produce the final signal. If you can't start with the information being conveyed, then you haven't shown a valid way to achieve modulation. I could simply record a signal and play it back, does that mean that I have "modulated" the signal? You can take ANY signal and break it into separate frequency components and then add those back together to get something that (at least in the limit) is indistinquishable from the original modulated signal, does that mean that this method describes in any way how the modulation works or can be said to be performing the modulation? How can it if it can't accomplish this task starting from the information that is supposed to be impressed into the final signal?

 

Lets try this from a different angle.

True or False,

Strip off the sidebands, and you are left with a clean RF carrier, with no modulation at all.

 

True.

I have never said otherwise.

The false dilemma you are trying to create (probably unknowingly) is to claim that I am saying that the carrier would still be varying in amplitude even if you removed the sidebands. It won't. Just because signal A (the full AM signal) has some property (varying amplitude of the carrier), does not mean that signal B, created by modifying signal A (by stripping the sidebands), will continue to have that same property.

I have stated that as soon as you start varying the amplitude, you create sidebands. You can't vary the amplitude without doing so. The sidebands embody the information about how the carrier amplitude is varying. I have said that you can view this from the reverse direction as adding in the sidebands and saying that the sidebands are interacting with and causing the variations in the amplitude of the carrier. It is also true that you can view the spectrum (where I am using the 1kHz signal on a 1MHz carrier as the example) as three separate and constant signals, one at 999kHz, one at 1000kHz, and one at 1001kHz.

So we have three different ways of viewing the signal and all three are equally correct. Which you use depends one what is most useful for what you are doing at the moment.

It's like one person claiming that current through an an inductor changes because of the voltage across it and someone else claiming that voltage appears across the inductor as a result of changes in the current through it. It fundamentally makes no sense to be dogmatic and say that one is correct and the other is merely a useful fiction.

My turn:

I have a waveform that is given by:

v(t) = A(t)sin(wt)

True or false: This is sinusoid with a time-varying amplitude.

 

To WBahn,

I have simply maintained it is the sidebands, not the varying amplitude, that is the information. The sidebands are separate signals, they can be stripped very simply. This has my point through this whole thread. The fact they are separate signals has major implications as to how you can process the information. How you view it. What else you can do with it. Whether you can synthesize it with two frequencies.

An oscope does not distinguish separate frequencies on the same display, it merely shows waveforms. It is the separate and distinct signals that creates the classic AM pattern, not the single RF frequency varying amplitude.

To me it seems you wish to approach this as a confrontation, or some form of one-upmanship. I don't, I was trying to make a specific point. As I am not enjoying this (quite the contrary, for the last 4 pages or so I have stayed various forms of frustrated and angry), this conversation is over. At this point I really don't care. My feeling is I'm being trolled, and I do not like it.

Anyone else wants to post and ask about the points I am trying to make, I will respond. WBahn and I are done here.

 

I 'm not feeling very comfortable doing this, but I feel I have to. I will post my view of the problem and the ways that I approach it, in hope that it can act as a catalyst to the conversation.

The following posts come from a private discussion with Bill_Marsden. I will post only my pots, since these are the only ones that belong to me and can defend.

 

To WBraun,

I have maintained it is the sidebands, not the varying amplitude, that is the information. The sidebands are separate signals, they can be stripped very simply. This was my whole point through this whole thread.

An oscope does not distinguish separate frequencies on the same display. It is the separate and distinct signals that creates the classic AM pattern, not the single RF frequency varying amplitude.

You wish to approach this as a confrontation, or some form of one up-manship. I don't, I was trying to make a specific point. My conversation with you is over, as I am not enjoying this (quite the contrary, for the last 4 pages or so), and at this point I really don't care what you think. My feeling is I'm being trolled, and I do not like it.

Anyone else wants to post, I will respond. You and I are done.

That's fine. You stepped back and asked a simple question asking for a one word answer. This was supposedly in an attempt to allow us to incrementally move toward a common understanding. I was more than wiliing to accommodate the attempt, despite that fact that you refused my earlier effort to likewise get us to a common starting point. None-the-less, I answered your question using a one word answer and then provided my reasoning and trying to speculate at where the miscommunication might be coming from. But if I ask a simple question and invite you to likewise answer it simply and then expand on your answer, you refuse to do so. This has happened several times. The impression I am left with is that, faced with a question that will force you to address a contradiction (either real or perhaps just apparent) in your assertions, you choose to ignore it and get defensive. Nothing I can do about that.

 

I generally feel I 'm closer to WBahn in this argument, purely based on what I have been taught in the University, which I have to take as right, or else my last 5 years have been in vein. Let me explain:

Point #1
First of all, let me define the Linear System. A linear system is an operation T of the variable x, over an input u(x). It has the property that \(T(a \cdot u_1(x)+b \cdot u_2 (x))=a \cdot T(u_1(x)) + b \cdot T(u_2(x))\). Here I have not mentioned neither the time nor the frequency domain, just a definition.
This is what I have been taught over and over and I am pretty confident about it.

Point #2
A signal x(t) in the time domain is a real function of the time variable t. It has only one dimension and can be plotted in a 2-D space.

A signal X(f) in the frequency domain is a complex function of the frequency domain f. It also has only one dimension and can be broken down in two 2-D plots, one for the amplitude and one for the phase.


I state the following with the given that we talk about analog procedures. Digital methods of producing signals are limitless and make things much more complex.


Part A
That said, let us the the AM modulation equation, as t_n_k wrote it in post #37 (http://forum.allaboutcircuits.com/showpost.php?p=500210&postcount=37) which is the one I know too.

We have
\(f(t)=\(1+m \cdot u(t) \)A_c \sin(\omega_ct)\)

This is a function in the time domain with input \(u(t)=\sin(\omega_m t)\). If we replace the input with \(u'(t)=a \cdot \sin(\phi) + b \cdot \sin(\phi)\) then the output will be:
\(f'(t)=(1+m \cdot u'(t)) \cdot A_c \cdot \sin(\omega_c t)=\\
(1+m \cdot a \cdot \sin(\phi)) \cdot A_c \cdot \sin(\omega_c t) + m\cdot \sin(\phi) \cdot A_c \cdot \sin(\omega_c t)\)

which is not equal to \(a \cdot f(t) + b \cdot f(t)\).
Thus, by definition, in the time domain, AM modulation isn't a linear function.

In the frequency domain I 'll let the wikipedia do the writing for me, as it's a bit more extensive (http://en.wikipedia.org/wiki/Amplitude_modulation#Spectrum).

With the input \(X'(\omega)=a\cdot X_1(\omega)+b\cdot X_2(\omega)\), the output of the modulator is
\(F'(\omega)=a \cdot F(X_1\(\omega)\)+b \cdot F\(X_2(\omega)\)\)

That means that in the frequency domain, AM modulation IS a linear process.

BUT, since analog technology does the operations using transistors and OpAmps, I would say that it operates on the time domain and thus AM modulation is a non-linear function.

So, in any case, the AM modulation is a non-linear operation and it can't be characterized as linear in any of its forms.
I believe that Bill's related claim about the sum of the Base Band signal and the Side Band signal isn't valid, because the initial signal isn't the one in the the side band (1.001MHz) but the one at 100kHz.


Part B
Now, about the varying amplitude of the carrier: A periodical, continuous time signal can be broken down in a sum of sinusoidal waves of various frequencies. Our AM signal can be broken down into a sine at \(\omega_c\)Hz and two sines at \(\omega_m+\omega_c\)Hz and \(\omega_m-\omega_c\)Hz. You can say that the carrier sine is in there intact, but in effect, while looking at an AM signal in an oscilloscope (not spectrometer), one must admit that we could describe what we see as sine wave with a variable peak amplitude. I know it's a matter of perception, but I believe that it is more intuitive that way.

However, strictly speaking, a sine wave with varying amplitude isn't a sine wave anymore. It is a different signal that can be broken down as a sum of sine waves of different frequencies. A sine wave has always constant amplitude. (S1)

In any approach though, it is a fact that the AM signal in the time domain is one, one dimensional signal x(t). You cannot extract any info about how it was built without making assumptions. Heck, I could even say that it's a pulse train with the right amplitudes.

End of theoretical basis.

In the aftermath, I must admit that I was kinda lost in the discussion, mainly due to the language barrier, but if I understood correctly the main points of disagreement were:

A. WBahn was saying that AM modulation was a non-linear function, Bill said that under certain conditions it is linear.
I explained why I agree with WBahn in Part A.

B. Bill said that the carrier sine remains intact while WBahn talks about a sine wave at the carrier frequency with varying amplitude.
Based on what I said in Part B, I state that all I see when I look at an AM signal in the oscilloscope is a signal that resonates at the carrier frequency and has its amplitude multiplied by the information signal. I don't see two signals in there. However, that signal isn't a sine wave, base on my statement (S1).

(Correct me where you believe I have misunderstood you.)

On the matter of linearity and the heterodyne processing, I 'd say more correctly that heterodyne processing IS a non-linear action. You can't do it in a linear way and fail in your quest for that reason.

Strip off the sidebands, and you are left with a clean RF carrier, with no modulation at all.

Now, about this statement, yes, I agree. But it lacks meaning. If you strip the sidebands, you simply select one of the many frequency components of the AM signal, which happens to be the carrier frequency.
A dual statement would be "Disregard the carrier frequency and you are unable to decode the modulated information".
A full set of information is required to decode the signal.

The way I see it, in order to build an AM frequency modulator, you need to involve at some point a sine wave at the carrier frequency. This makes the carrier sine vital for the modulating process.
The same holds for the demodulation.
Does the information have to "ride" the carrier to get to you? No, not really. It travels in a frequency near it, not identical with it. It wouldn't be distinguishable if it was the latter. Also, the transmission of the carrier isn't vital for the information extraction.
BUT, it characterizes the modulation since it provides the channel (frequency band) upon which the information is transmitted.

In short: I don't believe the carrier is unimportant, nor I believe that all you need for the decoding is the side band. However, you can choose not to send the carrier via the air and generate it on the receiver.

Ultimately, I think the confusion lies in the fact that other people look at the time domain, while others have the frequency in mind. Bill belongs to the latter group.

WBahn sees the AM signal as y(t)=(1+m*u(t))*c(t), where c(t) is the carrier signal and u(t) is the information signal. All descriptions lie in the time domain. This is a valid representation. In that aspect, you apply an operation on the carrier signal c(t).
This is what you would see in the oscilloscope screen, where the x-axis is the time.

Bll, on the other hand, sees a spectrogram, where the carrier frequency sticks out in the right and then, after the modulation appears to be "accompanied" by an area that corresponds to the information spectrum. No change whatsoever to the carrier frequency spike.

You guys are looking at the same phenomenon through different lenses.

But the object under observation is the same: The Fourier transformation isn't some lossy process. It is simply the same signal/equation/description, put under different axis (crudely).

If x(t) is a signal and X(ω) is its Fourier transformation, then those signals are completely the same. They carry the same information. They can hop from one domain to another without any loss or morphing of the information content.

The difference is purely visual: You can see how x(t) evolves in time more intuitively. You can see it as it actually progresses in real time.
On the other hand, you can easily distinguish the frequency content, which is, indeed, very useful for industrial applications.

The model of the AM modulation is the same in both cases. What changes is the domain that you choose to step on in order to describe it.

 

The point I have made, and continue to do so, is the simple model of varying the amplitude of the RF does not allow for understanding of Upper Side Band, Lower Side Band, or Suppression of the RF carrier. Using an oscope to explain the behaviors of AM in general falls short, dramatically.

I'll state it is a great explanation, but so lacking in substantive detail it is little more than a teaching tool. You can use it to measure modulation %, but when you start messing with other forms of AM, such as carrier suppression, it all falls apart. It is a very limited model, and there are much better explanations that cover all the details. By using a model that only explains the shape of a waveform you are missing some very important and subtle details.

When the model was developed I'm not sure the spectrum was well understood. As with Fourier analysis explaining how harmonics shape waveforms a complete explanation of AM must include the spectrum. It can not be simply ignored because it is inconvenient when all you have is an oscilloscope. It is one of the reasons I keep harping about a spectrum analyzer. A spectrum analyzer shows the complete picture.

Once sidebands are understood, and how they relate, you can do things that the simplistic explanation of varying the amplitude of the RF carrier can not explain. For example, you can now remove the LSB and carrier and stack channels in half the spectrum space required by conventional AM.

I am a technician. I have had some training in engineering, 43 years ago. Much of what I have learned has been forgotten. I have also learned a lot more since I graduated college. I don't use a lot of the same terms engineers use. This handicaps my ability to express myself sometimes, but I do consider myself fairly articulate. When you have a theory that does not explain major details or simply leaves important details out, it is a marker of a flawed and incomplete theory. Sticking with such a theory, unless you are trying to communicate with a beginner, does not make much sense. I feel it lazy.

 

Yes! That makes complete sense and finds me in complete agreement.

Isn't it unnerving when sometimes the written language is so stiff it inhibits us from understanding each other?

 

@Georacer: By and large I have no problems with what you have put forth and largerly agree with you. There are really just two points, one technical and the other, I guess personal is the best descriptor, which I would like to address:

In the frequency domain I 'll let the wikipedia do the writing for me, as it's a bit more extensive (http://en.wikipedia.org/wiki/Amplitude_modulation#Spectrum).

With the input \(X'(\omega)=a\cdot X_1(\omega)+b\cdot X_2(\omega)\), the output of the modulator is
\(F'(\omega)=a \cdot F(X_1\(\omega)\)+b \cdot F\(X_2(\omega)\)\)

That means that in the frequency domain, AM modulation IS a linear process.

If a system in linear in the time domain, then it is linear in the ffrequency domain, and vice versa. The problem with your description above is that, since the modulator is non-linear, the Fourier transform of the modulator (i.e., it's impulse response) is meaningless because the Fourier transform is only valid for linear systems.

To underscore this, if the input to the modular you talk about (X1 and X2) are sinusoids at 1kHz and 1MHz, then the outputs you mention, F(X1) and F(X2), would have to be signals at 1kHz and 1MHz with nothing at 999kHz or 1001kHz.

Now, if you are talking about combining a 1MHz signal with a 1.001MHz signal by simply adding them, then I agree that THAT is a linear process and the description you have holds. Given what you say in the next few sentences, perhaps that is what you were talking about.

WBahn sees the AM signal as y(t)=(1+m*u(t))*c(t), where c(t) is the carrier signal and u(t) is the information signal. All descriptions lie in the time domain. This is a valid representation. In that aspect, you apply an operation on the carrier signal c(t).
This is what you would see in the oscilloscope screen, where the x-axis is the time.

The clarification I would make here is that I'm not ONLY seeing an AM signal this way. I've actually described three different ways of viewing it and assert that all three are equally valid and that none of them are "useful, but wrong". Perhaps I haven't stated it in so many words, but I have certainly implied that I don't think any explanation of AM (or anything similar) could be considered 'complete' unless it includes BOTH the time-domain and the freqiency-domain descriptions. Neither, by itself, is really sufficient (though, technically, either by itself is sufficient, but the necessary math would be prohibitive as to make progress virtually impossible).

What I am seeing, as best I can tell, coming from Bill is that there is only one acceptable and true way of viewing it and that any other discription is, at best, a useful fiction that we saddle beginners with because they aren't quite ready to handle the truth, yet.

If x(t) is a signal and X(ω) is its Fourier transformation, then those signals are completely the same. They carry the same information. They can hop from one domain to another without any loss or morphing of the information content.

The difference is purely visual: You can see how x(t) evolves in time more intuitively. You can see it as it actually progresses in real time.
On the other hand, you can easily distinguish the frequency content, which is, indeed, very useful for industrial applications.

The model of the AM modulation is the same in both cases. What changes is the domain that you choose to step on in order to describe it.

I agree and have made this point a few times. Does x(t) 'cause' X(ω)? Or does X(ω) 'cause' x(t)? It's a fundamentally meaningless question because they are simply two descriptions of the same thing. Thus saying that one is 'correct' and the other is a 'useful fiction' is meaningless.

We tend to live and interact with things in the time-domain and thus tend to percieve most things from that perspective first. That can lead to the notion that the time-domain is 'real' and the frequency domain, while a useful mathematical construct to model the real world, is not really real. That we employ complex (i.e., both real and 'imaginary' numbers in the process tends to reinforce this notion).

But when you come right down to it, out time-domain descriptions are also simply mathematical constructs to model the real world. There are plenty of examples in nature where it is arguably more reasonable to percieve them in the frequency domain first and then consider the time-domain equivalent. Both are equivalent and both are correct.

 

Ah! I had a hunch I had gone wrong somewhere in the linearity of the Fourier transformation. You are correct, it isn't applicable to any non-linear system. Thanks.

Also, I just thought about a case where the Fourier representation doesn't provide us with enough tools whereas the time-domain is more useful:
The simple demodulation circuit uses a germanium diode to "filter" the signal envelope and do away with the carrier frequency.
As far as I know, the non-linear diode can't be easily coded into a frequency-response system.
Am I right in this or both ways work here?