AM - What's really happening?

[HIPAR]

I 'm new here.  My name is Charles and I have been a HAM since 1959.  I even have a commercial radiotelephone license.  That only means I have probably answered all the FCC questions about AM correctly.

' In my head are many facts T
hat, as a student, I have studied to procure,
In my head are many facts..
Of which I wish I was more certain I was sure! '

(King of Siam - The King and I)

When I question the details, I really don't understand how AM works .

I have heard AM modulation described as a process during which the amplitude of a carrier signal is varied in accordance with the amplitude of an information waveform.  This is not so.

A look at the equation describing the spectrum of an AM signal shows the three distinct components.  The origional carrier, with no variations in amplitude due to the modulation and the two sidebands created from the modulating power.

My first hurdle to overcome is the AM scope pattern.  Why does it show zero amplitude on the negative peaks of a 100% modulated signal when a carrier of nonvarying amplitude is continuously present?  I'm not sure if I'm correct, but here is my explaination:  

The AM scope pattern vector sum of the three components.  Since the sidebands are frequencies offset from the carrier, they are continuously varying in phase with respect to it and the additions and subtractions result in a pattern that represents some kind of envelope.

This is a hurdle that I haven't been able to rationalize:

The ARRL handbook shows the typical transformer isolated plate modulation circuit.  It states that for 100% modulation, modulation voltage developed accross the trasnsformer secondary will cause the final amplifier plate voltage will vary between zero and twice the B+ voltage.  That seems to make sense in the context of a carrier amplitude variation scheme.  But, how can the amplifier continue to deliver the always present carrier power when its plate voltage is zero?

Maybe there is energy stored in the tank circuit or something nonintuitive like that going on.

Tis a puzzlement!

---  CHAS  WA2DYA

[WA1GFZ]

The final amplifier is a mixer producing 2 side bands with the carrier and on a scope you know why you see the voltage change. Now fire up your spectrum analyzer and observe the carrier with the two side bands 6 db down from carrier at 100% modulation. That is why it takes 1/2 as much audio as carrier to produce 100% modulation.  !/4 power in each side band.  It all adds up.  fc

[wavebourn]

Chas, it is simple:

output = carrier X (1/2 + mudulation)
Use Fourier transformation, and you'll get your carrier and sidebands.
Why not zero? Because not exactly 100%. Period.

[Bacon, WA3WDR]

The varying instantaneous amplitude model, and the three-vector model, are two ways of looking at an amplitude modulated carrier modulated by a sine wave signal.  This signal has three frequency components that beat against each other.  It looks like a carrier varying in amplitude, and in fact that is how we usually synthesize it.  But it also looks like three discrete frequencies of different frequencies in a certain phase and time relationship.

If you had two unchanging carriers of the same amplitude and slightly different frequencies, the amplitude of their sum would vary up and down, from two times the amplitude of one of them, to zero, and so on.  It would have to.   The fact that they occasionally cancel is incidental.  Each is still always present.

In a sine-wave modulated AM signal, the carrier is joined by two slightly different frequencies, spaced exactly the same frequency distance from the carrier, but in opposite directions. One is x higher than the carrier frequency, and the other is x lower than the carrier frequency.  The frequency difference causes a beating - that is, the signals alternately add to and subtract from each other.  The timings are such that when the two side frequencies are in phase (0 degrees relative phase), they are either exactly in phase with the carrier (0 degrees relative phase), or exactly out of phase with the carrier (180 degree relative phase).  This is significant, because the main difference between AM and narrowband Phase Modulation is that in PM, when the two side frequencies are in phase, they are plus or minus 90 degrees to the carrier.

It is not intuitive that the varying amplitude composite signal should consist of a constant level carrier and two side frequencies, yet this is definitely the case.  In an amplitude modulated signal with 100% sine wave modulation, the carrier can be constant even though the instantaneous amplitude of the composite signal might vary from 2x carrier to 0.  The variations are the result of a complex beating between the steady carrier and the steady side frequencies.  In fact with more sophisticated modulator circuits, it is possible to go below zero into the negative!

Going back to PM, even with small side frequency amplitudes, the carrier would still vary slightly in amplitude because of the beating effect.  This is why in PM there are higher order side frequencies generated with sine-wave modulation.  These higher order side currents add to the complex beat pattern in such a way as to eliminate the amplitude variation.  The resulting spectrum is quite complex.

(Fixed a typo 1/16/2005 11:53AM)

[W8ER]

Chas Said:

The ARRL handbook shows the typical transformer isolated plate modulation circuit. It states that for 100% modulation, modulation voltage developed accross the trasnsformer secondary will cause the final amplifier plate voltage will vary between zero and twice the B+ voltage. That seems to make sense in the context of a carrier amplitude variation scheme. But, how can the amplifier continue to deliver the always present carrier power when its plate voltage is zero?

Chas ..  It can't and doesn't. The explanation in the ARRL Handbook is correct. The audio voltage developed across the secondary of the modulation transformer does vary between twice the resting plate voltage and zero, under 100% symetrical modulation. A scope will display the amplitude (vertical displacement) of the transmitter output (audio and RF mixed) versus time (horizontal displacement). Looking at that display you can confirm that the carrier is being completely cut off, for an instant, under 100% modulation. That is why you will see the carrier (output) fall to the baseline, at that time.

Remember that the display on the scope shows you the instantaneous amplitude of the tranmitter output, NOT the frequency. The scope display is a mix of the audio and RF (mixed together) at any one instant .. repeated many times across the oscilloscopes screen.

If you wish to sort out the individual components (LSB, carrier and USB) and see their amplitudes individually, you must use a spectrum analyzer, not an oscilloscope.

I sure hope that helps!

--Larry W8ER

[k4kyv]

From the beginnings of radiotelephony there has been a question whether sidebands exist as physical reality or only in the mathematics of modulation theory.  In the early 20's this was a hotly debated topic, with a noted group of British engineers maintaining that sidebands existed only in the maths, while an equally well-remembered group of American engineers argued that sidebands do, in fact physically exist.

Today, the issue seems settled once and for all.  We can tune our modern-day highly selective receivers through double-sideband and single-sideband voice signals, and tune in upper or lower sideband, and even adjust the selectivity to the point that we can tune in the carrier minus the sidebands.  Nearly everyone accepts the notion that sidebands do indeed exist physically.  But do they?  Maybe it's a matter of how we observe them, and the result is modified by our measuring instruments.  Remember the Heisenberg Uncertainty Principle, which states that you cannot observe something without affecting it to some degree, or more precisely, that it is impossible to know both the position (physical  location) and velocity (speed and direction) of a particle at the same time.

Imagine a cw transmitter equipped with an electronic keyer.  Also imagine that there is no shaping circuitry, so that the carrier is instantly switched between full output and zero output. Such a signal can be expected to generate extremely broad key clicks above and below the fundamental frequency because of the sharp corners of the keying waveform.  Set the keying speed up to max, and send a series of dits.  If the keyer is adjusted properly, the dits and spaces will be of equal length, identical to a full carrier AM signal 100 percent modulated by a perfect square wave.

Suppose the keyer is adjusted to send, say, 20 dits per second when the "dit" paddle is held down. The result is a 20 hZ square-wave-modulated AM signal.  Now turn the speed up. If the keyer has the capability, run it up to 100 dits per second.  If you tune in the signal using a receiver with very narrow selectivity (100 hZ or less, easily achievable using today's technology), you can actually tune in the carrier, and then as you move the dial slightly you can tune in sideband components 100, 300, 500 hZ, etc. removed from the carrier frequency. A square wave consists of a fundamental frequency plus an infinite series of odd harmonics of diminishing amplitude. Theoretically you would hear carrier components spaced every 200 Hz throughout the spectrum.  In a practical case, due to the finite noise floor, the diminishing amplitude of the sideband components and selectivity of the tuned circuits in the transmitter tank circuit and antenna itself, these sideband components eventually become inaudibly buried in the background noise as the receiver is tuned away from the carrier frequency.

Suppose we gradually slow down the keyer.  As we change to lower keying speed, it takes more and more selectivity to discriminate between carrier and sideband components, as the modulation frequency becomes lower and the sideband components become spaced more closely together. Let's observe what happens when we slow the dit rate down to 10 dits per second. Now the fundamental modulation frequency is 10 Hz, and there are sideband components at 30 Hz, 50 Hz, and 70 Hz removed from the carrier and continuing above and below the carrier frequency at intervals of 20 Hz until the signals disappear into the background noise.  In order to distinguish individual sideband components, we need selectivity on the order of 10 Hz, which is possible if we use resonant i.f. selectivity filters with extremely high "Q".  This can be accomplished using crystal filters, regenerative amplifiers or even conventional L-C tuned circuits if we carefully design the components to have high enough Q.  

As we achieve extreme selectivity with these high Q resonant circuits, we observe a sometimes annoying characteristic familiarly known as "ringing." This ringing effect is due to the "flywheel effect" of a tuned circuit, the same "flywheel effect" that allows a class-C tube type final or class-E solid state final to generate a harmonic-free sinewave rf carrier waveform.  The selective rf tank circuit stores energy which is re-released to fill in missing parts of the sinewave, thus filtering out the harmonics inherent to operation of these classes of amplifier.  CW operators are very aware of the ringing effect of very narrow filters, which can make the dits and dahs of high speed CW run together, causing the signal to be just as difficult to read with the narrow filter in line, as the same CW signal would be if one used a wider filter that admits harmful adjacent channel interference.  Kind of a damned if you do, damned if you don't scenario.
           
Let's now take our example of code speed and selectivity to a degree of absurdity.  We can slow down our keyer to a microscopic fraction of a Hertz, to the point where each dit is six months long, and the space between dits is also six months long.  In effect, we are transmitting an unmodulated carrier for six months, then shutting down the transmitter for six months. But still, this is only a matter of a degree of code speed; the signal waveform is still identical to the AM transmitter tone modulated with a perfect square wave, but whose frequency is one cycle per year, or 3.17 X (10 to the -8) Hz.  That means that in theory, the steady uninterrupted carrier is still being transmitted, along with a series of sideband components spaced every 6.34 X (10 to the -8) Hz.

Now, carriers spaced every 6.34 X (10 to the -8) Hz apart are inarguably VERY close together, to the point that building a filter capable of separating them would be of complexity on the order of a successful expedition to Mars, but it is still theoretically possible. Let us assume we are able to build such a filter.  We would undoubtedly have to resort to superconductivity in the tuned circuits, requiring components cooled to near absolute zero, and thoroughly shield every rf carrying conductor to prevent radiation loss, but here we are talking about something hypothetical, without the practical restraints of cost, construction time and availability of material.  Anyway, let us just assume we were able to successfully build the required selectivity filter.

The receiver would indeed be able to discriminate between sidebands and carrier of the one cycle/year or 3.17 X (10 to the -8) Hz modulated AM signal, identical to a CW transmitter with carrier on for six months and off for six months.  So how can we detect a steady carrier while the transmitter is shut off for six months?  The answer lies in our receiver.  In order to achieve high enough selectivity to separate carrier and sideband components at such a low modulating frequency and close spacing, the Q of the tuned circuit would have to be so high that the flywheel effect, or ringing of the filter, would maintain the missing rf carrier during the six-month key-up period.

This takes us back to the longstanding debate over the reality of sidebands.  If we use a wideband receiver such as a crystal set with little or no front-end selectivity, we can indeed think of the AM signal precisely as a steady carrier that varies in amplitude in step with the modulating frequency.  This is physically the case if the total bandwidth of the signal is negligible compared to the selectivity of the receiver.  Once we achieve selectivity of the same order as the bandwidth of the signal, which has been the norm for practical receivers dating from as early as the 1920's up to to the present, reception of the signal behaves according to the principle of a steady carrier with distinctly observable upper and lower sidebands.  The "holes" in the carrier at 100% modulation are inaudible due to the flywheel effect of the tuned circuits, even though the "holes" may be observable on the envelope pattern of the oscilloscope.

An oscilloscope set up for envelope pattern, with the deflection plates coupled directly to a sample of the transmitter's output, is a wideband device much like a crystal set. It allows us to physically observe the AM signal as a carrier of varying amplitude. A spectrum analyser on the other hand, is an instrument of high selectivity, namely a selective receiver programmed to sweep back and forth across a predetermined band of spectrum while visually displaying the amplitude of the signal. It clearly displays distinct upper and lower sidebands with a steady carrier in between.

Furthermore, it has often been observed that the envelope pattern of a signal as displayed from the i.f. of a receiver can be quite different from that of a monitor scope at the transmitter site.  This is yet another example of how the pattern is altered (distorted) by the selective components of the receiver.

In conclusion, there is no correct yes or no answer to the age-old question whether or not sidebands are physical reality, or exist only in the mathematics of modulation theory. It all depends on how you physically observe the signal.  Sidebands physically exist only if you use an instrument selective enough to observe them. Recall the Heisenberg uncertainty Principle.

[Steve - WB3HUZ]

I've used FFT based analyzers with micro-Hertz, yes, micro-Hertz resolution capability. Performing DFT/FFT analysis on a signal requires only that you collect an appropriately long time sample.

The sidebands exist just as much as the audio you hear coming out of your speaker. Otherwise one could say, you aren't really hearing that audio, it's just the way your are interacting with the signal. The audio doesn't really exist. :tfrag:  :tfrag:

[Philip, AB9IL]

Well there it is, Chas...a fine description of what we get as a result of mixing audio and a carrier.  

Just for fun, I tuned in WLS on my li'l gen coverage receiver, went to sideband mode, and looked at the signal in SpectrumLab.  Sure enough,  the relationship between the carrier and sidebands holds up.  It may be interesting to record WWV and use software to observe the phase of the carrier and those sidebands.

This discussion sheds some light on why a well designed sync detector is a great asset to an AMer's receive gear...The phasing must be considered along with the bandwidth and amplitudes of the incoming signal.


Thanks for posing the query, and 73!

[Bacon, WA3WDR]

I contend that we are synthesizing a double-sideband with carrier signal when we modulate amplitude by varying PA B+, or by other means.

You can draw a graph of the carrier, upper sideband and lower sideband sine waves (of course you must make the relative timing/phase correct), and add them together, and the resulting composite signal will be identical to the modulated signal we produce with our modulated stage.  Therefore I conclude that they are the same; that the sidebands exist, and that the carrier is continuous, even though the composite signal may vary from 0 to 2x the carrier level.

[W8ER]

Chas .. it is interesting .. and when you get something basic like this out to be examined, all of the different theories are fun to look at.

I guess I can't ignore what I see on the scope and that is a complete cutoff of the output (to include carrier)! If you superimpose (or mix) a 1khz tone on top of a 1 mhz carrier, there are going to be several cycles of RF that occur during the time when the audio tone is at or near complete cutoff of the plate supply. No output is seen at that time. To my mind it seems that the carrier is, in fact, missing, missing at an audio rate.  

So, Bacon I am having a hard time understanding your contention that the carrier is always present. I don't see how IT can be .. and we are talking about those teeny tiny slices of time when the final is not supplied with power.

HUZ what hath you to say about this?

--Larry

[Steve - WB3HUZ]

I say the carrier goes away, just like in Don's square wave example. But what the heck do I know?

[Bacon, WA3WDR]

Indeed, 'tis a puzzlement.  I can draw three continuous sine waves of apppropriate amplitude, frequency and phase, corresponding to carrier, upper sideband and lower sideband, and I can add them point by point, and I can duplicate a 100% modulated AM signal waveform as we understand it.  I have to believe that this works backwards; if I make this waveform by modulating a carrier in an AM transmitter, then I am synthesizing the same continuous sine waves.  It seems to me that an on-off sequence would only differ from the sine wave modulated condition in the complexity of the sideband spectrum.

But in the real world, sometimes theory does not produce results.  For example, if there was some FM audio subchannel modulating the on-off carrier, I would simply not be able to receive it while the transmitter was off, even if I was positive that the carrier was theoretically still present.  So as Dennis asked on the amradio mailing list - if a tree falls in the forest, and nobody sees it or hears it... did it really fall?

When I am receiving zero, I can only know that I am receiving zero.  And if there is interference,  I may not even be sure I am receiving zero!

And as Don pointed out, there is the question of reality, and the appearance of reality. If I took the time scale to infinity, then if a carrier was EVER transmitted, even for only a second, I would have to say that it existed for all of time.  That's ridiculous!  I only think that because I am integrating over all time, and then generalizing.

I guess it's off when it's off, and it's signalling us when it's on.  So the question becomes, at what time scale do we stop seeing beat notes, and start seeing variations and on-off switching?

[KL7OF]

And in a parallel universe.................. ................................................... Great topic

[k4kyv]

Indeed, 'tis a puzzlement.  I can draw three continuous sine waves of apppropriate amplitude, frequency and phase, corresponding to carrier, upper sideband and lower sideband, and I can add them point by point, and I can duplicate a 100% modulated AM signal waveform as we understand it.

It is very interesting to demonstrate it with  rotating vectors, letting one vector represent the carrier, one upper sideband and one lower sideband.  You can demonstrate both amplitude and phase variations in the carrier, duplicate the envelope pattern, and graphically demonstrate the quatrature distortion that exists with carrier and one sideband, and show that the quadrature distortion in the two sidebands cancels out with double sideband, leaving pure amplitude modulation.  You can demonstrate why the envelope of carrier plus one sideband has low distortion at low percentages of modulation, but becomes highly distorted as the percentage of modulation approaches 100%.

I used to do it with pencil and paper.  It would be very interesting to set up the vector diagrams on a computer where you could actually see the vectors rotate instead of having to imagine motion in still images drawn on paper.  Wish I had the computer savvy to do that.

[Bacon, WA3WDR]

I am convinced that sidebands exist.  But as for whether the carrier goes away at 100% negative modulation...

How's this: a carrier should be considered to exist during the period when it has recently been detected, and we are actually receiving modulation from it.  It should not be considered to exist when it is clearly not being generated.

So in the example of an AM signal at the instant of 100% negative modulation, we should consider the zero to represent zero percent of the carrier, averaged over time during recent history.  We can see that this will result in appropriate demodulation of the actual signal.

But when the AM transmission is over, either after the end of a transmission, or after the end of a broadcast day, or in most cases if the signal fades out due to propagation, etc, we should consider the carrier not to be present.  This gets a little tricky, because we can not really be certain (Don's Heisenberg Uncertainty Principle thought here) that the transmitter is not sending us a very, very long zero.  But we can be pretty sure, pretty quick.

A synchronous detector is a good flywheel that tracks the known carrier frequency and holds the reference for us during zeros and noise bursts.  We set up a synchronous detector to detect the carrier and hold the reference for a short period of time.  This works pretty well for the signals we actually transmit.

Now take the example of a 1 KHz sinewave transmitted as double sideband suppressed carrier.  You get pulses of alternating carrier polarity.  During any given pulse, for about 500 microseconds at a time, you get an AM signal with a carrier, transmitting one half of a sine wave.  When the pulse is over and the signal passes through zero, it goes negative and the next pulse appears - pretty much identical to the first pulse, but with reverse carrier polarity.  You would not know the carrier polarity reversed, except for your flywheel reference.  But demodulating with the flywheel reference gives you the 1 KHz sinewave.  It is evident to the observer that the flywheel reference was correct.

With a very low modulating frequency, a DSBSC signal would just look like a fading carrier.  The transmit frequency stability and the reference flywheel precision would have to be very high to determine that the carrier polarity had reversed.  At some point this becomes irrelevant, because the transmission path varies in length, there is drift and phase noise in TX and RX, and the signal is not received well enough to know for sure that the carrier polarity flipped.  (Uncertainty again.)

And although with DSBSC we keep getting pulses of carrier, they keep reversing in polarity, and on average they balance out to zero.  We accept that the carrier is suppressed, we don't hear a heterodyne where we would usually hear one, etc, yet we see the carrier dancing on the oscilloscope.  But over the appropriate integration time, its frequent polarity reversals cause it to balance out to zero.

For most real signals, a very long time base is inappropriate.  In some cases though, such as slow synchronous CW, a long time base is appropriate.  So the receiver time scale should be appropriate to the signal being sought.

So it's a reality-check issue.  Surely a carrier was not transmitted for all time, just because it existed for less than a second at some point.  But just as surely, the instant of 100% negative modulation should not be reproduced as a glitch.  The application of the appropriate time scale is the key, and it is up to the listener to determine what happened.

  Bacon, WA3WDR

[WA1GFZ]

A class C final and modulator is just a single balanced mixer, that simple.

[k4kyv]

A class C final and modulator is just a single balanced mixer, that simple.

Actually, isn't it an unbalanced mixer?   Neither signal (rf carrier or modulating audio) is nulled out.  When modulation exceeds 100% the rf output cuts off, and splatter is produced.  Audio is prevented from appearing at the antenna terminals of the transmitter by the selectivity of the rf tank cincuit.

A singly balanced mixer (balanced modulator) would generate double sideband, independent of carrier.  That is what the "upside down tube" circuit does.  Beyond 100% modulation, it continues to produce rf output, but with a 180 degree phase reversal.  It results in distortion with envelope detector, but (theoretically) no splatter.  Cut off the DC voltage altogether, and you have DSB suppressed carrier.  

A doubly balanced mixer is designed to null out both of the original signals so that only the sum and difference frequencies can appear at the output, theoretically without the need for tuned circuits.  There is no need for this in an AM transmitter because two frequencies (rf and audio) are so far removed.  The doubly balanced mixer is useful in rf mixer applications where the two frequencies are close enough together that both need to be nulled out so that they don't appear in the output, such a heterodyne type exciter, typical of SSB rigs.  For example, a 9 mHz xtal oscillator output is mixed with a 5.0-5.5 mHz VFO to generate SSB on 75 and 20 by tuning the output to the sum and difference frequencies.  The DBM makes it easier to eliminate throughput, avoiding undesired radiation of the 5 mHz vfo signal or the 9 mHz output from the SSB generator.

[Steve - WB3HUZ]

It is very interesting to demonstrate it with  rotating vectors, letting one vector represent the carrier, one upper sideband and one lower sideband.  You can demonstrate both amplitude and phase variations in the carrier, duplicate the envelope pattern, and graphically demonstrate the quatrature distortion that exists with carrier and one sideband, and show that the quadrature distortion in the two sidebands cancels out with double sideband, leaving pure amplitude modulation.  You can demonstrate why the envelope of carrier plus one sideband has low distortion at low percentages of modulation, but becomes highly distorted as the percentage of modulation approaches 100%.

I used to do it with pencil and paper.  It would be very interesting to set up the vector diagrams on a computer where you could actually see the vectors rotate instead of having to imagine motion in still images drawn on paper.  Wish I had the computer savvy to do that.

Here's a  phasor representation. Note the display is effectively "strobed" at the carrier rate, otherwise the carrier phasor would be rotating too. [Key: red - carrier, yellow - one sideband, green - the other sideband]. One of these days I will add an envelope display and show a time reference so you can correlate the phasor display to the proper point on the envelope display.

[WA1GFZ]

Now we need someone to add a scope and spectrum analyzer display.
Then add a feature to slow things down.
It could be saved somewhere so everytime this comes up the person could be directed to page 53 to see the movie.

Sounds like a job for our smart guy KE1GF

[k4kyv]

Here's a  phasor representation. Note the display is effectively "strobed" at the carrier rate, otherwise the carrier phasor would be rotating too. . One of these days I will add an envelope display and show a time reference so you can correlate the phasor display to the proper point on the envelope display.

The "resultant" vector, along the Y axis represents the instantaneous amplitude of the output.  The X axis component represents phase (deviation from the strobed carrier position).  Eliminate one of the sidebands and you can see that as the remaining sideband vector rotates, the  resultant vector wobbles from  left to  right as it varies in length (amplitude).  This shows  that  SSB is a combination of amplitude and phase modulation.  In DSB, the phase component of each sideband is equal and opposite;  therefore they cancel out,  resulting in purely amplitude variations.

[Steve - WB3HUZ]

Here are the phasors with the envelope display included.

[Bacon, WA3WDR]

Eliminate one of the sidebands and you can see that as the remaining sideband vector rotates, the resultant vector wobbles from left to right as it varies in length (amplitude). This shows that SSB is a combination of amplitude and phase modulation.

That is the basis of the "phasing" method of SSB generation - but for that, balanced modulators are normally used, and the carrier is suppressed.

[Steve - WB3HUZ]

Using the controls and sliders at the link below, you should be able to slow things down, reverse, etc.  No spectrum display though.

http://www.amwindow.org/misc/phwenvanim.mov

Now we need someone to add a scope and spectrum analyzer display.
Then add a feature to slow things down.
It could be saved somewhere so everytime this comes up the person could be directed to page 53 to see the movie.

Sounds like a job for our smart guy KE1GF

[Rob K2CU]

TO start, we eliminate magic.

Fact is, in DSB AM, that we all know and love, the carrier is ever present.

It seems that the confusion arises from the method of observation. The oscilloscope displays the algebraic sum of the signals present. The mixer, or modulator is in actuality a multiplier. In simple terms, it mulriplies two sinusoids to produce a product. Supose the carrier signal is represented by Sin(2*PI*Fc*t), where Fc is the carrier frequency, t is time and 2 *PI is necessary to conver the frequency in Hz to Radians per second.  The result is that over the course of one period = 1/t, the argument of the Sine function will go through 360 degrees.  IN a similar fashion, the modulating signal can alsp be a Sine function, Sin(2*PI*Fm*t), where Fm is the modulating frequency.  Sine's sister function Cosine, could also be used as it is Sine shifted by -90 degrees.   In a doubly balanced mixer/modulator, we multply thse two and get upper and lower sidebands...DSB.  For the trig equations for multiply sin and cosine, refer to:

http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

in particule equations 5 through 8.  

if the balance is upset by adding a constant term (DC) to the modulating signal, the original carrier will be added to the output.

ONe can take independantly generated signals and add them together to get the same result. For example, in the TR-7, AM is generated by first generating SSB and then, prior to the linear PA stage, adding back in the carrier. That rig produces A3H or SSB plus carrier. CHU does this as well.

So, this example of AM being gernerated by adding carrier to a sideband demonstrates it can be done this way.  Clearly, the carrier that is injected after the modulation process is continuous and goes right on out the linear PA section of the rig.

Another thought ot consider. earlier in this thread, a square wave was injected into the conversation as a possible source of modulation, effectively cutting off the carrier during the zero period. Well, I challenge you to examine the square wave itself! IT is agreed to be made up of the fundamental along with an infinite series sum of odd harmionics of decreasing amplitude. Added together, they produce a signal that goes to zero for one half the period. Yet, the individual components are continuous in nature! Observation by oscilloscope would tell us that all the components disappear for one half cycle, just as the carrier appears to  go to zero at the valley of the modulation.  Yet, we know that this is not true.   Pass that square wave through a filter and you can extract whatever component you may want to observe, and it will be continuous.

[Steve - WB3HUZ]

I can pass the output of my RX's detector through a filter all day long during the period when the carrier is cutoff, and the filter will reveal nothing continuous but noise.

I don't think your TR-7 example is also incorrect. IIRC, the carrier is not linearly added back into the SSB signal, it is mixed/multiplied. And at least one other mixing takes place to shift the IF to the RF output frequency.