The amplitudes of distinct Fourier components do add, at least they do in a linear system. If you take a 1.0005 kHz signal of unit amplitude, and multiply it by a unit-amplitude 0.5 Hz signal, by the usual sum and difference identities we get
One thing I forgot--I did say that the carrier frequency had to be 'correctly phased' in order to get a 100% AM signal, which I think is the point you're raising. Because the USB and LSB offsets are symmetrical about the carrier, if you get the phase right at t=0, it'll stay right.
Putting the bottle beween your knees ant pulling the cork out with your teeth isn't 'effortlessly'.
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** The above are simply names for things that exist in the world of RF - not technical descriptions.
" Single Sideband AM " refers to an RF signal that varies in amplitude in line with the information signal - but has no carrier and is really just a frequency up-shifted version of a low frequency ( ie audio) signal.
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I love to cook with wine Sometimes I even put it in the food
| >(On the topic of whether turning a volume knob create sidebands...) | >
| snip | >Footnotes: | >
| >[1] no fair using multiple sources that are 1/4 wave apart, 1/2 | >wave apart, etc. They have to be at the same point otherwise | >we will get bogged down thinking about interference patterns, | >lobes, etc. | >
| >[2] You may have noticed that I failed to specify the exact | >frequency. You tell me: is it 1000 Hz? 1001 Hz? 1000.5 Hz? | >Other? To keep the carrier at exactly 1000 Hz despite changes | >in the modulation signal, you really need a *double* sideband | >AM signal, which would have complicated my description and made | >the basic principle less clear. I also cheated a bit by assuming | >a pure sine wave for the modulation... A question for you to | >ponder: assuming that "~" means "approximately", would our radio | >engineer be able to look at a signal with a ~1KHz amplitude | >modulation of a ~1MHz carrier and tell you whether is was SSB AM | >or DSB AM? How about the case where the modulation was a human | >voice? | >
| >[3] I still recall being amazed a a small child when my dad | >showed me using all-tube test equipment and a blackboard | >that any periodic waveform can be created out of pure sine | >waves. As a teenager, when I learned about the Fourier | >series and the various Fourier-related transforms, it was the | >first time it really hit me that math wasn't just something | >teachers use to torture students but was instead a way to | >gain a deeper understanding of exactly how the world really | >works. | I got a slightly diffement theery |
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Since we\'re talking about spectral lines which _are_ mathematically pure
sine waves, you\'re cheating when you state that:
"If we choose the carrier phase such that its trough lines up with the
peaks of both sidebands at t\'...",
since there can be no sidebands generated if the system is linear.
JF
There are only two degrees of freedom in the problem: the time origin and the carrier phase. I assume that you'd agree that the most general expression for two unit-amplitude sine waves adding together is
Then (assuming the frequencies are not mathematically pure and not phase locked so as to prevent this) if we wait long enough, then regardless of phi_1 and phi_2, we will eventually slip enough phase such that the peaks of the two cosines line up to any accuracy we like.
This may take awhile, but we'll call that time t'.
If we choose the carrier phase such that its trough lines up with the peaks of both sidebands at t', and its amplitude to be 2 (the sum of the sideband amplitudes), we will indeed have a plain vanilla 100% AM signal.
** In practice, synthesising a steady AM wave from 3 sine waves is a near impossible task because of the *exact* phase frequency and phase alignment required of the central or carrier frequency.
Having just tried it with 3 low distortion sine wave generators , I know.
I'm showing that there's no deep magic to synthesizing an AM waveform out of three sine waves. The two tones we started with wind up being the sidebands when we're done, so I take the liberty of calling them that from the beginning. Shoot me.
Don't do that - we would lose one of the few people in the newsgroup who still talk about electronics:)
Maybe this will help. Here's a LTspice file showing the summation of three frequencies, labeled Upper Sideband, Carrier, and Lower Sideband. To show the effect of alignment, the carrier phase is shifted 90 degrees in the second example.
The waveforms are as you predicted.
Here is the ASC file:
Version 4 SHEET 1 880 720 WIRE 0 144 -224 144 WIRE 416 144 0 144 WIRE -224 160 -224 144 WIRE 0 176 0 144 WIRE 416 176 416 144 WIRE -224 256 -224 240 WIRE -144 352 -224 352 WIRE 0 352 0 256 WIRE 0 352 -64 352 WIRE 48 352 0 352 WIRE 272 352 192 352 WIRE 416 352 416 256 WIRE 416 352 352 352 WIRE 464 352 416 352 WIRE -224 368 -224 352 WIRE 192 368 192 352 WIRE 0 432 0 352 WIRE 416 432 416 352 WIRE -224 464 -224 448 WIRE 192 464 192 448 WIRE 0 544 0 512 WIRE 0 544 -224 544 WIRE 416 544 416 512 WIRE 416 544 0 544 WIRE -224 560 -224 544 WIRE -224 656 -224 640 FLAG -224 256 0 FLAG -224 464 0 FLAG -224 656 0 FLAG 192 464 0 FLAG 48 352 0DegCarrier FLAG 464 352 90DegCarrier SYMBOL Voltage -224 144 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V1 SYMATTR Value SINE(0 0.5 51e3) SYMBOL res -16 272 M180 WINDOW 0 36 76 Left 0 WINDOW 3 36 40 Left 0 SYMATTR InstName R1 SYMATTR Value 50 SYMBOL Voltage -224 352 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 WINDOW 3 -85 143 Left 0 SYMATTR Value SINE(0 1 50e3 0 0 0) SYMATTR InstName V2 SYMBOL res -48 336 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R2 SYMATTR Value 50 SYMBOL Voltage -224 544 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V3 SYMATTR Value SINE(0 0.5 49e3) SYMBOL Voltage 192 352 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 WINDOW 3 -86 145 Left 0 SYMATTR Value SINE(0 1 50e3 0 0 90) SYMATTR InstName V4 SYMBOL res 368 336 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R3 SYMATTR Value 50 SYMBOL res 400 272 M180 WINDOW 0 36 76 Left 0 WINDOW 3 36 40 Left 0 SYMATTR InstName R4 SYMATTR Value 50 SYMBOL res -16 528 M180 WINDOW 0 36 76 Left 0 WINDOW 3 36 40 Left 0 SYMATTR InstName R5 SYMATTR Value 50 SYMBOL res 400 528 M180 WINDOW 0 36 76 Left 0 WINDOW 3 36 40 Left 0 SYMATTR InstName R6 SYMATTR Value 50 TEXT -8 56 Left 0 !.tran 0 2e-3 0 1e-6 TEXT -8 80 Left 0 !.options plotwinsize=0 TEXT -8 32 Left 0 ;'AM Modulation Showing Carrier Phase TEXT -184 200 Left 0 ;Upper Sideband TEXT -184 600 Left 0 ;Lower Sideband
It is a *mathematical* identity that the sum of two sine waves can also be expressed as the product form although it would look nicer specified as 999Hz and 1001Hz added to give round numbers for CW and modulation.
sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2)
In this case you could interpret this linear mixture as the product
sin(1000t)cos(t) Where sin(1000t) would be the carrier frequency and cos(t) the amplitude modulation. The snag is here that a volume control cannot do negatives. But that isn't a fundamental barrier to creating new frequencies.
Now a volume control can only generate |cos(t)| or max(0, cos(t)) since it can only provide half the positive half cycle 0 < f(t) < 1, but this doesn't prevent it from generating sidebands it just puts more power into them. And it is much harder to analyse. Sharp discontinuities in the time domain always hurt in the frequency domain. You can haggle about naming the modulation type if you like but it doesn't alter the facts.
Taking a pure 1000Hz sine wave and modulating its amplitude with the volume control will generate new frequencies from the input.
The simplest case that can be realistically done within the limitations of the passive volume control hardware is modulation by (1
Next simplest to analyse is a linear ramp or even more brutally a square wave modulation (a la Dalek voice style sidebands). The latter with harmonics coming from periodic step changes in amplitude is related to the zipper noise that started the original thread.
Most likely due to finite windowing and/or inadequate anti aliassing of the data prior to doing the FFT. It is something of a black art to get the optimum DFT of real world sampled data in practical cases with an FFT where boundary discontinuities can be important.
Chances are your extra false sideband artifacts come from the implicit edge discontinuity when the start of the record is implicitly wrapped to the the end (which is a feature of the standard FFT). If it is from that source you should be able to make them vanish by choosing frequencies and a record length so that both datasets start and end on an exact multiple of cycles. It should be noticably better if it starts and ends on any zero (and that is easier to do).
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Well, I won\'t shoot you, but I will remind you that unless they\'re
created by mixing they\'re not properly called "sidebands" and neither
is the signal midway between them, spectrally, a "carrier".
JF
If all I gave you was a black box, how would you know how they were generated? Would you be extra careful and call them "sidebands, assuming this is a True AM process and not some vile counterfeit using phaselocked sine generators"?
And what is the _spectral_ distinction between a true carrier and that peak in the middle, since the instantaneous signal voltages are identical at all times? The definition of the spectrum is an infinite time integral of the Fourier kernel times the signal voltage, so if the voltages are always the same, the spectra are also identical. Spectral as in ghostly?
Yes, as you pointed out earlier, holding the correct phase relationship is vital. If the strong tone isn't at exactly (f_L+f_U)/2, the phase error will build up and mess up the waveform completely. If you were to phase lock it to half the sum frequency and adjust the phase and amplitude exactly, the nice AM waveform would be stable. Three DDSes would work too, provided that all the frequency spacings were exactly represented in the DDS word length.
This isn't the easiest way to make an AM signal, but the waveform you wind up with is the same.
Pity the way the UCC went down the tubes, even after their lunatic fringe split off to form the MCC. They were about 10 years ahead of the Episcopals--who have now caught up, unfortunately, which is why I'm in this Ugandan church plant. (It's also great fun.)
The grape juice thing is originally an American peculiarity--in some circles it has just been an understood thing that if you 'got relgion' you had to become a teetotaller. Originally the Baptists and Methodists and Puritans drank wine with dinner like sensible people everywhere.
Of course, Protestants in general don't have a sacramental Eucharistic theology, so messing around with the elements of the Sacrament like that doesn't worry them. I'm more Catholic than that, though not Roman.
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