My maths is probably worse than Jim's, but: it starts ok, but if you do go further with the "mishmash" it just reverts to sin(ω
c)sin(ω
a) plain and simple.
I agree that in practice you may get all sorts of other bits and pieces, IMO because the multipliers are not perfect and the original carrier may not be pure. So you do filtering as you say. But the maths says you could get a pair of perfect sidebands.
I agree totally with the voila, 2 sidebands. But this is DSB. Notice, there is no carrier present, just the two sidebands. No need to suppress nor eliminate a carrier. It simply is not there,
IF you can accurately multiply the carrier by the modulation.
For SSB there is also a similar expression which generates a single sideband, without carrier nor other sideband. It too relies on accurate multiplication, but also on being able to generate quadrature signals for both carrier and modulation, which was very difficult before DSP.
(In case anyone is interested, LSB = cos(ω
c)cos(ω
a) + sin(ω
c)sin(ω
a)
and USB = cos(ω
c)cos(ω
a) - sin(ω
c)sin(ω
a) )
The expression for AM is (carrier) x (1 + modulation) or sin(ω
c )(1+sin(ω
a))
This is easy for old guys to remember, if you think of the old amplitude modulation class A circuit. It was done by modulating the HT line to the PA. Since this can't go negative, you add the modulation to the standing HT, so that it swings (up to) from near zero to near double.
With no modulation, you get full carrier;
With modulation amplitude equal to the HT, you get 100% modulation with full carrier and two sidebands, carrying between them power equal to that of the carrier;
With greater modulation the valve cuts off during modulation peaks and you get splatter (mishmash?) and become very unpopular;
With modulation amplitude less than the HT, you get undermodulation, with full carrier and two sidebands carrying less power than the carrier. This is what is happening most of the time, because only peaks can be allowed to reach 100% modulation
A bit long winded, but
OP seems to know about this signal. What you notice here is that in every case you are putting at least half your power into the carrier and this carrier is
exactly the same whatever the modulation, if any. So it tells the receiver nothing by itself. The information about the modulation is all in the sidebands. But, as OP says, you can easily see the modulation in the envelope of the combined signal. And it is ridiculously easy to get the modulation back, just using a diode (or crystal and cats whisker.)
So one motivation for DSB is to not waste power on transmitting carrier. If you can transmit 100W, instead of sending 50W of carrier and 50W of sidebands, send just the sidebands at the full 100W, then let the receiver generate a carrier signal and add it to the sidebands it receives. That will regenerate that simple AM signal in the receiver, but at twice the power.
Doing this (in the receiver) was not trivial, which is one reason why DSB was not that common. You can work out what the carrier frequency is, because it is midway between the sidebands. And you can manually adjust your local oscillator to sit at the right frequency. But it is much more critical than tuning a simple AM signal.
Since the two sidebands are like mirror images of each other, which both contain exactly the same information just arranged in a different way (if the whiz kids will allow that fuzzy desciption), you only need one of them, hence Single Side Band.
One is sin(ω
c +ω
m) the other sin(ω
c -ω
m)
Again they don't look nor sound like the original modulation, but if you reinject the carrier sinewave at the receiver, then you get the modulation back.
Eg. Take the USB sin(ω
c +ω
m) and multiply by sin(ω
c)
you get 1/2[cos(ω
m) - cos(2ω
c + ω
m) ] which looks messy, until you realize the left cos is the modulation and the right cos is RF at nearly twice the carrier frequency, so can be filtered out to leave the modulation.
Again you need to know what the carrier frequency is and it is a bit more difficult than with DSB. If the receiver knows what frequency is used at the transmitter it has to set it's oscillator to that frequency (and keep it there, used to be the difficult part!) One (common) technique was simply to have a matching crystal in both transmitter and receiver. Even without knowing the exact frequency, as with DSB, a skilled listener can tune the oscillator until the signal becomes intelligible when you get close enough to the original carrier.
It also says one of the main fall backs of SSB and DSB is that on the receiving end it is hard to remove the carrier signal.
I think you mean "drawbacks" and "on the receiving end it is hard to
reinstate the carrier".
Yes. As said above, you need the carrier to make sense of an SSB or DSB signal. Recreating that was the problem. Crystal controlled oscillators was one solution. Phase locked loops was another. Reduced carrier, rather than suppressed carrier was a halfway house.
Removing the carrier at the transmitter was also a problem, mainly met by crystal filters and balanced mixers until DSP came along.
Typically in textbooks they always show things with sin waves, but if USB and LSB are different frequencies then how is the modulated signal shown with just a single frequency? (aside from showing the signals before they are joined)
The modulated signal is generally not a single frequency (could be for SSB), and I don't think anyone would show a modulated signal as a simple sinewave.
First using sinewaves in books, makes the maths easy to follow. They can write the formulae for any modulating signal - it just gets messier.
Secondly, about 200yrs ago Fourier showed that a more complex signal is mathematically identical to a collection of sinewaves. You can work out what happens to any signal waveform, by working out what happens to each of those sinewaves and adding the results together.
USB & LSB, are
bands of frequencies.
A single sinewave modulation produces an upper side frequency and a lower side frequency. Eg. 300Hz audio modulated onto a 1MHz carrier, gives 1.0003 MHz and 0.9997 MHz, spot frequencies. If you know that the carrier was 1MHz and you get either of these frequencies, you can work out what the audio frequency was. Try this: if you receive 1.0005 MHz, what was the audio frequency?
When a radio station broadcasts a pop record, they want to send a complicated waveform which is a mixture of frequencies (at varying levels). So for each frequency you get an upper and lower side frequency. Together these make the two bands of frequencies, called sidebands. Converting these side bands of frequencies back to their original sinewaves and adding them together gives us the original complicated waveform.
In the eg. above, if you had a sound containing a 300Hz tone and a 500Hz tone of twice the amplitude, it wouldn't look like a simple sinewave, but you'd get 1.0003MHz with one amplitude and 1.0005MHz with twice the amplitude. (and for DSB or AM, 0.9997 and 0.9995MHz) This modulated signal would not look like (nor be) a simple sinewave. When you added back the carrier frequency and recovered the 300Hz and 500Hz , you would get twice as much 500Hz as 300. When you added them together, they would again make the same waveform as the original mixture.
Dave
