The Shocking Truth!
There is no hand waving involved here and while you can say it is in the mathematics, that is absolutely TRUE! When you impress an Audio Signal upon an RF Carrier Signal in a Balanced Modulator, two Sidebands appear in the output. If your Balanced Modulator is of a good design, then the carrier is suppressed (not entirely eliminated) in many cases by as much as 50 dB (the evidence of having a good design) . A carrier at 50 dB down is like not there.
So lets start 1st by looking at 50 dB down. If you have an RF output of 1 milliwatt (0.6325 Volts PTP across 50 Ohms) a 50 dB stronger signal as referenced to 1 milliwatt (dBm) would have to be 100,000 times larger. 10*log(100K) = 50 dBm. So let us just say the carrier is way down there.
But all and I mean ALL Balanced Modulators (BM) (whether 4X1N914's or the NE602 or like me with the MC1496) produce a Double Sideband Output. It is only when you then pass that sideband signal through a Narrow Filter with fairly tight skirts does Single Sideband results. Now typically in a Filter Type SSB Rig, the RF Carrier Frequency is related to the Filter Frequency. Thusly picking a Beat Frequency Oscillator (BFO) Frequency ABOVE the Filter Center Frequency results in Lower Sideband (LSB) and picking a BFO Frequency BELOW results in Upper Sideband (USB).
Let us take a by the numbers journey. For those who have used a 9 MHz filter in a project, the usual suspects for the BFO Frequencies are 8.998500 MHz (USB) and 9.001500 MHz (LSB). There go the emails -- Stop, I am not taking into account the case of Sideband inversion.
If I had (God Forbid) a drifty 5 MHz Analog VFO along with my 9 MHz Filter and wanted a USB signal on 20 Meters then the 8.998500 MHz BFO would be used. But if I had a slick Si5351 in play and had the LO set to 23 MHz, then the incoming signal is subtracted from the LO and sideband inversion results so we now must use the 9.001500 MHz BFO for USB.
So let us assume we use the 8.988500 MHz BFO as the RF Carrier and in our Balanced Modulator impress a 1200 Hz tone into the Audio port. Two output signals will result one at 8.999700 MHz and the other at 8.997300 MHz. So if we had a receiver that would tune this range we would copy the signal at two places on the dial (8.9997 and 8.9973 MHz)
. That my friend is Double Sideband!
Our 9 MHz Filter has a Center Frequency of 9 MHz and lets say a BW of 3 KHz --so this filter would look Up to 9.001500 MHz and Down to 8.998500 MHz as the Pass Band. Since 8.998500 < 8.999700 < 9.001500 that signal will pass through the filter; but 8.997300 < 8.998500 < 9.001500 and so that won't pass.
Now let us look at the 9.001500 MHz Carrier and our 1200 Hz tone where out of the BM we get two signals with one at 9.002700 MHz and the other at 9.000300 MHz. Again our span is 8.998500 MHz to 9.001500 MHz. Thusly we have 8.998500 < 9.000300 < 9.001500 so this signal will pass. But we now find that 9.002700 > 9.001500 > 8.998500 where it won't pass.
Here is a notworthy observation: In the case of the 8.998500 MHz carrier only the UPPER Mixing Product will pass through the 9.0 MHz Filter and in the second case only the LOWER Mixing Product will pass through the 9.0 MHz Filter -- Is this how we got USB and LSB -- I simply don't know.
Some other fun with Math. Suppose we used our 8.998500 MHz Carrier but a 3 kHz Tone so the two sidebands would be 8.995500 and 9001500. Boom this tells us that the max audio BW is 3000 Hz to pass thrugh the filter and again it is the UPPER Mixed Frequency.
Now the other example with the 9.001500 Carrier and the 3 kHz Tone. The two mixing products are 8.998500 MHz and 9.004500. Only the 8.998500 MHz mixing prodcut will pass through the filter which is the LOWER Mixed Frequency. QED!
Back to DSB. For our DSB Transmitter the BFO is really the operating frequency so if we we are on 40 Meters and our LO read 7.200000 and we had the 1200 Hz tone we get outputs at 7201200 and one at 7.198800. So in tuning your DSB transceiever, tune the radio until you hear the proper sideband and you should be dead nuts on this frequency.
The are some real advantages to DSB as a 1st step. For the noob -- the circuits are far less complex to understand, build and align. The part count is minimal thus cost issues are addressed. With the carrier eliminated you get some power savings over AM ( In an AM signal 50% in the Carrier and 25% in each sideband ). When proeprly tuned it sounds just like another SSB signal. It is an authorized mode of operation.
The downside from the FLEX Police. It is Double Sideband! You are taking up a bit more spectrum space.