FOR RADIO AMATEURS - NO INTENTION TO TEACH MATHEMATICS
The modern amateur radio transceiver keyed to produce CW.
The key produces the keying signal, typically a voltage that is +5V for key up and zero for key down. The keying voltage is mixed with a tone oscillator at something like 700Hz in a double balanced mixer or some equivalent circuit producing an audio CW signal, a 700Hz tone that is present when the key is down and absent when the key is up.
Fig. 1. The CW "modulator". To generate a morse coded audio tone or a morse coded radio frequency signal one uses the same fundamental circuitry, a double balanced mixer (Or some equivalent circuit)
The audio CW signal is then mixed with a local oscillator at for example 10.7MHz, the one we traditionally regard as the carrier, producing a radio frequency signal.
The radio frequency signal contains two frequencies, the sum frequency and the difference frequency at 10.6993 and 10.7007MHz respectively. These two signals are the lower and the upper sidebands.
By means of a filter one of the sidebands is removed leaving a single morse coded radio frequency signal.
The RF telegraphy signal at i.e. 10.6993 is classified as a J2A signal with carrier frequency 10.700MHz. That is because it is generated with a SSB transmitter that would be capable of transmitting a voice signal!
The J2A signal can in no way be differentiated from a signal that is produced by the schematic of fig.1 using the frequency 10.6993 for the local oscillator. Produced directly, the signal would be classified as A1A with a carrier frequency of 10.6993MHz.
It is a good idea to forget "carrier" and "modulation". Thinking about signals in general terms is more fruitful and will make it easier to see the equivalence between digital signal processing and the corresponding analog radio receiver or transmitter.
Describing the CW signal
The CW signal discussed above really has a carrier in the sense that its spectrum is symmetric with the strongest component at the center, at 10.6993MHz.It is however perfectly valid to describe it as a SSB signal with a carrier frequency of 10.700MHz or any other arbitrary frequency that one would like to choose
As it turns out one needs complex numbers to describe the CW signal in case any other frequency than 10.6993MHz is regarded as the carrier frequency. Rather than carriers one could think about reference signals and baseband signals. The baseband signal has two components, I (in phase) and Q (quadrature, 90 degrees out of phase)
The reference signal is a sine wave and a cosine wave of fixed frequency, a complex pair.
In case we select 10.6993 as the reference frequency, I is the filtered keying signal while Q is zero. (or vice versa or some combination)
In case 10.700MHz is selected as the reference frequency, both I and Q are the morse coded audio signal at 700Hz. I and Q differ in that the phase of the 700Hz signal differs by exactly 90 degrees.
The reference frequency is the frequency "from which we observe the signal". To change the reference frequency for the oscilloscope display of a signal, a complex mixer is needed. The entire complex pair I and Q of the signal at the old reference frequency is mixed with the sine and cosine of the new reference frequency to produce a new complex pair I and Q. That is all. (no filters) This is the phasing method for SSB generation.
In the frequency domain (on a spectrum analyser display) changing the reference frequency is the same as moving the spectrum along the frequency axis (x-axis).
Changing the reference frequency, transposing the frequency or converting the frequency are different names for the same thing. It is a linear process that changes the frequency of a signal.
The reference frequency, our observation point is not some fictive mathematical concept. It is a physical reality and understanding receivers and transmitters this way may aviod some common misunderstandings.