The Hilbert Transform and SSB Modulation

(Below are some brief notes to myself on the Hilbert Transform's use for the "phasing" version of SSB modulation.)

The "Phasing" method of SSB generation was a popular way of generating SSB in the early days of Amateur Radio SSB operation.

Some transmitters, such as the Heathkit TX-1, were designed to utilize outboard phasing accessories, such as the Heathkit SB-10, shown below.  (The TX-1 and SB-10 were my first SSB station while in high school).

Other early SSB transmitters had their phasing networks built in, such as the Hallicrafters HT-37 and the Central Electronics CE-100V.

The phasing method requires that the audio frequencies in the voice signal be shifted by 90 degrees.  In those early transmitters, this shift was accomplished with an analog phase-shift network.  Typically its setup would involve a nulling process using several knobs.

Now, with digital signal processing, the requisite 90 degree shift of the audio signal can be accomplished much more accurately and without tuning using a Hilbert Transform.

Below are two visual representations of the math underlying SSB generation via the Hilbert Transform in terms of sines and cosines.  For visualization I find it useful to express sines and cosines in their complex-exponential form.  E.g:

cos(2πf_ot) = (e^j2πf_ot + e^−j2πf_ot)/2

and

sin(2πf_ot) = j*(e^-j2πf_ot - e^j2πf_ot)/2

Note that j =  e^jπ/2.   When multiplying (e^-j2πf_ot - e^j2πf_ot)/2 by j, the "π/2" term in j's complex-exponential representation results a +90 degree rotation of each of the two exponentials in (e^-j2πf_ot - e^j2πf_ot)/2.  The result is that the negative-frequency exponential (e^-j2πf_ot) is rotated by +90 degrees, and the positive-frequency exponential (- e^j2πf_ot) has a total rotation of 270 degrees: 90 degrees due to the multiplication with j, and an additional 180 degrees due to the minus sign in front of it.

Or, in other words, the positive-frequency exponential is rotated by -90 degrees rather than +90 degrees.

Here's a visual representation of LSB generation:

(Click on image to enlarge)

And second, USB generation:

(Click on image to enlarge)

If the input audio were a sawtooth waveform, the image below shows the signals at various stages of the modulation process.

(Click on image to enlarge)

Note, in the image above, that the resulting SSB signal can have amplitudes larger than the peak values of its input audio (due to the Hilbert Transform's phase shifting).

This difference in input versus output signal magnitudes can be seen more easily in the image, below.  The sawtooth input has had its peak magnitude defined to be 1.0.  The peak value of the modulator's output, however, is 1.8.

A significant difference!

(Click on image to enlarge)

Therefore, because the level of the modulator's output IQ signal level can differ dramatically from the input audio's signal level, feed-forward transmitter gain control is ideally accomplished using the magnitude of the modulator's output IQ signal, rather than the magnitude of the input audio signal.

This conclusion is also true for the Weaver method of SSB generation.

Resources:

http://flylib.com/books/en/2.729.1/hilbert_transform_definition.html

http://k6jca.blogspot.com/2017/02/sdr-notes-weaver-modulation-and.html


Standard Caveat:

I might have made a mistake in my designs, equations, schematics, models, etc.  If anything looks confusing or wrong to you, please feel free to comment below or send me an email.

Also, I will note:

This information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.