SDR Notes: Weaver Modulation and Demodulation

There are two techniques commonly used for modulating and demodulating SSB signals:  filtering out the unwanted sideband with a filter, and cancelling the unwanted sideband by using a phasing technique that includes shifting the audio frequencies by 90 degrees.

In 1956 Donald Weaver published a third method in the Proceedings of the IRE ("A Third Method of Generating and Detecting Single Side Band Signals", Dec. 1956).

Although this technique has now been around for more than 50 years, it was never widely utilized in commercial SSB products.

This lack of adoption by commercial manufacturers might be, in part, due to the requirements placed upon the analog design and implementation of a Weaver Modulator/Demodulator -- the low-pass filters should be matched, as any phase errors between the two will result in appearance of the opposite, unwanted sideband along with the desired sideband.  And imbalance or DC offset can cause the low frequency oscillator to "bleed through" into the audio passband.

Given these requirements, it is not too surprising that manufacturers preferred, for example, the straight-forward technique of simply filtering out the unwanted sideband with a crystal or mechanical filter at the radio's IF frequency.

But what might be difficult to implement in the analog domain becomes trivial in the digital domain -- filters are exactly matched, DC offsets can be eliminated, and thus Weaver modulation and demodulation become an excellent choice for implementation in a digital transceiver.

Now let's take a closer look at Weaver's technique...


Weaver Modulation:

Here's is the block diagram for Weaver's Modulator, as described in U.S. Patent 2,928,055:

And here is its implementation:

Three comments:

1.  The block diagram shows sine and cosine signals driving the balanced modulators.  Note that these signals are implemented in the schematic with two oscillators:  one at the AF frequency and one at the RF frequency, and these oscillators are each shifted + or - 45 degrees (depending upon which "branch" (upper or lower) of the modulator they drive) to achieve the requisite 90 degree phase shift between upper and lower branches.

2. Assuming the phase of the output transformers sum the signals from the two branches rather than subtract, the modulator above generates USB.  To generate LSB, either the phase of the audio into the "sine" branch must be inverted, or the RF output from the "sine" branch must be inverted.

3.  With respect to I and Q, the branch mixed with the cosine signals would be the I (In-phase) channel, while the branch mixed with the sine signals would be the Q (Quadrature) channel.


Weaver Modulator Theory:

Let's look at how the operation of a Weaver Modulator is typically described.  From a 73 Magazine article (Feb. 1977), here is a block diagram very similar to Weaver's original:

And here is its associated spectrum diagram showing the step-by-step conversion from Audio to USB:

Note that the outputs of Balanced Modulators A2 and B2 each contain two versions of the shifted audio spectrum.  The spectrum of Modulator A2's output has two "positive" spectrums, one whose frequencies are reversed from the other.  The spectrum of Modulator B2's output also has two spectrums, one with reversed frequencies from the other, but in addition to the reversed frequencies one also has reversed amplitudes ("upside-down").

If these outputs are then added, spectrums that have a common frequency orientation and identical amplitudes will add, while spectrums with a common frequency orientation but reversed amplitudes will cancel (sum to zero), thus cancelling one of the sidebands.

I was curious how the Q channel (i.e. modulator B2 in the block diagram, above) inverted the spectrum amplitude, so I thought I'd look into circuit operation a bit more deeply.  As a visual representation I came up with this diagram:

This diagram represents a USB modulator.  Note that every time a signal is multiplied by a sine, its phase shifts by 90 degrees (I represent this as a 90 degree rotation), and this  phase shift is key to understanding how the "upside-down" spectrum is created.

In other words, if one were to think of Weaver Modulator operation in terms of sines and cosines, where the Audio input is a cosine signal, then, the first Q mixer's output would consist solely of sines:

The second Q mixer would convert these sines back to cosines:

(Equations from here)

But note!  Now there is a negative sign in front of the final cosine term.  If you were to write the equations of signal transformation for both I and Q branches, from audio to RF, and then sum them (per the block diagram), it is this negative sign result in the Q channel that causes the cancellation of one sideband.

(Personally, I prefer my visual representation over the equations, as the latter, as one expands them, can quickly become cumbersome.)

Here's a similar diagram for LSB generation:

Note that the LSB generator inverts the Audio signal in the Q path, prior to the first Q-channel mixer.  To create LSB, this inversion can either be at the beginning of the Q-channel path (prior to the first Q-channel mixer, as shown above), or at the end of the Q-channel path (after the second Q-channel mixer, but prior to the final adder).

The diagrams above, representing how a Weaver Modulator might be implemented in an FPGA, can look daunting.  For a different approach, let's look at Weaver Modulation from the perspective of "Complex" signals, which allows us to examine Weaver Modulation without the complication of spectrums folding back upon themselves:

USB Generation:

And LSB Generation:

Notes:

1.  The signs of the oscillator frequencies for the two final multipliers have been swapped (negative for positive, positive for negative) in the LSB version, compared to the USB version.

2.  The Complex Conjugate of a complex exponential function simply changes the sign of that function's frequency, from positive to negative, or from negative to positive, as shown below:

(For a very useful tutorial on Complex signals, go here).

Forward to demodulation!


Weaver Demodulation:

(This will be a very short section.)

From Weaver's Patent:

Looks very similar to the modulator, doesn't it?

Because there are no phase inversions in the diagram above of  RF prior to the first modulators or Audio following the final modulators, then this demodulator would demodulate a USB signal.

To demodulate an LSB signal, instead of adding top ("I") and bottom (Q") branches to create audio, you would instead subtract the bottom branch (Q) from the top branch (I).

As you can see, demodulation is essentially just the reverse of the modulation process.  And with that, I now end my discussion!


Resources:

"A Third Method of Generation and Detection of Single-Sideband Signals," Weaver, Proceedings of the IRE, Dec., 1956.  The original article!
Weaver's Patent 2928055
A 9 MHz Digital SSB Modulator, IV3NWV
AN1981, Philips.  Contains description of Weaver Modulator and Demodulator.
SSB Demodulation, Pandora SDR.  A nice visual on Weaver Demodulation.
"The Third Method of S.S.B.", Wright, W1PNB, QST, Sept., 1957
"SSB:  The Third Method," Wilson, WB0JXY/0, 73 Magazine, Feb., 1977

Quadrature Signals:  Complex, but not Complicated, Richard Lyons


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 design and any associated 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.