The phasing method

In radio communications, one way to generate single sideband (SSB) signals is to make a double sideband signal by feeding a carrier and a modulation signal into a balanced modulator to create a double sideband (DSB) signal and then filter out one of the two resulting sidebands.

If you filter out the lower sideband, you’re left with the upper sideband and if you filter out the upper sideband, you’re left with the lower sideband. However, another way to generate SSB without that filtering has been called “the phasing method.”

Let’s look at that in the following sketch in Figure 1.

Figure 1 Phasing method of generating an SSB signal where the outputs of Fc and Fm are 90° apart with respect to each other

The outputs of the carrier (Fc) quadrature phase shifter and the modulating signal (Fm) quadrature phase shifter need only be 90° apart with respect to each other. The phase relationships to their respective inputs are irrelevant.

Four cases of SSB generation

In the following equations, those two unimportant phase shifts are called “phi” and “chi” for no particular reason other than their pronunciations happen to rhyme. Mathematically, we examine four cases of SSB generation.

Case 1, where “Fc at 90°” and “Fm at 90°” are both +90°, or in the same directions (Figure 2). Case 2, where “Fc at 90°” and “Fm at 90°” are both -90°, or in the same directions (Figure 3).

Figure 2 Mathematically solving for upper and lower side bands where “Fc at 90°” and “Fm at 90°” are both +90°, or in the same directions.

Figure 3 Mathematically solving for upper and lower side bands where “Fc at 90°” and “Fm at 90°” are both -90°, or in the same directions.

Case 3, where “Fc at 90°” is -90°and “Fm at 90°” is +90°, or in the opposite directions (Figure 4). Case 4, where “Fc at 90°” is +90°and “Fm at 90°” is -90°, or in the opposite directions (Figure 5).

Figure 4 Mathematically solving for upper and lower side bands where “Fc at 90°” is -90°and “Fm at 90°” is +90°, or in the opposite directions

Figure 5 Mathematically solving for upper and lower side bands where “Fc at 90°” is +90°and “Fm at 90°” is -90°, or in the opposite directions.

The quadrature phase shifter for the carrier signal only needs to operate at one frequency, which is that of the carrier itself and which we have called “Fc”. The quadrature phase shifter for the modulating signal however has to operate over a range of frequencies. That device has to develop 90° phase shifts for all the frequency components of that modulating signal and therein lies a challenge.

90° phase shifts for all frequency components

There is a mathematical operator called the Hilbert transform which is described here. There, we find an illustration of the Hilbert transformation of a square wave. From that page, we present the sketch in Figure 6.

Figure 6 A square wave and its Hilbert transform, bringing about a 90° phase shift of each frequency component of the input signal in its own time base.

The underlying mathematics of the Hilbert transform is described in terms of a convolution integral but in another sense, you can look at the result as bringing about a 90° phase shift of each frequency component of the input signal in its own time base, in the above case, of a square wave. This phase shift property is the very thing we want for our modulating signal in SSB generation.

In the case of Figure 7, I took each frequency component of a square wave—by which I mean the fundamental frequency plus a large number of properly scaled odd harmonics—and phase shifted each of them by 90° in their respective time frames. I then added up those phase-shifted terms.

Figure 7 A square wave and the result of 90° phase shifts of each harmonic component in that square wave.

Please compare Figure 6 to the result in Figure 5. They look very much the same. The finite number of 90° phase shift and summing steps very nicely approximate the Hilbert transform.

The ideal case for SSB generation can be expressed as starting with a carrier signal, you create a second carrier signal at the same frequency as the first, but phase shifted by 90°. Putting this another way, the first carrier signal and the second carrier signal are in quadrature with respect to one another.

You then take your modulating signal and generate its Hilbert transform. You now have two modulating signals in which each frequency component of the one is in quadrature with the corresponding frequency component of the other.

Using two balanced modulators, you apply one carrier and one modulating signal to one balanced modulator and apply the other carrier and the other modulating signal to the other balanced modulator. The outputs of the two balanced modulators are then either added to each other or subtracted from each other. Based on the four mathematical examples above, you end up with either an upper sideband SSB signal or a lower sideband SSB signal.

This offers high performance and thus the costly filters described in the first paragraph above are not needed.

Practically applying a Hilbert transform

As a practical matter however, instead of actually making a true Hilbert transformer (I have no idea how or even if that could be done.), we can make a variety of different circuits which will give us the 90° phase shifts we need for our modulating signals over some range of operating frequencies with each frequency component 90° shifted in its own time frame.

One of the earliest purchasable devices for doing this over the range of speech frequencies was a resistor-capacitor network called the 2Q4 which was made by a company called Barker and Williamson. The 2Q4 came in a metal can with a vacuum-tube-like octal base. Its dimensions were very close to that of a 6J5 vacuum tube, but the can of the 2Q4 was painted grey instead of black. (Yes, I know that I’m getting old.)

Another approach to obtaining the needed 90° phase relationships of the modulating signals is by using cascaded sets of all-pass filters. That technique is described in “All-pass filter phase shifters.”

One thing to note is that the Hilbert transformation itself and our approximation of it can lead to some really spiky signals. The spikiness we see for the square wave arises for speech waveforms too. This fact has an important practical implication.

SSB transmitters tend to have high peak output powers versus their average output power levels. This is why in amateur radio, while there is an FCC-imposed operating power limit of 1000 watts, the limit for SSB transmission is 2000 watts peak power.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

Related Content

The post Single sideband generation appeared first on EDN.

Source link