## Sunday, December 20, 2020

### A Ham Radio Series 20 – Modulation, Constellations and Bandwidth, Oh My!

The previous series on radio that I put together, back in the winter and spring of ‘19, included a very wide-ranging post on modulation.  It’s a worthwhile refresher to go back to read to get a feel for the whole process of modulation, if some of this seems like it comes out of the blue.  Nevertheless, there were some important concepts that I dropped in the effort to keep the post length a bit more reasonable.

The most important part of the posting is the idea of a “universal modulator.”  A universal modulator is nothing more than a multiplier circuit that takes an RF input to be modulated splits the signal in two paths which then go to two mixers.  One side of the split is shifted in phase by 90 degrees.  The modulating signals, shown here as i and q (both as time-varying functions), are applied to the low frequency ports of those mixers.

Let’s start with this concept.  Any characteristic of the radio signal can be modulated.  Amplitude can be modulated (AM or SSB); the simplest Amplitude Modulation is presence or absence of the transmit signal, On-Off Keying.  Frequency can be modulated (FM, FSK).  Phase can be modulated (analog PM, BPSK, QPSK, 8PSK, 16PSK).  There are even types of modulation that combine more than one kind; the most common of those being Quadrature AM, like 64 or 128QAM (pronounced “kw-ahm”).  Quadrature AM modulates both the amplitude and phase of the signal to transfer data.

The only difference between generating AM, or SSB or 64 QAM is the way the i(t) and q(t) signals are created and processed.  If they’re kept identical in phase and a DC offset added, full-carrier AM is generated.  If they’re left in quadrature, single sideband is produced.  If Q is left constant and I varies 180 degrees for modulation, BPSK is generated.  Everything is produced by the way those i and q signals are created.

Because of the concept of modulation by applying I and Q signals, Q can be plotted versus I, producing what’s referred to as constellation.  BPSK, or Bi-Phase Shift Keying is simply shifting the phase of the carrier from no shift to 180 degree shift for each symbol.  The signal shifts from +1 to -1 constantly to transfer the information.  QPSK is Quadri-Phase Shift Keying, shown here as varying I and Q at the same time, giving constellation values at (1,1), (-1,1), (-1,-1) and (1,-1), or simply changing one axis at a time (1,0), (0,1), (-1,0), (0,-1).  8PSK and 16PSK are what the names imply: 8 and 16 values of amplitude and phase.

This isn’t the only interpretation.  16QAM is modulating both amplitude and phase.  There is 16 phase shift keying (16PSK), where the amplitude remains constant and the constellation points are all on a circle of constant amplitude but with (360 / 16 or) 22.5 degrees between the symbols.

This is 64QAM as it would be seen on a piece of test equipment.  Each of those dots represents a transmission and their random looking appearance (spread out and not just a dot) is from the introduction of noise in the channel.  If it were pure phase noise each of those would be short arc centered on the origin (0,0) in the graph with the points farthest into the corners being the longest arcs.  These look to be amplitude noise and not phase noise.

There are subtle changes to the constellations that affect not just the number of constellation points but also the trajectories that the signal takes to get between them.  Below is a math simulation of a differential 8-phase shift keying (D8PSK), which has become widely adopted in the aviation world.  In one application, data at 31.5 KBPS is used in a 16 kHz wide AM channel previously used for voice, which has a 6 kHz bandwidth.

As I’ve been saying in almost every post, physics doesn’t allow something for nothing.  Perhaps in your wanderings around the ‘net you’ve come across the concept that information is proportional to bandwidth.  Simply, the more information you’re sending in the same amount of time, the more bandwidth the signal takes.  I bet you know this even if you’ve never seen it stated online.  Voice channels take up more bandwidth than on-off keying; video takes up more bandwidth than voice.  Lower data rates take less bandwidth than higher data rates.

The trade between the modulation types is that they convey more information in the same bandwidth, but require higher amplitudes for the same bit error rate.  That means more signal power at the receiver, so transmit powers may have to be higher and marginal signal strength might not work.  The complexity (read that as cost) of the transmitter and receiver go up.  For that cost, the D8PSK signal provides more digital data in the transmit bandwidth than plain AM voice.

These curves are generically referred to as “waterfall curves.”  It’s not quite as vivid in this graph but for some of these modes, as the signal to noise ratio (horizontal axis) goes up, the bit error rate (BER) falls off like going over a waterfall.  The signal to noise ratio is expressed as Eb/N0energy per bit to noise power spectral density ratio”  It’s calculated from the signal to noise ratio (SNR).

This is read by choosing an error rate you’re comfortable with; I’ll choose 10-4 because all three curves are on the chart.  That means 1 bit in 10,000 is wrong.  For BPSK, you’ll see where the blue curve intersects that line, go down to the X axis and see it’s a little over 8 dB; I’ll call it 8.5.  For the 16-PSK curve, you see it’s just over 16 dB.  You get a higher data rate but it costs you 7.5 dB more in the hardware complexity.  Adding 7-1/2 dB transmit power is expensive in the ham world, depending on the frequency and other requirements.  On the other hand, if the receiver is reasonably well-designed, it's physically impossible to make one 7-1/2 dB more sensitive.