S5-1.5 Digital Modulation Techniques

What this is and why it exists

The bits now have to cross a real channel. This topic asks two questions about that crossing. What is the best receiver you could build, and how often will it be wrong?

The answer to the second question ranks the schemes, and that ranking is what the topic exists for.

The vocabulary

• Signal space — a picture in which each transmitted waveform is a point. Distance between points shows how likely two signals are to be confused.
• Optimum receiver — the receiver that picks the transmitted signal nearest to what it received. It uses a matched filter or a correlator.
• Energy per bit — the signal energy spent on each bit. It is the fair basis for comparing schemes.
• Bit error rate — the probability that a received bit is decided wrongly.
• ASK — amplitude shift keying. The carrier is switched on and off.
• PSK — phase shift keying. The carrier phase flips by half a cycle between the two symbols.
• DPSK — differential PSK. The change in phase carries the bit, not the phase itself.
• FSK — frequency shift keying. Two different carrier frequencies stand for the two bits.
• Coherent detection — detection using a locally generated carrier locked in phase.
• M-ary signalling — sending more than one bit per symbol by using more than two waveforms.
• QPSK — quadrature PSK. Four phases, and two bits per symbol.

The mental model

Draw the signals as points rather than as waveforms. Noise pushes the received point away from the transmitted one. The optimum receiver decides for the nearest candidate point. So the error rate depends on one thing. It depends on how far apart the candidate points are, compared with the size of the noise. Once you see the topic that way, the formulas stop needing memorising.

Now place the three binary schemes on that picture, at equal average energy per bit.

PSK puts its two points on opposite sides of the origin. They are as far apart as the energy allows. ASK puts one point at the origin and one away from it, so half the symbols carry no energy at all. FSK puts its two points at right angles, because the two tones are orthogonal.

Work out the distances and the ranking falls out. The two PSK points are separated by twice the square root of the bit energy. The two orthogonal FSK points are separated by the square root of twice that energy. Square both and PSK has twice the squared distance of FSK. Twice is about three decibels. So coherent PSK reaches a given error rate with about three decibels less energy per bit than coherent FSK. ASK compares with FSK rather than with PSK.

DPSK is the engineering bargain of the topic. Coherent PSK needs a carrier recovered in the right phase, which costs hardware and can slip. DPSK carries the bit in the change of phase between consecutive symbols, so the previous symbol serves as the reference. The receiver needs no coherent carrier. It pays a small penalty in error rate for a large simplification. You will meet that shape of trade repeatedly.

FSK has a second property worth noting. Its envelope is constant, so it survives amplitude disturbances and nonlinear amplifiers better than ASK does.

M-ary signalling is the doorway out of the binary world. QPSK uses four phases instead of two, so each symbol carries two bits. It occupies the same bandwidth as binary PSK while carrying twice the data. Read it as two independent binary PSK streams. One rides a carrier and the other rides a carrier a quarter cycle later. Once the constellation reads that way, larger constellations are a small step.

Line coding sits underneath all of this. Before any carrier is involved, bits must become voltages. That choice sets the spectrum and decides whether a receiver can recover its timing at all. It is overlooked often and expensive to get wrong.

What you should now be able to explain or do

Describe the optimum receiver as a nearest-point decision, and name the matched filter and correlator forms. Place ASK, PSK and FSK in signal space and derive the ordering of their error rates. Say why coherent PSK saves about three decibels over coherent FSK. Explain what DPSK gives up and what it gains. Read QPSK as two binary streams and say why it doubles the rate without doubling the bandwidth.

Check yourself