Digital modulation
Turning voice into bits is half the problem; the other half is fitting those bits onto a radio carrier without using more spectrum than necessary. Digital modulations solve exactly that trade-off between speed, bandwidth and robustness.
Sampling and its conditions are covered in sampling theorem, and analog modulations and their comparison in modulation systems. Here the chain is completed: quantize, encode and modulate.
01PCM: from sample to number
A telephone channel digitizes at 8 bits and an audio CD at 16 bits.
- Telephony: SNR = 6.02 × 8 + 1.76 = 49.9 dB. More than enough for voice.
- Audio: SNR = 6.02 × 16 + 1.76 = 98.1 dB. This is what separates a CD from a tape.
- With an input range of 2 V and 8 bits, the step is 2/256 = 7.8 mV: any difference smaller than that is lost forever.
And the detail that matters in practice: that SNR is the value at full scale. A signal that uses only a tenth of the range loses 20 dB, because the quantization noise is the same and the signal is smaller. That is why adjusting the input level is not a cosmetic detail.
Voice spends most of its time at low levels, so a quantizer with equal steps wastes resolution on the peaks and skimps on it where it is most noticeable. Companding uses small steps near zero and large ones at the extremes: with 8 bits it achieves a quality equivalent to about 12 linear bits.
There are two laws in use: A in Europe and in Argentina, and μ in North America and Japan. They are incompatible, which is why international links carry a conversion between the two.
02Digital modulations
| Modulation | What varies | Bits per symbol | Behavior |
|---|---|---|---|
| ASK | The amplitude of the carrier. | 1 | The simplest and the most sensitive to noise. Hardly ever used alone in radio. |
| FSK | The frequency, between two or more values. | 1 or more | Very robust and easy to generate. Telemetry, radio modems, remote control. |
| BPSK | The phase, between two opposite states. | 1 | The most noise-resistant of all. Low-level satellite links. |
| QPSK | The phase, between four states. | 2 | Doubles the speed with very little penalty. It is the workhorse. |
| 16-QAM | Amplitude and phase: 16 points. | 4 | A good compromise. Requires a decent signal-to-noise ratio. |
| 64-QAM | 64 points. | 6 | Digital TV and links with a good signal. |
| 256-QAM | 256 points. | 8 | Maximum speed, only on very clean and stable links. |
A link has 7 MHz of bandwidth available and works at 6 Mbaud. What data rate is obtained with each modulation?
| Modulation | Bits/symbol | Data rate | Required SNR |
|---|---|---|---|
| QPSK | 2 | 12 Mbit/s | ≈ 14 dB |
| 16-QAM | 4 | 24 Mbit/s | ≈ 21 dB |
| 64-QAM | 6 | 36 Mbit/s | ≈ 27 dB |
| 256-QAM | 8 | 48 Mbit/s | ≈ 33 dB |
The data rate is multiplied by four between QPSK and 256-QAM, but the required signal-to-noise ratio rises by 19 dB: it takes almost a hundred times more power, or a much larger antenna. That is the central trade-off of all digital radio.
Modern links do not pick one modulation and stick with it: they change it on the fly according to the instantaneous quality. In good weather they work in 256-QAM at maximum speed; when it rains and the signal-to-noise ratio drops, they step down to 64-QAM, then to 16-QAM and finally to QPSK. The link becomes slower, but it does not drop, which is what matters.
03Measuring the quality of a digital link
| Indicator | What it tells you |
|---|---|
| BER, bit error rate | How many bits arrive wrong out of the total. A data link requires better than 10⁻⁶; with error correction 10⁻¹² is reached. |
| Eb/N₀ | Energy per bit over noise density. It is the correct way to compare modulations, because it does not depend on the data rate. |
| MER | How much the received points are scattered relative to their ideal position in the constellation. It is the digital equivalent of the signal-to-noise ratio. |
| Eye diagram | Superposition of many symbol periods. An open “eye” means decision margin; closed, imminent errors. |
| Measured constellation | Visual diagnosis: round clouds indicate noise, rotation indicates phase error, deformation indicates amplifier nonlinearity. |
Redundant bits are added, calculated so that the receiver can reconstruct the damaged bits without requesting retransmission. It costs capacity —a rate-3/4 code uses a quarter of the throughput on redundancy— and in return gives several decibels of coding gain: the link behaves as if it had more power.
On a satellite link, where every decibel is very expensive, that gain is the difference between a possible link and an impossible one.
04In the lab
Digitize a voice recording with 12, 8, 6 and 4 bits and compare the result by ear and in the spectrum. Measure the signal-to-noise ratio in each case and compare it with the formula. Repeat with the signal well below full scale and check the loss.
With a microcontroller, transmit data by FSK between two audio tones and receive it with another. Measure the error rate as a function of the added noise. It is a complete radio modem, built with two boards and a speaker.
With a software-defined radio, capture a real digital signal —digital TV or a data link— and observe its constellation and its MER. Compare the cloud with the antenna well and poorly aimed.
On a baseband digital signal, trigger the oscilloscope from the symbol clock and observe the eye diagram. Add noise and limit the channel bandwidth, and see how the eye closes for each cause.
05Common mistakes
| Mistake | Consequence |
|---|---|
| Confusing baud with bits per second | They are equal only with one bit per symbol. In 64-QAM they differ by a factor of six. |
| Choosing the fastest modulation available | If the signal-to-noise ratio is not enough, the link drops with any degradation. |
| Working well below full scale | Decibels of SNR are lost: the quantization noise is fixed and the signal got smaller. |
| Mixing A-law with μ-law | Voice sounds distorted even though the link is perfect. |
| Power amplifier working saturated | It deforms the constellation of modulations with amplitude variation, such as QAM. |
| Ignoring error correction | Several decibels of free gain are wasted. |
| Measuring only received power | The power can be good and the quality terrible. You have to look at MER, BER or the eye diagram. |
06Self-assessment
How much does the signal-to-noise ratio improve for each bit added?
About 6 dB. The full expression is SNR ≈ 6.02·n + 1.76 dB.
Calculate the SNR of a 12-bit converter.
6.02 × 12 + 1.76 = 74 dB.
What is quantization error?
The difference between the actual value of the sample and the step it is approximated to. It behaves like noise added in the conversion that can no longer be removed.
What is companding for?
To use small steps at low levels —where voice spends most of its time— and large ones at the peaks. With 8 bits a quality equivalent to about 12 linear bits is achieved.
How many bits per symbol does 16-QAM carry?
log₂(16) = 4 bits per symbol.
At 4 Mbaud with 64-QAM, what is the data rate?
4 Mbaud × 6 bits = 24 Mbit/s.
Why is QPSK more robust than 256-QAM?
Because its four points are much farther apart in the constellation: it takes far more noise to confuse one with another. In exchange it carries 2 bits per symbol instead of 8.
What is adaptive modulation and what problem does it solve?
Changing the modulation on the fly according to the instantaneous quality of the link. In bad weather the link slows down but does not drop.
What information does an eye diagram give?
The receiver's decision margin. An open eye indicates there is margin; a closed one, that errors are imminent due to noise or bandwidth limitation.
What is gained with forward error correction?
Several decibels of coding gain: the receiver reconstructs damaged bits without requesting retransmission. It is paid for with part of the throughput used for redundancy.