Digital modulation
With minimum-bandwidth pulses, a 10 % error in the sampling instant closes the eye by 40 %. With twice the bandwidth, the same error closes it by less than 10 %.
01Bits, symbols and pulses
A digital system groups bits \( k \) at a time and transmits one of \( M = 2^k \) possible symbols. Each symbol is a pulse of known shape whose amplitude, phase or frequency depends on the bits. Two rates must be distinguished:
The introduction to the modulations with the equipment is in digital modulation, in the technician program. Here we work with the models: the pulse shape, the bandwidth and the complex envelope of each modulation.
02Intersymbol interference and the Nyquist criterion
A band-limited pulse lasts, in theory, an infinite time, and overlaps with its neighbors. This intersymbol interference (ISI) does no harm if, at the sampling instant of each symbol, the other pulses pass through zero. This is the Nyquist criterion: the pulse \( p(t) \) must equal 1 at \( t = 0 \) and 0 at all multiples of \( T = 1/R_s \). The narrowest pulse in frequency that meets it is the sinc, which occupies \( R_s/2 \); in practice the raised cosine is used, which falls off faster in time in exchange for an excess bandwidth α:
The eye diagram is the tool for seeing ISI: on the oscilloscope, segments of the signal two symbol periods long are superimposed, triggering on the clock. The vertical opening of the eye is the margin against noise; the horizontal one, against errors in the sampling instant.
240 random binary symbols (±1) with raised-cosine pulses, passed through a first-order lowpass channel. The yellow line is the sampling instant, which can be shifted from the optimum. The opening is calculated from the worst case over all symbols, relative to the mean amplitude.
03Bandpass modulations
To transmit by radio, the pulse modulates a carrier. Written as a complex envelope, each modulation is a set of points in the I/Q plane, the constellation, and the signal jumps from point to point at the symbol rate:
| Modulation | Bits per symbol | Constellation | Typical use |
|---|---|---|---|
| ASK / OOK | 1 | Two points on the I axis, one at the origin | Remote controls, short-range optical fiber |
| FSK | 1 or more | Different frequencies: not a flat constellation | Low-speed links, Bluetooth (GFSK) |
| BPSK | 1 | Two opposite points | Very noisy links, GPS |
| QPSK | 2 | Four points on a circle | Satellite, cellular telephony |
| M-QAM | \( \log_2 M \) | A grid of \( \sqrt{M} \times \sqrt{M} \) | Wi-Fi, cable modem, microwave |
A bandpass modulation with raised-cosine pulses occupies twice as much as in baseband, \( (1 + \alpha) R_s \), because the spectrum is split to both sides of the carrier. Bits are assigned to the points with a Gray code: neighboring points differ in a single bit, and the most probable error, mistaking a point for its neighbor, costs a single bit.
04Spectral efficiency
That is why modern systems adapt the modulation to the state of the channel: a phone near the antenna uses 256-QAM and far away drops to QPSK. The relationship among error rate, power and number of points is the subject of the unit on noise in digital modulations. M-PSK and FSK constellations, with constant or nearly constant envelope, are preferred when the power amplifier has to work close to saturation.
05In the lab
Transmit a pseudorandom sequence over a long cable and observe the eye diagram on the oscilloscope, triggering on the data clock. Increase the rate until the eye closes.
With a vector signal generator and an analyzer, observe the QPSK and 16-QAM constellations and the error vector magnitude (EVM).
Measure the occupied bandwidth of a QPSK signal for several values of α and compare it with \( (1 + \alpha)R_s \).
06Common mistakes
- Confusing bauds with bit/s. They coincide only in binary modulations.
- Believing that the sinc is the best choice. It occupies the minimum band, but its eye closes with the slightest timing error.
- Forgetting that the channel adds ISI. Even if the pulse meets the Nyquist criterion, a band-limited channel deforms it, and an equalizer is needed.
- Assigning bits to the points in natural binary order. Without a Gray code, an error between neighbors can cost several bits.
07Self-assessment
At how many bauds are 60 Mbit/s transmitted with 64-QAM?
At 10 Mbaud: each symbol carries 6 bits.
What bandwidth does a 20 Mbit/s QPSK with α = 0.2 occupy?
\( R_s \) = 10 Mbaud and \( B = 1.2 \cdot 10 = 12 \) MHz.
What does the vertical opening of the eye measure?
The amplitude margin between the symbols at the sampling instant: how much noise can be added before a mistake is made.
Why does a larger α tolerate timing errors better?
Because the tails of the pulse decay faster and the ISI grows more slowly as one moves away from the optimum instant.
08Further reading
- John G. Proakis and Masoud Salehi. Digital Communications. 5th ed., McGraw-Hill, 2008. The reference on digital modulations, the Nyquist criterion and equalization.
- Bernard Sklar. Digital Communications: Fundamentals and Applications. 2nd ed., Prentice Hall, 2001. ISI, the raised cosine and bandpass modulations with numerical examples.
- B. P. Lathi and Zhi Ding. Modern Digital and Analog Communication Systems. 5th ed., Oxford University Press, 2019. Baseband and bandpass digital transmission in a single framework.