Linear and nonlinear modulations
Binary and M-ary modulations:
Digital modulations and wireless standards
Two main factors influencing the choice of a digital modulation scheme are spectral efficiency (minimum bandwidth occupancy) and power efficiency(minimum required transmitted power).
Other requirements are:
Often conflicting requirements, tradeoffs are needed (depending on the application).
The ability of a modulation technique to preserve the fidelity of the digital message at low power level.
The amount of signal power required to obtain a certain level of fidelity (i.e., an acceptable BER) depends on the particular type of modulation.
Power efficiency can be measured by the minimum energy contrast γb required for a certain BER (e.g., BER=10-3).
The ability of a modulation scheme to accommodate data within a limited bandwidth:
Evaluated in bps/Hz, it measures how many bps can be transmitted in 1 Hz of bandwidth.
A modulation scheme with M waveforms is:
Bandwidth efficiency and power efficiency are conflicting requirements:
Desirable in wireless communications for several reasons:
However, constant envelope modulations generally occupy a larger bandwidth.
Linear modulations (ASK, PSK, QAM)
Non linear modulations (FSK and variants)
Binary modulations (M=2) are the simplest to analyze.
Binary Amplitude Shift Keying (M=2) → the amplitude of the carrier signal is varied to represent binary 1 or 0.
Disadvantage: amplitude is very susceptible to fading.
Unipolar BASK is also called OOK (On-Off Keying).
Bipolar BASK is more common due to its zero DC.
The bit rate of BASK is Rb = 1/Tb.
The bandwidth of BASK is well approximated by B ≈ 1/Ts = 1/Tb.
The spectral efficiency is:
Example: if B=200 kHz → the bit-rate is Rb=200 kb/s.
Binary Phase Shift Keying (M=2) → the phase of the carrier signal is varied to represent binary 1 or 0.
Advantage: phase is less susceptible to fading, constant envelope property.
Disadvantage: more complex demodulation.
The bit rate of BPSK is Rb = 1/Tb.
The bandwidth of BPSK is well approximated by B ≈ 1/Ts = 1/Tb.
The spectral efficiency is:
To demodulate BPSK accurate knowledge at the receiver of the carrier phase is required → coherent demodulation.
Carrier phase recovery is obtained by using special circuits at the receiver (PLL, phase locked loop).
In many cases it is difficult/expensive to obtain a precise phase reference → one resorts to differential modulation/demodulationtechniques (differential BPSK=DBPSK):
Binary Frequency Shift Keying (M=2) → the frequency of the carrier signal is varied to represent binary 1 or 0.
Advantage: less susceptible to noise and fading, constant envelope property, simpler demodulation.
Disadvantage: larger bandwidth when M>2.
The signals s1(t) and s2(t) can be made orthogonal by appropriate choice of the carrier separation Δf= f2 – f1
Orthogonality simplifies demodulation of BFSK signals.
The minimum carrier separation is Δf = 0.5/Tb → the bandwidth of BFSK is well approximated by B ≈ 2Δf = 1/Tb.
The bandwidth efficiency is:
When the channel is AWGN (Gaussian noise) the performance of binary modulation techniques can be easily derived: see the table.
Since Q(.) is an exponentially decreasing function, the power efficiency of BPSK/BASK (bipolar) is 3dB better (a factor of 2) with respect to BFSK/BASK (unipolar).
In Rayleigh fading the performance can be obtained by averaging the AWGN results with respect to fading statistics: see the table.
BPSK/BASK (bipolar) has still a 3dB advantage over BFSK/BASK(unipolar) and DBPSK.
In AWGN Pb is exponentially decreasing with γb.In Rayleigh fading Pb decreases linearly with γb.
heavy performance degradation.
Evaluate the energy contrast γb needed at the receiver to assure BER = 10-3 for BPSK modulation:
(a) over an AWGN channel
(b) over a Rayleigh fading channel
23.98 – 6.82=17.16 dB is the excess power required to combat Rayleigh fading!
In the previous examples the modulation employed rectangular waveforms:
A drawback of rectangular waveforms is the high level of out-of-band radiation:
By appropriately shaping the pulse waveform the sidelobes are reduced → better spectral properties.
The considered binary modulations are all approximately equivalent in terms of spectral efficiency (1 bps/Hz).
In terms of performance, BPSK/ BASK (bipolar) exhibits the best performance both in AWGN and Rayleigh fading channel.
However, in terms of receiver complexity, DBPSK and BFSK are preferable.
Due to their low spectral efficiency, binary modulation are used only in low-rate applications → M-ary modulations with M>2 are needed to implement high-speed modems.
Quadrature Phase Shift Keying.
M=4 → each symbol carries two bits
QPSK modulation can be regarded as two BPSK modulations with orthogonal (sin/cos) carriers:
Since the bandwidth of QPSK is well approximated by B ≈ 1/Ts = 0.5*1/Tb, the spectral efficiency is doubled:
QPSK modulation has a constant envelope.
Offset QPSK solve this problem by delaying the Q-channel of a half-symbol period in order to constrain the maximum phase shift to π/2 radians.
Uses two different QPSK signal constellation shifted by π/4 and moves from one to the other in every symbol interval:
Easily amenable to differential mo/demodulation (π/4-DQPSK).
Minimum Shift Keying:
The spectrum of MSK is significantly better than BPSK/ QPSK/OQPSK but still too large to satisfy typical bandwidth requirements of wireless communications.
Gaussian Minimum Shift Keying.
Employed in GSM and DECT.
Obtained from MSK by filtering the data before modulation with a Gaussian-shaped filterwith bandwidth B
Main advantage is high spectral efficiency coupled with constant envelope property.
In M-PSK the constellation points are equispaced on a circle.
In M-QAM the constellation points are spaced on a square “lattice”.
Increasing the value of M improves spectral efficiency → spectrally efficient modulations.
M-QAM is more power efficient than M-PSK since the distance between the constellation points is higher
A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (selected parts of chaps. 5 and 6)
Supplementary material eventually available on the website