# Giacinto Gelli » 10.Digital modulation schemes

### Outline

Design considerations:

• power efficiency
• spectral efficiency
• constant envelope

Linear and nonlinear modulations

Binary and M-ary modulations:

• signal constellations
• modulation/demodulation
• performances in AWGN and Rayleigh fading
• pulse shaping and spectrum

Digital modulations and wireless standards

### Choice of a digital modulation

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:

• robustness to channel impairments
• low power/low cost implementation
• constant envelope

Often conflicting requirements, tradeoffs are needed (depending on the application).

### Power efficiency

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.

$\mbox{BER} = f( \gamma_b)$

Power efficiency can be measured by the minimum energy contrast γb required for a certain BER (e.g., BER=10-3).

### Spectral efficiency

The ability of a modulation scheme to accommodate data within a limited bandwidth:

• increasing the data rate implies decreasing the signaling interval Ts, which increases bandwidth
• some modulation schemes perform better than others in this tradeoff

$\eta = \frac{R_b}{B} = \frac{\log_2 M}{B T_s}$

where:

• Rb = 1/Tb bit-rate [bps=b/s]
• M = cardinality of the modulation scheme
• Ts = symbol interval
• B = one-sided bandwidth of s(t)

Evaluated in bps/Hz, it measures how many bps can be transmitted in 1 Hz of bandwidth.

### Bandwidth vs. power efficiency

A modulation scheme with M waveforms is:

• bandwidth-efficient if the spectral efficiency η increases with M
• power-efficient if the power efficiency γb for a given BER level decreases with M

Bandwidth efficiency and power efficiency are conflicting requirements:

• linear modulations are bandwidth-efficient (but not power-efficient)
• nonlinear modulations are power-efficient (but not bandwidth-efficient)

### Constant envelope property

Desirable in wireless communications for several reasons:

• power efficient class C (nonlinear) amplifiers can be used without introducing degradation in the spectrum occupancy of the transmitted signal
• low out-of-band radiation of the order of -60 to -70 dB can be achieved
• limiter-discriminator detection can be used → simplified receiver design

However, constant envelope modulations generally occupy a larger bandwidth.

### Linear and nonlinear modulations

• information coded in amplitude and/or phase
• spectrally efficient
• less robust to channel impairments and amplifier nonlinearities
• more complicated to demodulate

Non linear modulations (FSK and variants)

• information coded in frequency
• power efficient
• more robust to channel impairments and amplifier nonlinearities
• Ssmpler to generate and demodulate

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.

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:

$\eta_{\rm BASK} = \frac{R_b}{B} = \frac{1/T_b}{1/T_b} = 1 \:\mbox{bps/Hz}$
Example: if B=200 kHz → the bit-rate is Rb=200 kb/s.

### BPSK

Binary Phase Shift Keying (M=2) → the phase of the carrier signal is varied to represent binary 1 or 0.

### BPSK (cont’d)

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:

$\eta_{\rm BPSK} = \frac{R_b}{B} = \frac{1/T_b}{1/T_b} = 1 \: \mbox{bps/Hz}$

### DBPSK

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):

• the phase in the previous signaling interval is used as phase reference for the present symbol → an absolute phase reference at the receiver is not needed
• the channel phase must remain stable at least over two consecutive signaling intervals (slow fading)

### BFSK

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.

### BFSK (cont’d)

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:

$\eta_{\rm BFSK} = \frac{R_b}{B} = \frac{1/T_b}{1/T_b} = 1 \: \mbox{bps/Hz}$

### Performances in AWGN

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).

### DBPSK vs. BPSK

DBPSK pays a penalty < 1 dB for large yb

In Rayleigh fading the performance can be obtained by averaging the AWGN results with respect to fading statistics: see the table.

In AWGN Pb is exponentially decreasing with γb.In Rayleigh fading Pb decreases linearly with γb.

### Example

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

Solution:

(a)  $\mbox{invert} \: P_b = Q( \sqrt{2\gamma_b} ) = 0.001 \Longrightarrow \sqrt{2\gamma_b} = 3.10 \Longrightarrow \gamma_b = 6.82 \: \mbox{dB}$.

(b) $\mbox{invert} \: P_b \approx 1/(4 \gamma_b) = 0.001 \Longrightarrow4\gamma_b = 1000 \Longrightarrow \gamma_b = 23.98 \: \mbox{dB}$

23.98 – 6.82=17.16 dB is the excess power required to combat Rayleigh fading!

### Pulse shaping

In the previous examples the modulation employed rectangular waveforms:

• simplified implementation of transmitter and receiver
• constant envelope property

A drawback of rectangular waveforms is the high level of out-of-band radiation:

• rectangular waveforms are characterized by sinc spectra in the frequency domain with very strong and slowly decaying sidelobes (20dB/decade)

By appropriately shaping the pulse waveform the sidelobes are reduced → better spectral properties.

### Example: BPSK

BPSK with rectangular shaping

BPSK with raised cosine shaping (β=0.5)

### Comparison of binary modulations

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.

### QPSK

M=4 → each symbol carries two bits   $s_i(t) = A \, (2/T_s)^{1/2} \cos( 2 \pi f_c t + \theta_i)$

• θi=2π(i-1)/4+Φ0, i=1,…,4
• Φ0constellation displacement

$s_i(t) = A \, (2/T_s)^{1/2} \cos( 2 \pi f_c t) \cos(\theta_i) -A \, (2/T_s)^{1/2} \sin( 2 \pi f_c t) \sin(\theta_i)$

$\phi_1(t) &=& (2/T_s)^{1/2} \cos( 2 \pi f_c t) \nonumber\\$

$\phi_2(t) &=& -(2/T_s)^{1/2} \sin( 2 \pi f_c t) \nonumber$

$\mathbf{s}_i = (s_{i1}, s_{i2} ) = [ A \, \cos(\theta_i), A \sin(\theta_i) ]$

### QPSK (cont’d)

QPSK modulation can be regarded as two BPSK modulations with orthogonal (sin/cos) carriers:

• the BER of QPSK is practically the same at BPSK
• QPSK carries two bits per symbol, hence Ts = 2*Tb

Since the bandwidth of QPSK is well approximated by B ≈ 1/Ts = 0.5*1/Tb, the spectral efficiency is doubled:

$\eta_{\rm QPSK} = \frac{R_b}{B} = \frac{1/T_b}{1/T_s} = \frac{T_s}{T_b} = 2 \: \mbox{bps/Hz}$

$\eta_{\rm BPSK} = \frac{R_b}{B} = \frac{1/T_b}{1/T_b} = 1 \: \mbox{bps/Hz}$

### OQPSK

QPSK modulation has a constant envelope.

• desirable feature to prevent nonlinear amplifier distortions
• occasional phase shift of π radians cause the envelope to pass from zero, which destroys the constant envelope properties

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.

QPSK

OQPSK

### π/4-QPSK and π/4-DQPSK

Uses two different QPSK signal constellation shifted by π/4 and moves from one to the other in every symbol interval:

• pseudo-octonary: uses 8 phases to carry 2 information bits per symbol (not 3)
• maximum phase transition between two adjacent symbols of 135°
• one phase transition of at least π/4 in each interval→ eases symbol synchronization

Easily amenable to differential mo/demodulation (π/4-DQPSK).

### MSK

Minimum Shift Keying:

• derived from OQPSK by replacing the rectangular pulse with a half-cycle sinusoidal pulse
• can be regarded also as a form of continuous-phase FSK with minimum frequency spacing Δf = 0.5/Tb
• constant envelope modulation.

The spectrum of MSK is significantly better than BPSK/ QPSK/OQPSK but still too large to satisfy typical bandwidth requirements of wireless communications.

### GMSK

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

• B Tb=0.3 in GSM, B Tb =0.5 in DECT

Main advantage is high spectral efficiency coupled with constant envelope property.

### M-PSK and M-QAM

In M-PSK the constellation points are equispaced on a circle.

In M-QAM the constellation points are spaced on a square “lattice”.

$\eta_{\rm M-PSK} = \eta_{\rm M-QAM} = \frac{R_b}{B} = \frac{\log_2 M /T_s}{1/T_s} = \log_2 M \: \mbox{bps/Hz}$

Increasing the value of M improves spectral efficiency → spectrally efficient modulations.

### Performances in AWGN

M-QAM is more power efficient than M-PSK since the distance between the constellation points is higher

Note: only M-QAM performances are reported, those of M-PSK are slightly worse

### I materiali di supporto della lezione

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (selected parts of chaps. 5 and 6)

Supplementary material eventually available on the website

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion