CHAPTER 7
M-ARY DIGITAL MODULATION
SCHEMES
M -ary signaling schemes can be used for reducing the bandwidth requirements of
baseband PAM data transmission systems as well as bandpass digital modulation
techniques. Here, one of M (M > 2) signals s1 (t), s2 (t), · · · , sM (t) is transmitted
during each signaling interval of duration Ts . These signals are generated by changing the amplitude, phase, or frequency of a carrier in M discrete steps. Thus we can
have M -ary ASK, M -ary PSK, M -ary FSK, and M-ary quadrature amplitude modulation (QAM) digital modulation schemes. M -ary digital modulation schemes are
preferred over binary digital modulation schemes for transmitting digital information
over bandpass channels when one wishes to conserve bandwidth (at the expense of
increasing power requirements).
In our discussion of M -ary schemes, we will assume that the input to the modulator is an independent sequence of equiprobable binary digits. We will further
assume that the modulator takes blocks of N binary digits and assigns one of M
possible waveforms to each block (M = 2N ).
Communication Engineering II, First Edition.
By Osama A. Alkishriwo Copyright c 2017 John Wiley & Sons, Inc.
119
120
M-ARY DIGITAL MODULATION SCHEMES
7.1
Definition of Spectral Efficiency
Spectral efficiency, ηs , is defined as the rate of information transmission per unit of
occupied bandwidth, i.e.
ηs =
Rs H
, (bits/s/Hz)
B
(7.1)
where Rs is the symbol rate, H is entropy, i.e. the average amount of information (measured in bits) conveyed per symbol, and B is occupied bandwidth. (For
an alphabet containing M , statistically independent, equiprobable symbols, H =
log2 (M ) bit/symbol.
Since Rs = 1/Ts and H = log2 (M ), then ηs can be expressed as:
ηs =
log2 (M )
, (bits/s/Hz)
Ts B
(7.2)
It is apparent from Eq. (7.2) that spectral efficiency is maximised by making the symbol alphabet size, M , large and the Ts B product small. This is exactly the strategy
employed by spectrally efficient modulation techniques.
7.2
M -ary Phase Shift Keying (MPSK)
In M -ary PSK systems, the phase of the carrier is allowed to take on one of M
possible values φk = 2πk/M (k = 0, 1, 2, · · · , M − 1). Thus the M possible
signals that would be transmitted during each signaling interval of duration Ts are
sk (t) = A cos(ωc t + 2πk/M ),
k = 0, 1, , 2, · · · , M − 1 and 0 ≤ t ≤ Ts (7.3)
The probability of symbol error for MPSK systems is found by integrating the
two dimensional pdf of the noise centred on the tip of each signal phasor in turn
over the corresponding error region and averaging the results. A simple but good
approximation which is useful for M ≥ 4 is:
s
!
π
2Es
2
Ps ≈ 2Q
sin
, M ≥4
(7.4)
η
M
If a Gray code is used to map binary symbols to phasor states this type of error
results in only a single decoded bit error. In this case, providing the probability of
errors other than this type is negligible, then the bit error probability is:
Pe =
Ps
log2 (M )
(7.5)
In order to compare the performance of different modulation schemes on an equitable basis it is useful to express performance in terms of Pe as a function of average
M -ARY PHASE SHIFT KEYING (MPSK)
121
energy per bit, Eb . Since the energy, Es , of all symbols in an MPSK system are
identical the average energy per bit is:
Eb =
Es
log2 (M )
(7.6)
Substituting Eqs. (7.5) and (7.6) into Eq. (7.4) we have

s
π
2E
log
(M
)
2
b
2

Q
sin2
Pe =
log2 (M )
N0
M
7.2.1
(7.7)
Quadrature Phase Shift Keying (QPSK)
When M = 4 in Eq. (7.3), the expression for QPSK may be written as
sk (t) = A cos(ωc t + 2πk/4),
Figure 7.1
k = 0, 1, , 2, 3 and 0 ≤ t ≤ Ts
(7.8)
Block diagram of a QPSK modulator.
The block diagram of a QPSK modulator consists of a serial–to–parallel converter, a pair of product modulators, an oscillator for carrier waves, a summer and
a phase shifter as shown in Fig. 7.1. The function of serial–to–parallel converter to
represent each successive pair of bits of incoming binary data stream as two separate bits with one bit applied to the in–phase channel and the other bit applied to the
quadrature channel. So, the source data is split into two data streams, with each data
stream running at half the rate of the input data stream as given in Fig. 7.2.
The receiver for the system is shown in Fig. 7.3. The receiver requires two local
reference waveforms A cos(ωc t + 45o ) and A cos(ωc t − 45o ) that are derived from
a coherent earner reference A cos(ωc t).
Figure 7.4 shows s01 and s02 corresponding to each of the four possible signals
s1 (t), s2 (t), s3 (t), and s4 (t).
122
M-ARY DIGITAL MODULATION SCHEMES
Figure 7.2
Examples of aI (t) and aQ (t) in QPSK modulation.
Figure 7.3
Figure 7.4
7.3
Receiver for QPSK scheme.
Output signal levels at sampling times.
M -ary Quadrature Amplitude Modulation (MQAM)
We can represent the source symbols by combining amplitude and phase modulation
to form M -ary QAM. M -ary QAM is also called M -ary amplitude–phase keying
(APK). It is useful for channels having limited bandwidth and provides lower error
rates than other M -ary systems with keyed modulation operating at the same symbol
rate.
M -ARY QUADRATURE AMPLITUDE MODULATION (MQAM)
123
Figure 7.5 shows the time domain waveforms for a 16-state QAM constellation
which are obtained by encoding binary data in 4-bit sequences.
Figure 7.5
16-state quadrature amplitude modulated (QAM) signal constellations.
A simple approximation for the probability of symbol error for MQAM (M even)
signalling in Gaussian noise is
!
!
√
r
M −1
3
Es
√
Q
(7.9)
Ps = 4
M − 1 N0
M
where (Es ) is the average energy per QAM symbol.
For Gray code mapping of bits along the inphase and quadrature axes of the QAM
constellation the probability of bit error Pe is given approximately by Eq. (7.5).
Denoting the average energy per bit by Eb = Es / log2 (M ), Eq. (7.10) can be
written as:

! s
√
4
M −1
3
log
(M
)
E
b
2

√
Pe =
Q
(7.10)
log2 (M )
M − 1 N0
M
Like MPSK all the symbols in a QAM (or APK) signal occupy the same spectral space. The spectral efficiency is therefore identical to MPSK and is given (for
statistically independent equiprobable symbols) by Eq. (7.2).