Contents
5 Signal Constellations
5.1 Pulse-amplitude Modulation (PAM) . . . . . . . . . . . . . . .
5.1.1 Performance of PAM in Additive White Gaussian Noise
5.2 Phase-shift Keying . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Performance of PSK in Additive Gaussian Noise . . . .
5.3 Quadrature Amplitude-modulation (QAM) . . . . . . . . . . .
5.4 Performance of QAM in Additive Gaussian Noise . . . . . . . .
5.5 Frequency-shift keying . . . . . . . . . . . . . . . . . . . . . . .
5.6 Performance in additive Gaussian noise . . . . . . . . . . . . .
5.7 Continuous-phase modulation . . . . . . . . . . . . . . . . . . .
5.8 The Power Spectrum of Linearly Modulated Signals . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
4
5
7
8
9
10
12
13
14
A Error probability for PAM Signals in AWGN
17
B Error probability for PSK Signals in AWGN
18
C Error probability for PSK Signals in AWGN -Alternative Form
19
1
List of Figures
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
The signal space representation of binary PAM, 4-PAM and 8-PAM
constellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Symbol error probability for 2, 4 and 8-PAM as a function of SNR
per bit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Signal space representation of various PSK constellations. . . . . . .
The function f (θ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Symbol error probability for BPSK, QPSK, 8-PSK and 16-PSK as a
function of the signal-to-noise ratio per bit. . . . . . . . . . . . . . .
Signal-space representation of various QAM constellations. . . . . .
Symbol error probability as a function of SNR per bit for 4, 16, and
64-QAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Error probability comparison between coherent and incoherent FSK.
2
3
5
6
8
9
10
11
13
Chapter 5
Signal Constellations
In this chapter we study a number of signal constellations often used in practice.
5.1
Pulse-amplitude Modulation (PAM)
As the name implies, pulse-amplitude modulation (PAM) conveys information by
assigning k = log2 (M ) bits to a set of M discrete amplitudes of a transmitted
signal (baseband or passband). M -ary PAM is a one-dimensional signaling scheme
described mathematically by
si (t) = ai φ(t),
i = 1, 2, · · · , M,
0 ≤ t ≤ T,
(5.1)
where
ai = (2i − 1 − M )A,
i = 1, 2, · · · , M,
(5.2)
φ(t) is a unit-energy signal, and A is half the spacing between adjacent signals in
the constellation (i.e. dmin = 2A where dmin is the minimum Euclidean distance of
the contellation). Figure 5.1 shows the signal-space representation of PAM signals
assuming A = 1. Clearly not every signal in the constellation has the same energy.
8-PAM
4-PAM
2-PAM
-7
-5
-3
1
-1
3
5
7
Figure 5.1: The signal space representation of binary PAM, 4-PAM and 8-PAM
constellations.
3
CHAPTER 5. SIGNAL CONSTELLATIONS
4
The average energy of the constellation is
Eav
M
M
1 X
A2 X
=
a2i =
(2i − 1 − M )2 =
M i=1
M i=1
Ã
!
M2 − 1
A2 ,
3
(5.3)
and the peak-energy is
Ep = (M − 1)2 A2 .
(5.4)
Everything else being the same, it is desirable to have constellations whose peakenergy is not much different than their average energy. This is desirable in practice,
since the transmitter must be built to accommodate the peak power required and
if the latter is much larger than the average (which determines performance), then
the higher cost of hardware does not result in a proportional performance gain. A
figure of merit for signal constellations is the peak-to-average ratio, which for PAM
constellations is
6
Ep
=3−
,
(5.5)
Eav
M +1
which tends to 3 as M → ∞. This is a large peak-to-average energy ratio, and it is
one of the reasons multi-level PAM is rarely used in communication systems1 .
5.1.1
Performance of PAM in Additive White Gaussian Noise
Based on the data r(t) received (as given in (??)), the maximum-likelihood receiver
for PAM signaling chooses as the most likely signal transmitted the signal that
maximizes
a2
`i = ai · r − i ,
2
or equivalently the signal that minimizes (r − ai )2 , where
r=
Z T
0
r(t)φ(t)dt.
In signal space, the decision boundaries for this receiver are midway between constellation points, and a decision is made accordingly, based on where r falls on the
real line.
The error probability for M -ary PAM signals is derived in Appendix ?? and is
given by
Ãs
!
(M − 1)
3
Eav
PPAM (e) =
erfc
.
(5.6)
M
M 2 − 1 N0
The error probability for various PAM constellations is shown in Figure 5.2 as a
function of SNR per bit. Cleraly, even for the same SNR per bit, as M increases,
performance degrades. The gain, of course, is in increased bit-rate for the same
bandwidth.
1
One notable exception is in PCM modems, where the system makes use of the 256 PCM levels
in the digital part of the telephone network
CHAPTER 5. SIGNAL CONSTELLATIONS
5
10 0
Error-Probability
10 -1
10 -2
8-PAM
4-PAM
10 -3
2-PAM
10
-4
10 -5
0
2
4
6
8
10
12
14
16
18
SNR, dB
Figure 5.2: Symbol error probability for 2, 4 and 8-PAM as a function of SNR per
bit.
5.2
Phase-shift Keying
Under phase-shift keying (PSK), the information bits determine the phase of a
carrier, which takes values from a discrete set in accordance with the information
bits. The general form of M -ary PSK signals is given by
s
si (t) =
2E
cos(2πfc t + θi ),
T
where
θi =
i = 1, 2, · · · , M,
2π(i − 1)
, i = 1, 2, · · · , M
M
and
E=
Z T
0
s2i (t)dt
0 ≤ t ≤ T,
(5.7)
CHAPTER 5. SIGNAL CONSTELLATIONS
6
is the signal energy (the same for all signals). Equation (5.7) can be re-written in a
slightly different form as
"
r
√
si (t) =
E cos(θi )
r
2
cos(2πfc t) − sin(θi )
T
√
E [cos(θi )φ1 (t) − sin(θi )φ2 (t)]
=
where
#
2
sin(2πfc t)
T
r
2
cos(2πfc t)
(5.8)
rT
2
φ2 (t) =
sin(2πfc t)
(5.9)
T
are easily seen to be orthonormal. Thus, PSK signals are points in a two-dimensional
space spanned by φ1 (t) and φ2 (t). Figure 5.3 illustrates various PSK signal constellations, including binary PSK (BPSK), and 4-ary PSK, also known as quadrature
PSK (QPSK). The figure also illustrates the mapping of information bits to each
φ1 (t) =
ϕ2 (t )
ϕ2 (t )
01
11
00
ϕ1 (t )
E
E
ϕ1 (t )
10
BPSK: d min = 2 E
QPSK: d min = 2E
ϕ2 (t )
ϕ2 (t )
011
010
001
110
000
E
ϕ1 (t )
E
ϕ1 (t )
100
111
101
8-PSK: d min = 2 E sin(
π
)
8
16-PSK: d min = 2 E sin(
π
)
16
Figure 5.3: Signal space representation of various PSK constellations.
signal in the constellation for 4-PSK and 8-PSK. The illustrated mapping, known as
Gray coding, has the property that adjacent signals are assigned binary sequences
that differ in only one bit. This is desirable in practice, because, when a detection
error is made, it is more likely to be to a signal adjacent to the transmitted signal.
Then Gray coding results in a single bit error for the most likely signal errors.
CHAPTER 5. SIGNAL CONSTELLATIONS
5.2.1
7
Performance of PSK in Additive Gaussian Noise
For PSK signals, the optimum receiver decides which of the M possible PSK signals
was transmitted by finding the signal that maximizes
`i =
Z T
r(t)si (t)dt = rc ai − rs bi ,
0
(5.10)
where
rc =
rs =
ai
bi
Z T
0
Z T
0
r(t)φ1 (t)dt
r(t)φ2 (t)dt
·
¸
√
2π(i − 1)
=
E cos
M
·
¸
√
2π(i − 1)
=
E sin
.
M
For binary PSK, which corresponds to antipodal signaling, the probability that the
optimal receiver makes a decision error is
1
PBPSK (e) = erfc
2
Ãs
E
N0
!
.
(5.11)
The performance of QPSK is also derived easily and is given by
PQPSK (e) = PBPSK (e)[2 − PBPSK (e)]
where PBPSK (e) is as given in Eq. (5.11). An exact expression for the error probability of larger PSK constellations also exists and is derived in Appendix B. It is
given by
PMPSK (e) = 1 −
Z π/M
−π/M
f (θ)dθ,
(5.12)
where
f (θ) =
³ √
´i
√
1 −SN R h
2
e
1 + π · SN R cos(θ)eSN R·cos (θ) erfc − SN R cos(θ) , (5.13)
2π
where, as defined earlier, SN R = E/N0 . Figure 5.4 plots the function f (θ) for
various SNR values.
Figure 5.5 shows the error probability of various PSK constellations as a function
of the SNR per information bit.
Another interesting expression for the error-probability of two-dimensional signal
constellations, and in particular M-PSK, was obtained by Craig [1]:
1
PMPSK (e) =
π
Z
0
(M −1)π
M
"
¡
¢#
π
sin2 M
exp −SNR
sin2 (θ)
dθ,
(5.14)
CHAPTER 5. SIGNAL CONSTELLATIONS
8
1.2
6dB
1
0.8
3dB
f (θ)
0.6
0.4
0.2
0
0dB
-3
-2
-1
0
1
2
3
θ, Radians
Figure 5.4: The function f (θ).
The derivation is given in Appendix C. From (5.14) it is easy to derive an upperbound to the probability of error for arbitrary PSK signals by noting the the integrand is maximized at θ = π/2. Thus:
·
µ
π
(M − 1)
exp −SNR · sin2
PMPSK (e) ≤
M
M
¶¸
(5.15)
An interesting implication of the expression in (5.14) is that
1
1
erfc(x) =
2
π
5.3
Z
π
2
0
e
−
x2
sin2 (θ)
dθ.
Quadrature Amplitude-modulation (QAM)
Quadrature amplitude modulation (QAM) is a popular scheme for high-rate, high
bandwidth efficiency systems. QAM is a combination of both amplitude and phase
modulation. Mathematically, M -ary QAM is described by
√
si (t) = Ep(t) [ai cos(2πfc t) + bi sin(2πfc t)] , 0 ≤ t ≤ T, i = 1, 2, · · · , M (5.16)
CHAPTER 5. SIGNAL CONSTELLATIONS
9
2-PSK
-2
10
Symbol Error Probability
4-PSK
16-PSK
10
-4
8-PSK
10
10
-6
-8
-10
10
0
5
10
15
SNR per bit, dB
20
Figure 5.5: Symbol error probability for BPSK, QPSK, 8-PSK and 16-PSK as a
function of the signal-to-noise ratio per bit.
where ai and bi take values from the set {±1, ±3, ±5, · · ·} and E and p(t) are as
defined earlier. The signal-space representation of QAM signals is shown in Figure
5.6 for various values of M which are powers of 2, that is, M = 2k , k = 2, 3, · · ·.
For even values of k, the constellations are square, whereas for odd values of k
the constellations have a cross shape and are thus called cross constellations. For
square constellations, QAM corresponds to the independent amplitude modulation
of an in-phase carrier (i.e., the cosine carrier) and a quadrature carrier (i.e., the sine
carrier).
5.4
Performance of QAM in Additive Gaussian Noise
The optimum receiver for QAM signals chooses the signal that maximizes
√ ³
´
E 2
`i = ai rc + bi rs −
ai + b2i
4
CHAPTER 5. SIGNAL CONSTELLATIONS
10
64-QAM
32-QAM
16-QAM
8-QAM
4-QAM
Figure 5.6: Signal-space representation of various QAM constellations.
where
rc =
and
rs =
Z T
0
Z T
0
r(t)p(t) cos(2πfc t)dt
r(t)p(t) sin(2πfc t)dt
For square constellations which correspond to independent PAM of each carrier,
an exact error probability is derived easily and is given by
"
µ
1
PQAM (e) = 1 − 1 − 1 − √
M
Ãs
¶
erfc
3
Eav
·
2(M − 1) N0
!#2
.
For cross constellations, tight upper-bounds and good approximations are available.
Figure 5.7 plots the symbol-error probability of various square QAM constellations
as a function of SNR per bit.
5.5
Frequency-shift keying
As the name implies, frequency-shift keying (FSK) modulates the frequency of a
carrier to convey information. FSK is one of the oldest digital modulation techniques
CHAPTER 5. SIGNAL CONSTELLATIONS
11
-1
10
Error-Probability
64-QAM
-2
10
16-QAM
-3
10
-4
10
4-QAM
-5
10
0
2
4
6
8
10
12
SNR per bit, dB
14
16
18
Figure 5.7: Symbol error probability as a function of SNR per bit for 4, 16, and
64-QAM.
and was the modulation of choice for the first, low-rate modems. Its main attribute
that makes it of interest in some applications is that it is detected incoherently (and
coherently), which reduces the cost of the receiver. Mathematically, the modulated
M -ary FSK signal is described by
s
si (t) =
2E
cos[2π(fc + fi )t], 0 ≤ t ≤ T,
T
where
µ
fi =
2i − 1 − M
2
i = 1, 2, · · · , M
¶
∆f.
∆f is the minimum frequency separation between modulation tones. For orthogonal signaling (i.e. when the correlation between all pairs of distinct signals is zero),
the minimum tone spacing is 1/2T . This is often a condition imposed in practice.
Orthogonal signaling performs well as a function of energy per bit, but it is also
rather bandwidth-inefficient, making it impractical for high-speed, band-limited applications.
CHAPTER 5. SIGNAL CONSTELLATIONS
5.6
12
Performance in additive Gaussian noise
FSK is detected coherently or incoherently. Coherent detection requires a carrier
phase synchronization subsystem at the receiver which generates locally a carrier
which is phase-locked to the received carrier. The optimum receiver for coherent
detection makes decisions by maximizing the following (implementation assumes
phase-coherence)
`i =
Z T
0
r(t)si (t)dt.
For binary (orthogonal) signaling, the error probability is given simply by
1
PFSK (e) = erfc
2
Ãs
E
2N0
!
,
(coherent FSK)
which is 3dB worse than BPSK. For M -ary signaling, an exact expression exists in
integral form and is found, for example, in [?].
Incoherent detection does not assume phase coherence and does not attempt to
phase-lock the locally generated carrier to the received signal. In this case, it is easy
to argue that the phase difference between the LO carrier and the received carrier
is completely randomized. An optimum receiver is also derived in this case and it
is one that maximizes over the set of frequency tones
2
2
`i = rci
+ rsi
where
2
rci
=
and
2
rsi
=
Z T
0
Z T
0
r(t) cos[2π(fc + fi )t]dt
r(t) sin[2π(fc + fi )t]dt
The exact error-probability performance of this incoherent receiver is available
in analytical form, but it is rather complicated to compute for the general M -ary
case (see, for example, [?]). For the binary case, the error probability has a simple
form given by
1 − E
PFSK (e) = e 2N0 , (incoherent FSK)
2
Figure 5.8 compares the performance of coherent and Incoherent binary FSK.
At an error probability of about 10−6 , incoherent detection is inferior only slightly
more than half a dB compared with coherent detection. However, this small loss is
well compensated for by the fact that no carrier phase synchronization is needed for
the former.
CHAPTER 5. SIGNAL CONSTELLATIONS
13
10-1
Error-Probability
Incoherent
Detection
10-2
10-3
Coherent
Detection
10-4
10-5
0
2
4
6
8
SNR per bit, dB
10
12
14
Figure 5.8: Error probability comparison between coherent and incoherent FSK.
5.7
Continuous-phase modulation
All of the modulation schemes described so far are memoryless, in the sense that
the signal transmitted in a certain symbol interval does not depend on any past
(or future) symbols. In many cases, for example, when there is a need to shape
the transmitted signal spectrum to match that of the channel, it is necessary to
constrain the transmitted signals in some form. Invariably, the imposed constraints
introduce memory into the transmitted signals. One important class of modulation
signals with memory are continuous phase modulation (CPM) signals. These signals
constrain the phase of the transmitted carrier to be continuous, thereby reducing
the spectral sidelobes of the transmitted signals. Mathematically, the modulation
signals for CPM are described by the expression
u(t) = A cos [2πfc t + φ(t; d)]
where
φ(t; d) = 2π
n
X
k=−∞
dk hk q(t − kT ),
nT ≤ t ≤ (n + 1)T.
CHAPTER 5. SIGNAL CONSTELLATIONS
14
The dk are the modulation symbols and hk are the modulation indices, which may
vary from symbol to symbol. For binary modulation, the modulation symbols are
either 1 or -1. Finally, q(t)is the integral of some baseband pulse p(t) containing no
impulses (thus guaranteeing that q(t) is continuous)
q(t) =
Z t
−∞
p(τ )dτ.
When p(t) is zero for t ≥ T , we have what is called full response CPM, otherwise we
have partial-response CPM. In general, partial-response CPM achieves better spectral sidelobe reduction than full-response CPM. A special case of CPM in which the
modulation indices are all equal and p(t) is a rectangular pulse of duration T seconds is called continuous-phase FSK (CPFSK). If, further, h = 1/2, we have what
is called minimum-shift keying (MSK). A variation of MSK, in which the rectangular baseband pulse is first passed through a filter with a Gaussian-shape impulse
response for further reduction in the spectral sidelobes, is called Gaussian MSK
(GMSK). Various simple ways for detecting GMSK are available, which, combined
with its spectral efficiency, has made it a popular modulation scheme. In particular,
it is the modulation scheme used for the European digital cellular radio standard,
known as GSM.
For more information on CPM signaling, including spectral characteristics and
performance in noise, refer to [?].
5.8
The Power Spectrum of Linearly Modulated Signals
Since the discrete modulation symbols which are mapped into transmitted signals are
random, the resulting transmitted signal is a stochastic process. It is of interest to
determine the frequency content of a transmitted stochastic signal, which is given by
its power spectrum, for a number of reasons. One important reason is to determine
its bandwidth and thus determine the channel bandwidth required for its undistorted
transmission. Another use of the power spectrum of a modulated signal is in timesynchronization systems. A class of time synchronization systems extract their
timing information by tracking a discrete frequency component, which is at some
multiple of the signaling rate, in the spectrum of the modulated signal.
A general, linearly modulated signal can be expressed as
x(t) =
X
dk p(t − kT ),
(5.17)
k
where {dk } is a sequence of complex symbols, 1/T is the signaling rate, and p(t)
is the transmitted pulse. We will assume that the sequence {dk } is wide-sense
stationary having mean µ and autocorrelation function
1
rdd (k) = E[d∗j dj+k ].
2
(5.18)
CHAPTER 5. SIGNAL CONSTELLATIONS
15
In computing the power spectrum of x(t), we first derive an expression for the
autocorrelation function of x(t)
1
E[x∗ (t)x(t + τ )]
2
1 XX
E[d∗k dj ]p∗ (t − kT )p(t + τ − jT )
2 k j
Rxx (t, t + τ ) =
=
XX
=
k
X
=
rdd (j − k)p∗ (t − kT )p(t + τ − jT )
j
k
XX
=
(5.19)
rdd (m)p∗ (t − kT )p (t + τ − (k + m)T )
m
rdd (m)
m
X
p∗ (t − kT )p (t + τ − (k + m)T )
(5.20)
k
Clearly, Rxx (t, t + τ ) is periodic in t with period T . In addition, the mean of x(t)
E[x(t)] = µ
X
p(t − kT )
k
is periodic in t with period T . Thus, x(t) is what is referred to as a cyclostationary
process. For cyclostationary processes, the dependence on t is removed (and thus the
resulting autocorrelation becomes only a function of a single variable τ ) by averaging
Rxx (t, t + τ ) over a period:
Rxx (τ ) =
=
1
T
Z T /2
X
−T /2
Rxx (t, t + τ )dt
X 1 Z T /2
p∗ (t − kT )p (t + τ − (k + m)T ) dt
rdd (m)
m
=
X
m
=
k
rdd (m)
T
−T /2
X 1 Z T /2−kT
k
T
−T /2−kT
p∗ (t)p (t + τ − mT ) dt
Z ∞
1X
rdd (m)
p∗ (t)p (t + τ − mT ) dt
T m
−∞
Letting
Rpp (τ ) =
Z ∞
−∞
p∗ (t)p (t + τ ) dt
(5.21)
(5.22)
be the autocorrelation function of p(t) we have
Rxx (τ ) =
1X
rdd (m)Rpp (τ − mT )
T m
(5.23)
Taking Fourier transforms on both sides above we obtain
SX (f ) =
1
|P (f )|2 Sd (f )
T
(5.24)
CHAPTER 5. SIGNAL CONSTELLATIONS
where
Sd (f ) =
X
rdd (k)e−j2πf kT
k
is the discrete Fourier transform of {rdd (k)}.
16
(5.25)
Appendix A
Error probability for PAM
Signals in AWGN
17
Appendix B
Error probability for PSK
Signals in AWGN
18
Appendix C
Error probability for PSK
Signals in AWGN -Alternative
Form
19
Bibliography
[1] J.W. Craig, “A New, Simple and Exact Result for Calculating the Probability of Error for Two-Dimensional Signal Constellations,” Proceedings IEEE
MILCOM’91, Boston, MA, pp. 25.5.1-25.5.5.
20