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Post-graduate Course in Radio Communications
2001-2002
Fitting Signals into Given Spectrum
Modulation Methods
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Abstract
Modulation is the process where the message information is embedded into the radio
carrier. Message information can be transferred in the amplitude, frequency or the phase
component of the carrier signal. Modulation methods are categorised according to which
component is used for transmitting the information.
To achieve high spectral efficiency, modulation schemes need to have high bandwidth
efficiency. Three properties need to be satisfied, when digital modulation techniques are
chosen for wireless systems. First, compact power density spectrum with a narrow main
lobe and fast roll-off of side-lobes is required to minimise the channel interference.
Secondly a good error rate performance in all environments is required. Finally a constant
envelope is important in mobile applications, where battery power is a limited sourceand
amplifiers are typically non-linear.
In this study, different modulation methods and the bit error performance with different
modulation methods and signal sets are evaluated.
One important factor in bit error performance is the shape of the pulse. To prevent
intersymbol interference the selected pulse shape has to satisfy the Nyquist criterion. The
ideal Nyquist pulse, however, has slow time decay. Therefore other pulses that satisfy the
criterion has to be constructed.
Different modulation methods are evaluated. The signal constellation is an important factor
when error probability is calculated. In coherent demodulation of two equally likely signals
transmitted on AWGN channel the error probability depends only on the Euclidean
distance between the two signals.
Any digital modulation aims at realising the best possible trade-off in a given situation
among the bit error probability, the bandwidth efficiency, the signal to noise ratio and the
complexity of the equipment. In the end the performance of these modulation methods’ are
compared. The power density function is not in the scope of this study.
The background material consists of three books. All of them descibes digital modulation
methods and could be used as such. The most part in this study is refers to St über [1], but
the presentation of Nyquist criterion is mainly based on Lee [2] and in the evaluation of
error performance I used Benedetto [3].
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Table of Contents
Abstract............................................................................................................................... 3
Table of Contents................................................................................................................ 5
Abbreviations ...................................................................................................................... 7
1
Digital Modulation ........................................................................................................ 9
2
Nyquist Pulse Shaping............................................................................................... 11
3
Error Probability Evaluation ....................................................................................... 17
3.1.1 Symbol Error Probability for Binary Signals............................................... 18
3.1.2 Symbol Error Probability for Rectangular Signal Sets ............................... 21
4
Digital Modulation Schemes ...................................................................................... 24
4.1 Quadrature Amplitude Modulation ...................................................................... 24
4.2 Phase Shift Keying ............................................................................................. 25
4.2.1 Offset Quadrature Phase Shift Keying....................................................... 27
4.2.2 ’π/4’-DQPSK .............................................................................................. 29
4.3 Orthogonal Modulation ....................................................................................... 30
4.4 Orthogonal Frequency Division Multiplexing....................................................... 31
4.4.1 Multiresolution Modulation......................................................................... 31
4.4.2 FFT-based OFDM System ........................................................................ 33
4.5 Continuous Phase Modulation ............................................................................ 33
4.5.1 Full Response CPM .................................................................................. 34
4.5.2 Minimum Shift Keying................................................................................ 35
4.5.3 Partial Response CPM .............................................................................. 37
5
Digital Modulation Trade-Offs .................................................................................... 38
Litterature.......................................................................................................................... 41
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Abbreviations
AWGN
BER
FFT
ISI
ML
LAN
PDS
Additive White Gaussian Noise
Bit Error Rate
Fast Fourier Transform
Intersymbol Interference
Maximum Likelihood
Local Area Network
Power Density Spectrum
Modulation methods:
π/4-DQPSK π/4-Differential QPSK
CPM
Continuous Phase Modulation
CPFSK
Continuous Phase Frequency Shift Keying
DCPSK
Differentially Coherent Phase Shift Keying
FSK
Frequency Shift Keying
GMSK
Gaussian Minimum Shift Keying
MRM
Multiresolution Modulation
MSK
Minimum Shift Keying
OFDM
Orthogonal Frequency Division Multiplexing
OQPSK
Offset QPSK
PAM
Pulse Amplitude Modulation
PSK
Phase Shift Keying
QAM
Quadrature Amplitude Modulation
QPSK
Quadrature Phase Shift Keying
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1 Digital Modulation
Modulation is the process where the message information is embedded into the radio
carrier. Message information can be transferred in
1. amplitude,
2. frequency or
3. the phase
1
3
2
of the carrier or a combination of these in either analog or digital form. In digital cellular
systems digital modulation is used because of its bandwidth efficiency.
To achieve high spectral efficiency, modulation schemes need to have high bandwidth
efficiency, measured in units of bits per second Hertz of bandwidth (bits/s/Hz). When
digital modulation techniques are chosen for wireless systems following three properties
need to be satisfied:
Compact Power Density Spectrum: To minimise the effect of adjacent channel
interference, the power radiated into the adjacent band should be 60 to 80 dB below
that in the desired band. Hence, modulation techniques with a narrow main lobe and
fast roll-off of side-lobes are needed.
Good Bit Error Rate Performance: A low bit error probability must be achieved in the
presence of fading, Doppler spread, intersymbol interference, adjacent and cochannel interference and thermal noise. In this presentation only intersymbol
interference and noise are considered.
Envelope Properties: Portable and mobile applications typically employ non-linear
power amplifiers to minimise battery drain. Non-linear amplification may degrade the
bit error rate (BER) performance of modulation schemes that transmit information in
the amplitude of the carrier. Also, spectral shaping is usually performed prior to upconversion and non-linear amplification. To prevent the regrowth of spectral sidelobes during non-linear amplification, relatively constant envelope modulation
schemes are preferred.
Two of the more widely used digital modulation techniques for cellular mobile radio are
π/4-DQPSK and GMSK. In both modulation methods the information is carried in the
phase component of the carrier signal.
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2 Nyquist Pulse Shaping
Example:
If the channel is an ideal bandlimited channel B ( jω ) = 1 , when ω < W and B ( jω ) = 0 ,
when ω ≥ W , then the ideally bandlimited pulse can be used which in time domain is a
sinc pulse as shown below.
Now consider two successive symbols with values a 0 = 1 and a1 = 2 . The contribution of
these two symbols to the signal is shown below
If the channel is ideally bandlimited, then the receiver only needs to sample at 0 and T .
Neighboring symbols do not interfere with one another at the proper sampling time, so
there is no intersymbol interference (ISI).
Consider a modulation scheme where the complex envelope has the form
~
s (t ) = A∑ x n p (t − nT )
n
Where p(t ) is a shaping pulse, {x n } is the complex data symbol sequence, and T is the
baud period. Now suppose the complex envelope is sampled every T seconds to yield the
sample sequence {y n },
yk = ~
s (kT + t 0 ) = A∑ x n p (kT + t 0 − nT )
n
Where t 0 is a timing offset assumed to lie in the interval [0, T ) . When t 0 = 0 and
p m = p(mT ) is the sampled pulse
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y k = A∑ x n p k −n = Ax k p 0 + A∑ x n p k −n
n
n≠k
The first term is equal to the data symbol transmitted at the k th baud epoch, scaled by the
factor p 0 . The second term is the contribution of all other data symbols on the sample y k .
This term is called intersymbol interference (ISI). To avoid the appearance of ISI, the
sampled pulse response {p k } must satisfy the condition
p k = δ k 0 p0
Therefore, to avoid ISI the pulse p(t ) must have equally spaced zero crossings at intervals
of T seconds. This requirement is known as the (first) Nyquist criterion. An equivalent
requirement in the frequency domain is
PΣ ( f ) =ˆ
1 ∞ 
n
P f +  = p 0
∑
T n = −∞ 
T
This allows us to design pulses in frequency domain that will yield to zero ISI. First
consider the pulse
P( f ) = T ⋅ rect ( fT ) ,
Figure 1 Pulse rect(fT)
This pulse yields a flat folded spectrum. In the time domain
p(t) = sinc(t/T)
This pulse achives the Nyquist criterion because it has equally spaced zero crossings at T
second intervals. Furthermore, from the requirement of a flat folded spectrum, it achieves
zero ISI while occupying the smallest possible bandwidth, hence, it is called an ideal
Nyquist pulse. However, the problem with this pulse is that the roll-off of side-lobes is slow.
Better roll-off factors are given by raised cosine and root raised cosine pulses which also
achieves Nyquist criterion, see figure 2. Raised cosine pulses are given by
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 sin (πt T )  cos(απt T ) 


p (t ) = 
2 

 πt T  1 − (2αt T ) 
where α is the so called roll-off factor. For α = 0, the pulse is identical to the ideally
bandlimited pulse. For other values of α, the energy rolls off more gradually with increasing
frequency. The pulse for α = 0 is the pulse with the smallest bandwidth that has zero
crossings at multiples of π W ; larger values of α require excess bandwidth varying from
0% to 100% as α varies from 0 to 1. In the time domain, the tails of the pulses are infinite
in extent. However, as α increases, the size of the tails diminishes.
Figure 2 Raised cosine pulses with different roll-off factors
There are an infinite number of pulses that satisfy the Nyquist criterion and hence have
zero crossings at multiples of π W . Some of these are shown in figure 3.
Figure 3 The Fourier transform of some pulses that satisfy the Nyquist criterion
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3 Error Probability Evaluation
It is assumed that the analog channel connecting the modulator output to the demodulator
input is an additive white Gaussian noise (AWGN channel with an infinite bandwidth. The
demodulator is a maximum likelihood (ML) demodulator and operates according to
minimum distance rule.
Figure 4 Geometry of the minimum distance rule
The signal in figure 4 is a complex signal with three possible symbols. When symbol sI is
received, the receiver observes signal r. While the channel adds noise to the transmitted
signal, the observed signal is r = sI + n ≠ si as in figure 4.
The minimum distance detector chooses the nearest value of the possible signal set. For
correct detection, the received signal has to be observed in the correct decision area.
Noise is assumed to be zero-mean Gaussian noise with variance N0/2.
Having a random signal, i.e. all symbols are equally likely, the symbol error probability is
expressed as
P(e ) = 1 − P(c ) = 1 −
1
M
∑ P(c | s ),
M
j =1
j
( )
where P c s j is the probability of a correct decision given that the signal vector s j ,
corresponding to the symbol m j , was transmitted.
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3.1.1 Symbol Error Probability for Binary Signals
Figure 5 Detection decision regions for binary signals
For a binary signal in figure 5 b), the symbol error probability can be determined as
follows. Signal s1 is detected with error, if noise element causes the detection to recognise
a value less than 0. This happens when the additive noise equals n < − d / 2 . Now we can
write the symbol error probability as
d

P(e ) = P n < − 
2

Using the definition of error function
P(ξ > x ) =
1
 x−m
erfc
,
2
 2σ 
and remembering that noise is zero-mean with variance N 0 / 2 , symbol error probability
can be written
P(e ) =
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2 N
2
0


.


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Figure 6 Error probability for antipodal and orthogonal binary signals
This means, that coherent demodulation of two equally likely signals transmitted on the
AWGN channel, the error probability depends only on the Euclidean distance between the
two signals. I.e. the selection of the set of symbols has an impact on error probability.
Comparing antipodal signals to orthogonal signals in figure 6 shows, that there is a 3 dB
penalty in the signal energy to be paid with orthogonal signals with respect to antipodal
signals which are shown in figure 5.
3.1.2 Symbol Error Probability for Rectangular Signal Sets
The binary representation can be applied to signal sets that have a “rectangular”
configuration. This means the cases where the decision regions are 2D-hyperplanes.
Figure 7 Received samples perturbed by additive Gaussian noise form a Gaussian cloud
around each of the points in the signal constellation
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Figure 8 2D signal set with 16 signals, a 16-QAM signal constellation
By studying the different decision regions in the signal set in figure 8, we can see that
there are only three different types of decision areas. First we need to define the
probabilities for correct decisions.
p1 =ˆ P(c s1 )
p 2 =ˆ P(c s 2 )
p3 =ˆ P(c s3 )
When different noise-components are independent of each other, we can define by using
the results of binary case

s −s  
s −s 
p1 = P  n1 < 1 2  ⋅ P  n2 > 1 5 
2  
2 

d 
d

= P  n1 <  ⋅ P n2 < 
2 
2

= (1 − p )
2
where p is the symbol error probability for binary signals with Euclidean distance d
between the different symbols. With similar calculation we can obtain
p2 = (1 − 2 p )(1 − p )
p3 = (1 − 2 p )
2
, where
p =ˆ
 d
1
erfc
2 N
2
0





Finally we can obtain the total symbol error probability for signal set in figure 8
P ( e ) = 4 p1 + 8 p2 + 4 p3
= 1−
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4 Digital Modulation Schemes
4.1
Quadrature Amplitude Modulation
When using quadrature amplitude modulation (QAM) the data information is transmitted in
the amplitude component of the signal. The quadrature representation is a special case of
pulse amplitude modulation (PAM). In QAM, two PAM-signals are combined in, and the
combination of these determines the transferred signal.
With QAM, the complex envelope is
~
s (t ) = A∑ b(t − nT , x n )
n
where
b(t , x n ) = x n ha (t )
ha ( t ) is the amplitude shaping pulse and xn = xI ,n + jxQ , n is the complex data symbol that is
transmitted at epoch n. It is apparent that with the amplitude and the phase of a QAM
signal depend on the complex symbol. QAM has the advantage of high bandwidth
efficiency, but amplifier nonlinearities will degrade its performance due to the non-constant
envelope.
Figure 9 Complex signal-space diagram for square QAM constellation
A variety of QAM signal constellations may be constructed. Square constellations can be
constructed when M is a power of 4, as shown in figure 9. When M is not a power of 4,
the signal constellation is not a square. Usually, the constellation is given the shape of a
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cross to minimise the average energy in the constellation for a given minimum Euclidean
distance between signal vectors.
Error probability for 16-QAM was calculated in previous chapter.
4.2
Phase Shift Keying
In PSK modulation the information is signalled in the phase-component. The complex
envelope is
~
s (t ) = A∑ b(t − nT , x n )
n
where
b(t , x n ) = ha (t )e jθ
The carrier phase takes on values
θn =
2π
xn + θ 0
M
Figure 10 Complex signal-space diagram QPSK, OQPSK and π/4-DQPSK
The source binary symbols are Gray-coded. As a consequence, adjacent phase signals
differ by only one binary digit.
4.2.1 Offset Quadrature Phase Shift Keying
QPSK or 4-PSK is equivalent to 4-QAM. The QPSK signal can have either ±90° or ±180°
phase shifts from one baud interval to the next. With offset QPSK (OQPSK) signals the
possibility of ±180° phase shifts is eliminated. In fact the phase can change by only ±90°
every Tb seconds. This corresponds to the bit rate of the signal.
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Figure 11 In-phase and quadrature baseband components in QPSK, OQPSK and MSK
signals
With OQPSK two bits are transferred every baud epoch as in QPSK, but the quadrature
component is delayed by half of the baud rate. With this shift, the two separate
components never change at the same time. The difference between QPSK and OQPSK
is shown in figure 9. The in-phase components are the same, but the quadrature
component is shifted by half of a symbol in OQPSK.
Note from figure 10 that the phase trajectories do not pass through the origin. This
property reduces the peak-to-average ratio of the complex envelope, making the OQPSK
signal less sensitive to amplifier non-linearities than the QPSK signal.
What is gained from OQPSK with respect to QPSK: Both methods have same error
performance, since signal constellation is equal. The gain of choosing OQPSK is on power
density spectrum. In OQPSK the ±180° phase shifts are eliminated and hence the pds is
more compact.
4.2.2 ’π/4’-DQPSK
QPSK transmits 2 bits/baud by transmitting sinusoidal pulses having one of 4 absolute
phases. π/4-DQPSK also transmits 2 bits/baud, but information is encoded into the
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differential carrier phase and sinusoidal pulses having one of 8 absolute carrier phases are
transmitted at each baud epoch.
The phase differences are ±π/4 and ±3π/4. The absolute carrier phase during the even and
odd baud intervals belongs to the sets {0, π/2, π, 3π/2} and {π/4, 3π/4, 5π/4, 7π/4}. With
π/4-DQPSK the amplitude shaping pulse is often chosen to be the root raised cosine
pulse.
The signal space diagram is shown in figure 10, where the dotted lines show allowable
phase transitions. Note that the phase trajectories do not pass through the origin. Like in
OQPSK, this property reduces the peak-to-average ratio of the complex envelope, making
the π/4-DQPSK signal less sensitive to amplifier non-linearities. The error performance is
equal to QPSK, since signal constellation during one symbol is same (or shifted by π/4).
4.3
Orthogonal Modulation
Orthogonal modulation schemes transmit information by using a set of waveforms,
{sm (t )}mM=−01 that are orthogonal in time. FSK modulation technique provides simple means of
generating an orthogonal signal set. Orthogonal M-ary frequency shift keying (MFSK)
modulation uses a set of M waveforms that have different frequencies.
For coherent demodulation orthogonality is met when the correlation coefficients of the
real signal is zero. This condition is fulfilled when the frequency separation between
adjacent signals is such that
m
,
m any integer .
2
Thus the minimum frequency separation for orthogonality with coherent detection is such
that 2 f d T = 0,5 .
2 fdT =
The demodulation of of these types of signals increases the complexity in the receiver.
The need for a bank of perfectly coherent oscillators renders it rather impractical. The bit
error performance is though different from amplitude and phase modulation techniques.
There exists an improvement in performance when M is increased, which is exactly the
opposite behaviour of PAM and PSK signals. However, this improvement is obtained at
the expense of a larger bandwidth. Increasing M requires more frequencies and therefore
more bandwidth.
For incoherent demodulation the orthogonality condition need to be fulfilled independently
of the phases of the signals. The condition is fulfilled when the frequency separation
between adjacent signals is
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2 fdT = m ,
m any integer
which is twice as much frequency separation as of coherent demodulation. The
performance is somewhat inferior to the coherent case, but this is traded off by the easier
implementation.
4.4
Orthogonal Frequency Division Multiplexing
Orthogonal Frequency division multiplexing (OFDM) is a modulation technique that has
been suggested for use in cellular radio, digital audio broadcasting, digital video
broadcasting and wireless LAN systems. OFDM is a block modulation scheme where data
symbols are transmitted in parallel by employing a (large) number of orthogonal subcarriers.
4.4.1 Multiresolution Modulation
Multiresolution modulation (MRM) refers to a class of modulation techniques where
multiple classes of bit streams are transmitted simultaneously that differ in their rates and
error probabilities.
Figure 12 16-QAM embedded MRM signal constellation, defining two priority classes
Figure 12 shows and example of a 16-QAM MRM signal constellation, that can be used to
transmit two diferent classes of bit streams, called low priority (LP) and high priority (HP).
Two HP bits are used to select the quadrant of the transmitted signal point, while two LP
bits are used to select the signal point within the selected quadrant. In order to control the
relative error probability between the two priorities, a parameter λ = d l d h is used. In
general, λ should be less than 0,5, since the MRM constellation becomes symmetric 16QAM at this point. As λ becomes smaller, more power is allocated to the HP bits and they
are received with a smaller error probability.
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4.4.2 FFT-based OFDM System
A key advantage of using OFDMis that the modulation and demodulation can be achieved
in the discrete-domain by using a discrete Fourier tranform. The fast Fourier transform
(FFT) algorithm efficiently implements the discrete Fourier transform. When FFT is used,
the rectangular shaping pulses turn into non-causal pulses in figure 13.
Figure 13 Time domain OFDM amplitude shaping pulse
Another key advantage of OFDM is the ease by which the effects of ISI can be mitigated.
This can be done, by using a cyclic guard interval. The guard is appended to the
generated signal in the transmitter. On the receiver it is assumed that the first sample is
corrupted by ISI and therefore replaces the ISI-component with the guard.
4.5
Continuous Phase Modulation
Continuous Phase modulation (CPM) refers to a broad class of frequency modulation
techniques where the carrier phase varies in a continuous manner. CPM schemes are
attractive because they have constant envelope and excellent spectral characteristics, i.e.
narrow main lobe and fast roll-off of side lobes.
The complex envelope of a general CPM waveform has the form
~
s (t ) = Ae j (φ (t )+θ 0 )
where φ (t ) is called the excess phase and defined as
t
∞
φ (t ) = 2π ∫ ∑ hk x k h f (τ − kT )dτ
0 k =0
h f (t ) is the frequency shaping function, that is zero for t < 0 and t > LT . A full response
CPM has L = 1 , while partial response CPM has L > 1 . The phase is continuous in CPM
signals so long as the frequency shaping function does not contain impulses, which
accounts for all practical cases. hk is the modulation index.
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4.5.1 Full Response CPM
Continuous phase frequency shift keying (CPFSK) is a special type of full response CPM
obtained by using the rectangular frequency shaping function with L = 1 .
h f (t ) =
1
u LT (t )
2 LT
Figure 14 Phase tree of binary CPFSK with modulation index h .
CPM signals can be visualised by sketching the evolution of the excess phase Φ(t) for all
possible data sequences. This plot is called a phase tree and a typical phase tree is shown
in figure 14 for binary CPFSK. In each baud interval, the phase increases by πh if the data
symbol is +1 and decreases by πh if the data symbol is -1.
4.5.2 Minimum Shift Keying
Minimum shift keying (MSK) is a special case of CPFSK, with modulation index h = 1 2
and number of levels M = 2 . The MSK signal can be described in terms of the phase tree
in figure 14 with h = 1 2 . At the end of each symbol interval the excess phase φ (t ) takes on
values that are integer multiplies of π 2 and a phase trellis may be plotted.
Figure 15 Phase trellis diagram for MSK.
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Consider the MSK band-pass waveform in the interval [nT , (n + 1)T ] , given by
 
x  π n −1
πn 
s (t ) = A cos 2π  f c + n t + ∑ x k −
xn  .
4T  2 k =0
2 
 
Observe that the MSK signal has one of two possible frequencies
f L = fc −
1
4T
and
fU = f c +
1
4T
The difference between these frequencies is ∆f = f U − f L = 1 2T . This is the minimum
frequency separation to ensure orthogonality between two co-phased sinusoids of duration
T and, hence, the name minimum shift keying.
By viewing figure 15 this type of modulation can be thought of as a special case of OQPSK
in which the rectangular waveform is replaced by a sinusoidal pulse, which is shown in
figure 11.
Figure 16 Performance of different CPFSK signals
Figure 16 shows performances of different CPFSK signals compared to MSK. By
increasing the number of levels, the bandwidth efficiency is increased clearly.
4.5.3 Partial Response CPM
Partial response CPM signals have a frequency shaping pulse hf(t) with duration LT where
L > 1. Partial response signals have better spectral characteristics than full response CPM
signals, i.e., a narrower main lobe and faster roll-off of side lobes.
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5 Digital Modulation Trade-Offs
Any digital modulation aims at realising the best possible trade-off in a given situation
among the bit error probability Pb(e), the bandwidth efficiency Rs/W, the ratio εb/N0 and the
complexity of the equipment. Following results are taken from Benedetto [3].
Figure 17 Comparison of different modulation methods on the bandwidth-efficiency plane for
-5.
a bit error probability Pb(e) = 10
A comparison of different modulation methods is illustrated in figure 17 where a bit error
probability Pb(e) =10-5 has been fixed. The Shannon capacity bound shows the maximum
bandwidth efficiency, which can teoretically be achieved.
The graph shows the fact that amplitude modulation (ASK), and phase modulation (CPSK
and DCPSK) systems are bandwidth-efficient signalling techniques, since they cover the
region of the plane where Rs/W > 1. In this region, the system bandwidth is limited and it
can be traded for power (i.e. εb/N0). In fact, for a fixed bandwidth, the bandwidth efficiency
can be increased with an increase in the number of levels M. The price paid to achieve the
same Pb(e) is an increase in εb/N0.
On the other hand, FSK signals make an inefficient use of bandwidth, since they cover the
region of the plane where Rs/W < 1. But these systems trade bandwidth for a reduction of
the εb/N0 required to achieve the same Pb(e).
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Litterature
[1]
Stüber, G. Principles of Mobile Communication. Second edition. Norwell,
Massachusetts, USA. Kluwer Academic Publishers. 2001. p. 751.
[2]
Lee, E.A. Messerschmitt, D.G. Digital Communication. Second edition.
Norwell, Massachusetts, USA. Kluwer Academic Publishers. 1994. p. 893.
[3]
Benedetto, S. Biglieri, E. Castellani, V. Digital Transmission Theory.
Englewood Cliffs, New Jersey, USA. Prentice-Hall, Inc. 1987. p. 639.
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