Fitting Signals into Given Spectrum Modulation Methods · Fitting Signals into Given Spectrum...

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S-72.333Post-graduate Course in Radio Communications2001-2002

Fitting Signals into Given SpectrumModulation Methods

Lars Maura41747e

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[email protected] 2(39)

Abstract

Modulation is the process where the message information is embedded into the radiocarrier. Message information can be transferred in the amplitude, frequency or the phasecomponent of the carrier signal. Modulation methods are categorised according to whichcomponent is used for transmitting the information.

To achieve high spectral efficiency, modulation schemes need to have high bandwidthefficiency. Three properties need to be satisfied, when digital modulation techniques arechosen for wireless systems. First, compact power density spectrum with a narrow mainlobe 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 constantenvelope is important in mobile applications, where battery power is a limited sourceandamplifiers are typically non-linear.

In this study, different modulation methods and the bit error performance with differentmodulation methods and signal sets are evaluated.

One important factor in bit error performance is the shape of the pulse. To preventintersymbol interference the selected pulse shape has to satisfy the Nyquist criterion. Theideal Nyquist pulse, however, has slow time decay. Therefore other pulses that satisfy thecriterion has to be constructed.

Different modulation methods are evaluated. The signal constellation is an important factorwhen error probability is calculated. In coherent demodulation of two equally likely signalstransmitted on AWGN channel the error probability depends only on the Euclideandistance between the two signals.

Any digital modulation aims at realising the best possible trade-off in a given situationamong the bit error probability, the bandwidth efficiency, the signal to noise ratio and thecomplexity of the equipment. In the end the performance of these modulation methods’ arecompared. The power density function is not in the scope of this study.

The background material consists of three books. All of them descibes digital modulationmethods and could be used as such. The most part in this study is refers to St über [1], butthe presentation of Nyquist criterion is mainly based on Lee [2] and in the evaluation oferror 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 Additive White Gaussian NoiseBER Bit Error RateFFT Fast Fourier TransformISI Intersymbol InterferenceML Maximum LikelihoodLAN Local Area NetworkPDS Power Density Spectrum

Modulation methods:π/4-DQPSK π/4-Differential QPSKCPM Continuous Phase ModulationCPFSK Continuous Phase Frequency Shift KeyingDCPSK Differentially Coherent Phase Shift KeyingFSK Frequency Shift KeyingGMSK Gaussian Minimum Shift KeyingMRM Multiresolution ModulationMSK Minimum Shift KeyingOFDM Orthogonal Frequency Division MultiplexingOQPSK Offset QPSKPAM Pulse Amplitude ModulationPSK Phase Shift KeyingQAM Quadrature Amplitude ModulationQPSK Quadrature Phase Shift Keying

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1 Digital Modulation

Modulation is the process where the message information is embedded into the radiocarrier. Message information can be transferred in

1. amplitude,2. frequency or3. the phase

of the carrier or a combination of these in either analog or digital form. In digital cellularsystems digital modulation is used because of its bandwidth efficiency.

To achieve high spectral efficiency, modulation schemes need to have high bandwidthefficiency, measured in units of bits per second Hertz of bandwidth (bits/s/Hz). Whendigital modulation techniques are chosen for wireless systems following three propertiesneed to be satisfied:

Compact Power Density Spectrum: To minimise the effect of adjacent channelinterference, the power radiated into the adjacent band should be 60 to 80 dB belowthat in the desired band. Hence, modulation techniques with a narrow main lobe andfast roll-off of side-lobes are needed.

Good Bit Error Rate Performance: A low bit error probability must be achieved in thepresence of fading, Doppler spread, intersymbol interference, adjacent and co-channel interference and thermal noise. In this presentation only intersymbolinterference and noise are considered.

Envelope Properties: Portable and mobile applications typically employ non-linearpower amplifiers to minimise battery drain. Non-linear amplification may degrade thebit error rate (BER) performance of modulation schemes that transmit information inthe amplitude of the carrier. Also, spectral shaping is usually performed prior to up-conversion and non-linear amplification. To prevent the regrowth of spectral side-lobes during non-linear amplification, relatively constant envelope modulationschemes are preferred.

Two of the more widely used digital modulation techniques for cellular mobile radio areπ/4-DQPSK and GMSK. In both modulation methods the information is carried in thephase component of the carrier signal.

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2 Nyquist Pulse Shaping

Example:

If the channel is an ideal bandlimited channel ( ) 1B jω = , when Wω < and ( ) 0B jω = ,

when Wω ≥ , 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 10 =a and 21 =a . 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, sothere is no intersymbol interference (ISI).

Consider a modulation scheme where the complex envelope has the form

( ) ( )∑ −=n

n nTtpxAts~

Where ( )tp is a shaping pulse, { }nx is the complex data symbol sequence, and T is the

baud period. Now suppose the complex envelope is sampled every T seconds to yield thesample sequence { }ny ,

( ) ( )∑ −+=+=n

nk nTtkTpxAtkTsy 00~

Where 0t is a timing offset assumed to lie in the interval [ )T,0 . When 00 =t and

( )mTppm = is the sampled pulse

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∑∑≠

−− +==kn

nknkn

nknk pxApAxpxAy 0

The first term is equal to the data symbol transmitted at the k th baud epoch, scaled by thefactor 0p . The second term is the contribution of all other data symbols on the sample ky .

This term is called intersymbol interference (ISI). To avoid the appearance of ISI, thesampled pulse response { }kp must satisfy the condition

00 pp kk δ=

Therefore, to avoid ISI the pulse ( )tp must have equally spaced zero crossings at intervals

of T seconds. This requirement is known as the (first) Nyquist criterion. An equivalentrequirement in the frequency domain is

( ) 0

1ˆ p

T

nfP

TfP

n

=

+= ∑

∞

−∞=Σ

This allows us to design pulses in frequency domain that will yield to zero ISI. Firstconsider the pulse

( ) ( )fTrectTfP ⋅= ,

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 Tsecond intervals. Furthermore, from the requirement of a flat folded spectrum, it achieveszero ISI while occupying the smallest possible bandwidth, hence, it is called an idealNyquist 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 alsoachieves Nyquist criterion, see figure 2. Raised cosine pulses are given by

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( ) ( ) ( )( )

−

=

221

cossin

Tt

Tt

Tt

Tttp

ααπ

ππ

where α is the so called roll-off factor. For α = 0, the pulse is identical to the ideallybandlimited pulse. For other values of α, the energy rolls off more gradually with increasingfrequency. The pulse for α = 0 is the pulse with the smallest bandwidth that has zerocrossings at multiples of Wπ ; 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 infinitein 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 havezero crossings at multiples of Wπ . 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 demodulatorinput is an additive white Gaussian noise (AWGN channel with an infinite bandwidth. Thedemodulator is a maximum likelihood (ML) demodulator and operates according tominimum 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 isreceived, the receiver observes signal r. While the channel adds noise to the transmittedsignal, 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. Forcorrect 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 isexpressed as

( ) ( ) ( )∑=

−=−=M

jjscP

McPeP

1

|1

11 ,

where ( )jP c s is the probability of a correct decision given that the signal vector js ,

corresponding to the symbol jm , 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 asfollows. Signal 1s is detected with error, if noise element causes the detection to recognise

a value less than 0. This happens when the additive noise equals / 2n d< − . Now we canwrite the symbol error probability as

( )

−<=

2

dnPeP

Using the definition of error function

( )

−=>

σξ

22

1 mxerfcxP ,

and remembering that noise is zero-mean with variance 0 / 2N , symbol error probability

can be written

( )

=

022

1

N

derfceP .

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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 theAWGN channel, the error probability depends only on the Euclidean distance between thetwo 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 dBpenalty in the signal energy to be paid with orthogonal signals with respect to antipodalsignals 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 cloudaround 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 thatthere are only three different types of decision areas. First we need to define theprobabilities for correct decisions.

( )11 ˆ scPp =

( )22 ˆ scPp =

( )33 ˆ scPp =

When different noise-components are independent of each other, we can define by usingthe results of binary case

( )

1 2 1 51 1 2

1 2

2

2 2

2 2

1

s s s sp P n P n

d dP n P n

p

− −= < ⋅ >

= < ⋅ <

= −

where p is the symbol error probability for binary signals with Euclidean distance d

between the different symbols. With similar calculation we can obtain

( )( )( )

2

2

3

1 2 1

1 2

p p p

p p

= − −

= −, where

=

022

1ˆ

N

derfcp

Finally we can obtain the total symbol error probability for signal set in figure 8

( )

( )1 2 3

2

4 8 4

11 2 3

4

P e p p p

p

= + +

= − −

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4 Digital Modulation Schemes

4.1 Quadrature Amplitude Modulation

When using quadrature amplitude modulation (QAM) the data information is transmitted inthe amplitude component of the signal. The quadrature representation is a special case ofpulse amplitude modulation (PAM). In QAM, two PAM-signals are combined in, and thecombination of these determines the transferred signal.

With QAM, the complex envelope is

( ) ( )∑ −=n

nnTtbAts x,~

where

( ) ( )thxtb ann =x,

( )ah t is the amplitude shaping pulse and , ,n I n Q nx x jx= + is the complex data symbol that is

transmitted at epoch n. It is apparent that with the amplitude and the phase of a QAMsignal depend on the complex symbol. QAM has the advantage of high bandwidthefficiency, but amplifier nonlinearities will degrade its performance due to the non-constantenvelope.

Figure 9 Complex signal-space diagram for square QAM constellation

A variety of QAM signal constellations may be constructed. Square constellations can beconstructed 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 Euclideandistance 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 complexenvelope is

( ) ( )∑ −=n

nnTtbAts x,~

where

( ) ( ) θjan ethtb =x,

The carrier phase takes on values

0

2 θπθ += nn xM

Figure 10 Complex signal-space diagram QPSK, OQPSK and π/4-DQPSK

The source binary symbols are Gray-coded. As a consequence, adjacent phase signalsdiffer 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 thepossibility of ±180° phase shifts is eliminated. In fact the phase can change by only ±90°every bT 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 MSKsignals

With OQPSK two bits are transferred every baud epoch as in QPSK, but the quadraturecomponent is delayed by half of the baud rate. With this shift, the two separatecomponents never change at the same time. The difference between QPSK and OQPSKis shown in figure 9. The in-phase components are the same, but the quadraturecomponent is shifted by half of a symbol in OQPSK.

Note from figure 10 that the phase trajectories do not pass through the origin. Thisproperty reduces the peak-to-average ratio of the complex envelope, making the OQPSKsignal less sensitive to amplifier non-linearities than the QPSK signal.

What is gained from OQPSK with respect to QPSK: Both methods have same errorperformance, since signal constellation is equal. The gain of choosing OQPSK is on powerdensity spectrum. In OQPSK the ±180° phase shifts are eliminated and hence the pds ismore compact.

4.2.2 ’π/4’-DQPSK

QPSK transmits 2 bits/baud by transmitting sinusoidal pulses having one of 4 absolutephases. 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 aretransmitted at each baud epoch.

The phase differences are ±π/4 and ±3π/4. The absolute carrier phase during the even andodd baud intervals belongs to the sets {0, π/2, π, 3π/2} and {π/4, 3π/4, 5π/4, 7π/4}. Withπ/4-DQPSK the amplitude shaping pulse is often chosen to be the root raised cosinepulse.

The signal space diagram is shown in figure 10, where the dotted lines show allowablephase transitions. Note that the phase trajectories do not pass through the origin. Like inOQPSK, this property reduces the peak-to-average ratio of the complex envelope, makingthe π/4-DQPSK signal less sensitive to amplifier non-linearities. The error performance isequal 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,10{ ( )}M

m ms t −= 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 thereal signal is zero. This condition is fulfilled when the frequency separation betweenadjacent signals is such that

22d

mf T = , m any integer .

Thus the minimum frequency separation for orthogonality with coherent detection is suchthat 2 0,5df T = .

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 biterror performance is though different from amplitude and phase modulation techniques.There exists an improvement in performance when M is increased, which is exactly theopposite behaviour of PAM and PSK signals. However, this improvement is obtained atthe expense of a larger bandwidth. Increasing M requires more frequencies and thereforemore bandwidth.

For incoherent demodulation the orthogonality condition need to be fulfilled independentlyof the phases of the signals. The condition is fulfilled when the frequency separationbetween adjacent signals is

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2 df T m= , m any integer

which is twice as much frequency separation as of coherent demodulation. Theperformance is somewhat inferior to the coherent case, but this is traded off by the easierimplementation.

4.4 Orthogonal Frequency Division Multiplexing

Orthogonal Frequency division multiplexing (OFDM) is a modulation technique that hasbeen suggested for use in cellular radio, digital audio broadcasting, digital videobroadcasting and wireless LAN systems. OFDM is a block modulation scheme where datasymbols are transmitted in parallel by employing a (large) number of orthogonal sub-carriers.

4.4.1 Multiresolution Modulation

Multiresolution modulation (MRM) refers to a class of modulation techniques wheremultiple classes of bit streams are transmitted simultaneously that differ in their rates anderror 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 totransmit 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 LPbits 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 l hd dλ = is used. In

general, λ should be less than 0,5, since the MRM constellation becomes symmetric 16-QAM at this point. As λ becomes smaller, more power is allocated to the HP bits and theyare 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 achievedin 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 thegenerated signal in the transmitter. On the receiver it is assumed that the first sample iscorrupted 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 modulationtechniques where the carrier phase varies in a continuous manner. CPM schemes areattractive 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

( ) ( )( )0~ θφ += tjAets

where ( )tφ is called the excess phase and defined as

( ) ( )∫∑∞

=−=

t

kfkk dkThxht

0 0

2 ττπφ

( )th f is the frequency shaping function, that is zero for 0<t and LTt > . A full response

CPM has 1=L , while partial response CPM has 1>L . The phase is continuous in CPMsignals so long as the frequency shaping function does not contain impulses, whichaccounts for all practical cases. kh 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 CPMobtained by using the rectangular frequency shaping function with 1=L .

( ) ( )tuLT

th LTf 2

1=

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 allpossible data sequences. This plot is called a phase tree and a typical phase tree is shownin figure 14 for binary CPFSK. In each baud interval, the phase increases by hπ if the datasymbol is +1 and decreases by hπ 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 21=h

and number of levels 2M = . The MSK signal can be described in terms of the phase treein figure 14 with 21=h . At the end of each symbol interval the excess phase ( )tφ takes on

values that are integer multiplies of 2π 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 ( )[ ]TnnT 1, + , given by

( )

−+

+= ∑

−

=n

n

kk

nc x

nxt

T

xfAts

2242cos

1

0

πππ .

Observe that the MSK signal has one of two possible frequencies

Tff cL 4

1−= andT

ff cU 4

1+=

The difference between these frequencies is Tfff LU 21=−=∆ . This is the minimum

frequency separation to ensure orthogonality between two co-phased sinusoids of durationT and, hence, the name minimum shift keying.

By viewing figure 15 this type of modulation can be thought of as a special case of OQPSKin which the rectangular waveform is replaced by a sinusoidal pulse, which is shown infigure 11.

Figure 16 Performance of different CPFSK signals

Figure 16 shows performances of different CPFSK signals compared to MSK. Byincreasing 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 whereL > 1. Partial response signals have better spectral characteristics than full response CPMsignals, 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 situationamong the bit error probability Pb(e), the bandwidth efficiency Rs/W, the ratio εb/N0 and thecomplexity of the equipment. Following results are taken from Benedetto [3].

Figure 17 Comparison of different modulation methods on the bandwidth-efficiency plane fora bit error probability Pb(e) = 10-5.

A comparison of different modulation methods is illustrated in figure 17 where a bit errorprobability Pb(e) =10-5 has been fixed. The Shannon capacity bound shows the maximumbandwidth efficiency, which can teoretically be achieved.

The graph shows the fact that amplitude modulation (ASK), and phase modulation (CPSKand DCPSK) systems are bandwidth-efficient signalling techniques, since they cover theregion of the plane where Rs/W > 1. In this region, the system bandwidth is limited and itcan be traded for power (i.e. εb/N0). In fact, for a fixed bandwidth, the bandwidth efficiencycan be increased with an increase in the number of levels M. The price paid to achieve thesame Pb(e) is an increase in εb/N0.

On the other hand, FSK signals make an inefficient use of bandwidth, since they cover theregion of the plane where Rs/W < 1. But these systems trade bandwidth for a reduction ofthe 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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