theory of telecommunications tts networks...

Theory of Telecommunications Networks

Anton ČižmárJán Papaj Department of electronics

and multimedia telecommunications

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CONTENTS Preface ....................................................................................................................................... 5

1 Introduction ...................................................................................................................... 6

1.1 Mathematical models for communication channels .................................................... 8

1.2 Channel capacity for digital communication ............................................................ 10

1.2.1 Shannon Capacity and Interpretation ................................................................ 10

1.2.2 Hartley Channel Capacity .................................................................................. 12

1.2.3 Solved Problems ................................................................................................. 13

1.3 Noise in digital communication system ..................................................................... 15

1.3.1 White Noise ........................................................................................................ 17

1.3.2 Thermal Noise .................................................................................................... 18

1.3.3 Solved Problems ................................................................................................. 19

1.4 Summary .................................................................................................................... 20

1.5 Exercises .................................................................................................................... 21

2 Signal and Spectra .......................................................................................................... 23

2.1 Deterministic and random signals ............................................................................. 23

2.2 Periodic and nonperiodic signals .............................................................................. 23

2.3 Analog and discrete Signals ...................................................................................... 23

2.4 Energy and power Signals ......................................................................................... 23

2.5 Spectral Density ......................................................................................................... 25

2.5.1 Energy Spectral Density ..................................................................................... 25

2.5.2 Power Spectral Density ...................................................................................... 25

2.5.3 Solved Problems ................................................................................................. 26

2.6 Autocorrelation .......................................................................................................... 27

2.6.1 Autocorrelation of an Energy Signal ................................................................. 27

2.6.2 Autocorrelation of a Periodic Signal ................................................................. 27

2.7 Baseband versus Bandpass ........................................................................................ 28

2.8 Summary .................................................................................................................... 29

2.9 Exercises .................................................................................................................... 30

3 Probability and stochastic processes ............................................................................ 31

3.1 Probability ................................................................................................................. 31

3.1.1 Joint Events and Joint Probabilities .................................................................. 31

3.1.2 Conditional Probabilities ................................................................................... 32

3.1.3 Statistical Independence ..................................................................................... 33

3.1.4 Solved Problems ................................................................................................. 33

3.2 Random Variables, Probability Distributions, and probability Densities ................ 36

3.2.1 Statistically Independent Random Variables ..................................................... 37

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3.2.2 Statistical Averages of Random Variables ......................................................... 37

3.2.3 Some Useful Probability Distributions .............................................................. 38

3.3 Stochastic processes .................................................................................................. 41

3.3.1 Stationary Stochastic Processes ......................................................................... 41

3.3.2 Statistical Averages ............................................................................................ 41

3.3.3 Power Density Spectrum .................................................................................... 43

3.3.4 Response of a Linear Time-Invariant System (channel) to a Random Input Signal ............................................................................................................................ 43

3.3.5 Sampling Theorem for Band-Limited Stochastic Processes .............................. 44

3.3.6 Discrete-Time Stochastic Signals and Systems .................................................. 45

3.3.7 Cyclostationary Processes ................................................................................. 46

3.3.8 Solved Problems ................................................................................................. 47

3.4 Summary .................................................................................................................... 50

3.5 Exercises .................................................................................................................... 52

4 Signal space concept ....................................................................................................... 55

4.1 Representation Of Band-Pass Signals And Systems .................................................. 55

4.1.1 Representation of Band-Pass Signals ................................................................ 55

4.1.2 Representation of Band-Pass Stationary Stochastic Processes ......................... 58

4.2 Introduction of the Hilbert transform ........................................................................ 59

4.3 Different look at the Hilbert transform ...................................................................... 59

4.3.1 Hilbert Transform, Analytic Signal and the Complex Envelope ........................ 59

4.3.2 Hilbert Transform in Frequency Domain .......................................................... 61

4.3.3 Hilbert Transform in Time Domain ................................................................... 62

4.3.4 Analytic Signal ................................................................................................... 64

4.3.5 Solved Problems ................................................................................................. 66

4.4 Signal Space Representation ..................................................................................... 69

4.4.1 Vector Space Concepts ....................................................................................... 69

4.4.2 Signal Space Concepts ....................................................................................... 70

4.4.3 Orthogonal Expansions of Signals ..................................................................... 70

4.4.4 Gram-Schmidt procedure ................................................................................... 71

4.4.5 Solved Problems ................................................................................................. 74

4.4.6 Summary ............................................................................................................. 78

4.5 Exercises .................................................................................................................... 79

5 Digital modulation schemes ........................................................................................... 82

5.1 Signal Space Representation ..................................................................................... 82

5.2 Memoryless Modulation Methods ............................................................................. 82

5.2.1 Pulse-amplitude-modulated (PAM) signals (ASK) ............................................ 83

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5.2.2 Phase-modulated signal (PSK) .......................................................................... 85

5.2.3 Quadrature Amplitude Modulation (QAM) ........................................................ 86

5.3 Multidimensional Signals .......................................................................................... 88

5.3.1 Orthogonal multidimensional signals ................................................................ 88

5.3.2 Linear Modulation with Memory ....................................................................... 92

5.3.3 Non-Linear Modulation Methods with Memory ................................................. 95

5.4 Spectral Characteristic Of Digitally Modulated Signals ........................................ 101

5.4.1 Power Spectra of Linearly Modulated Signals ................................................ 101

5.4.2 Power Spectra of CPFSK and CPM Signals .................................................... 102

5.4.3 Solved Problems ............................................................................................... 106

5.5 Summary .................................................................................................................. 110

5.6 Exercises .................................................................................................................. 110

6 Optimum Receivers for the AWGN Channel ............................................................ 113

6.1 Optimum Receivers For Signals Corrupted By Awgn ............................................. 113

6.1.1 Correlation demodulator .................................................................................. 114

6.1.2 Matched-Filter demodulator ............................................................................ 116

6.1.3 The Optimum detector ...................................................................................... 118

6.1.4 The Maximum-Likelihood Sequence Detector ................................................. 120

6.2 Performance Of The Optimum Receiver For Memoryless Modulation .................. 123

6.2.1 Probability of Error for Binary Modulation .................................................... 123

6.2.2 Probability of Error for M-ary Orthogonal Signals ........................................ 126

6.2.3 Probability of Error for M-ary Biorthogonal Signals ...................................... 127

6.2.4 Probability of Error for Simplex Signals ......................................................... 129

6.2.5 Probability of Error for M-ary Binary-Coded Signals .................................... 129

6.2.6 Probability of Error for M-ary PAM ................................................................ 130

6.2.7 Probability of Error for M-ary PSK ................................................................. 130

6.2.8 Probability of Error for QAM .......................................................................... 132

6.3 Solved Problems ...................................................................................................... 134

6.4 Summary .................................................................................................................. 141

6.5 Exercises .................................................................................................................. 142

7 Performance analysis of digital modulations ............................................................. 144

7.1 Goals Of The Communications System Designer .................................................... 144

7.2 Error Probability Plane ........................................................................................... 144

7.3 Nyquist Minimum Bandwidth .................................................................................. 146

7.4 Shannon-Hartley Capacity Theorem ....................................................................... 146

7.4.1 Shannon Limit .................................................................................................. 148

7.5 Bandwidth-Efficiency Plane .................................................................................... 150

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7.5.1 Bandwidth Efficiency of MPSK and MFSK Modulation .................................. 151

7.5.2 Analogies Between Bandwidth-Efficiency and Error-Probability Planes ....... 152

7.6 Modulation And Coding Trade-Offs ........................................................................ 153

7.7 Defining, Designing, And Evaluating Digital Communication Systems ................. 154

7.7.1 M-ary Signaling ................................................................................................ 154

7.7.2 Bandwidth-Limited Systems ............................................................................. 155

7.7.3 Power-Limited Systems .................................................................................... 156

7.7.4 Requirements for MPSK and MFSK Signaling ................................................ 157

7.7.5 Bandwidth-Limited Uncoded System Example ................................................ 158

7.7.6 Power-Limited Uncoded System Example ....................................................... 160

7.8 Solved Problems ...................................................................................................... 162

7.9 Summary .................................................................................................................. 165

7.10 Exercise ................................................................................................................... 166

8 Why use error-correction coding ................................................................................ 167

8.1 Trade-Off 1: Error Performance versus Bandwidth ............................................... 167

8.2 Trade-Off 2: Power versus Bandwidth .................................................................... 168

8.3 Coding Gain ............................................................................................................ 168

8.4 Trade-Off 3: Data Rate versus Bandwidth .............................................................. 168

8.5 Trade-Off 4: Capacity versus Bandwidth ................................................................ 169

8.6 Code Performance at Low Values of Eb/N0 ............................................................. 169

8.7 Solved problem ........................................................................................................ 170

8.8 Exercise ................................................................................................................... 171

Appendix A ........................................................................................................................... 173

The Q-function .................................................................................................................... 173

The Error Function ............................................................................................................ 174

Appendix B ............................................................................................................................ 175

Comparison of M-ary signaling techniques ....................................................................... 175

Error performance of M-ary signaling techniques ............................................................ 175

References ............................................................................................................................. 176

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PREFACE Providing the theory of digital communication systems, this textbook prepares senior undergraduate and graduate students for the engineering practices required in the real word.

With this textbook, students can understand how digital communication systems operate in practice, learn how to design subsystems, and evaluate end-to-end performance.

The book contains many examples to help students achieve an understanding of the subject. The problems are at the end of the each chapter follow closely the order of the sections.

The entire book is suitable for one semester course in digital communication.

All materials for teaching texts were drawn from sources listed in References.

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6 OPTIMUM RECEIVERS FOR THE AWGN CHANNEL In the previous chapter, we described various types of modulation methods that may be used to transmit digital information through a communication channel. As we have observed, the modulator at the transmitter performs the function of mapping the digital sequence into signal waveforms.

This chapter deals with the design and performance characteristics of optimum receivers for the various modulation methods, when the channel corrupts the transmitted signal by the addition of Gaussian noise.

6.1 OPTIMUM RECEIVERS FOR SIGNALS CORRUPTED BY AWGN Let us begin by developing a mathematical model for the signal at the input to the receiver. We assume that the transmitter sends digital information by use of M signal waveforms ,1,2, … , . Each waveform is transmitted within the symbol (signaling) interval of duration T. To be specific, we consider the transmission of information over the interval 0 .

The channel is assumed to corrupt the signal by the addition of white Gaussian noise as illustrated in Figure 6.1.

Figure 6.1 Model for received signal passed through an AWGN channel

Thus, the received signal in the interval 0 may be expressed as

(6.1)

with power spectral densisty of

Φ / (6.2)

Based on the observation of ) over the signal interval, we wish to design a receiver that is optimum in the sense that it minimizes the probability of making an error.

It is convenient to subdivide the receiver into two parts – the signal demodulator and the detector – Figure 6.2.

Figure 6.2 Receiver configuration

ChannelTransmitted

signalsm(t)

Noise n(t)

Received signal

r(t)=sm(t)+n(t)

Signaldemodulator DetectorReceived

signalOutput decesion

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114

The function of the signal demodulator is to convert the received waveform into N-dimensional vector … , where N is the dimension of the transmitted signal waveforms. The function of the detector is to decide which of M possible signal waveforms was transmitted based of the vector r.

Two realizations of the signal demodulator are described in the next two sections. One is based on the used of signal correlators. The second is based on the use of matched filters. The optimum detector that follows the signal demodulator is designed to minimize the probability of error.

6.1.1 Correlation demodulator

We describe a correlation demodulator that decomposes the received signal and noise into N-dimensional vectors. The signal and the noise are expanded into a series of linearly weighted orthonormal basis functions . It is assumed that the N basis functions span the signal space, so that every one of the possible transmitted signal of the set , can be represented as a linear combination of . In the case of the noise, the functions do not span the noise space. However, the noise terms that fall outside the signal space are irrelevant to the detection of the signal. Suppose the received signal is passed through a parallel bank of N cross correlators which basically compute the projection of onto the N basis functions , as illustrated in Figure 6.3.

Figure 6.3 Correlation-type demodulator

Thus we have

, k 1, 2, … , N

(6.3)

Where

, k 1, 2, … , N (6.4)

Receivedsignal

r(t)

. . . . . .

f1(t)

f2(t)

fN(t)

∫0( )dt

T r1

r2

rN

Sample at t=T

. . . . . .

∫0( )dt

T

∫0( )dt

T

}To detector

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The signal is now represented by the vector with components , 1,2, … , . Their values depend on which of the M signals was transmitted. We can express the received signal in the interval as 0 as

(6.5)

Where

(6.6)

is a zero mean Gaussian noise process that represents the difference between the original noise process and the part corresponding to the projection of onto the basis functions and is

irrelevant to the decision as to which signal was transmitted. The decision may be based entirely on the correlator output signal and the noise components , k 1,2, … , N.

The correlator outputs conditioned on the mth signal being transmitted are Gaussian random variables with mean and variance

12

(6.7)

Since, the noise components are uncorrelated Gaussian random variables, they are also statistically independent. As a consequence, the correlator outputs conditioned on the m-th signal being transmitted are statistically independent Gaussian variables. Hence the conditional probability density functions of the random variables … are simply

, 1, 2, … , (6.8)

Where

1

exp , 1, 2, … , (6.9)

By substituting 6.8 into 6.9, we obtain the joint conditional PDFs

1

exp , 1, 2, … , (6.10)

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The correlator outputs are sufficient statistics for reaching a decision on which of the M signals was transmitted. All the relevant information is contained in the correlator outputs . Hence may be ignored.

6.1.2 Matched-Filter demodulator

Instead of using a bank of N correlators to generate the variables , we may use a bank of N linear filters. Suppose that the impulse responses of the N filters are

, 0 (6.11)

where are the N basis functions and 0 outside of the interval 0 . The outputs of these filters are

, 0 (6.12)

Now, if we sample the outputs of the filters at , we obtain

(6.13)

Hence, the sampled outputs of the filters at time are exactly the set of values obtained from the N linear correlators.

A filter whose impulse response is called matched filter to the signal .

Figure 6.4 Signal s(t) and filter matched to s(t)

The response of to the signal ) is

(6.14)

s(t)

t0 T

A

h(t)=s(T-t)

t0 T

A

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Figure 6.5 The matched filter output is autocorrelation function of s(t)

which is basically the time-autocorrelation function of the signal . Figure 6.5 illustrates for the triangular signal pulse shown in Figure 6.4. Note thet the autocorrelation function is an even function of , which attains a peak at .

In the case of the demodulator described above, the N matched filters are matched to the basis functions . Figure 6.6 illustrates the matched filter demodulator that generates the observed variables .

Figure 6.6 Matched filter demodulator

Properties of the matched filter:

If a signal s(t) is corrupted by AWGN, the filter with an impulse response matched to s(t) maximizes the output signal-to-noise ratio (SNR), which is as follows

2 2 (6.15)

Note that the output SNR from the matched filter depends on the energy of the waveform but not on the detailed characteristics of . This is another interesting property of the matched filter.

0 T

y(T)

2Tt

( ) ( ) ( )0

t

y t s s T t dτ τ τ= − +∫

Receivedsignal

r(t)

. . . . . .

r1f1(T-t)

f2(T-t)

fN(T-t)

r2

rN

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6.1.3 The Optimum detector

We have demonstrated that, for a signal transmitted over an AWGN channel, either a correlation demodulator or a matched filter demodulator produces the vector … , which contains all the relevant information in the received signal waveform. In this section we describe the optimum decision rule based on the observation vector . We assume that there is no memory in signals transmitted in successive signal intervals.

We wish to design a signal detector that makes a decision on the transmitted signal in each signal interval based on the observation of the vector r in each interval such that the probability of a correct decision is maximized.

We consider a decision rule based on the computation of the posteriori probabilities defined as

| , 1, 2, 3, … , (6.16)

which we abbreviate as | . The decision criterion is based on selecting the signal corresponding to the maximum of the set of posterior probabilities | (minimize the probability of error and maximize the probability of correct decision). This decision criterion is called the maximum a posteriori probability (MAP) criterion.

Using Bayes’ rule, the posterior probabilities may be expressed as

||

(6.17)

where | is the conditional PDF of the observed vector given and is the a priori probability of the m-th signal being transmitted. The denominator of 6.17 may be expressed as

| (6.18)

Some simplification occurs in the MAP criterion when the M signals are equally probable apriori, i.e.

for all M. Furthermore, we note that the denominator in 6.17 is independent of which

signal is transmitted. Consequently, the decision rule based on finding the signal that maximize | is equivalent to finding the signal that maximize | .

The conditional PDF | or any monotonic function of it is usually called the likelihood function. The decision criterion based on maximum of | over the M signals is called the maximum-likelihood (ML) criterion. We observe that a detector based on the MAP criterion and one that is based on the ML criterion make the same decision as long as the a priori probabilities are all equal. i.e., the signals are equiprobable.

In the case of an AWGN channel, the likelihood function | is given by Equation 6.10. To simplify the computations, we may work with the natural logarithm of | , which is a monotonic function. Thus

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ln |12

1 (6.19)

The maximum of ln | over is equivalent to finding the signal that minimizes the Euclidean distance (minimum distance detection)

, (6.20)

We call , , 1, 2, … , , the distance metrics. Hence, for the AWGN channel, the decision rule based on the ML criterion reduces to finding the signal that is closest in distance to the received signal vector . We shall refer to this decision rule as minimum distance detection.

Another interpretation of the optimum decision rule based on the ML criterion is obtained by expanding the distance metrics in Equation 6.20 as

, 2 ‖ ‖ 2 ‖ ‖ , 1,2, … , (6.21)

The term ‖ ‖ is common to all distance metrics, and, hence, it may be ignored in the computations of the metrics. The result is a set of modified distance metrics

, 2 ‖ ‖ (6.22)

Note that selecting the signal that minimizes , is equivalent to selecting the signal that maximizes the metric , , i.e.,

, 2 ‖ ‖ (6.23)

The term . represents the projection of the received signal vector onto each of the M possible transmitted signal vectors. The value of each of these projections is a measure of the correlation between the received vector and the m-th signal. For this reason, we call , , 1, 2… , , the correlation metrics for deciding which of the M signals was transmitted. Finally, the terms ‖ ‖

, 1, 2, … , may be viewed as bias terms that serve as compensation for signal sets that have unequal energies, such as PAM. If all signals have the same energy, ‖ ‖ may also be ignored in the computation of the correlation metrics , and the distance metrics , or , .

It is easy to show (Solved problem 5) that the correlation metrics , can also be expressed as

, 2 , 1,2, … , (6.24)

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Therefore, these metrics can be generated by a demodulator that coss-correlates the received signal with each of the M possible transmitted signals and adjust each correlator output for the bias in

the case of enequal signal energies. Equivalently, the received signal may be passed through a bank of M filters matched to the possible transmitted signals and sampled at , the end of the symbolic interval. Consequently, the optimum receiver (demodulator and detector) can be implemented in the alternative configuration illustrated in Figure 6.7.

Figure 6.7 An alternative realization of the optimum AWGN receiver

In summary, we have demonstrated that the optimum ML detector computes a set of M distances , or , and selects the signal corresponding to the smallest (distance) metrics.

Equivalently, the optimum ML detector computes a set of M correlator metrics , and selects the signal corresponding to the largest correlation metric.

The above development for the optimum detector treated the important case in which all signals are equally probable. In this case, the MAP criterion is equivalent to the ML criterion. However, when the signals are not equally probable, the optimum MAP detector bases its decision on the probabilities P(sm|r), m=1,2…,M, given by Equation 6.17 or, equivalently, on the metrics

, |

6.1.4 The Maximum-Likelihood Sequence Detector

When the signal has no memory, the symbol-by-symbol detector described in the preceding section is optimum in the sense of minimizing the probability of a symbol error. On the other hand, when the transmitted signal has memory, i.e., the signal transmitted in successive symbol intervals are interdependent, the optimum detector is a detector that bases its decisions on observation of a sequence of received signals over successive signal intervals. In this section, we describe a maximum-likelihood sequence detection algorithm that searches for the minimum Euclidean distance path through the trellis that characterizes the memory in the transmitted signal. Let us consider, as an example, the NRZI signal. Its memory is characterized by the trellis shown in Figure 6.8.

Select the

largest

Output decision

Receivedsignal

r(t)

s1(t)

s2(t)

sM(t)

Sample at t=T

112

ε

212

ε

M12

ε

-

-

-

∫0( )dt

T

∫0( )dt

T

∫0( )dt

T

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Figure 6.8 The Trellis diagram for NRZI signal.

The signal transmitted in each signal interval is binary PAM. Hence, there are two possible transmitted

signals corresponding to the signal points , where is the energy per bit. The output

of the matched-filter or correlation demodulator for binary PAM in the k-th signal interval may be expressed as

(6.25)

where is a zero-mean Gaussian random variable with variance . Consequently, the

conditional PDFs for the two possible transmitted signals are

1

√2exp

2

1

√2exp

2

(6.26)

For any given transmitted sequence , the joint PDF of , , … , may be expressed as a product of K marginal PDFs, i.e.,

, , … 1

√2exp

2

1

√2exp

2

(6.27)

where either or . Then, given the received sequence , , … , at the output of

the matched filter or correlation demodulator, the detector determines the sequence

, , … , = that maximizes the conditional PDF , , … . Such detector is

called the maximum-likelihood (ML) sequence-detector.

By taking the logarithm of Equation 6.29 and neglecting the terms that are independent of

, , … , we find that an equivalent ML sequence detector selects the sequence that minimizes the Euclidean distance metric

S0=0

S1=1

1/-s(t)

1/s(t)

0/s(t)

0/-s(t)

1/-s(t) 1/-s(t) 1/-s(t)

1/s(t) 1/s(t) 1/s(t)

0/-s(t) 0/-s(t) 0/-s(t)

0/s(t)0/s(t)0/s(t)

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, (6.28)

In searching through the trellis for the sequence that minimizes the Euclidean distance , , it

may appear that we must compute the distance , for every possible sequence. For the NRZI

example, which employs bojary modulation, the total numer of sequences is 2 , where K is the numer of outputs obtained from the demodulator. However, this is not the case. We may redukce the numer of sequences in the trellis search by usány the Viterbi algorithm to eliminace sequences as new data is received from the demodulator.

The Viterbi algorithm is a sequential trellis search algorithm for performing ML sequence detection. We assume that the search process begins initially at state . The corresponding trellis is shown in Figure 6.9.

Figure 6.9 Trellis for NRZI signal.

At time , we receive from the demodulator, and at 2 , we receive

. Since the signal memory is one bit, which we denote by 1, we observe that the trellis

reaches its regular (steady state) form after two transitions. Thus upon receipt of at 2 (and thereafter), we observe that there are two signal paths entering each of the nodes and two signal paths leaving each node. The two paths entering node at 2 correspond to the information bits 0,0

and 1,1 or, equivalently, to the signal points , and , , respectively. The

two paths entering node at 2 correspond to the information bits 0,1 and 1,0 or,

equivalently, to the signal points , and , , respectively.

For the two paths entering node , we compute the two Euclidean distance metrics

0,0

0,0 (6.29)

by using the outputs and from the demodulator. The Viterbi algorithm compares these two metrics and discards the path having the larger (greater-distance) metric. The other path with the lower metric is saved and is called the survivor at 2 .

Similarly, for the two paths entering node at 2 , we compute the two Euclidean distance metrics

0,1

1,0

(6.30)

S0

0/ bE 0/ bE 0/ bE 0/ bE

1/ bE1/ bE 1/ bE 1/ bE 1/ bE

0/ bE 0/ bE 0/ bES1

t=T t=2T t=3T t=T

1/

bE

1/

bE

1/

bE

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by using the outputs and from the demodulator. The two metrics are compared and the signal path with the larger metric is eliminated. Thus, at 2 , we are left with two survivor paths, one at node and the other at node , and their corresponding metrics. The signal paths at nodes and are then extended along the two survivor paths. Upon receipt of at 3 , we compute the metrics of the two paths entering state . Suppose the survivors at 2 are the paths 0,0 at and (0,1) at . Then, the two metrics for the paths entering at 3 are

0,0,0 0,0

0 0,1,1 0,1

(6.31)

These two metrics are compared and the path with the larger (greater-distance) metric is eliminated. Similarly, the metrics for the two paths entering at 3 are

0,0,1 0,0

0,1,0 0,1

(6.32)

These two metrics are compared and the path with the larger (greater-distance) metric is eliminated. This process is continued as each new signal sample is received from the demodulator. Thus, the Viterbi algorithm computes two metrics for the two signal paths entering a node at each stage of the trellis search and eliminates one of the two paths at each node. The two survivor paths are then extended forward to the next state. Therefore, the number of paths searched in the trellis is reduced by a factor of 2 at each stage.

6.2 PERFORMANCE OF THE OPTIMUM RECEIVER FOR MEMORYLESS MODULATION

In this section we study signaling schemes that are mainly characterized by their low bandwidth requirements. These signaling schemes have low dimensionality which is independent from the number of transmitted signals, and, as we will see, their power efficiency decreases when the number of messages increases.

6.2.1 Probability of Error for Binary Modulation

PAM (antipodal)

Let us consider binary PAM signals, where the two signal waveforms are ) and ), and is an arbitrary pulse that is nonzero in the interval 0 and zero elsewhere.

The energy in the pulse is .

Figure 6.10 Signal points for antipodal signal.

Let us assume that the two signals are equally likely and that signal was transmitted. Then the received signal from the (matched filter or correlation) demodulator is

(6.33)

s2

0

s1

EbEb

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where represents the additive Gaussian noise component, which has zero mean and variance

. In this case, the decision rule based on the correlation metric given by Equation 6.23

compares with the threshold zero. If 0, the decision is made favor of , and if 0, the decision is made that , was transmitted. The two conditional PDFs of are

1

exp (6.34)

1

exp (6.35)

These two conditional PDFs are shown in Figure 6.11.

Figure 6.11 Conditional PDF’s of two signals.

Given that , was transmitted, the probability of error is simply the probability that 0, i.e.,

2

(6.36)

Where is the Q-function. Similarly, if we assume that , was transmitted,

and the probability that 0 is also .

Since the signals and are equally likely to be transmitted, the average probability of error is

12

12

2

(6.37)

We should observe two important characteristics of this performance measure. First, the probability of

error depends only on the ratio and not on any other detailed characteristics of the signals and the

noise. Secondly, we note that is also the output from the matched-filter (and correlation)

demodulator. The ratio is usually called the signal-to-noise ratio per bit.

The probability may be also expressed in terms of the distance between the two signals and .

From Figure 6.10, we observe that the two signals are separated by the distance 2 . By

substituting into Equation 6.37, we obtain

2 (6.38)

- 0

p(r|s2) p(r|s1)

Eb Eb

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Binary orthogonal signals

Signal vectorss and s are two-dimensional, as shown in Figure 6.12, and may be expressed as

, 0

0, (6.39)

Figure 6.12 Signal points for binary orthogonal signals.

Figure 6.13 Error probability for binary antipodal and binary orthogonal signaling.

The average error probability for binary orthogonal signals is

(6.40)

where by definition, is the SNR per bit.

The error probability versus 10 log for these two types of signals is shown in Figure 6.13. As

observed from this figure, at any given error probability, the required for orthogonal signals is 3 dB

more than for antipodal signals.

f2(t)

f1(t)

Eb

Eb2Eb

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6.2.2 Probability of Error for M-ary Orthogonal Signals

The probability of a (k-bit) symbol error is

1

√21

1

√2

12

2

(6.41)

The same expression for the probability of error is obtained when any one of the other 1 signals is transmitted. Since all the M signals are equally likely, the expression for given in Equation 6.46 is the average probability of a symbol error.

In comparing the performance of various digital modulation methods, it is desirable to have the

probability of error expressed in terms of the SNR per bit, , instead of the SNR per symbol, . With

2 . each symbol conveys k bits of information, and hence . Thus, Equation 6.41 may

be expressed in terms of by substituting for .

Sometimes, it is also desirable to convert the probability of a symbol error into an equal probability of a binary digit error. For equiprobable orthogonal signals, all symbols errors are equiprobable and occur with probability

1 2 1 (6.42)

Furthermore, there are ways in which n bits out of k may be in error. Hence, the average number

of bit errors per k-bit symbol is

2 122 1

(6.43)

and the average bit error probability is just the result in Equation 6.43 divided by k, the number of bits per symbol. Thus,

22 1 2

, ≫ 1 (6.44)

The graph of the probability of a binary digit error as a function of the SNR per bit, , are shown in

Figure 6.14 for M=2, 4, 6, 16, 32 and 64.

This figure illustrates that, by increasing the number M of waveforms, one can reduce the SNR per bit

required to achieve a given probability of a bit error. For example, to achieve a 10 , the required SNR per bit is a little more than 12 dB for 2, but if M is increased to 64 signal waveforms ( 6 / ), the required SNR per bit is approximately 6 dB (savings of over 6 dB !!!).

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What is the minimum required to achieve an arbitrarily small probability of error as → ∞?

ln 2 0,693 1,6 .This minimum SNR is called the Shannon limit for an AWGN

channel.

Figure 6.14 Probability of bit error for coherent detection of orthogonal signaling.

6.2.3 Probability of Error for M-ary Biorthogonal Signals

A set of 2 biorthogonal signals is constructed from orthogonal signals by including the

negatives of the orthogonal signals. Thus, we achieve a reduction in the complexity of the demodulator for the biorthogonal signals relative to that for orthogonal signals, since the former is

implemented with cross-correlators or matched filters, whereas the latter requires M matched

filters, or cross-correlators. In biorthogonal signaling , and the vector representation for

signals are given by

√ , 0, … ,0

0, √ ,… ,0 ⋮ ⋮ ⋮

0, 0, … , √

(6.45)

To evaluate the probability of error for the optimum detector, let us assume that the signal

corresponding to the vector √ , 0, … ,0 was transmitted. Then the received signal vector is

√ , , … , (6.46)

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where the are zero-mean, mutually statistically independent and identically distributed Gaussian

random variables with variance . Since all signals are equiprobable and have equal energy,

the optimum detector decides in favor of the signal corresponding to the largest in magnitude of the cross-correlators

, . ; 112

(6.47)

while the sign of this largest term is used to decide whether or was transmitted. According to this decision rule, the probability of a correct decision is equal to the probability that

√ and exceeds | | | | 2, 3, … , . But

| | | 01

1

√2

(6.48)

Then the probability of a correct decision is

1

√2

(6.49)

from which, upon substitution for , we obtain

1

√2

1

√2

(6.50)

where we have used the PDF of as a Gaussian random variable with mean equal to √ and variance

. Finally, the probability of a symbol error 1 . , and hence, may be evaluated

numerically for different values of M from Equation 6.50. The graph shown in Figure 6.15 illustrates

as a function of , where , for M = 2, 4, 8, 16, and 32. We observe that this graph is

similar to that for orthogonal signals (see Figure 6.14). However, in this case, the probability of error for 4 is greater than that for 2. This is due to the fact that we have plotted the symbol error probability in Figure 6.15. If we plotted the equivalent bit error probability, we should find

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that the graphs for M = 2 and M = 4 coincide. As in the case of orthogonal signals, as → ∞ or

→ ∞), the minimum required to achieve an arbitrarily small probability of error is 1.6 , the

Shannon limit.

Figure 6.15 Probability of symbol error for biorthogonal signals.

6.2.4 Probability of Error for Simplex Signals

The probability of error for simplex signals is identical to the probability of error for orthogonal signals, but this performance is achieved with a saving of

10 log 1 10 log1

(6.51)

in SNR. For 2, the saving is 3 . However, as M is increased, the saving in SNR approaches 0 .

6.2.5 Probability of Error for M-ary Binary-Coded Signals

If is the minimum Euclidean distance of the M signal waveforms, then the probability of a

symbol error is upper-bounded as

1 12

24

(6.52)

The value of the minimum Euclidean distance will depend on the selection of the code words, i.e., design of the code.

P e

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6.2.6 Probability of Error for M-ary PAM

The constellation for an ASK signaling scheme is shown in Figure 6.16. In this constellation the minimum distance between any two points is

Figure 6.16 The PAM or ASK constellation.

Probability of a symbol error

2 1 6 log1

(6.53)

Figure 6.17 Probability of symbol error for PAM signals.

6.2.7 Probability of Error for M-ary PSK

The constellation for an M-ary PSK signaling is shown in Figure 6.18. In this constellation the decision region is also shown. Note that since we are assuming the messages are equiprobable, the decision regions are based on the minimum-distance detection rule. By symmetry of the constellation,

the error probability of the system is equal to the error probability when √ , 0 is transmitted.

dmin . . . . . .

6 4 2 0 2 4 8

2

5

2

5

2

5

2

5

2

5

6 10 12 14 16 18 20 22SNR per bit, γb [dB]

10 6

10 5

10 4

10 3

10 2

10 1

Prob

abili

ty o

f a sy

mbo

l err

or P

e

M 2

M 4

M 8

M 16

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Figure 6.18 The constellation for PSK signaling.

The received vector r is given by

, √ , (6.54)

Figure 6.19 Probability of symbol error for PSK signals.

Probability of a symbol error

2 2 sin 2 2 sin (6.55)

s2

sM

s1D1

P e

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6.2.8 Probability of Error for QAM

Recall, that QAM signal waveforms may be expressed as

cos 2 sin 2 (6.56)

where and are the information-bearing signal amplitudes of the quadrature carriers and is the signal pulse. The vector representation of these waveforms is

12

;12

(6.57)

To determine the probability of error for QAM, we must specify the signal point constellation. We begin with QAM signal sets that have 4 points. Figure 6.20 illustrates two four-point signal sets. The first is a four-phase modulated signal and the second is a QAM signal with two amplitude levels, labeled and , and four phases. Because the probability of error is dominated by the minimum

distance between pairs of signal points, let us impose the condition that 2 for both signal

constellations and let us evaluate the average transmitter power, based on the premise that all signal points are equally probable.

Figure 6.20 Two four-point QAM signal constellations.

For the four-phase signal, we have

144 2 2 (6.58)

For the two-amplitude, four-phase QAM, we place the points on circles of radii and √3 .

Thus, 2 , and

142 3 2 2 (6.59)

which is the same average power as the M=4 phase signal constellation. Hence, for all practical purposes, the error rate performance of the two signal sets is the same. In other words, there is no advantage of the two-amplitude QAM signal set over M=4-phase modulation.

b)a)

d=2A2A

A2

A1

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Next, let us consider 8 QAM, as shown in Figure 6.21 2 . Assuming that the signal

points are equally probable, the average transmitted signal power is

1 (6.60)

where , , are the coordinates of the signal points, normalized by A.

Figure 6.21 Four eight-point QAM signal constellations.

Therefore, the fourth signal set requires approximately 1 dB less power than the first two and 1.6 dB less power than the third to achieve the same probability of error.

For 16 , there are many more possibilities for selecting the QAM signal points in the two-dimensional space. For example, we may choose a circular multiamplitude constellation for M=16. In

this case, the signal points at a given amplitude level are phase-rotated by relative to the signal

points at adjacent amplitude levels. However, the circular 16-QAM is not the best 16-point QAM signal constellation for the AWGN channel.

Rectangular QAM signal constellation have the distinct advantage of being easily generated as two PAM signals impressed on phase-quadrature carriers. In addition, they are easily demodulated. Although they are not the best M-ary QAM signal constellation for 16 , the average transmitted power required to achieve a given minimum distance is only slightly greater than the average power required for the best M-ary QAM signal constellation. For these reasons, rectangular M-ary QAM signals are most frequently used in practice.

If we employ the optimum detector that bases its decision on the optimum distance metrics, it is relatively straightforward to show that the symbol error probability is tightly upper-bounded as

43

1 (6.61)

b)a)

d)c)

Pav=6A2

Pav=6A2

Pav=6,83A2

Pav=4,73A2

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for any 1 , where is the average SNR per bit. The probability of a symbol error is plotted in

Figure 6.22.

Figure 6.22 Four eight-point QAM signal constellations.

6.3 SOLVED PROBLEMS Problem 1

Consider an M-ary baseband PAM signal set in which the basic pulse shape g(t) is rectangular in the period (0,T) with the amplitude α (see Figure). The additive noise is zero mean white Gaussian noise process. Let us determine the basis function and the output of the correlation-type demodulator. The energy in the rectangular pulse is

Since the PAM signal set has dimension N = 1, there is only one basis function . This is given as

1

√

1

√, 0

0,

P e

g(t)

t0 T

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The output of the correlation-type demodulator is

1

√

It is interesting to note that the correlator becomes a simple integrator when is rectangular. If we substitute for , we obtain

1

√

where the noise term 0 and .

The probability density function for the sampled output is

1

exp

Problem 2

Consider the case of binary PAM signals in which the two possible signal points are

, where is the energy per bit. The prior probabilities are and 1 . Let

us determine the metrics for the optimum map detector when the transmitted signal is corrupted with AWGN.

Solution

The received signal vector (one-dimensional) for binary PAM is

(6.62)

where is a zero-mean Gaussian random variable with variance . Consequently, the

conditional PDFs | for the two signals are

1

√2exp

2 (6.63)

1

√2exp

2 (6.64)

Then the metrics , and , are

, √2

exp2

(6.65)

, √2

exp2

(6.66)

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If , , , we select as the transmitted signal; otherwise, we select . This

decision rule may be expressed as

,,

1 (6.67)

But

,, 1

exp2

(6.68)

So that Equation 6.30 may be expressed as

2ln1

(6.69)

or equivalently,

12

ln1 1

4ln1

(6.70)

Figure 6.23 Signal space representation illustrating the operation of the optimum detector for binary PAM modulation.

This is the final form for the optimum detector. It computes the correlation metric ,

and compares it with threshold ln . Figure 6.23 illustrates the two signal points and

. The threshold, denoted by hτ, divides the real line into two regions, say and , where consists of the set of points that are greater than , and consists of the set of points that are less

than . If , the decision is made that was transmitted, and if , the decision is

made that was transmitted. The threshold depends on and p. If , 0. If p>1/2, the

signal point is more probable, and, hence, 0. In this case, the region is larger than , so

that is more likely to be selected than . If , the opposite is the case. Thus, the average

probability of error is minimized.

It is interesting to note that in the case of unequal prior probabilities, it is necessary to know not only the values of the prior probabilities but also the value of the power spectral density , or,

equivalently, the noise-to-signal ration , in order to compute the threshold. When , the

threshold is zero, and the knowledge of is not required by the detector.

Problem 3

A matched filter has the frequency response

s2

Region R2

rth0

Region R1s1

Eb Eb

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12

a) Determine the impulse response h(t) corresponding to H(f). b) Determine the signal waveform to which the filter characteristic is matched.

Solution

a) Taking the impulse h(t) corresponding to H(f)

12 2

2 2

Where is the signum signal (1if x>0, -1 if x<0 and 0 if x=0) and ∏ is a rectangular pulse of unit heigh and widt, centered at x=0.

b) The signal waveform, to which h(t) is matched is:

2 2

Where we have used the symmetry of ∏ with respect to the axis.

Problem 4

Consider the signal

2 , 0 t

0,

a) Determine the impulse response of the matched filter for the signal. b) Determine the output of the matched filter at t=T. c) Suppose the signal s(t) is passed through a correlator that correlates the input s(t) with s(t).

Determine the value of the correlator output at t=T. Compare your result with than in (b).

Solution

a) The impulse response of the matched filter is

2 , 0 t

0,

b) The output of the matched filter at t=T is

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∗ |

2

2

2

6 4 ∗ 21

8 ∗ 2sin 4

sin 44 2

|

6 4 ∗ 21

8 ∗ 2sin 4

sin 44 2

a) The output of the correlator at t=T is

2

However this is the same expression with the case of the output of the matched filter sampled at t=T. Thus, the correlator can substitute the matched filter in a demodulation system and vice versa.

Problem 5

The correlation metrics are

, 2 , 1,2, . . ,

Where

Show that the correlation metrics are equivalent to the metrics

, 2

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Solution

Since constitutes an orthonormal basis for the signal space: ∑ ,∑ . Hence for any m:

, 2

2

2

2

Where : . The last form is indeed the original form of the correlation

metrics , .

Problem 6

Consider that octal signal point constellations in Figure.

a) The nearest-neighbor signal points in the 8-QAM signal constellation are separated in distance by A units. Determine the radii a and b of the inner and outer circles.

b) The adjacent signal points in the 8-PSK are separated by a distance of A units. Determine the radius r of the circle.

c) Determine the average transmitter powers for two signal constellations and compare the two powers. What is the relative power advantage of one constellation over the other? (Assume that all signal points are equally probable.)

Solution

a) Consider the QAM constellation of Figure. Using the Pythagorean theorem we can find the radius of the inner circle as

8-PSK 8-QAM

r ba

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⟹1

√2

The radius of the outer circle can be found using the cosine rule. Since b is the third side of a triangle with a and A the two other sides and angle between then equal to Θ 75°,we obtain:

2 75° ⟹1 √32

b) If we denote by r the radius of the inner circle, then using the cosine theorem we obtain:

2 45° ⟹2 √2

c) Then average transmitted power of the PSK constellation is:

8 ∗18∗

2 √2⟹

2 √2

Whereas the average transmitted power of the QAM constellation:

18

42

41 √3

4⟹

2 1 √38

The relative power advantage of the PSD constellation over the QAM constellation is: 8

2 1 √3 2 √21,5927

Problem 7

Consider the 8-point QAM signal constellation shown in Figure.

a) Is it possible to assign three data bits to each point of the signal constellation such that nearest (adjacent) points differ in only one bit position?

b) Determine the symbol rate if the desired bit rate is 90 Mbits/s.

Solution

a) Although it is possible to assign three bits to each point of the 8-PSK signal constellation so that adjacent points differ in only one bit, (e.g. going in a clockwise direction: 000, 001, 011, 010, 110, 111, 101, 100). This is not the case for the 8-QAM constellation in figure. This is because there are fully connected graphs consisted of three points. To see this consider an equilateral

8-PSK 8-QAM

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triangle with vertices A, B, and C. If, without loss of generality, we assign the all zero sequence 0,0, … ,0 to point A, then point B and C should have the form

0,… ,0,1,0, … ,0 0, … ,0,1,0, … ,0 Where the position of the 1 in the sequences is not the same, otherwise B=C. Thus the sequences of B and C differ in two bits

b) Since each symbol conveys 3 bits of information, the resulted symbol rate is:

90 ∗ 103

30 ∗ 10 /

6.4 SUMMARY Two general classes of optimization problems are signal detection and parameter estimation.

Although both detection and estimation are often involved simultaneously in signal reception, from an analysis standpoint, it is easiest to consider them as separate problems.

Bayes detectors are designed to minimize the average cost of making a decision. They involve testing a likelihood ratio, which is the ratio of the aposteriori (posterior) probabilities of the observations, against a threshold, which depends on the a priori (prior) probabilities of the two possible hypotheses and costs of the various decision–hypothesis combinations. The performance of a Bayes detector is characterized by the average cost, or risk, of making a decision.

It was shown that a minimum-probability-of-error detector is really a Bayes detector with zero costs for making right decisions and equal costs for making either type of wrong decision. Such a receiver is also referred to as a maximum a posteriori (MAP) detector, since the decision rule amounts to choosing as the correct hypothesis the one corresponding to the largest a posteriori probability for a given observation.

The introduction of signal space concepts allowed the MAP criterion to be expressed as a receiver structure that chooses as the transmitted signal the signal whose location in signal space is closest to the observed data point. Two examples considered were coherent detection of M-ary orthogonal signals and noncoherent detection of binary FSK in a Rayleigh fading channel.

For M-ary orthogonal signal detection, arbitrarily small probability of error can be achieved as → ∞ provided the ratio of energy per bit to noise spectral density is greater than 1,6 .

This perfect performance is achieved at the expense of infinite transmission bandwidth, however.

For the Rayleigh fading channel, the probability of error decreases only inversely with the SNR rather than exponentially, as for the nonfading case. A way to improve performance is by using diversity.

Bayes estimation involves the minimization of a cost function, as for signal detection. The squared-error cost function results in the a posteriori conditional mean of the parameter as the optimum estimate, and a square-well cost function with infinitely narrow well results in the maximum of the a posteriori pdf of the data, given the parameter, as the optimum estimate (MAP estimate). Because of its ease of implementation, the MAP estimate is often employed even though the conditional-mean estimate is more general, in that it minimizes any symmetrical, convex-upward cost function as long as the posterior pdf is symmetrical about a single peak.

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6.5 EXERCISES 1. A binary digital communication system employs the signals

0, 0

, 0

for transmitting the information. This is called on-off signaling. The demodulator cross-correlates the received signal r(t) with s(t) and samples the output of the correlator at t=T. a) Determine the optimum detector for an AWGN channel and the optimum threshold,

assuming that the signals are equally probable. b) Determine the probability of error as a function of the SNR. How does on-off signaling

compare with antipodal signaling? 2. Supposed that binary PSK is used for transmitting information over an AWGN with a power

spectral density of N 10 . The transmitted signal energy is E A T where T is the

bit interval and A is the signal amplitude. Determine the signal amplitude required to achieve an error probability of 10 when the data rate is a) 10 kbits/s. b) 100 kbits/s c) 1 Mbit/s

3. Two quadrature carriers cos 2 and sin 2 are used to transmit digital information through an AWGN channel at two different data rates, 10 kbits/s and 100 kbits/s. Determine

the relative amplitudes of the signals for the two carriers so that the for the two channels is

identical. 4. Consider a digital communication system that transmits information via QAM over a voice-

band telephone channel at a rate of 2400 symbols/s. The additive noise is assumed to be white and Gaussian.

a) Determine the required to achieve an error probability of 10 at 4800 bits/s.

b) Repeat (a) for a rate of 9600 bits/s c) Repeat (a) for a rate of 19,200 bits/s d) What conclusion do you reach from these results?

5. Consider that four-phase and eight-phase signal constellations show in Figure. Determine the radii and of the circles such that the distance between two adjacent points in the two constellations is d. From these result, determine the additional transmitted energy required in the in the 8-PSK signal to achieve the same error probability as the four-phase signal at the high SNR, where the probability of error is determined by errors in selecting adjacent points.

M=8

r2

d

r2d

M=4

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6. Consider the two 8-point QAM signal constellations shown in Figure. The minimum distance between adjacent points is 2A. Determine the average transmitted power for each constellation, assuming that the signal points are equally probable. Which constellation is more power-efficient?

7. Digital information is to be transmitted by carrier modulation through an additive Gaussian

noise channel with a bandwidth of 100 kHz and 10 W/Hz. Determine the maximum rate that can be transmitted through the channel for four-phase PSK, binary FSK, and four-frequency orthogonal FSK, which is detected noncoherently.

8. A speech signal is sampled at a rate of 8 kHz, logarithmically compressed and encoded into a PCM format using 8 bits per sample. The PCM data is transmitted through an AWGN baseband channel via M-level PAM. Determine the bandwidth required for transmission when a) M=4. b) M=8. c) M=16.

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173

APPENDIX A

THE Q-FUNCTION Computation of probabilities that involve a Gaussian process require finding the area under the tail of the Gaussian (normal) probability density function as shown in Figure.

Figure A.1 Gaussian probability density function. Shaded area is 𝑃𝑟(𝑥 ≥ 𝑥0) for a Gaussian random variable.

Figure A.l illustrates the probability that a Gaussian random variable x exceeds 𝑥0 , 𝑃𝑟(𝑥 ≥ 𝑥0), which is evaluated as:

𝑃𝑟(𝑥 ≥ 𝑥0) = �1

𝜎√2𝜋𝑒−�

𝑥−𝑚𝑥𝜎√2

�2∞

𝑥0

(A.1)

The Gaussian probability density function in Equation A.1 cannot be integrated in closed form. Any Gaussian probability density function may be rewritten through use of the substitution:

𝑦 =𝑥 −𝑚𝑥

𝜎 (A.2)

To yield

𝑃𝑟 �𝑦 > 𝑥0 − 𝑚𝑥

𝜎� = �

1√2𝜋

𝑒−�𝑦√2�2∞

𝑥0−𝑚𝑥𝜎

(A.3)

where the kernel of the integral on the right-hand side of Equation A.3 is the normalized Gaussian probability density function with mean of 0 and standard deviation of 1. Evaluation of the integral in Equation A.3 is designated as the Q-function, which is defined as

𝑄(𝑧) = �1

√2𝜋𝑒−�

𝑦√2�2∞

𝑧

(A.4)

Hence

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APPENDIX A

174

𝑃 �𝑦 > 𝑥0 − 𝑚𝑥

𝜎� = 𝑄 �

𝑥0 − 𝑚𝑥

𝜎� = 𝑄(𝑧) (A.5)

The Q-function is bounded by two analytical expressions as follows:

�1 −1𝑧2�

1𝑧√2𝜋

𝑒−�𝑦√2�2

≤ 𝑄(𝑧) ≤1

𝑧√2𝜋𝑒−�

𝑦√2�2

(A.6)

For values of z greater 3.0, both of these bounds closely approximate Q(z). Two important properties of Q(z) are:

𝑄(−𝑧) = 1 − 𝑄(𝑧)

𝑄(0) =12

(A.6)

THE ERROR FUNCTION

The error function, denoted by erf(x), is defined in a number of different ways in the literature. We shall use the following definition:

erf (𝑥) =2√𝜋

�𝑒−(𝑦)2𝑑𝑦𝑥

0

(A.7)

The error function has two useful properties:

• Symmetry property: 𝑒𝑟𝑓(−𝑥) = −𝑒𝑟𝑓(𝑥) • Asymptote property: As x approaches infinity, erf(x) approaches unity; that is,

erf(𝑥) =2√𝜋

� 𝑒−(𝑦)2𝑑𝑦∞

0

= 1

The complementary error function is defined by

erf (𝑥) =2√𝜋

� 𝑒−(𝑦)2𝑑𝑦∞

𝑥

(A.8)

The complementary error function is related to the error function as follows:

erfc(𝑥) = 1 − erf (𝑥) (A.9) The Q-function defines the area under the standardized Gaussian tail. Inspection of Equation A.4 and A.8 reveals that the Q-function is related to the complementary error function as follows:

Q(x) =12

erfc �𝑥√2� (A.10)

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175

APPENDIX B

COMPARISON OF M-ARY SIGNALING TECHNIQUES

Modulation M-ASK and QAM

r𝐵

= 2 log2 𝑀 (B.1)

Modulation M-PSK

r𝐵

= log2 𝑀 (B.2)

Modulation M-FSK

r𝐵

=2 log2 𝑀

𝑀 (B.4)

ERROR PERFORMANCE OF M-ARY SIGNALING TECHNIQUES

Modulation M-ASK

𝑃𝑒(𝑠) =

2(𝑀 − 1)𝑀

𝑄��6 log2 𝑀𝑀2 − 1

𝐸𝑏𝑁0�

𝑃𝑒(𝑏) =𝑃𝑒(𝑠)

log2 𝑀

(B.5)

Modulation B-PSK

𝑃𝑒(𝑠) = 𝑄��

2𝐸𝑏𝑁0

� (B.6)

Modulation M-PSK

𝑃(𝑏𝑖𝑡𝑒𝑟𝑟𝑜𝑟) ≅

1log2 𝑀

𝑄��𝑘𝑠𝑖𝑛2 �𝜋𝑀�

2𝐸𝑏𝑁0

�

(B.7)

Modulation M-FSK

𝑃𝑒 < 𝑀𝑒−

𝐸𝑠2𝑁0 (B.8)

Modulation M-QAM

𝑃𝑒 < 4𝑄 ��

3𝑘(𝑀 − 1)

𝐸𝑏𝑁0� (B.8)

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176

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Ziemer, Rodger E.: Principles of communications: systems, modulation, and noise, 2009, ISBN: 978-0-470-25254-3

Apurba, D.: Digital Communication: Principles and System Modelling, 2010, ISBN 9783642-12743-4

Ha, Tri T.: Theory and Design of Digital Communication Systems, 2011, ISBN: 978-0-521-76174-1

Lecture notes available at: http://kemt.fei.tuke.sk/tts/teoria-telekomunikacnych-systemov-o-predmete/

KEMTTTS

Department of electronics and multimedia telecommunications

KEMT TTS

theory of telecommunications tts networks...

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