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Semester 2, 2020

ELEC3540 Analog and Digital Communications

School of Electrical Engineering & Computing EXAMINATION Semester 2, 2020

ELEC3540 Analog and Digital Communications This paper is for all students.

Examination Duration:

130 minutes

This exam has __6__ questions.

Exam Conditions: This is a TIMED EXAM (Fixed Start) This is an OPEN book examination - While there are no restrictions placed on resources available to you, the University’s academic integrity rules, including those relating to assistance from other people, will apply. After you have attempted this exam, you may be required to take part in a viva (or oral exam) for quality assurance purposes. A viva will consist of an interview with one or perhaps two staff conducted via Zoom, and will last approximately 10 minutes. During this time, you will be asked questions about the answers you have provided in the exam.

Special Instructions: Answer ALL questions. Students are given extra 20 minutes to scan and upload their written solutions onto Blackboard.

Page 1 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

Question 1. (21 marks) Answer the following questions. You must justify your answers. (a) Assume we communicate over an additive white Gaussian noise (AWGN) channel with a signal-to-noise ratio Eb /N0 = 20dB. What is the bit error rate for the coherently detected BPSK modulation scheme? [3 marks] (b) Consider binary signalling over an AWGN channel. Is the probability of bit error for antipodal signalling always half of that for orthogonal signalling? Assume both signalling schemes have the same average energy per bit. [3 marks] (c) Explain the advantages and disadvantages of using M-ary digital transmissions over binary transmissions. [4 marks] (d) For a given M-ary signalling scheme, is the bit error probability always smaller than or equal to the symbol error probability? [3 marks] (e) Explain why the raised-cosine pulse shaping filter is preferred over the rectangular filter in dealing with intersymbol interference. [4 marks] (f) The two constellation diagrams in Figure 1 represent two different modulation schemes. In both diagrams, the outer circle’s radius is double the inner circle’s radius. However, the scale of the left diagram could be different from the scale of the right diagram. Is the following statement True or False? : To achieve the same given minimum distance among the signal points, the constellation on the right diagram requires less power than that on the left diagram. [4 marks]

Figure 1: Two constellations for Question 1(f)

Question 2. (25 marks) The two orthonormal basis functions shown in Figure 2 are used to construct a binary signal set as follows: s1 (t) = −φ1 (t) − φ2 (t), s2 (t) = 2φ1 (t) + 0.5φ2 (t). Here, s1 (t) and s2 (t) are used for transmitting bit “0” and bit “1”, respectively. Let P [“0” transmitted] = P [“1” transmitted] = 0.5. The additive white Gaussian noise is zero-mean and has a two-sided power spectral density of N0 /2 (watts/hertz). Page 2 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

φ1 ( t )

φ2 ( t ) rect

2 Tb

1 Tb

half − cos

Tb 2

0

Tb

0

Tb

−

2 Tb

Figure 2: Two orthonormal basis functions for Question 2 (a) Represent s1 (t) and s2 (t) on the signal space diagram spanned by φ1 (t) and φ2 (t). Draw the optimum decision regions. Write an expression for the optimum decision rule and simplify it as much as you can. [8 marks] (b) What is the maximum spectral density level of the noise that can be tolerated for a probability of bit error P [error] ≤ 10−5 ? [4 marks] (c) On the signal space diagram in Part (a), clearly show how you perform axis rotation to obtain new basis functions φˆ1 (t) and φˆ2 (t) so that the optimum receiver can be implemented with only one matched filter (You MUST specify the direction and angle of rotation). Draw the block diagram of such an optimum receiver. Determine and plot the impulse response of the matched filter. Note: Determining the exact value of the threshold for the comparison block is optional. [11 marks] (d) Assume that you cannot change s1 (t), but you can freely move s2 (t) in the same signal space as long as the energy of s2 (t) stays the same. Modify s2 (t) so that the bit error probability is as small as possible (You must explain your answer.). [2 marks]

Question 3. (12 marks) In antipodal signalling, two signals s(t) and −s(t) are used to transmit equally likely bits “0” and “1”, respectively. Consider two communication systems, called system (i) and system (ii), that use two different time-limited waveforms as shown in Figure 3. Assume both systems are subjected to additive white Gaussian noise with zero mean and the same noise power spectral density of N0 /2. (a) If the two communication systems are to have the same error performance, what is the relationship between the parameters A and B of the two waveforms? You must explain your answer. [7 marks] (b) Consider system (i) in Figure 3. The bit rate is 5 Mbps and the additive white Gaussian noise has a power spectral density of N0 /2 = 10−7 (watts/hertz). How high does the voltage level A need to be for system (i) to achieve a bit error probability P [error] ≤ 10−5 ? [5 marks] Page 3 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

s( t )

s( t )

B A

T

0

t

System (i)

0

T T 2 System (ii)

t

Figure 3: The two waveforms for antipodal signalling for Question 3 Question 4. (14 marks) Consider the 16-QAM constellation shown in Figure 4. There are two concentric circles, where 8 signals are equally spaced on the inner circle and 8 signals are equally spaced on the outer circle. Assume that the radii R1 and R2 are adjusted such that the distance between the adjacent signal points on the inner circle is equal to the distance between the nearest points across two circles, i.e., abc is an equilateral triangle with a side of ∆.

Figure 4: The 16-QAM constellation for Question 4 (a) Determine the average symbol energy of this constellation. [12 marks] (b) Is it possible to perform a Gray mapping for this constellation? You must explain your answer. [2 marks] Question 5. (14 marks) Consider the two signal constellations shown in Figure 5. (a) Determine the radii r1 and r2 of the circles, such that the distance between two adjacent points in each constellation is d. [7 marks] Page 4 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

(b) From the result in (a), determine the additional transmitted energy (in dB) required in the 8-PSK signal to achieve the same error probability as the 4PSK signal at high signal-to-noise ratios, where the probability of error is determined by errors in selecting adjacent points. [7 marks]

d

r2

r1 d

M⫽4 M⫽8 Figure 5: Two PSK constellations for Question 5

Question 6. (14 marks) Satellite channels can be well modelled as band-limited additive white Gaussian noise channels. Consider a 30MHz satellite channel in the Ku-band, whose transfer function is shown in Figure 6. This channel has been used for digital video broadcasting via satellites. The first-generation standard, namely DVB-S, specifies only the QPSK modulation and a roll-off factor β = 0.35. What is the maximum bit rate supported by DVB-S that can avoid intersymbol interference? Explain your answer by clearly plotting the overall spectrum of the signal (i.e., after the receiver’s matched filter). Identify all relevant frequencies in the plot.

Figure 6: The satellite channel transfer function for Question 6

#END OF EXAMINATION# Page 5 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

Fourier Transform of Some Common Functions

Trigonometry and Exponentials sin(x ± y) = sin x cos y ± cos x sin y cos(x ± y) = cos x cos y ∓ sin x sin y 1 sin x sin y = [cos(x − y) − cos(x + y )] 2 1 cos x cos y = [cos(x − y) + cos(x + y )] 2 1 sin x cos y = [sin(x + y) + sin(x − y )] 2 1 cos x sin y = [sin(x + y) − sin(x − y )] 2 ejx = cos x + j sin x ejx − e−j x sin x = 2j jx e + e−j x cos x = 2

Page 6 of 7

Semester 2, 2020

ELEC3540 Analog and Digital Communications

Table of Q(x) x

0.00

0.01

0.02

0.03

0.04

0.05

0.0

0.5000000

0.4960106

0.4920217

0.4880335

0.4840466

0.4800612

0.4760778

0.4720968

0.4681186

0.464143

0.1

0.4601722

0.4562047

0.4522416

0.4482832

0.4443300

0.4403823

0.4364405

0.4325051

0.4285763

0.424654

0.2

0.4207403

0.4168338

0.4129356

0.4090459

0.4051651

0.4012937

0.3974319

0.3935801

0.3897388

0.385908

0.3

0.3820886

0.3782805

0.3744842

0.3707000

0.3669283

0.3631693

0.3594236

0.3556912

0.3519727

0.348268

0.4

0.3445783

0.3409030

0.3372427

0.3335978

0.3299686

0.3263552

0.3227581

0.3191775

0.3156137

0.312066

0.5

0.3085375

0.3050257

0.3015318

0.2980560

0.2945985

0.2911597

0.2877397

0.2843388

0.2809573

0.277595

0.6

0.2742531

0.2709309

0.2676289

0.2643473

0.2610863

0.2578461

0.2546269

0.2514289

0.2482522

0.245097

0.7

0.2419637

0.2388521

0.2357625

0.2326951

0.2296500

0.2266274

0.2236273

0.2206499

0.2176954

0.214763

0.8

0.2118554

0.2089701

0.2061081

0.2032694

0.2004542

0.1976625

0.1948945

0.1921502

0.1894297

0.186732

0.9

0.1840601

0.1814113

0.1787864

0.1761855

0.1736088

0.1710561

0.1685276

0.1660232

0.1635431

0.161087

1.0

0.1586553

0.1562476

0.1538642

0.1515050

0.1491700

0.1468591

0.1445723

0.1423097

0.1400711

0.137856

1.1

0.1356661

0.1334995

0.1313569

0.1292381

0.1271432

0.1250719

0.1230244

0.1210005

0.1190001

0.117023

1.2

0.1150697

0.1131394

0.1112324

0.1093486

0.1074877

0.1056498

0.1038347

0.1020423

0.1002726

0.098525

1.3

0.0968005

0.0950979

0.0934175

0.0917591

0.0901227

0.0885080

0.0869150

0.0853435

0.0837933

0.082264

1.4

0.0807567

0.0792698

0.0778038

0.0763585

0.0749337

0.0735293

0.0721450

0.0707809

0.0694366

0.068112

1.5

0.0668072

0.0655217

0.0642555

0.0630084

0.0617802

0.0605708

0.0593799

0.0582076

0.0570534

0.055917

1.6

0.0547993

0.0536989

0.0526161

0.0515507

0.0505026

0.0494715

0.0484572

0.0474597

0.0464787

0.045514

1.7

0.0445655

0.0436329

0.0427162

0.0418151

0.0409295

0.0400592

0.0392039

0.0383636

0.0375380

0.036727

1.8

0.0359303

0.0351479

0.0343795

0.0336250

0.0328841

0.0321568

0.0314428

0.0307419

0.0300540

0.029379

1.9 2.0

0.0287166 0.0227501

0.0280666 0.0222156

0.0274289 0.0216917

0.0268034 0.0211783

0.0261898 0.0206752

0.0255881 0.0201822

0.0249979 0.0196993

0.0244192 0.0192262

0.0238518 0.0187628

0.023295 0.018308

2.1

0.0178644

0.0174292

0.0170030

0.0165858

0.0161774

0.0157776

0.0153863

0.0150034

0.0146287

0.014262

2.2

0.0139034

0.0135526

0.0132094

0.0128737

0.0125455

0.0122245

0.0119106

0.0116038

0.0113038

0.011010

2.3

0.0107241

0.0104441

0.0101704

0.0099031

0.0096419

0.0093867

0.0091375

0.0088940

0.0086563

0.008424

2.4

0.0081975

0.0079763

0.0077603

0.0075494

0.0073436

0.0071428

0.0069469

0.0067557

0.0065691

0.006387

2.5

0.0062097

0.0060366

0.0058677

0.0057031

0.0055426

0.0053861

0.0052336

0.0050849

0.0049400

0.004798

2.6

0.0046612

0.0045271

0.0043965

0.0042692

0.0041453

0.0040246

0.0039070

0.0037926

0.0036811

0.003572

2.7

0.0034670

0.0033642

0.0032641

0.0031667

0.0030720

0.0029798

0.0028901

0.0028028

0.0027179

0.002635

2.8

0.0025551

0.0024771

0.0024012

0.0023274

0.0022557

0.0021860

0.0021182

0.0020524

0.0019884

0.001926

2.9

0.0018658

0.0018071

0.0017502

0.0016948

0.0016411

0.0015889

0.0015382

0.0014890

0.0014412

0.001394

3.0

0.0013499

0.0013062

0.0012639

0.0012228

0.0011829

0.0011442

0.0011067

0.0010703

0.0010350

0.001000

3.1

0.0009676

0.0009354

0.0009043

0.0008740

0.0008447

0.0008164

0.0007888

0.0007622

0.0007364

0.000711

3.2

0.0006871

0.0006637

0.0006410

0.0006190

0.0005976

0.0005770

0.0005571

0.0005377

0.0005190

0.000500

3.3

0.0004834

0.0004665

0.0004501

0.0004342

0.0004189

0.0004041

0.0003897

0.0003758

0.0003624

0.000349

3.4

0.0003369

0.0003248

0.0003131

0.0003018

0.0002909

0.0002803

0.0002701

0.0002602

0.0002507

0.000241

3.5

0.0002326

0.0002241

0.0002158

0.0002078

0.0002001

0.0001926

0.0001854

0.0001785

0.0001718

0.000165

3.6

0.0001591

0.0001531

0.0001473

0.0001417

0.0001363

0.0001311

0.0001261

0.0001213

0.0001166

0.000112

3.7

0.0001078

0.0001036

0.0000996

0.0000957

0.0000920

0.0000884

0.0000850

0.0000816

0.0000784

0.000075

3.8

0.0000723

0.0000695

0.0000667

0.0000641

0.0000615

0.0000591

0.0000567

0.0000544

0.0000522

0.000050

3.9

0.0000481

0.0000461

0.0000443

0.0000425

0.0000407

0.0000391

0.0000375

0.0000359

0.0000345

0.000033

4.0

0.0000317

0.0000304

0.0000291

0.0000279

0.0000267

0.0000256

0.0000245

0.0000235

0.0000225

0.000021

4.1

0.0000207

0.0000198

0.0000189

0.0000181

0.0000174

0.0000166

0.0000159

0.0000152

0.0000146

0.000013

4.2

0.0000133

0.0000128

0.0000122

0.0000117

0.0000112

0.0000107

0.0000102

0.0000098

0.0000093

0.000008

4.3

0.0000085

0.0000082

0.0000078

0.0000075

0.0000071

0.0000068

0.0000065

0.0000062

0.0000059

0.000005

4.4

0.0000054

0.0000052

0.0000049

0.0000047

0.0000045

0.0000043

0.0000041

0.0000039

0.0000037

0.000003

4.5

0.0000034

0.0000032

0.0000031

0.0000029

0.0000028

0.0000027

0.0000026

0.0000024

0.0000023

0.000002

4.6

0.0000021

0.0000020

0.0000019

0.0000018

0.0000017

0.0000017

0.0000016

0.0000015

0.0000014

0.000001

4.7

0.0000013

0.0000012

0.0000012