# Appendix H. Frequency Sampling Filter Design Tables

In Section 7.1 we described the so-called Type-IV frequency sampling filter (FSF). The tables in this appendix provide a list of optimum transition coefficient values for the Case I (see Figure 7-25) Type-IV lowpass FSFs of various passband bandwidths, over a range of values of N. Table H-1 provides the H(k) single transition coefficient and two transition coefficients for even values of N. Table H-2 provides the H(k) three transition coefficients for even N. Table H-3 provides the H(k) single transition coefficient and two transition coefficients for odd values of N, while Table H-4 provides the H(k) three transition coefficients for odd N.

The passband bandwidth in these tables, signified by the BW parameter, is the number of FSF sections having unity-valued H(k) gain factors. For example, an N = 32 lowpass FSF using six passband sections and a single transition region coefficient (T1) would have the H(k) gain values shown in Figure H-1(a). In this case, the T1 coefficient would be found in Table H-1 for N = 32 at a bandwidth BW = 6. An N = 23 lowpass FSF with five passband sections and two transition region coefficients (T1 and T2) would have the H(k) gain values shown in Figure H-1(b). In this case, the T1 and T2 coefficients are found in Table H-2 for N = 23 at a bandwidth BW = 5. An additional parameter in the tables is the maximum stopband sidelobe attenuation levels (Atten).

Figure H-1. Transition coefficient examples: (a) one coefficient for N = 32 and BW = 6; (b) two coefficients for N = 23 and BW = 5.

Table H-1. Lowpass Type-IV FSF for Even N (One and Two Coefficients)

BW

Atten

T1

BW

Atten

T1

T2

N = 16

N = 16

1

–44.9

0.41924081

1

–76.5

0.56626687

0.07922718

2

–45.8

0.38969818

2

–77.2

0.55487263

0.08012238

3

–47.3

0.36942214

3

–81.2

0.53095099

0.07087993

4

–49.6

0.34918551

4

–87.7

0.49927622

0.05813368

N = 24

N = 24

1

–44

0.42452816

1

–73.6

0.57734042

0.08641861

2

–44.1

0.40042889

2

–72.5

0.57708274

0.09305238

3

–44.9

0.38622106

3

–72.9

0.56983709

0.09177956

4

–45.7

0.37556064

4

–73.8

0.55958351

0.08770698

5

–46.5

0.36663149

5

–75.6

0.54689579

0.08202772

N = 32

N = 32

1

–43.6

0.42638815

1

–72.6

0.58109341

0.08892320

2

–43.6

0.40407598

2

–71

0.58466392

0.09771906

3

–44

0.39197681

3

–70.8

0.58101218

0.09823378

4

–44.5

0.38377786

5

–71.6

0.57002693

0.09442029

6

–45.5

0.37172559

7

–73.2

0.55593774

0.0879846

7

–45.8

0.38912443

9

–76.2

0.53661082

0.07884406

8

–46.6

0.36087271

13

–106

0.43242657

0.03643965

N = 48

N = 48

1

–43.4

0.42772741

1

–72

0.58385966

0.09079945

2

–43.2

0.40654471

2

–70

0.58999731

0.10107095

3

–43.5

0.39569517

3

–69.5

0.58898579

0.1030238

4

–43.8

0.38879556

5

–69.6

0.58356869

0.10204926

6

–44.4

0.37994174

7

–70

0.57749269

0.09959325

8

–44.9

0.37394938

9

–70.6

0.57143299

0.09683486

10

–45.3

0.3690437

N = 64

N = 64

1

–43.3

0.42815077

1

–71.8

0.58480329

0.09144087

2

–43.1

0.40742967

2

–69.7

0.59178518

0.10221701

3

–43.4

0.39690507

3

–69.2

0.59168291

0.10467406

4

–43.6

0.39043339

4

–68.9

0.5899207

0.10496398

5

–43.9

0.38583162

5

–68.9

0.58788109

0.10457886

6

–44.1

0.38244173

9

–69.4

0.58010661

0.10157302

10

–44.7

0.37382147

13

–70.2

0.57272483

0.09810828

14

–45.3

0.36813961

17

–71.1

0.56417336

0.09386963

N = 96

N = 96

1

–43.2

0.42856954

1

–71.6

0.585424

0.09185794

2

–43

0.40790815

2

–69.5

0.59305878

0.10302346

3

–43.3

0.3977603

3

–68.8

0.59332978

0.10571202

4

–43.5

0.39154291

4

–68.4

0.59202942

0.10623854

5

–43.7

0.38719365

5

–68.3

0.59062246

0.10623287

6

–43.9

0.38409245

9

–68.5

0.58514671

0.10445521

10

–44.4

0.37686993

13

–68.9

0.58087019

0.10253761

14

–44.7

0.37270333

17

–69.3

0.57751103

0.10097157

18

–45

0.36984146

21

–65.3

0.59419612

0.11180709

N = 128

N = 128

1

–43.2

0.42864273

1

–71.6

0.58574352

0.09208128

2

–43

0.40823844

2

–69.4

0.59357535

0.10335703

3

–43.2

0.39811024

3

–68.6

0.59383385

0.10601544

4

–43.4

0.39188378

4

–68.3

0.59279342

0.10671242

5

–43.7

0.38774353

6

–68.1

0.59024028

0.10655014

7

–44

0.38236389

9

–68.2

0.5868017

0.10542034

10

–44.3

0.37771128

17

–68.7

0.58071332

0.10271656

18

–44.8

0.3717883

25

–69.2

0.57652818

0.1006787

26

–45.1

0.36862161

33

–69.8

0.57265344

0.09871098

N = 192

N = 192

1

–43.2

0.42881027

1

–71.5

0.58589507

0.09218797

2

–43

0.40831822

2

–69.4

0.59383365

0.1035231

3

–43.2

0.39830476

3

–68.6

0.59433749

0.10635017

4

–43.4

0.39219024

4

–68.2

0.5933049

0.10702144

5

–43.6

0.38797095

6

–67.9

0.59098287

0.10700026

7

–43.9

0.3828125

9

–68

0.58790082

0.10604544

10

–44.2

0.37845008

17

–68.4

0.58295021

0.10399738

18

–44.7

0.37302517

25

–68.7

0.57978921

0.10242432

26

–44.9

0.37049755

33

–69

0.57773363

0.10143253

34

–45

0.36908526

41

–69.2

0.57597277

0.10055134

42

–45.2

0.36764286

49

–69.5

0.57407637

0.09954949

N = 224

N = 224

1

–43.2

0.42874481

1

–71.5

0.58591935

0.09220841

2

–43

0.40836093

2

–69.4

0.593894

0.10355622

3

–43.2

0.39831237

3

–68.5

0.59435536

0.10634325

4

–43.4

0.3921651

4

–68.1

0.59354863

0.10719315

5

–43.6

0.38807204

6

–67.9

0.59106856

0.1070268

7

–43.9

0.38281226

9

–67.9

0.58816185

0.10619806

10

–44.2

0.37847605

10

–67.9

0.58726527

0.10582934

11

–44.3

0.37742038

17

–68.3

0.58317185

0.10407347

18

–44.6

0.37324982

33

–68.8

0.57860121

0.10189345

34

–45

0.36946431

49

–69.2

0.5757377

0.10043439

50

–45.2

0.36753866

57

–69.4

0.57440527

0.09975201

N = 256

N = 256

1

–43.2

0.42874481

1

–71.5

0.58599793

0.09225917

2

–43

0.40844072

2

–69.4

0.59395199

0.10360852

3

–43.2

0.39839019

3

–68.5

0.59445009

0.10641045

4

–43.4

0.39231838

4

–68.1

0.59358715

0.10721801

5

–43.6

0.38814788

6

–67.9

0.59121865

0.10712227

7

–43.9

0.38296192

9

–67.9

0.58824601

0.10622746

10

–44.2

0.37855003

10

–67.9

0.58727091

0.10580213

11

–44.3

0.37756795

17

–68.2

0.58347729

0.10424412

18

–44.6

0.3733958

33

–68.7

0.57913354

0.1021842

34

–44.9

0.36982575

49

–69

0.57667852

0.10093887

50

–45.1

0.36804233

57

–69.2

0.57568486

0.10043603

58

–45.2

0.3673716

65

–69.3

0.57469205

0.09993526

Table H-2. Lowpass Type-IV FSF for Odd N (One and Two Coefficients)

BW

Atten

T1

BW

Atten

T1

T2

N = 15

N = 15

1

–45.1

0.41802444

1

–77.3

0.56378193

0.07767395

2

–46.2

0.38716426

2

–78.9

0.54831544

0.07646082

3

–48

0.36461603

3

–83.4

0.52139051

0.06627593

4

–51.4

0.34005655

4

–96.3

0.47732772

0.0487037

N = 23

N = 23

1

–44

0.42412451

1

–73.8

0.57648809

0.08585489

2

–44.3

0.3995718

2

–72.7

0.57569102

0.09221862

3

–45.1

0.38497937

3

–73.2

0.56732401

0.09032595

4

–46

0.37367551

4

–74.6

0.55575797

0.08563268

5

–46.8

0.36445788

5

–76.5

0.54245814

0.07981527

N = 33

N = 33

1

–43.6

0.42659097

1

–72.6

0.58141103

0.08913503

2

–43.5

0.40433257

2

–70.9

0.58529197

0.09811292

3

–44

0.39239983

3

–70.7

0.58187597

0.09874619

4

–44.4

0.3843389

5

–71.3

0.57154421

0.09525674

6

–45.3

0.37272147

7

–72.9

0.55826179

0.08918146

8

–46.4

0.36256798

9

–75.3

0.54128598

0.08116809

N = 47

N = 47

1

–43.4

0.42768264

1

–72

0.58376507

0.09073592

2

–43.2

0.40649692

2

–70.1

0.58975381

0.1009182

3

–43.5

0.39553589

3

–69.6

0.5886934

0.10284475

4

–43.8

0.38859729

5

–69.6

0.58311582

0.10178496

6

–44.4

0.37968179

7

–70.1

0.57687412

0.09925266

8

–44.9

0.37362421

9

–70.7

0.57045477

0.0963056

10

–45.4

0.36853968

11

–71.6

0.56319305

0.09280396

N = 65

N = 65

1

–43.3

0.4281872

1

–71.7

0.58480896

0.09144011

–43.1

0.40743541

2

–69.7

0.59188395

0.10227409

3

–43.4

0.39694541

3

–69.1

0.59158717

0.10459977

4

–43.6

0.39043991

4

–68.9

0.59001405

0.10501496

5

–43.9

0.3859325

5

–68.8

0.58797401

0.10462176

6

–44.1

0.38249194

9

–69.4

0.58031079

0.10167681

10

–44.7

0.37399454

13

–70.1

0.57312044

0.09830111

14

–45.2

0.36841874

17

–71

0.56506463

0.0943328

18

–45.8

0.36324429

21

–72.5

0.55369626

0.08867447

N = 95

N = 95

1

–43.2

0.42852251

1

–71.6

0.58559589

0.09199196

2

–43

0.40799464

2

–69.5

0.59306419

0.10302347

3

–43.3

0.39772511

3

–68.8

0.59323668

0.10563966

4

–43.5

0.39143867

4

–68.5

0.59207559

0.10628422

5

–43.7

0.38722579

5

–68.3

0.5905046

0.10614086

6

–43.9

0.38400287

9

–68.5

0.58506537

0.10440925

10

–44.4

0.3766755

13

–68.9

0.58094477

0.10262083

14

–44.8

0.37268027

17

–69.3

0.57725045

0.10081244

18

–45

0.369842

21

–69.8

0.57370707

0.09903691

N = 125

N = 125

1

–43.2

0.42857276

1

–71.5

0.5856764

0.0920373

2

–43

0.40819419

2

–69.5

0.59346193

0.10328234

3

–43.2

0.39806479

3

–68.7

0.59389103

0.1060686

4

–43.5

0.39191493

5

–68.1

0.59144981

0.1067222

6

–43.8

0.38458541

7

–68.1

0.58887274

0.10611287

8

–44.1

0.38046599

9

–68.2

0.58670301

0.10535739

10

–44.3

0.37761366

17

–68.7

0.5805297

0.10262003

18

–44.8

0.37166577

25

–69.3

0.57610034

0.10041636

26

–45.1

0.36836415

33

–69.9

0.57206269

0.0984002

N = 191

N = 191

1

–43.2

0.42865655

1

–71.5

0.5859143

0.09220684

2

–43

0.40839373

2

–69.4

0.5937664

0.10346999

3

–43.2

0.39822053

3

–68.5

0.59423193

0.1062604

4

–43.4

0.39214502

5

–68

0.59213475

0.1071269

6

–43.8

0.38503751

7

–67.9

0.589875

0.10671096

8

–44

0.38095089

9

–68

0.58788996

0.10603382

10

–44.2

0.37828083

17

–68.4

0.58282022

0.10390276

18

–44.7

0.37305805

25

–68.7

0.57971044

0.10236319

26

–44.9

0.37048161

33

–69

0.57760031

0.10133385

34

–45

0.36904678

41

–69.2

0.57578133

0.10042123

42

–45.1

0.3676021

49

–69.5

0.57404729

0.09954845

N = 223

N = 223

1

–43.2

0.42874481

1

–71.5

0.58589267

0.0921919

2

–43

0.40836093

2

–69.4

0.59379277

0.10347913

3

–43.2

0.39835128

3

–68.5

0.59435536

0.10634325

4

–43.4

0.3921651

4

–68.1

0.59354863

0.10719315

5

–43.6

0.38807204

6

–67.9

0.59114264

0.10708575

7

–43.9

0.38288709

9

–67.9

0.58797899

0.10606475

10

–44.2

0.37851304

10

–67.9

0.58726527

0.10582934

11

–44.3

0.37742038

17

–68.3

0.58329541

0.10416575

18

–44.6

0.37324982

33

–68.8

0.57849661

0.10182631

34

–45

0.36946431

49

–69.2

0.57568307

0.10040604

50

–45.2

0.3674667

57

–69.4

0.5744086

0.09975962

N = 255

N = 255

1

–43.2

0.42874481

1

–71.5

0.58590294

0.09218854

2

–43

0.40836093

2

–69.3

0.59392035

0.10356223

3

–43.2

0.39831237

3

–68.5

0.59440493

0.10637653

4

–43.4

0.39224174

4

–68.1

0.59363182

0.10725379

5

–43.6

0.38814788

6

–67.9

0.59121586

0.10712951

7

–43.9

0.38296192

9

–67.9

0.58822523

0.10621751

10

–44.2

0.37855003

10

–67.9

0.58747402

0.10596604

11

–44.3

0.37756795

17

–68.2

0.58351595

0.10427857

18

–44.6

0.3733958

33

–68.7

0.57906429

0.10213183

34

–44.9

0.36982575

49

–69

0.57660259

0.10088411

50

–45.1

0.36804233

57

–69.2

0.57562728

0.10040259

58

–45.2

0.3673716

65

–69.4

0.57461162

0.09987875

Table H-3. Lowpass Type-IV FSF for Even N (Three Coefficients)

BW

Atten

T1

T2

T3

N = 16

1

–112.6

0.64272445

0.15442399

0.00880089

2

–95.9

0.70487291

0.22419597

0.01947599

3

–100.7

0.70063089

0.21872748

0.01757096

4

–115.9

0.68531929

0.19831357

0.01270197

N = 24

1

–103.8

0.6599002

0.17315143

0.01189889

2

–101.8

0.67718249

0.19461557

0.01429673

3

–95.5

0.69609682

0.21773826

0.01860944

4

–104.1

0.66830223

0.18959572

0.01339907

N = 32

1

–99.6

0.6642114

0.17798254

0.01278046

2

–98.5

0.68804961

0.20639825

0.01646338

4

–87.4

0.73378289

0.26142233

0.02762054

6

–100.5

0.67913658

0.20169658

0.01554611

8

–105.3

0.65936975

0.18380663

0.01270743

N = 48

1

–93.8

0.68361144

0.2010237

0.01735969

2

–96

0.69534463

0.21480253

0.01812435

4

–87.2

0.73314865

0.26098449

0.02762804

6

–86.4

0.73802064

0.26732823

0.02900775

8

–95

0.69703503

0.2211425

0.01909109

10

–90

0.71746809

0.24474881

0.02420421

N = 64

1

–96.6

0.67620503

0.19208214

0.01551621

2

–94.9

0.69693984

0.21653685

0.01842226

3

–89.7

0.72079468

0.24569738

0.02432222

4

–92.3

0.7068141

0.22927121

0.02042893

8

–91.4

0.70957119

0.23498487

0.02215407

12

–93.8

0.7026052

0.22772953

0.02059288

16

–85.3

0.74439511

0.27543213

0.03085705

N = 96

1

–98.5

0.6720933

0.18712559

0.01449609

2

–92.9

0.70471821

0.22591053

0.02048075

3

–93

0.70905096

0.23165702

0.02121954

4

–88.7

0.72625477

0.25269331

0.02574193

8

–90.8

0.71369108

0.23929089

0.02281527

12

–90.8

0.71110318

0.23715671

0.02248568

16

–85.2

0.74356072

0.27478153

0.03080406

20

–85.8

0.74022029

0.27104418

0.02999046

N = 128

1

–98.3

0.67221636

0.18725564

0.01451885

2

–94.4

0.70015724

0.22042278

0.01929075

3

–92.6

0.70981704

0.23257905

0.02143209

5

–90.6

0.71933148

0.24480839

0.02391897

8

–89.8

0.72190475

0.24869701

0.02481883

16

–88.5

0.72569265

0.25405918

0.02615712

24

–87.4

0.7301942

0.25964746

0.02748522

N = 192

1

–98.1

0.67216994

0.1871603

0.01447431

2

–94.3

0.70064573

0.22097713

0.01939796

3

–92.6

0.71046628

0.23329177

0.02156244

5

–90.6

0.71933299

0.24477507

0.0238993

8

–89.8

0.72185688

0.24857861

0.02477626

16

–88.5

0.72617255

0.2545026

0.02622728

24

–87.7

0.72957884

0.25880678

0.02726692

32

–90.9

0.71321929

0.24041037

0.02328586

40

–91.2

0.71133926

0.23853571

0.02293979

48

–91.4

0.70862489

0.2357226

0.0224067

N = 224

1

–98.2

0.67256687

0.18767169

0.01459779

2

–94.2

0.70077254

0.22112728

0.01942992

3

–90.4

0.7026477

0.22304697

0.01885735

5

–91.1

0.71677647

0.24176238

0.0232377

8

–89.9

0.72168089

0.24837531

0.02473386

9

–89.8

0.71675825

0.24253218

0.02331464

16

–90.3

0.71805244

0.24514888

0.0241623

32

–90.6

0.71429115

0.24150812

0.0234885

48

–91.1

0.71133746

0.23857357

0.02295657

N = 256

1

–95.7

0.67780153

0.19398356

0.01590119

2

–94

0.70138048

0.22187281

0.01959708

3

–90.3

0.70235664

0.22265441

0.01875372

5

–91

0.71654134

0.24139758

0.02311255

8

–89.9

0.72167623

0.24835995

0.02472548

9

–89.7

0.71676546

0.24249377

0.02328724

16

–90.2

0.71841628

0.24555225

0.02424786

32

–90.5

0.71523646

0.24257287

0.02372755

48

–90.8

0.71282545

0.2402303

0.02331467

56

–90.7

0.71353605

0.24104134

0.02347778

Table H-4. Lowpass Type-IV FSF for Odd N (Three Coefficients)

BW

Atten

T1

T2

T3

N = 15

1

–99.1

0.67276446

0.18765467

0.01448603

2

–109

0.65109591

0.16879848

0.01032845

3

–103

0.64743501

0.16597437

0.00887322

4

–129.8

0.58430569

0.11830427

0.0041067

N = 23

1

–98.6

0.67021235

0.18516808

0.01420518

2

–96.9

0.6596023

0.17408826

0.00996838

3

–95.3

0.64635467

0.16260027

0.0077623

4

–86.1

0.60390729

0.12509768

0.00296913

N = 33

1

–98.4

0.67150869

0.18654613

0.01442309

2

–97.9

0.68975409

0.20844142

0.01686088

4

–93.8

0.70392025

0.22717012

0.02035858

6

–92.1

0.70836197

0.23374423

0.02185812

8

–92.9

0.70271751

0.22868478

0.02098636

N = 47

1

–99.7

0.66933083

0.18386234

0.01384901

2

–94.7

0.69037782

0.20845236

0.01654889

4

–94.4

0.70435781

0.22714301

0.02019897

6

–94.5

0.70200706

0.22582668

0.01999782

8

–95.4

0.69662478

0.22082819

0.01907788

10

–97

0.69029654

0.21493063

0.01807728

12

–88.8

0.64107819

0.16254219

0.0076571

N = 65

1

–98.8

0.67071168

0.18547676

0.01416929

2

–94.8

0.69743725

0.21725

0.01864255

3

–93.6

0.70659336

0.22882367

0.02062804

4

–93

0.70962871

0.23307976

0.02141978

8

–92.6

0.70884359

0.23403175

0.02173011

12

–88.7

0.72402053

0.25220517

0.02579854

16

–89.9

0.71679306

0.24461807

0.02427071

N = 95

1

–98.6

0.67204252

0.18706788

0.0144861

2

–94.5

0.69934889

0.21945399

0.01908355

3

–93

0.70879388

0.23134381

0.02114855

4

–92.2

0.71277193

0.23666779

0.0221755

8

–89.2

0.72434568

0.25160197

0.02547937

12

–88.8

0.72479877

0.25277864

0.02583655

16

–90.1

0.71754976

0.24492389

0.02419794

20

–92.6

0.70606236

0.23250776

0.02165584

N = 125

1

–98.3

0.67220923

0.18724753

0.01451708

2

–94.4

0.7001615

0.22041952

0.01928486

4

–90.8

0.71816438

0.24299337

0.02355056

6

–90.5

0.71950797

0.24538851

0.02405304

8

–90.5

0.71546288

0.2410237

0.02303409

16

–89.5

0.72139717

0.24910324

0.02505485

24

–89.9

0.71817491

0.24597097

0.02448715

32

–88.9

0.72170356

0.25030101

0.02550214

N = 191

1

–98.1

0.67223344

0.1872747

0.01452257

2

–94.3

0.70075621

0.22112031

0.01943466

4

–90.5

0.7196321

0.24471727

0.02393159

6

–90.9

0.71816017

0.24379217

0.02370239

8

–90.4

0.71833134

0.24451461

0.02390559

16

–90.4

0.71734228

0.244362

0.02399541

24

–90.1

0.71838476

0.24599127

0.02444213

32

–89

0.72239379

0.25075541

0.02551853

40

–89.7

0.7188511

0.24690437

0.0247143

48

–91.6

0.70888027

0.23598149

0.02243978

N = 223

1

–98.2

0.67257273

0.18768014

0.01460038

2

–94.2

0.70072996

0.22108243

0.01942305

3

–90.4

0.70262204

0.22302806

0.0188601

5

–91

0.71641456

0.24126979

0.02309323

8

–89.9

0.72166528

0.24835901

0.02473006

9

–89.8

0.71674758

0.24252389

0.02331481

16

–90.3

0.7179882

0.24507651

0.02414698

32

–90.6

0.71403002

0.24119297

0.02341017

48

–91.2

0.71143573

0.2386827

0.02297579

56

–90.7

0.71338506

0.24097504

0.02348867

N = 255

1

–97.6

0.67176623

0.18670931

0.01441482

2

–94.2

0.70070534

0.22105163

0.01941457

3

–90.3

0.70233551

0.22263226

0.01874791

5

–91

0.71654995

0.24140965

0.02311617

8

–89.9

0.72164196

0.24831241

0.02471143

9

–89.7

0.71673449

0.2424607

0.02328277

16

–90.1

0.71885073

0.24604614

0.02435372

32

–89.6

0.71549778

0.24342724

0.02411907

48

–90.9

0.71268842

0.23999548

0.02323405

56

–90.7

0.71349454

0.240994

0.02346651

64

–91.3

0.71035623

0.23761455

0.02277658

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