DIGITAL FILTERS


Authors Lars Wanhammar, Håkan Johansson

Please report errors to the authors

Table of contents


DISCRETE-TIME AND DIGITAL SYSTEMS

DIGITAL FILTERS

1 DISCRETE-TIME AND DIGITAL SYSTEMS


1.1 INTRODUCTION 11
1.2 COMMUNICATION SYSTEMS 12
1.2.1 MULTIMEDIA COMMUNICATION SYSTEMS 13
1.2.2 SATELLITE NETWORKS 14
1.2.3 WIDEBAND RADIO RELAY SYSTEMS 14
1.3 FILTERS 15
1.3.1 IN THE BEGINNING- 15
1.4 DISCRETE-TIME AND DIGITAL SYSTEMS 16
1.5 DISCRETE-TIME FILTERS 19
1.5.1 CTD FILTERS 19
1.5.2 SWITCHED-CAPACITOR FILTERS 20
1.5.3 SWITCHED-CAPACITOR CONVERTERS 20
1.5.4 SWITCHED-CURRENT FILTERS 21
1.5.5 SWITCHED-CURRENT CONVERTERS 21
1.6 DIGITAL FILTERS 22
1.6.1 PROPERTIES OF DIGITAL FILTERS 26
1.6.2 SENSITIVITY 27
1.6.3 ROBUSTNESS 28
1.7 FREQUENCY SELECTIVE FILTERS 29
1.7.1 DIFFERENCE EQUATIONS 29
1.7.2 FIR AND IIR FILTERS 29
1.7.3 TRANSFER FUNCTION 30
1.7.4 FREQUENCY RESPONSE 31
1.7.5 MAGNITUDE RESPONSE 32
1.7.6 ATTENUATION 33
1.7.7 PHASE RESPONSE 34
1.7.8 PHASE DELAY 35
1.7.9 GROUP DELAY 35
1.8 FILTER ALGORITHMS 37
1.9 SIGNAL-FLOW GRAPHS 38
1.10 FILTER STRUCTURES 39
1.11 DIRECT FORMS 39
1.11.1 DIRECT FORM I 39
1.11.2 DIRECT FORM II 40
1.12 TRANSPOSITION 42
1.13 CASCADE FORM STRUCTURES 44
1.14 PARALLEL FORM STRUCTURES 46
1.15 STATE-SPACE STRUCTURES 48
1.16 THE FAST FOURIER TRANSFORM 50

2 ANALOG-DIGITAL CONVERSION

2.1 INTRODUCTION 51
2.2 SAMPLING OF ANALOG SIGNALS 51
2.3 POISSON'S SUMMATION FORMULA 52
2.4 SELECTION OF the SAMPLing FREQUENCY 55
2.5 A/D CONVERSION 57
2.5.1 HIGH-SPEED A/D CONVERTERS 58
2.5.2 OVERSAMPLED A/D CONVERTERS 58
2.6 D/A CONVERSION 62

3 DSP ALGORITHMS

3.1 INTRODUCTION 65
3.2 DSP ALGORITHMS 65
3.2.1 ARITHMETIC OPERATIONS 67
3.2.2 PRECEDENCE GRAPHS 67
3.2.3 PARALLELISM IN ALGORITHMS 68
3.2.4 SEQUENTIALLY COMPUTABLE ALGORITHMS 69
3.3 SFGS IN PRECEDENCE FORM 70
3.4 DIFFERENCE EQUATIONS 74
3.5 COMPUTATION GRAPHS 77
3.5.1 FULLY SPECIFIED SIGNAL-FLOW GRAPHS 78
3.5.2 CRITICAL PATH 78
3.5.3 EQUALIZING DELAY 78
3.5.4 SHIMMING DELAY 79
3.5.5 MAXIMAL SAMPLE FREQUENCY 80
3.5.6 LOW POWER CONSUMPTION 82
3.5.7 High speed versus low Power consumption 82
3.6 EQUIVALENCE TRANSFORMATIONS 83
3.7 PIPELINING 85

4 FINITE WORD LENGTHS EFFECTS

4.1 INTRODUCTION 89
4.2 PARASITIC OSCILLATIONS 90
4.2.1 ZERO-INPUT OSCILLATIONS 91
4.2.2 OVERFLOW CHARACTERISTICS 93
4.2.3 PERIODIC-INPUT OSCILLATIONS 95
4.2.4 NON-OBSERVABLE OSCILLATIONS 96
4.2.5 PARASITIC OSCILLATIONS IN ALGORITHMS USING FLOATING-POINT ARITHMETIC 97
4.3 STABILITY 97
4.4 SCALING OF SIGNAL LEVELS 97
4.4.1 SAFE SCALING 99
4.4.2 Lp-norms 101
4.4.3 SCALING OF WIDE-BAND SIGNALS 103
4.4.4 SCALING OF NARROW-BAND SIGNALS 107
4.4.5 SCALING OF SIGNAL LEVELS IN STATE-SPACE STRUCTURES 109
4.4.6 SCALING TRANSFORMATIONS 109
4.5 ROUNDOFF NOISE 110
4.5.1 ROUNDOFF NOISE IN STATE-SPACE STRUCURES 113
4.5.2 MINIMUM NOISE STATE-SPACE STRUCTURES 115
4.5.3 ERROR SPECTRUM SHAPING 116
4.6 MEASURING ROUNDOFF NOISE 117
4.7 THE INFLUENCE OF COEFFICIENT ERRORS 119
4.7.1 COEFFICIENT SENSITIVITY 119
4.7.2 POLE SENSITIVITY 121
4.7.3 ROOT DENSITY 123
4.7.4 MINIMUM SENSITIVITY STATE-SPACE STRUCTURES 124
4.7.5 MINIMUM-NORM AND NORMAL STATE-SPACE STRUCTURES 124
4.7.6 COEFFICIENT WORD LENGTHS 124
4.7.7 SENSITIVITY AND ROUNDOFF NOISE 125

5 FIR FILTERS

5.1 INTRODUCTION 127
5.2 FIR FILTERS 127
5.3 LINEAR-PHASE FIR FILTERS 128
5.3.1 FREQUENCY RESPONSE 129
5.3.2 EXPRESSIONS FOR HR(wT) 131
5.4 DESIGN OF FIR FILTERS USING WINDOWING TECHNIQUES 131
5.4.1 RECTANGULAR WINDOW 132
5.4.2 von HANN's WINDOW 134
5.4.3 HAMMING's WINDOW 135
5.4.4 BARTLETT's WINDOW 137
5.4.5 BLACKMAN's WINDOW 138
5.4.6 KAISER's WINDOW 139
5.4.7 SARAMÄKI's aAND DOLPH-CHEBYSHEVS WINDOWS 141
5.5 FIR FILTER STRUCTURES 141
5.5.1 DIRECT FORM 141
5.5.2 TRANSPOSED DIRECT FORM 142
5.5.3 LINEAR-PHASE STRUCTURE 143
5.5.4 CASCADE FORM FIR FILTERS 144
5.5.5 COMPLEMENTARY FIR FILTERS 144
5.5.6 LATTICE FIR FILTERS 145
5.5.7 FIR FILTERs BASED ON THE FFT 146
5.5.8 FREQUENCY SAMPLING FIR FILTER STRUCTURES 146
5.6 FREQUENCY TRANSFORMATIONS 146
5.7 MAXIMALLY FLAT FIR FILTERS 147
5.8 MINIMUM-PHASE FIR FILTERS 147
5.9 DIFFERENTIATORS 148
5.10 HILBERT TRANSFORMERS 148
5.11 RAISED COSINE FILTERS 149
5.12 NYQUIST FILTERS 149
5.12.1 HALF-BAND FIR FILTERS 149
5.13 FRACTIONAL DELAY FILTERS 150
5.14 FIR CHIPS 150

6 DESIGN OF FIR FILTERS USING OPTIMIZATION

6.1 INTRODUCTION 153
6.2 SPECIFICATION OF LINEAR-PHASE FIR FILTERS 153
6.2.1 ESTIMATION OF FILTER ORDER 155
6.3 McCLELLAN-PARKS-RABINERS DESIGN 156
6.3.1 APPROXIMATION PROBLEM 156
6.3.2 ALGORITHM 157
6.4 LINEAR PROGRAMMING DESIGN 162
6.4.1 LINEAR PROGRAMMING - LP 163
6.4.2 SOLVING LP PROBLEMS USING MATLAB 164
6.4.3 APPROXIMATION PROBLEMS 166
6.5 DESIGN EXAMPLES 169
6.6 QUADRATIC PROGRAMMING DESIGN 175
6.6.1 QUADRATIC PROGRAMMING - QP 175
6.6.2 SOLVING QP PROBLEMS USING MATLAB 176
6.6.3 APPROXIMATION PROBLEM 177
6.6.4 DESIGN EXAMPLES 180

7 SYNTHESIS OF ANALOG LOWPASS FILTERS

7.1 INTRODUCTION 189
7.2 FREQUENCY SELECTIVE FILTERS 189
7.2.1 IDEAL LOWPASS FILTER 191
7.2.2 FILTER SPECIFICATION 192
7.3 THE APPROXIMATION PROBLEM 195
7.3.1 SQUARED MAGNITUDE FUNCTION 196
7.3.2 ANALOG FILTER APPROXIMATIONS 197
7.3.3 NOTATIONS 198
7.4 STANDARD APPROXIMATIONS 198
7.4.1 BUTTERWORTH FILTERS 199
7.4.2 FILTER ORDER - BUTTERWORTH FILTERS 199
7.4.3 TRANSFER FUNCTION - BUTTERWORTH FILTERS 200
7.5 CHEBYSHEV I FILTERS 211
7.5.1 TRANSFER FUNCTION - CHEBYSHEV I FILTERS 214
7.5.2 CHEBYSHEV I FILTERS TYPE C 221
7.6 CHEBYSHEV II FILTERS 221
7.6.1 TRANSFER FUNCTION - CHEBYSHEV II FILTERS 223
7.6.2 CHEBYSHEV II FILTERS OF TYPE B 229
7.7 CAUER FILTERS 229
7.7.1 TRANSFER FUNCTION - CAUER FILTERS 231
7.7.2 CAUER FILTERS OF TYPE B AND C 239
7.8 BESSEL FILTERS 239
7.9 GAUSS FILTERS 239

8 SYNTHESIS OF ANALOG HIGHPASS, BANDPASS, AND BANDSTOP FILTERS

8.1 INTRODUCTION 241
8.2 FREQUENCY TRANSFORMATIONS 241
8.2.1 LOWPASS-TO-HIGHPASS TRANSFORMATION 242
8.2.2 LOWPASS-TO-BANDPASS TRANSFORMATION 247
8.2.3 LOWPASS-TO-BANDSTOP TRANSFORMATION 253
8.3 TRANSFORMED VARIABLES 256
8.3.1 POLEPLACER 257

9 SYNTHESIS OF IIR FILTERS

9.1 INTRODUCTION 259
9.1.1 DESIGN OF DIGITAL FILTERS FROM ANALOG
PROTOTYPES 259
9.1.2 DESIGN OF DIGITAL FILTERS USING OPTIMIZATION 260
9.2 IMPULSE- AND STEP-INVARIANT TRANSFORMATIONS 260
9.2.1 IMPULSE-INVARIANT TRANSFORMATION 260
9.2.2 STEP-INVARIANT TRANSFORMATION 261
9.3 BILINEAR TRANSFORMATION 261
9.3.1 BILINEARLY TRANSFORMED FILTERS USING MATLAB 264
9.4 FREQUENCY TRANSFORMATIONS 266
9.4.1 LOWPASS-TO-LOWPASS TRANSFORMATION 267
9.4.2 LOWPASS-TO-HIGHPASS TRANSFORMATION 267
9.4.3 LOWPASS-TO-BANDPASS TRANSFORMATION 268
9.4.4 LOWPASS-TO-BANDSTOP TRANSFORMATION 268
9.5 DESIGN OF IIR FILTERS USING OPTIMIZATION 269
9.5.1 DESIGN OF IIR FILTERS USING OPTIMIZATION OF ANALOG FILTERS AND THE BILINEAR TRANSFORM 269
9.5.2 NONLINEAR-PROGRAMMING DESIGN 271

10 ANALOG FILTER STRUCTURES WITH LUMPED ELEMENTS

10.1 INTRODUCTION 279
10.2 SENSITIVITY IN PASSIVE FILTERS 279
10.2.1 DOUBLY TERMINATED LC FILTERS 279
10.2.2 FELDTKELLER's EQUATION 280
10.2.3 ELEMENT SENSITIVITY 282
10.3 LC LADDER STRUCTURES 291
10.3.1 LC LADDER STRUCTURES FOR LOWPASS FILTERS 291
10.3.2 DESIGN OF LOWPASS FILTERS 292
10.3.3 FREQUENCY TRANSFORMATIONS 293
10.4 LATTICE STRUCTURES 295
10.4.1 TRANSFER FUNCTION 296
10.4.2 SCATTERING PARAMETERS 297
10.4.3 REALIZATION OF REACTANCES 299
10.4.4 BARTLETTS AND BRUNES THEOREM 302

11 ANALOG FILTER STRUCTURES WITH DISTRIBUTED ELEMENTS

11.1 INTRODUCTION 305
11.2 TRANSMISSION LINES 305
11.2.1 COMMENSURATE TRANSMISSION LINES 305
11.2.2 RICHARDS' VARIABLE 307
11.2.3 UNIT ELEMENTS 308
11.3 TRANSMISSION LINE FILTERS 309
11.3.1 RICHARDS' THEOREM 311
11.4 SYNTHESIS OF RICHARDS' STRUCTURES 312
11.4.1 MAXIMALLY FLAT RICHARDS' FILTERS 314

12 INTRODUCTION TO WAVE DIGITAL FILTERS

12.1 INTRODUCTION 315
12.2 DIGITAL FILTER STRUCTURES IMITATING ANALOG FILTERS 315
12.2.1 TOPOLOGICAL SIMULATION 315
12.2.2 IMMITANCE SIMULATION 316
12.2.3 WAVE DIGITAL FILTERS 317
12.3 WAVE DESCRIPTIONS 317
12.3.1 POWER WAVES 318
12.3.2 CURRENT WAVES 318
12.3.3 VOLTAGE WAVES 318
12.3.4 REFLECTANCE FUNCTION 319
12.4 WAVE-FLOW BUILDING BLOCKS 319
12.4.1 CIRCUIT ELEMENTS 320
12.5 INTERCONNECTION NETWORKS 323
12.5.1 SYMMETRIC TWO-PORT ADAPTOR 324
12.5.2 SERIES ADAPTORS 325
12.5.3 PARALLEL ADAPTORS 327
12.5.4 DIRECT INTERCONNECTION OF ADAPTORS 330
12.6 DESIGN OF WAVE DIGITAL FILTERS 332
12.6.1 FELDTKELLER's EQUATION 334
12.7 SUPPRESSION OF PARASITIC OSCILLATIONS IN WAVE DIGITAL FILTERS 336

13 WAVE DIGITAL FILTERS BASED ON RICHARDS' STRUCTURES

13.1 INTRODUCTION 339
13.2 RICHARDS' WAVE DIGITAL FILTERS 339
13.3 REFLECTANCE FUNCTION OF RICHARDS' WDFs 341
13.4 GRAY AND MARKEL' STRUCTURES 343

14 WAVE DIGITAL FILTERS BASED ON LADDER STRUCTURES

14.1 INTRODUCTION 345
14.2 LADDER WAVE DIGITAL FILTERS USING SEPARATING UNIT ELEMENTS 345
14.2.1 KURODA-LEVY IDENTITIES 345
14.3 LADDER WAVE DIGITAL FILTERS USING DIRECTLY INTERCONNECTED ADAPTORS 351
14.3.1 MINIMIZING THE NUMBER OF DELAY ELEMENTS 354

15 WAVE DIGITAL FILTERS BASED ON LATTICE S STRUCTURES

15.1 INTRODUCTION 355
15.2 LATTICE WAVE DIGITAL FILTERS 355
15.2.1 TRANSFER FUNCTION 356
15.2.2 LATTICE WDF STRUCTURES 357
15.3 LATTICE WDF STRUCTURES BASED ON RICHARDS' AND CIRCULATOR STRUCTURES 358
15.3.1 RICHARDS' STRUCTURES 358
15.3.2 CIRCULATOR STRUCTURES 360
15.3.3 LATTICE WAVE DIGITAL FILTERS BASED ON RICHARDS' AND CIRCULATOR STRUCTURES 361
15.3.4 MAXIMAL SAMPLE FREQUENCY 363
15.4 DESIGN OF LATTICE WDF'S 364
15.5 LINEAR-PHASE LATTICE WDFS 369
15.5.1 FILTERS WITH A PURE DELAY BRANCH 369
15.5.2 GENERAL LINEAR-PHASE LATTICE WDF STRUCTURE 371
15.6 BIRECIPROCAL LATTICE WDFs 374
15.6.1 BIRECIPROCAL LINEAR-PHASE LATTICE WDFs 377
15.7 WAVE CIRCULATOR FILTERS 380
15.8 COMPLEX WAVE DIGITAL FILTERS 381
15.8.1 WAVE DIGITAL HILBERT TRANSFORMERS 381
15.10 ROUNDOFF NOISE IN WDFS 384

16 FREQUENCY MASKING TECHNIQUES

16.1 INTRODUCTION 385
16.2 NARROW-BAND FIR FILTERS 385
16.2.1 LOWPASS FILTERS 386
16.2.2 ARITHMETIC COMPLEXITY 388
16.2.3 HIGHPASS AND BANDPASS FILTERS 390
16.2.4 MULTI-STAGE FILTERS 391
16.3 WIDE-BAND FIR FILTERS 391
16.4 ARBITRARY BANDWIDTH FIR FILTERS 393
16.5 INTRODUCTION TO FREQUENCY MASKING IIR FILTERs 395
16.6 NARROW-BAND IIR FILTERS 397
16.7 WIDE-BAND IIR FILTERS 400
16.8 ARBITRARY BANDWIDTH IIR FILTERS 402

17 INTERPOLATION AND DECIMATION FILTERS

17.1 INTRODUCTION 407
17.2 SAMPLING RATE CONVERSION 408
17.2.1 INTERPOLATION 408
17.2.2 DECIMATION 410
17.2.3 CONVERSION BY RATIONAL NUMBERS 412
17.3 POLYPHASE STRUCTURES 413
17.3.1 IDENTITIES 413
17.3.2 POLYPHASE REPRESENTATION 414
17.3.3 POLYPHASE INTERPOLATION AND DECIMATION STRUCTURES 417
17.3.4 CONVERTERS WITH TIME-VARIANT COEFFICIENTS 419
17.3.5 STRUCTURES FOR CONVERSION WITH RATIONAL FACTORS 420
17.4 INTERPOLATION AND DECIMATION FILTERS 422
17.4.1 SPECIFICATION 422
17.4.2 FIR FILTERS 423
17.4.3 IIR FILTERS 426
17.5 MULTISTAGE REALIZATIONS 429
17.5.1 TWO-STAGE REALIZATION FOR CONVERSION WITH PRIME NUMBERS 430
17.6 HSP43220? 434

REFERENCES 437


INDEX 453