Preface ........................................................ xi
Acknowledgements .............................................. xvi
Glossary of terms ............................................ xvii
1 Introduction to waveform generation ........................ 1
1.1 Preliminaries .............................................. 3
1.1.1 Outline chapter content ............................. 3
1.1.2 Digital signal processing ........................... 5
1.1.3 Periodic and aperiodic waveforms ....................6
1.1.4 Introducing the sine wave - properties and
parameters .......................................... 9
1.1.5 Instantaneous phase and frequency .................. 10
1.1.6 From phase to amplitude - the waveform function .... 12
1.1.7 Signal definition - waveform or spectrum? .......... 14
1.1.8 Signal specification - time or frequency domain? ... 15
1.1.9 A brief history of digital waveform generation ..... 17
1.2 A taxonomy of electronic waveform generation .............. 19
1.2.1 Background ......................................... 19
1.2.2 Analogue waveform generation ....................... 19
1.2.3 Hybrid waveform generation ......................... 23
1.2.4 Digital waveform generation ........................ 24
1.2.5 Arbitrary waveform generation ...................... 25
1.3 Analogue waveform generation .............................. 31
1.3.1 Analogue waveform generation in test
instrumentation .................................... 31
1.3.2 The Wien bridge oscillator ......................... 32
1.3.3 The state variable oscillator ...................... 34
1.3.4 The voltage-controlled switched integration
oscillator ......................................... 36
1.3.5 The charge-balancing voltage to frequency
converter .......................................... 38
1.3.6 The shortcomings of analogue waveform
generation ........................................ 38
1.4 Applications of arbitrary waveform generation ............. 41
1.4.1 Test and diagnostic applications ................... 41
1.4.2 Signal emulation applications ...................... 42
1.4.3 Bespoke embedded applications ...................... 43
1.4.4 RF up-conversion applications ...................... 43
1.4.5 Unique DDS AWG attributes .......................... 45
1.4.6 A simple example - testing a PLL ................... 45
1.5 Summary ................................................... 47
1.6 References ................................................ 48
2 The foundations of digital waveform generation ............ 49
2.1 Mathematical preliminaries ................................ 49
2.1.1 Continuous-and discrete-time signals ............... 49
2.1.2 The phase-frequency relationship of a
discrete-time sinusoid ............................. 52
2.1.3 Analogue reconstruction and alias image
rejection .......................................... 54
2.1.4 Quantisation and digital to analogue conversion .... 54
2.2 A fundamental concept - sampling a tabulated signal ....... 55
2.2.1 Tabulating a signal - introducing the wavetable .... 55
2.2.2 Regenerating a tabulated signal .................... 56
2.2.3 A sample rate conversion view ...................... 57
2.2.4 Frequency control resolution and wavetable
length ............................................. 58
2.3 Tabulating a waveform function ............................ 59
2.3.1 The wavetable revisited ............................ 59
2.3.2 Tabulating the sine function ....................... 60
2.3.3 Time domain arbitrary waveform specification ....... 62
2.3.4 Frequency domain arbitrary waveform
specification ...................................... 63
2.4 The foundations of direct digital synthesis (DDS) ......... 64
2.4.1 Phase accumulation frequency synthesis ............. 64
2.4.2 Phase-amplitude mapping ............................ 67
2.4.3 Phase continuity ................................... 68
2.4.4 Phase-amplitude mapping and fractional
addressing ......................................... 70
2.4.5 The aliasing problem ............................... 73
2.5 Control parameters and their characteristics .............. 74
2.5.1 Desirable control characteristics .................. 75
2.5.2 Frequency control .................................. 76
2.5.3 Phase offset control ............................... 78
2.5.4 Amplitude control .................................. 79
2.5.5 Waveshape control .................................. 79
2.5.6 Spectrum control ................................... 81
2.6 Performance metrics ....................................... 83
2.6.1 Phase truncation noise ............................. 83
2.6.2 The amplitude error spectrum ....................... 85
2.6.3 Waveshape error .................................... 85
2.6.4 Phase noise ........................................ 86
2.6.5 Arithmetic processing requirements ................. 88
2.7 Summary ................................................... 89
2.8 References ................................................ 89
3 Recursive sine wave oscillators ........................... 90
3.1 Mathematical preliminaries ................................ 91
3.2 The direct-form ........................................... 93
3.3 The coupled-form .......................................... 99
3.4 The modified coupled-form ................................ 101
3.5 The waveguide-form ....................................... 101
3.6 Implementation considerations ............................ 102
3.6.1 Initial conditions ................................ 102
3.6.2 Arithmetic quantisation effects ................... 104
3.6.3 Frequency control ................................. 106
3.7 Summary .................................................. 107
3.8 References ............................................... 108
4 DDS sine wave generation ................................. 109
4.1 Phase accumulation, truncation and arithmetic
processing ............................................... 109
4.1.1 Ideal sinusoidal DDS .............................. 110
4.1.2 Optimal phase-amplitude mapping ................... 112
4.1.3 Phase truncation and fractional wavetable
addressing ........................................ 114
4.1.4 Linear phase interpolation ........................ 122
4.1.5 Static and dynamic phase offset ................... 124
4.1.6 Quadrature sine wave generation ................... 125
4.1.7 Introducing phase domain processing ............... 127
4.2 Theoretical analysis of DDS phase truncation errors ...... 128
4.2.1 The DDS error model ............................... 128
4.2.2 Phase truncation errors ........................... 129
4.2.3 Wavetable sample quantisation errors .............. 132
4.3 Sinusoidal phase-amplitude mapping ....................... 134
4.3.1 SNR and oversampling .............................. 134
4.3.2 Phase truncated wavetable indexing ................ 135
4.3.3 Phase dithered wavetable indexing and noise
shaping ........................................... 138
4.3.4 Linear phase interpolated wavetable indexing ...... 144
4.3.5 Trigonometric identity phase interpolation ........ 148
4.3.6 Sinusoidal wavetable compression .................. 152
4.3.7 Direct computation phase-amplitude mapping -
the CORDIC algorithm .............................. 155
4.4 Summary .................................................. 159
4.5 References ............................................... 161
5 DDS arbitrary waveform generation ........................ 162
5.1 From sinusoidal to arbitrary waveforms ................... 162
5.1.1 Phase truncation and the need for
interpolation ..................................... 163
5.1.2 Introducing Lagrange polynomial interpolation ..... 165
5.1.3 The harmonic aliasing problem ..................... 170
5.1.4 Time domain specification of an arbitrary
waveform .......................................... 172
5.1.5 Frequency domain specification of an arbitrary
waveform .......................................... 178
5.1.6 Defining DDS AWG simulation parameters ............ 181
5.1.7 Optimal phase-amplitude mapping ................... 184
5.2 Interpolated phase-amplitude mapping ..................... 186
5.2.1 Phase truncated wavetable indexing ................ 186
5.2.2 Phase dithered wavetable indexing ................. 187
5.2.3 Linear phase interpolated wavetable indexing ...... 192
5.2.4 Lagrange polynomial phase interpolated
wavetable indexing ................................ 195
5.3 Analogue waveshaping ..................................... 203
5.3.1 The phase accumulator as a digital frequency
synthesiser ....................................... 205
5.3.2 Analogue waveshaping principles and some
fundamental problems .............................. 207
5.3.3 Triangle waveform shaping ......................... 211
5.4 DDS clock generation ..................................... 213
5.4.1 Hard-limited sinusoidal DDS ....................... 214
5.4.2 Reconstruction filter bandwidth ................... 216
5.5 Variable sample frequency AWG ............................ 217
5.5.1 Basic architecture and implementation
limitations ....................................... 217
5.5.2 Harmonic aliasing and reconstruction filter
considerations .................................... 218
5.6 Computer music and audio test applications of DDS
AWG ...................................................... 219
5.6.1 Harmonics and partials ............................ 220
5.6.2 Frequency control, musical pitch and tuning ...... 221
5.6.3 Audio measurement - harmonic and
intermodulation distortion ........................ 222
5.7 Summary .................................................. 225
5.8 References ............................................... 227
6 Dynamic waveshape and spectrum control ................... 228
6.1 Paged wavetable memory - piecewise stepped
waveshaping .............................................. 229
6.1.1 Page indexing a group of wavetables ............... 229
6.1.2 Specifying a group of wavetables .................. 231
6.1.3 Paged wavetable interpolation - piecewise
continuous waveshaping ............................ 233
6.1.4 Frequency domain behaviour ........................ 235
6.2 Linear combination of multiple wavetables ................ 236
6.2.1 Single phase accumulator case - harmonic
signal synthesis .................................. 237
6.2.2 The IDFT as a limiting case ....................... 239
6.2.3 Multiple phase accumulator case - band-pass
signal synthesis .................................. 241
6.3 Modulation ............................................... 242
6.3.1 Frequency, phase and amplitude modulation ......... 243
6.3.2 FSK and PSK modulation ............................ 244
6.3.3 Frequency sweep and chirp generation .............. 246
6.4 Summary .................................................. 247
6.5 References ............................................... 248
7 Phase domain processing - DDS and the IDFT ............... 249
7.1 Fundamental concepts ..................................... 249
7.1.1 The inverse discrete Fourier transform (IDFT) ..... 250
7.1.2 Weighted phase sequence summation ................. 252
7.1.3 Dynamic spectrum control and the metaparameter .... 254
7.1.4 Aliasing control and the mitigation of Gibbs
phenomenon ........................................ 256
7.1.5 Time division multiplexing (TDM) .................. 257
7.2 Contiguous harmonic phase generation and the IDFT ........ 258
7.2.1 Harmonic phase accumulation ....................... 258
7.2.2 Harmonic phase offset ............................. 260
7.2.3 Amplitude accumulation ............................ 261
7.3 Non-contiguous harmonic phase generation ................. 262
7.3.1 Phase multiplication .............................. 263
7.3.2 Modulo integer multiplication ..................... 263
7.4 Summary .................................................. 264
7.5 References ............................................... 265
8 Hardware implementation architectures .................... 266
8.1 Processing techniques and design considerations .......... 267
8.1.1 Pipelined and parallel processing ................. 267
8.1.2 Sample and block pipelining ....................... 268
8.1.3 Time division multiplexing ........................ 272
8.1.4 Arithmetic and modulo accumulation ................ 273
8.1.5 High speed binary addition ........................ 274
8.1.6 Wavetable memory architectures .................... 275
8.1.7 Vector memory ..................................... 277
8.2 Partitioned phase accumulation ........................... 280
8.3 Sinusoidal phase-amplitude mapping ....................... 283
8.3.1 Linear phase interpolation ........................ 284
8.3.2 Trigonometric identity phase interpolation ........ 286
8.4 Interpolated vector memory ............................... 289
8.4.1 Linear interpolation architecture ................. 289
8.4.2 Cubic interpolation architecture .................. 289
8.4.3 Interpolated vector memory design
considerations .................................... 293
8.4.4 A paged wavetable interpolation memory ............ 296
8.5 Waveform generation by direct computation ................ 296
8.5.1 A contiguous harmonic IDFT waveform
synthesiser ....................................... 299
8.5.2 An arbitrary harmonic IDFT waveform s
ynthesiser ........................................ 302
8.5.3 A band-pass spectrum waveform synthesiser ......... 302
8.6 Summary .................................................. 304
8.7 References ............................................... 306
9 Digital to analogue conversion ........................... 307
9.1 Error mechanisms ......................................... 311
9.1.1 Offset error ...................................... 313
9.1.2 Gain error ........................................ 315
9.1.3 Settling time ..................................... 315
9.1.4 Transition or'glitch'errors ....................... 317
9.1.5 Linearity errors .................................. 317
9.1.6 DAC quantisation noise ............................ 318
9.2 DAC dynamic specification considerations ................. 319
9.2.1 Harmonic and intermodulation distortion ........... 319
9.2.2 Spurious-free dynamic range (SFDR) ................ 321
9.2.3 Signal-to-noise ratio (SNR) ....................... 321
9.3 The reconstruction filter ................................ 322
9.3.1 Frequency and phase response ...................... 323
9.3.2 Filter implementation - passive or active? ........ 325
9.3.3 Signal distortion pitfalls - maintaining SFDR ..... 327
9.3.4 Reconstruction filter design considerations -
hints and tips .................................... 328
9.4 Supplementary analogue design considerations ............. 330
9.4.1 The DAC reference voltage and output amplifier .... 330
9.4.2 Switched attenuation and output level control ..... 332
9.4.3 DC offset and output dynamic range
considerations .................................... 335
9.4.4 The output drive amplifier ........................ 336
9.4.5 Square and pulse waveforms ........................ 337
9.5 Summary .................................................. 338
9.6 References ............................................... 340
Index ......................................................... 341
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