CONTENTS 4 7 Digital Filter Realizations 265 7.1 Direct Form,265 7.2 Canonical Form,271 7.3 Cascade Form,277 7.4 Cascade to Canonical,284 7.5 Hardware Realizations and Circular Buffers,293 7.6 Quantization Effects in Digital Filters,305 7.7 Problems,306 8 Signal Processing Applications 316 8.1 Digital Waveform Generators,316 8.1.1 Sinusoidal Generators,316 8.1.2 Periodic Waveform Generators,321 8.1.3 Wavetable Generators,330 8.2 Digital Audio Effects,349 8.2.1 Delays,Echoes,and Comb Filters,350 8.2.2 Flanging,Chorusing,and Phasing,355 8.2.3 Digital Reverberation,362 8.2.4 Multitap Delays,374 8.2.5 Compressors,Limiters,Expanders,and Gates,378 8.3 Noise Reduction and Signal Enhancement,382 8.3.1 Noise Reduction Filters,382 8.3.2 Notch and Comb Filters,398 8.3.3 Line and Frame Combs for Digital TV,409 8.3.4 Signal Averaging,421 8.3.5 Savitzky-Golay Smoothing Filters*,427 8.4 Problems,453 9 DFT/FFT Algorithms 464 9.1 Frequency Resolution and Windowing,464 9.2 DTFT Computation,475 9.2.1 DTFT at a Single Frequency,475 9.2.2 DTFT over Frequency Range,478 9.2.3DFT,479 9.2.4 Zero Padding,481 9.3 Physical versus Computational Resolution,482 9.4 Matrix Form of DFT,486 9.5 Modulo-N Reduction,489 9.6 Inverse DFT,496 9.7 Sampling of Periodic Signals and the DFT,499 9.8FFT,504 9.9 Fast Convolution,515 9.9.1 Circular Convolution,515 9.9.2 Overlap-Add and Overlap-Save Methods,520 9.10 Problems,523CONTENTS ix 7 Digital Filter Realizations 265 7.1 Direct Form, 265 7.2 Canonical Form, 271 7.3 Cascade Form, 277 7.4 Cascade to Canonical, 284 7.5 Hardware Realizations and Circular Buffers, 293 7.6 Quantization Effects in Digital Filters, 305 7.7 Problems, 306 8 Signal Processing Applications 316 8.1 Digital Waveform Generators, 316 8.1.1 Sinusoidal Generators, 316 8.1.2 Periodic Waveform Generators, 321 8.1.3 Wavetable Generators, 330 8.2 Digital Audio Effects, 349 8.2.1 Delays, Echoes, and Comb Filters, 350 8.2.2 Flanging, Chorusing, and Phasing, 355 8.2.3 Digital Reverberation, 362 8.2.4 Multitap Delays, 374 8.2.5 Compressors, Limiters, Expanders, and Gates, 378 8.3 Noise Reduction and Signal Enhancement, 382 8.3.1 Noise Reduction Filters, 382 8.3.2 Notch and Comb Filters, 398 8.3.3 Line and Frame Combs for Digital TV, 409 8.3.4 Signal Averaging, 421 8.3.5 Savitzky-Golay Smoothing Filters∗, 427 8.4 Problems, 453 9 DFT/FFT Algorithms 464 9.1 Frequency Resolution and Windowing, 464 9.2 DTFT Computation, 475 9.2.1 DTFT at a Single Frequency, 475 9.2.2 DTFT over Frequency Range, 478 9.2.3 DFT, 479 9.2.4 Zero Padding, 481 9.3 Physical versus Computational Resolution, 482 9.4 Matrix Form of DFT, 486 9.5 Modulo-N Reduction, 489 9.6 Inverse DFT, 496 9.7 Sampling of Periodic Signals and the DFT, 499 9.8 FFT, 504 9.9 Fast Convolution, 515 9.9.1 Circular Convolution, 515 9.9.2 Overlap-Add and Overlap-Save Methods, 520 9.10 Problems, 523