## DSP for MATLAB and LabVIEW: Digital filter designThis book is Volume III of the series DSP for MATLABâ„˘ and LabVIEWâ„˘. Volume III covers digital filter design, including the specific topics of FIR design via windowed-ideal-lowpass filter, FIR highpass, bandpass, and bandstop filter design from windowed-ideal lowpass filters, FIR design using the transition-band-optimized Frequency Sampling technique (implemented by Inverse-DFT or Cosine/Sine Summation Formulas), design of equiripple FIRs of all standard types including Hilbert Transformers and Differentiators via the Remez Exchange Algorithm, design of Butterworth, Chebyshev (Types I and II), and Elliptic analog prototype lowpass filters, conversion of analog lowpass prototype filters to highpass, bandpass, and bandstop filters, and conversion of analog filters to digital filters using the Impulse Invariance and Bilinear Transform techniques. Certain filter topologies specific to FIRs are also discussed, as are two simple FIR types, the Comb and Moving Average filters. The entire series consists of four volumes that collectively cover basic digital signal processing in a practical and accessible manner, but which nonetheless include all essential foundation mathematics. As the series title implies, the scripts (of which there are more than 200) described in the text and supplied in code form (available via the internet at www.morganclaypool.com/page/isen) will run on both MATLABâ„˘ and LabVIEWâ„˘.The text for all volumes contains many examples, and many useful computational scripts, augmented by demonstration scripts and LabVIEWâ„˘ Virtual Instruments (VIs) that can be run to illustrate various signal processing concepts graphically on the user's computer screen. Volume I consists of four chapters that collectively set forth a brief overview of the field of digital signal processing, useful signals and concepts (including convolution, recursion, difference equations, LTI systems, etc), conversion from the continuous to discrete domain and back (i.e., analog-to-digital and digital-to-analog conversion), aliasing, the Nyquist rate, normalized frequency, sample rate conversion and Mu-law compression, and signal processing principles including correlation, the correlation sequence, the Real DFT, correlation by convolution, matched filtering, simple FIR filters, and simple IIR filters. Chapter four of Volume I, in particular, provides an intuitive or "first principle" understanding of how digital filtering and frequency transforms work. Volume II provides detailed coverage of discrete frequency transforms, including a brief overview of common frequency transforms, both discrete and continuous, followed by detailed treatments of the Discrete Time Fourier Transform (DTFT), the z-Transform (including definition and properties, the inverse z-transform, frequency response via z-transform, and alternate filter realization topologies including Direct Form, Direct Form Transposed, Cascade Form, Parallel Form, and Lattice Form), and the Discrete Fourier Transform (DFT) (including Discrete Fourier Series, the DFT-IDFT pair, DFT of common signals, bin width, sampling duration, and sample rate, the FFT, the Goertzel Algorithm, Linear, Periodic, and Circular convolution, DFT Leakage, and computation of the Inverse DFT). Volume IV, the culmination of the series, is an introductory treatment of LMS Adaptive Filtering and applications, and covers cost functions, performance surfaces, coefficient perturbation to estimate the gradient, the LMS algorithm, response of the LMS algorithm to narrow-band signals, and various topologies such as ANC (Active Noise Cancelling) or system modeling, Periodic Signal Removal/Prediction/Adaptive Line Enhancement (ALE), Interference Cancellation, Echo Cancellation (with single- and dual-H topologies), and Inverse Filtering/Deconvolution/Equalization. |

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### Contents

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163 ZERO LOCATION IN LINEAR PHASE FILTERS | 10 |

17 LINEAR PHASE FIR FREQUENCY CONTENT AND RESPONSE | 13 |

322 CONVERGENCE | 125 |

323 RELATION TO FOURIER TRANSFORM | 126 |

324 RELATION TO zTRANSFORM | 127 |

33 PROTOTYPE ANALOG FILTERS | 129 |

332 SYSTEM FUNCTION AND PROPERTIES | 130 |

333 COMPUTED FREQUENCY RESPONSE | 132 |

334 GENERAL PROCEDURE FOR ANALOGDIGITAL FILTER DESIGN | 134 |

342 DESIGN BY STANDARD PARAMETERS | 140 |

18 DESIGN METHODS | 18 |

182 THREE DESIGN METHODS | 19 |

183 THE COMB AND MOVING AVERAGE FILTERS | 20 |

19 FIR REALIZATION | 24 |

192 CASCADE FORM | 26 |

194 CASCADED LINEAR PHASE FORM | 29 |

110 REFERENCES | 31 |

111 EXERCISES | 32 |

FIR Design Techniques | 40 |

23 SUMMARY OF DESIGN METHODS | 41 |

25 FIR DESIGN VIA WINDOWED IDEAL LOWPASS FILTER | 44 |

251 WINDOWS | 45 |

252 NET FREQUENCY RESPONSE | 48 |

253 WINDOWED LOWPASS FILTERSPASSBAND RIPPLE AND STOP BAND ATTENUATION | 52 |

255 IMPROVING STOPBAND ATTENUATION | 56 |

256 MEETING DESIGN SPECIFICATIONS | 59 |

26 FIR DESIGN VIA FREQUENCY SAMPLING | 61 |

261 USING THE INVERSE DFT | 67 |

262 USING COSINESINE SUMMATION FORMULAS | 71 |

263 IMPROVING STOPBAND ATTENUATION | 74 |

264 FILTERS OTHER THAN LOWPASS | 79 |

265 HILBERT TRANSFORMERS | 82 |

266 DIFFERENTIATORS | 92 |

27 OPTIMIZED FILTER DESIGN | 93 |

272 DESIGN GOAL | 94 |

273 ALTERNATION THEOREM | 96 |

274 A COMMON DESIGN PROBLEM FOR ALL LINEAR PHASE FILTERS | 97 |

275 WEIGHTED ERROR FUNCTION | 99 |

276 REMEZ EXCHANGE ALGORITHM | 100 |

28 REFERENCES | 108 |

Classical IIR Design | 124 |

35 LOWPASS ANALOG CHEBYSHEVTYPEI FILTERS | 141 |

351 DESIGN BY ORDER CUTOFF FREQUENCY AND EPSILON | 142 |

352 DESIGN BY STANDARD PARAMETERS | 147 |

36 LOWPASS ANALOG CHEBYSHEVTYPEII FILTERS | 149 |

362 DESIGN BY STANDARD PARAMETERS | 151 |

37 ANALOG LOWPASS ELLIPTIC FILTERS | 153 |

371 DESIGN BY STANDARD PARAMETERS | 154 |

38 FREQUENCY TRANSFORMATIONS IN THE ANALOG DOMAIN | 156 |

382 LOWPASS TO HIGHPASS | 159 |

383 TRANSFORMATION VIA CONVOLUTION | 161 |

384 LOWPASS TO BANDPASS | 164 |

385 LOWPASS TO BANDSTOP NOTCH | 166 |

39 ANALOG TO DIGITAL FILTER TRANSFORMATION | 168 |

391 IMPULSE INVARIANCE | 169 |

392 THE BILINEAR TRANSFORM | 176 |

310 MATHSCRIPT FILTER DESIGN FUNCTIONS | 186 |

311 PRONYS METHOD | 189 |

312 IIR OPTIMIZATION PROGRAMS | 195 |

Software for Use with this Book | 208 |

A2 DOWNLOADING THE SOFTWARE | 209 |

A5 MULTILINE MCODE EXAMPLES | 210 |

A6 HOW TO SUCCESSFULLY COPYANDPASTE MCODE | 211 |

A7 LEARNING TO USE MCODE | 212 |

VectorMatrix Operations in MCode | 214 |

B22 OUTER PRODUCT | 215 |

B4 MATRIX INVERSE AND PSEUDOINVERSE | 216 |

FIR Frequency Sampling Design Formulas | 218 |

C14 EVEN LENGTH SYMMETRIC TYPE IV | 219 |

Biography | 220 |

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### Common terms and phrases

amplitude analog ﬁlter b,a,G,NetRp,NetAs band edges bandpass ﬁlter bandstop ﬁlter Bilinear transform Butterworth ﬁlter Chebyshev Type-I circular convolution coefﬁcients comb ﬁlter complex conjugate correlator cosine cutoff frequency deﬁned design speciﬁcation desired digital ﬁlter Direct Form DTFT Elliptic equiripple Example Figure ﬁle filter ﬁlter designed ﬁlter impulse response ﬁlter length FIR design ﬁrst following code Freq frequency response Frequency Sampling highpass Hilbert Transformer ideal lowpass ﬁlter imaginary implemented Impulse Invariance impulse response Kaiser window LabVIEW Laplace transform linear phase linear phase ﬁlter m-code Magnitude dB Magnitude of frequency Magnitude response MathScript MATLAB matrix maximum method Normalized Frequency notch ﬁlter obtain odd length output parameters passband ripple Phase response poles and zeros rad/s radians realized values result roll-off shown in Fig symmetrical system function Test calls test signal transition band truncated Units of pi values of RP waveform Write a script z-domain z-transform Z,P,K,NetRp,NetAs