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Tables of Contents for Numerical Methods for Dsp Systems in C
Chapter/Section Title
Page #
Page Count
Preface
xiii
4
Introduction
xvii
1 From the Beginning
1
38
How a Signal Becomes an Equation in the Continuous Universe
2
11
A Brief Overview
2
3
Superposition
5
1
Convolution
5
1
A Simple Harmonic Oscillator
6
3
Impulse Response XXX(x)
9
1
Transfer Functions
10
1
The S-Plane
11
1
Poles and Zeros
12
1
Laplace
13
6
Laplace Transform
13
1
Basic Theorems of the Laplace Transform
14
5
Laplace Transform Pairs
15
4
Hilbert Spaces
19
8
Inner Product Spaces
20
1
The Norm
21
1
Normed Spaces
21
1
Four More Important Concepts Associated with Inner Products
22
1
Cauchy-Schwartz
22
1
Triangle Inequality
22
1
Parallelogram Law or Polarization
22
1
Cauchy Sequences
22
1
Subsets
23
1
Vector
23
1
Frame
23
1
The Basis
24
2
Orthagonality
26
1
Subspaces That Form a Direct Sum
26
1
Fourier Series
27
4
The Gibbs Phenomenon
29
1
The Fourier Series As a Least-Squares Approximation
30
1
Fourier Integral
31
5
Laplace Representation of Sample Sequence
36
1
Z-Transform Representation
36
3
2 Linear Algebra
39
28
The Matrix
39
19
Transposition and Transjugation
45
1
Special Forms of Matrices
45
2
Biorthogonal Matrices
47
2
Orthogonal Matrix
49
3
Matrix Adjoint
52
1
Paraunitary Matrix
52
1
Cross Product and Dot Product
52
1
Transforms
53
1
Determinant and Matrix Inversion
54
4
GE.C
58
3
Ill-Conditioned Operations
61
1
Writing Matrix Code
61
2
Scalar Addition
61
1
Vector Multiplication
62
1
Vector-Matrix Multiplication
62
1
Matrix Multiplication
63
1
Signal Processing Forms
63
4
Transposition
63
1
Convolution and Correlation
64
1
Interpolation and Decimation
64
3
3 An Analog Viewpoint
67
46
Networks
68
3
Ohm's Law
69
1
Reactive Elements
69
1
Kirchhoff's Current Law
70
1
Kirchhoff's Voltage Law
71
1
Network Analysis
71
25
Natural Current
71
1
Forced Current
72
1
Alternating Current
73
1
Network Theorems
74
3
Superposition
75
1
Reciprocity
76
1
Substitution
76
1
Transfer Function
77
2
Analog Filters
79
17
Poles and Zeros
80
1
First-Order Filter Networks
80
1
The RC Network
80
1
The RL Network
81
1
Second-Order Analog Poles and Zeros
82
4
Second-Order Filter Transfer Functions
86
2
Reading and Understanding Templates
88
3
Higher-Order Filters
91
1
The Basis for Templates
92
1
Sallen and Key
93
3
The Digital Signal
96
3
Sampling and Sequences
96
3
Digital IIR Filters
99
9
Analog Filters and Digital IIR Filters
99
5
Creating the Digital Filter
104
4
The Bilinear Transform
104
2
A Caveat: Warping
106
1
Difference Equations
107
1
Analog-to-Digital Conversion Example
108
5
Sliding Filters
108
5
4 Filters and Filter Banks
113
66
Linear Time-Invariant Systems
114
1
Matrix Representations of Filters
115
2
Direct Form Matrix Representations
116
1
Biorthogonal Filters
117
1
Orthogonal Filters
117
1
FIR/IIR Filter Design
118
6
Designing an Ideal Lowpass Filter
119
2
Finite Transition Band
121
1
Windowing
122
2
Filter Implementations
124
9
Complementary Filters
124
1
Mirror Image Filters
125
2
Halfband Filters
127
2
Halfband Filters: FIR Design
129
1
Lagrange Halfband Filters
130
1
Halfband Filters IIR Design
131
1
Allpass Filters
131
2
M-Band Lowpass Filters
133
1
Rate Changing
133
2
Polyphase Systems
135
27
Linear Time-Variant Systems
136
1
Polyphase Filter Notation
137
4
Modulation
141
2
Polyphase Matrices
143
4
Decimation
147
3
Interpolation
150
5
IIR Polyphase Networks
155
1
Allpass Functions for Polyphase Networks
155
1
Butterworth Filters
155
1
IIR Halfband Lowpass Filters
156
1
Decimation
157
1
Interpolation
158
1
Frequency Domain Effects
158
2
Upsampling Followed by Downsampling
160
1
Rational Sampling Converters
160
2
Interpolated Finite Impulse Response Filters
162
2
Filter Banks
164
15
Dyadic Cascades
165
2
Two-Band Subband Coding
167
1
Equations for Two-Band Coding
168
1
Spectral Factorization
169
3
Perfect Reconstruction
172
3
Orthogonal, Unitary, or Paraunitary Filter Banks
175
1
Conjugate Quadrature Filter Banks
176
2
FIR, Linear Phase, and Orthogonality
178
1
5 Multiresolution Analysis and Wavelets
179
58
Stochastic Analysis
180
2
Fourier Analysis
182
2
Time, Frequency, Localization
184
9
Short-Time Fourier Transform (Gabor Transform)
190
3
Discrete STFT
193
1
Multiresolution Analysis
193
3
Wavelet Transform
196
1
Intuitive Overview
197
5
Dilation
197
5
Wavelets and L(2)(R)
202
1
The Continuous Scaling Function
203
6
The Discrete Scaling Function
204
1
Regularity
205
1
Iterating the Scaling Function
205
4
The Continuous Wavelet Transform
209
1
Discrete Wavelet Transform
210
5
Inverse Discrete Wavelet Transform
214
1
Constructing Wavelets
215
7
Orthogonal Wavelet Conditions
215
1
Daubechies' Orthonormal Wavelets
216
2
Daubechies 4: A Second Derivation
218
4
Fast Wavelet Transform
222
15
The Transform
223
1
FWVT()
223
2
IFWVT ()
225
9
Two-Dimensional FWT
234
1
Biorthogonal Filter Banks and Wavelets
234
3
Appendix A
237
34
The Machine
237
2
Machine Epsilon
239
3
The Dynamic Range of the Arithmetic
240
2
The Choice of an Arithmetic Format
242
2
Fixed-Point Arithmetic
243
1
Signed Magnitude
243
1
One's Complement
244
1
Two's Complement
244
1
The Results of Computation
244
3
Floating Point
244
2
Floating-Point Arithmetic
246
1
Block Floating-Point Arithmetic
247
1
Good Algorithms and III-Conditioned Numerics
247
2
The Meaning of Variance, Mean, Standard Deviation, and Probability Density
247
2
Quantization
249
15
Analog-to-Digital Converters
251
7
Reduction of Quantization Noise by Oversampling
256
1
Digital-to-Analog Converters
257
1
Coefficient Quantization
258
4
Roundoff/Truncation Noise in Digital Filters
262
2
Noise in Sum of Products Routines
264
3
Integer
264
1
Method One
265
1
Method Two
266
1
Method Three
266
1
Poles and Zeros
267
2
Differences In Implementation
269
2
Direct Form FIR Systems
269
1
Floating-Point Realizations
269
1
Limit Cycles
269
2
Appendix B
271
42
Numerical Methods
271
1
Methods of Computing Functions
272
8
Horner's Rule
272
1
Series
273
1
Continued Fractions
274
1
Infinite Products
275
1
Range Reduction
275
2
Evaluation
277
1
Iteration
277
2
Table Formats
279
1
Series Approximation: Function Generation
280
1
Polynomial Approximation
280
4
Optimum Approximation
284
6
Chebyshev Polynomials
285
5
Orthogonal Polynomials
290
1
Interpolating Polynomials
290
13
Lagrange
291
3
Minmax (Chebyshev) Interpolating Polynomial
294
3
Extrapolation
297
1
Least-Squares Approximation
297
5
Trigonometric Approximation
302
1
CORDIC Functions
303
10
Appendix C
313
8
More Filter Templates
313
1
Infinite Gain Filter Templates
313
2
Simple Biquadratic Filter Elements and Templates
315
2
State Variable Templates
317
4
References
321
2
Index
323