Preface ....................................................... vii
Chapter 1 Introduction ......................................... 1
Why are digital filters associated with nonlinearities? ......... 1
Challenges for the analysis and design of digital filters
associated with nonlinearities .................................. 6
An overview ..................................................... 7
Chapter 2 Reviews .............................................. 8
Mathematical preliminary ........................................ 8
Backgrounds on signals and systems ............................. 13
Backgrounds on sampling theorem ................................ 25
Backgrounds on bifurcation theorem ............................. 27
Absolute stability theorem ..................................... 28
Exercises ...................................................... 28
Chapter 3 Quantization in Digital Filters ..................... 32
Model of quantizer ............................................. 32
Quantization noise analysis .................................... 36
Optimal code design ............................................ 42
Summary ........................................................ 51
Exercises ...................................................... 52
Chapter 4 Saturation in Digital Filters ....................... 53
System model ................................................... 53
Oscillations of digital filters associated with saturation
nonlinearity ................................................... 54
Stability of oscillations of digital filters associated with
saturation nonlinearity ........................................ 58
Summary ........................................................ 59
Exercises ...................................................... 60
Chapter 5 Autonomous Response of Digital Filters with Two's
Complement Arithmetic .......................................... 61
System model ................................................... 61
Linear and affine linear behaviors ............................. 63
Limit cycle behavior ........................................... 70
Chaotic behavior ............................................... 75
Summary ........................................................ 77
Exercises ...................................................... 77
Chapter 6 Step Response of Digital Filters with Two's
Complement Arithmetic .......................................... 78
Affine linear behavior ......................................... 78
Limit cycle behavior ........................................... 92
Fractal behavior .............................................. 104
Summary ....................................................... 109
Exercises ..................................................... 109
Chapter 7 Sinusoidal Response of Digital Filters with Two's
Complement Arithmetic ......................................... 114
No overflow case .............................................. 114
Overflow case ................................................. 127
Summary ....................................................... 138
Exercises ..................................................... 138
Chapter 8 Two's Complement Arithmetic in Complex Digital
Filters ....................................................... 139
First order complex digital filters ........................... 139
Second order complex digital filters .......................... 144
Summary ....................................................... 150
Exercises ..................................................... 151
Chapter 9 Quantization and Two's Complement Arithmetic in
Digital Filters ............................................... 152
Nonlinear behavioral differences of finite and infinite
state machines ................................................ 152
Nonlinear behavior of unstable second order digital filters ... 156
Nonlinear behaviors of digital filters with arbitrary orders
and initial conditions ........................................ 162
Summary ....................................................... 170
Exercises ..................................................... 171
Chapter 10 Properties and Applications of Digital Filters
with Nonlinearities ........................................... 172
Admissibility of symbolic sequences ........................... 172
Statistical property .......................................... 179
Computer cryptography via digital filters associated
with nonlinearities ........................................... 188
Summary ....................................................... 195
Exercises ..................................................... 197
Further Reading ............................................... 199
Index ......................................................... 205
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