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Preface | |

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General Introduction | |

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Signals | |

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Analog Signals | |

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Discrete-time and Digital Signals | |

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Deterministic and Random Signals | |

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Signal Processing Systems | |

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Analog Systems | |

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Discrete and Digital Systems | |

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The Fourier Series in Spectral Analysis and Function Approximation | |

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Introduction | |

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Fundamentals | |

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Simplifying the Notation | |

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Periodicity | |

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Derivation of the Expressions for the Coefficients | |

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Even and Odd Functions | |

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Signals as Functions of Time | |

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A Glimpse at the Applications of the Fourier Series | |

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Dirichlet's Conditions | |

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The Significance of Convergence | |

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The Complex Fourier Series | |

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The Time and Discrete Frequency Domains | |

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Line Spectra | |

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Convolution | |

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Bessel's Inequality, Parseval's Theorem and Power Spectrum | |

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The 'Least Squares' Approximation Property of the Fourier Series | |

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The Convergence Problem | |

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An Integral Formula for the Partial Sum | |

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The (sin [psi])/[psi] Function and the Sine Integral | |

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Convergence for Differentiable and Sectionally Differentiable Functions | |

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Well Convergent Series of Particularly Smooth Functions | |

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The Gibbs' Phenomenon | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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The Fourier Transformation and Generalized Signals | |

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Introduction | |

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From the Fourier Coefficients to the Fourier Integral | |

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Real Signals | |

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Causal Signals | |

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Basic Properties of the Fourier Transform | |

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Symmetry | |

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Conjugate Functions | |

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Linearity | |

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Scaling | |

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Shifting in Time | |

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Shifting in Frequency | |

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Modulation | |

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Differentiation with Respect to Time | |

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Differentiation with Respect to Frequency | |

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Convolution | |

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Time-convolution | |

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Frequency-convolution | |

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Parseval's Theorem and Energy Spectra | |

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Correlation Functions and the Wiener-Kintchine Theorem | |

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The Unit Impulse and Generalized Functions | |

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Motivation | |

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A Heuristic Introduction | |

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The Impulse as a Distribution (Generalized Function) | |

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The Unit Impulse as the Limit of a Sequence | |

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Properties of the Unit Impulse | |

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The Impact of Distributions on the Fourier Transform | |

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The Impulse Response and System Function | |

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Causal Functions and the Hilbert Transform | |

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The Impulse Train and its Applications | |

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Periodic Signals | |

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The Fourier Transform of a Periodic Signal | |

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Power Spectra and Correlation Functions | |

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Poisson's Sum Formula | |

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Schwartz's Inequality | |

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The Uncertainty Principle | |

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Time-limited and Band-limited Signals | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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The Laplace Transformation and Some of its Applications | |

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Introduction | |

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Definition and General Considerations | |

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The Laplace Transform of Some Elementary Functions | |

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Existence and Analyticity of the Laplace Transform | |

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Properties of the Laplace Transform | |

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Linearity | |

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Scaling | |

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Differentiation in the Time Domain | |

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Integration in the Time Domain | |

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Differentiation in the Frequency Domain | |

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Integration in the Frequency Domain | |

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Translation in the Time Domain | |

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Translation in the Frequency Domain | |

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Periodic Signals | |

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Convolution | |

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The Inverse Laplace Transform of a Rational Function | |

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A Preliminary Step | |

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The Partial Fraction Expansion of a Proper Fraction | |

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Formulae for the Residues | |

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The Complex Inversion Integral | |

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The Significance of Poles and Zeros | |

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The Initial and Final Value Theorems | |

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The Initial Value Theorem | |

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The Final Value Theorem | |

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Solution of Linear Differential Equations | |

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Network Analysis Using the Laplace Transform | |

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The Basic Building Blocks | |

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The Transformed Network | |

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The General Analysis Technique | |

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Source Transformations | |

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Network Theorems and Special Techniques | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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Analog Signal Processing Systems | |

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Introduction | |

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The System Function | |

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System Stability | |

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A Stability Test--Hurwitz Polynomials | |

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Elementary Signal Processing Systems | |

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Realization of System Functions | |

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Filters | |

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Ideal Filters | |

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Amplitude-oriented Design | |

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Frequency Transformations | |

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Phase-oriented Design | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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Digitization of Analog Signals | |

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Introduction | |

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Sampling | |

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Ideal Impulse Sampling | |

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The Sampling Theorem | |

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Practical Sampling Functions | |

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Quantization | |

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Encoding and Binary Number Representation | |

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Fixed-point Numbers | |

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Number Quantization | |

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Floating-point Numbers | |

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Comparison between Fixed-point and Floating-point Arithmetic | |

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Conclusion | |

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Problems | |

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Discrete Signals and Systems | |

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Introduction | |

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Sequences and Discrete Systems | |

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The z-Transformation: Definition and General Considerations | |

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Relationship between the z-Transform and the Laplace Transform | |

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Properties of the z-Transform | |

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Linearity | |

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Shifting | |

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Convolution of Sequences | |

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Convergence | |

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The Inverse z-Transform | |

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Use of the Inversion Integral | |

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Use of Long Division (Series Expansion) | |

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Use of Partial Fractions | |

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Complex Convolution | |

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Description of Discrete Systems | |

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The Difference Equation and Transfer Function | |

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The Frequency Response | |

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Stability | |

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The Bilinear Variable | |

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Digital Filters | |

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Digital Filter Building Blocks | |

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A Rudimentary Digital Processor | |

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Classification of Digital Filters | |

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Realization Structures of IIR Filters | |

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Non-recursive Realization of FIR Transfer Functions | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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Design of Digital Filters | |

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Introduction | |

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Amplitude-oriented Design of IIR Filters | |

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Low-pass Filters | |

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High-pass Filters | |

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Band-pass Filters | |

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Band-stop Filters | |

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Phase-oriented Design of IIR Filters | |

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Maximally Flat Group-delay Response | |

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FIR Filters | |

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The Exact Linear Phase Property | |

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Fourier-coefficient Filter Design | |

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Fourier-coefficient Design of Differentiators | |

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Monotonic Amplitude Response with the Optimum Number of Constraints | |

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Optimum Equiripple Response in both Passband and Stopband | |

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Comparison Between IIR and FIR Filters | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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The Fast Fourier Transform and Its Applications | |

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Introduction | |

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Periodic Functions | |

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Non-Periodic Functions | |

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The Discrete Fourier Transform (DFT) | |

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The Fast Fourier Transform (FFT) Algorithms | |

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Decimation-in-time Fast Fourier Transform | |

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Decimation-in-frequency Fast Fourier Transform | |

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Radix 4 Fast Fourier Transform | |

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Properties of the Discrete Fourier Transform | |

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Linearity | |

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Cyclic Convolution | |

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Shifting | |

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Symmetry and Conjugate Pairs | |

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Parseval's Relation and Power Spectrum | |

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Cyclic Correlation | |

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Relation to the z-Transform | |

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Spectral Analysis Using the Fft | |

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Evaluation of the Fourier Integral | |

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Evaluation of the Fourier Coefficients | |

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Spectral Windows | |

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Continuous-time Signals | |

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Discrete-time Signals | |

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Fast Convolution, Filtering and Correlation Using the FFT | |

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Cyclic (Periodic) Convolution | |

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Non-Periodic Convolution | |

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Filtering and Sectioned Convolution | |

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Fast Correlation | |

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Conclusion | |

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Summary | |

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Problems | |

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Computer Exercises | |

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Stochastic Signals and Power Spectra | |

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Introduction | |

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Random Variables | |

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Probability Distribution Function | |

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Probability Density Function | |

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Joint Distributions | |

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Statistical Parameters | |

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Analog Stochastic Processes | |

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Statistics of Stochastic Processes | |

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Stationary Processes | |

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Time Averages | |

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Ergodicity | |

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Power Spectra of Stochastic Signals | |

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Signals Through Linear Systems | |

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Discrete-time Stochastic Processes | |

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Statistical Parameters | |

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Stationary Processes | |

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Power Spectrum Estimation | |

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Continuous-time Signals | |

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Discrete-time Signals | |

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Conclusion | |

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Problems | |

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Finite Word-length Effects in Digital Signal Processors | |

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Introduction | |

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Input Signal Quantization Errors | |

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Coefficient Quantization Effects | |

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Effect of Round-off Accumulation | |

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Round-off Accumulation without Coefficient Quantization | |

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Round-off Accumulation with Coefficient Quantization | |

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Auto-Oscillations: Overflow and Limit Cycles | |

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Overflow Oscillations | |

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Limit Cycles and the Dead-Band Effect | |

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Conclusion | |

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Problems | |

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Selected Applications and Advanced Topics | |

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Introduction | |

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Mean-Square Approximation | |

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Analog Signals | |

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Discrete Signals | |

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Linear Estimation, Modelling and Optimum Filters | |

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Optimum Minimum Mean-Square Error Analog Estimation | |

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Smoothing by Non-Causal Wiener Filters | |

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Causal Wiener Filters | |

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The Matched Filter | |

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Discrete-Time Linear Estimation | |

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Non-recursive Wiener Filtering | |

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Adaptive Filtering Using the Minimum Mean-Square (Mms) Error Gradient Algorithm | |

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The Least Mean-square (Lms) Error Gradient Algorithm | |

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Adaptive IIR Filtering and System Modelling | |

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An Application of Adaptive Filters | |

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Sigma-delta Analog-to-digital Converters | |

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General Considerations | |

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The First-order Converter | |

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The Second-order Converter | |

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The Wavelet Transformation | |

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Conclusion | |

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Functions of a Real Variable | |

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Functions of a Complex Variable | |

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References | |

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Answers to Selected Problems | |