DSP First

Introduction 1-1 Mathematical Representation of Signals 1-2 Mathematical Representation of Systems 1-3 Systems as Building Blocks 1-4 The Next Step Sinusoids 2-1 Tuning Fork Experiment 2-2 Review of Sine and Cosine Functions 2-3 Sinusoidal Signals 2-3.1 Relation of Frequency to Period 2-3.2 Phase and Time Shift 2-4 Sampling and Plotting Sinusoids 2-5 Complex Exponentials and Phasors 2-5.1 Review of Complex Numbers 2-5.2 Complex Exponential Signals 2-5.3 The Rotating Phasor Interpretation 2-5.4 Inverse Euler Formulas Phasor Addition 2-6 Phasor Addition 2-6.1 Addition of Complex Numbers 2-6.2 Phasor Addition Rule 2-6.3 Phasor Addition Rule: Example 2-6.4 MATLAB Demo of Phasors 2-6.5 Summary of the Phasor Addition Rule Physics of the Tuning Fork 2-7.1 Equations from Laws of Physics 2-7.2 General Solution to the Differential Equation 2-7.3 Listening to Tones 2-8 Time Signals: More Than Formulas Summary and Links Problems Spectrum Representation 3-1 The Spectrum of a Sum of Sinusoids 3-1.1 Notation Change 3-1.2 Graphical Plot of the Spectrum 3-1.3 Analysis vs. Synthesis Sinusoidal Amplitude Modulation 3-2.1 Multiplication of Sinusoids 3-2.2 Beat Note Waveform 3-2.3 Amplitude Modulation 3-2.4 AM Spectrum 3-2.5 The Concept of Bandwidth Operations on the Spectrum 3-3.1 Scaling or Adding a Constant 3-3.2 Adding Signals 3-3.3 Time-Shifting x.t/ Multiplies ak by a Complex Exponential 3-3.4 Differentiating x.t/ Multiplies ak by .j 2nfk/ 3-3.5 Frequency Shifting Periodic Waveforms 3-4.1 Synthetic Vowel 3-4.3 Example of a Non-periodic Signal Fourier Series 3-5.1 Fourier Series: Analysis 3-5.2 Analysis of a Full-Wave Rectified Sine Wave 3-5.3 Spectrum of the FWRS Fourier Series 3-5.3.1 DC Value of Fourier Series 3-5.3.2 Finite Synthesis of a Full-Wave Rectified Sine Time-Frequency Spectrum 3-6.1 Stepped Frequency 3-6.2 Spectrogram Analysis Frequency Modulation: Chirp Signals 3-7.1 Chirp or Linearly Swept Frequency 3-7.2 A Closer Look at Instantaneous Frequency Summary and Links Problems Fourier Series Fourier Series Derivation 4-1.1 Fourier Integral Derivation Examples of Fourier Analysis 4-2.1 The Pulse Wave 4-2.1.1 Spectrum of a Pulse Wave 4-2.1.2 Finite Synthesis of a Pulse Wave 4-2.2 Triangle Wave 4-2.2.1 Spectrum of a Triangle Wave 4-2.2.2 Finite Synthesis of a Triangle Wave 4-2.3 Half-Wave Rectified Sine 4-2.3.1 Finite Synthesis of a Half-Wave Rectified Sine Operations on Fourier Series 4-3.1 Scaling or Adding a Constant 4-3.2 Adding Signals 4-3.3 Time-Scaling 4-3.4 Time-Shifting x.t/ Multiplies ak by a Complex Exponential 4-3.5 Differentiating x.t/ Multiplies ak by .j!0k/ 4-3.6 Multiply x.t/ by Sinusoid Average Power, Convergence, and Optimality 4-4.1 Derivation of Parseval's Theorem 4-4.2 Convergence of Fourier Synthesis 4-4.3 Minimum Mean-Square Approximation Pulsed-Doppler Radar Waveform 4-5.1 Measuring Range and Velocity Problems Sampling and Aliasing Sampling 5-1.1 Sampling Sinusoidal Signals 5-1.2 The Concept of Aliasing 5-1.3 Spectrum of a Discrete-Time Signal 5-1.4 The Sampling Theorem 5-1.5 Ideal Reconstruction Spectrum View of Sampling and Reconstruction 5-2.1 Spectrum of a Discrete-Time Signal Obtained by Sampling 5-2.2 Over-Sampling 5-2.3 Aliasing Due to Under-Sampling 5-2.4 Folding Due to Under-Sampling 5-2.5 Maximum Reconstructed Frequency Strobe Demonstration 5-3.1 Spectrum Interpretation Discrete-to-Continuous Conversion 5-4.1 Interpolation with Pulses 5-4.2 Zero-Order Hold Interpolation 5-4.3 Linear Interpolation 5-4.4 Cubic Spline Interpolation 5-4.5 Over-Sampling Aids Interpolation 5-4.6 Ideal Bandlimited Interpolation The Sampling Theorem Summary and Links Problems FIR Filters 6-1 Discrete-Time Systems 6-2 The Running-Average Filter 6-3 The General FIR Filter 6-3.1 An Illustration of FIR Filtering The Unit Impulse Response and Convolution 6-4.1 Unit Impulse Sequence 6-4.2 Unit Impulse Response Sequence 6-4.2.1 The Unit-Delay System 6-4.3 FIR Filters and Convolution 6-4.3.1 Computing the Output of a Convolution 6-4.3.2 The Length of a Convolution 6-4.3.3 Convolution in MATLAB 6-4.3.4 Polynomial Multiplication in MATLAB 6-4.3.5 Filtering the Unit-Step Signal 6-4.3.6 Convolution is Commutative 6-4.3.7 MATLAB GUI for Convolution Implementation of FIR Filters 6-5.1 Building Blocks 6-5.1.1 Multiplier 6-5.1.2 Adder 6-5.1.3 Unit Delay 6-5.2 Block Diagrams 6-5.2.1 Other Block Diagrams 6-5.2.2 Internal Hardware Details Linear Time-Invariant (LTI) Systems 6-6.1 Time Invariance 6-6.2 Linearity 6-6.3 The FIR Case Convolution and LTI Systems 6-7.1 Derivation of the Convolution Sum 6-7.2 Some Properties of LTI Systems Cascaded LTI Systems Example of FIR Filtering Summary and Links ProblemsFrequency Response of FIR Filters 7-1 Sinusoidal Response of FIR Systems 7-2 Superposition and the Frequency Response 7-3 Steady-State and Transient Response 7-4 Properties of the Frequency Response 7-4.1 Relation to Impulse Response and Difference Equation 7-4.2 Periodicity of H.ej !O / 7-4.3 Conjugate Symmetry Graphical Representation of the Frequency Response 7-5.1 Delay System 7-5.2 First-Difference System 7-5.3 A Simple Lowpass Filter Cascaded LTI Systems Running-Sum Filtering 7-7.1 Plotting the Frequency Response 7-7.2 Cascade of Magnitude and Phase 7-7.3 Frequency Response of Running Averager 7-7.4 Experiment: Smoothing an Image Filtering Sampled Continuous-Time Signals 7-8.1 Example: Lowpass Averager 7-8.2 Interpretation of Delay Summary and Links Problems The Discrete-Time Fourier Transform DTFT: Discrete-Time Fourier Transform 8-1.1 The Discrete-Time Fourier Transform 8-1.1.1 DTFT of a Shifted Impulse Sequence 8-1.1.2 Linearity of the DTFT 8-1.1.3 Uniqueness of the DTFT 8-1.1.4 DTFT of a Pulse 8-1.1.5 DTFT of a Right-Sided Exponential Sequence 8-1.1.6 Existence of the DTFT 8-1.2 The Inverse DTFT 8-1.2.1 Bandlimited DTFT 8-1.2.2 Inverse DTFT for the Right-Sided Exponential 8-1.3 The DTFT is the Spectrum Properties of the DTFT 8-2.1 The Linearity Property 8-2.2 The Time-Delay Property 8-2.3 The Frequency-Shift Property 8-2.3.1 DTFT of a Complex Exponential 8-2.3.2 DTFT of a Real Cosine Signal 8-2.4 Convolution and the DTFT 8-2.4.1 Filtering is Convolution 8-2.5 Energy Spectrum and the Autocorrelation Function 8-2.5.1 Autocorrelation Function Ideal Filters 8-3.1 Ideal Lowpass Filter 8-3.2 Ideal Highpass Filter 8-3.3 Ideal Bandpass Filter Practical FIR Filters 8-4.1 Windowing 8-4.2 Filter Design 8-4.2.1 Window the Ideal Impulse Response 8-4.2.2 Frequency Response of Practical Filters 8-4.2.3 Passband Defined for the Frequency Response 8-4.2.4 Stopband Defined for the Frequency Response 8-4.2.5 Transition Zone of the LPF 8-4.2.6 Summary of Filter Specifications 8-4.3 GUI for Filter Design Table of Fourier Transform Properties and Pairs Summary and Links Problems The Discrete Fourier Transform Discrete Fourier Transform (DFT) 9-1.1 The Inverse DFT 9-1.2 DFT Pairs from the DTFT 9-1.2.1 DFT of Shifted Impulse 9-1.2.2 DFT of Complex Exponential 9-1.3 Computing the DFT 9-1.4 Matrix Form of the DFT and IDFT Properties of the DFT 9-2.1 DFT Periodicity for XOek] 9-2.2 Negative Frequencies and the DFT 9-2.3 Conjugate Symmetry of the DFT 9-2.3.1 Ambiguity at XOeN=2] 9-2.4 Frequency Domain Sampling and Interpolation 9-2.5 DFT of a Real Cosine Signal Inherent Periodicity of xOen] in the DFT 9-3.1 DFT Periodicity for xOen] 9-3.2 The Time Delay Property for the DFT 9-3.2.1 Zero Padding 9-3.3 The Convolution Property for the DFT Table of Discrete Fourier Transform Properties and Pairs Spectrum Analysis of Discrete Periodic Signals 9-5.1 Periodic Discrete-time Signal: Fourier Series 9-5.2 Sampling Bandlimited Periodic Signals 9-5.3 Spectrum Analysis of Periodic Signals Windows 9-6.0.1 DTFT of Windows The Spectrogram 9-7.1 An Illustrative Example 9-7.2 Time-Dependent DFT 9-7.3 The Spectrogram Display 9-7.4 Interpretation of the Spectrogram 9-7.4.1 Frequency Resolution 9-7.5 Spectrograms in MATLAB The Fast Fourier Transform (FFT) 9-8.1 Derivation of the FFT 9-8.1.1 FFT Operation Count Summary and Links Problems z-Transforms Definition of the z-Transform Basic z-Transform Properties 10-2.1 Linearity Property of the z-Transform 10-2.2 Time-Delay Property of the z-Transform 10-2.3 A General z-Transform Formula The z-Transform and Linear Systems 10-3.1 Unit-Delay System 10-3.2 z-1 Notation in Block Diagrams 10-3.3 The z-Transform of an FIR Filter 10-3.4 z-Transform of the Impulse Response 10-3.5 Roots of a z-transform Polynomial Convolution and the z-Transform 10-4.1 Cascading Systems 10-4.2 Factoring z-Polynomials 10-4.3 Deconvolution Relationship Between the z-Domain and the !O -Domain 10-5.1 The z-Plane and the Unit Circle The Zeros and Poles of H.z/ 10-6.1 Pole-Zero Plot 10-6.2 Significance of the Zeros of H.z/ 10-6.3 Nulling Filters 10-6.4 Graphical Relation Between z and !O 10-6.5 Three-Domain Movies Simple Filters 10-7.1 Generalize the L-Point Running-Sum Filter 10-7.2 A Complex Bandpass Filter 10-7.3 A Bandpass Filter with Real Coefficients Practical Bandpass Filter Design Properties of Linear-Phase Filters 10-9.1 The Linear-Phase Condition 10-9.2 Locations of the Zeros of FIR Linear-Phase Systems Summary and Links Problems IIR Filters The General IIR Difference Equation Time-Domain Response 11-2.1 Linearity and Time Invariance of IIR Filters 11-2.2 Impulse Response of a First-Order IIR System 11-2.3 Response to Finite-Length Inputs 11-2.4 Step Response of a First-Order Recursive System System Function of an IIR Filter 11-3.1 The General First-Order Case 11-3.2 H.z/ from the Impulse Response 11-3.3 The z-Transform Method The System Function and Block-Diagram Structures 11-4.1 Direct Form I Structure 11-4.2 Direct Form II Structure 11-4.3 The Transposed Form Structure Poles and Zeros 11-5.1 Roots in MATLAB 11-5.2 Poles or Zeros at z D 0 or 1 11-5.3 Output Response from Pole Location Stability of IIR Systems 11-6.1 The Region of Convergence and Stability Frequency Response of an IIR Filter 11-7.1 Frequency Response using MATLAB 11-7.2 Three-Dimensional Plot of a System Function Three Domains The Inverse z-Transform and Some Applications 11-9.1 Revisiting the Step Response of a First-Order System 11-9.2 A General Procedure for Inverse z-Transformation Steady-State Response and Stability Second-Order Filters 11-11.1 z-Transform of Second-Order Filters 11-11.2 Structures for Second-Order IIR Systems 11-11.3 Poles and Zeros 11-11.4 Impulse Response of a Second-Order IIR System 11-11.4.1 Distinct Real Poles 11-11.5 Complex Poles Frequency Response of Second-Order IIR Filter 11-12.1 Frequency Response via MATLAB 11-12.23-dB Bandwidth 11-12.3 Three-Dimensional Plot of System Functions Example of an IIR Lowpass Filter Summary and Links Problems