DSP First

Content: Introduction 1-1 Mathematical Representation of Signals 1-2 Mathematical Representation of Systems 1-3 Systems as Building Blocks1-4 The Next Step Sinusoids 2-1 Tuning Fork Experiment 2-2 Review of Sine and Cosine Functions2-3 Sinusoidal Signals2-3.1 Relation of Frequency to Period2-3.2 Phase and Time Shift2-4 Sampling and Plotting Sinusoids2-5 Complex Exponentials and Phasors2-5.1 Review of Complex Numbers 2-5.2 Complex Exponential Signals2-5.3 The Rotating Phasor Interpretation2-5.4 Inverse Euler Formulas Phasor Addition2-6 Phasor Addition2-6.1 Addition of Complex Numbers2-6.2 Phasor Addition Rule2-6.3 Phasor Addition Rule: Example2-6.4 MATLAB Demo of Phasors2-6.5 Summary of the Phasor Addition Rule Physics of the Tuning Fork2-7.1 Equations from Laws of Physics2-7.2 General Solution to the Differential Equation2-7.3 Listening to Tones2-8 Time Signals: More Than FormulasSummary and LinksProblemsSpectrum Representation 3-1 The Spectrum of a Sum of Sinusoids3-1.1 Notation Change3-1.2 Graphical Plot of the Spectrum3-1.3 Analysis vs. SynthesisSinusoidal Amplitude Modulation3-2.1 Multiplication of Sinusoids3-2.2 Beat Note Waveform3-2.3 Amplitude Modulation3-2.4 AM Spectrum3-2.5 The Concept of BandwidthOperations on the Spectrum3-3.1 Scaling or Adding a Constant3-3.2 Adding Signals3-3.3 Time-Shifting x.t/ Multiplies ak by a Complex Exponential3-3.4 Differentiating x.t/ Multiplies ak by .j 2nfk/3-3.5 Frequency ShiftingPeriodic Waveforms3-4.1 Synthetic Vowel3-4.3 Example of a Non-periodic SignalFourier Series3-5.1 Fourier Series: Analysis3-5.2 Analysis of a Full-Wave Rectified Sine Wave3-5.3 Spectrum of the FWRS Fourier Series3-5.3.1 DC Value of Fourier Series3-5.3.2 Finite Synthesis of a Full-Wave Rectified SineTime-Frequency Spectrum3-6.1 Stepped Frequency3-6.2 Spectrogram AnalysisFrequency Modulation: Chirp Signals3-7.1 Chirp or Linearly Swept Frequency3-7.2 A Closer Look at Instantaneous FrequencySummary and LinksProblemsFourier Series Fourier Series Derivation4-1.1 Fourier Integral DerivationExamples of Fourier Analysis4-2.1 The Pulse Wave4-2.1.1 Spectrum of a Pulse Wave4-2.1.2 Finite Synthesis of a Pulse Wave4-2.2 Triangle Wave4-2.2.1 Spectrum of a Triangle Wave4-2.2.2 Finite Synthesis of a Triangle Wave4-2.3 Half-Wave Rectified Sine4-2.3.1 Finite Synthesis of a Half-Wave Rectified SineOperations on Fourier Series4-3.1 Scaling or Adding a Constant4-3.2 Adding Signals4-3.3 Time-Scaling4-3.4 Time-Shifting x.t/ Multiplies ak by a Complex Exponential4-3.5 Differentiating x.t/ Multiplies ak by .j!0k/4-3.6 Multiply x.t/ by SinusoidAverage Power, Convergence, and Optimality4-4.1 Derivation of Parseval's Theorem4-4.2 Convergence of Fourier Synthesis4-4.3 Minimum Mean-Square ApproximationPulsed-Doppler Radar Waveform4-5.1 Measuring Range and VelocityProblemsSampling and Aliasing Sampling5-1.1 Sampling Sinusoidal Signals5-1.2 The Concept of Aliasing5-1.3 Spectrum of a Discrete-Time Signal5-1.4 The Sampling Theorem5-1.5 Ideal ReconstructionSpectrum View of Sampling and Reconstruction5-2.1 Spectrum of a Discrete-Time Signal Obtained by Sampling5-2.2 Over-Sampling5-2.3 Aliasing Due to Under-Sampling5-2.4 Folding Due to Under-Sampling5-2.5 Maximum Reconstructed FrequencyStrobe Demonstration5-3.1 Spectrum InterpretationDiscrete-to-Continuous Conversion5-4.1 Interpolation with Pulses5-4.2 Zero-Order Hold Interpolation5-4.3 Linear Interpolation5-4.4 Cubic Spline Interpolation5-4.5 Over-Sampling Aids Interpolation5-4.6 Ideal Bandlimited InterpolationThe Sampling TheoremSummary and LinksProblemsFIR Filters 6-1 Discrete-Time Systems6-2 The Running-Average Filter6-3 The General FIR Filter6-3.1 An Illustration of FIR FilteringThe Unit Impulse Response and Convolution6-4.1 Unit Impulse Sequence6-4.2 Unit Impulse Response Sequence6-4.2.1 The Unit-Delay System6-4.3 FIR Filters and Convolution6-4.3.1 Computing the Output of a Convolution6-4.3.2 The Length of a Convolution6-4.3.3 Convolution in MATLAB6-4.3.4 Polynomial Multiplication in MATLAB6-4.3.5 Filtering the Unit-Step Signal6-4.3.6 Convolution is Commutative6-4.3.7 MATLAB GUI for ConvolutionImplementation of FIR Filters6-5.1 Building Blocks6-5.1.1 Multiplier6-5.1.2 Adder6-5.1.3 Unit Delay6-5.2 Block Diagrams6-5.2.1 Other Block Diagrams6-5.2.2 Internal Hardware DetailsLinear Time-Invariant (LTI) Systems6-6.1 Time Invariance6-6.2 Linearity6-6.3 The FIR CaseConvolution and LTI Systems6-7.1 Derivation of the Convolution Sum6-7.2 Some Properties of LTI SystemsCascaded LTI SystemsExample of FIR FilteringSummary and LinksProblemsFrequency Response of FIR Filters7-1 Sinusoidal Response of FIR Systems7-2 Superposition and the Frequency Response7-3 Steady-State and Transient Response7-4 Properties of the Frequency Response7-4.1 Relation to Impulse Response and Difference Equation7-4.2 Periodicity of H.ej !O /7-4.3 Conjugate Symmetry Graphical Representation of the Frequency Response7-5.1 Delay System7-5.2 First-Difference System7-5.3 A Simple Lowpass Filter Cascaded LTI SystemsRunning-Sum Filtering7-7.1 Plotting the Frequency Response7-7.2 Cascade of Magnitude and Phase7-7.3 Frequency Response of Running Averager7-7.4 Experiment: Smoothing an ImageFiltering Sampled Continuous-Time Signals7-8.1 Example: Lowpass Averager7-8.2 Interpretation of DelaySummary and LinksProblemsThe Discrete-Time Fourier Transform DTFT: Discrete-Time Fourier Transform8-1.1 The Discrete-Time Fourier Transform8-1.1.1 DTFT of a Shifted Impulse Sequence8-1.1.2 Linearity of the DTFT8-1.1.3 Uniqueness of the DTFT8-1.1.4 DTFT of a Pulse8-1.1.5 DTFT of a Right-Sided Exponential Sequence8-1.1.6 Existence of the DTFT8-1.2 The Inverse DTFT8-1.2.1 Bandlimited DTFT8-1.2.2 Inverse DTFT for the Right-Sided Exponential8-1.3 The DTFT is the SpectrumProperties of the DTFT8-2.1 The Linearity Property8-2.2 The Time-Delay Property8-2.3 The Frequency-Shift Property8-2.3.1 DTFT of a Complex Exponential8-2.3.2 DTFT of a Real Cosine Signal8-2.4 Convolution and the DTFT8-2.4.1 Filtering is Convolution8-2.5 Energy Spectrum and the Autocorrelation Function8-2.5.1 Autocorrelation FunctionIdeal Filters8-3.1 Ideal Lowpass Filter8-3.2 Ideal Highpass Filter8-3.3 Ideal Bandpass FilterPractical FIR Filters8-4.1 Windowing8-4.2 Filter Design 8-4.2.1 Window the Ideal Impulse Response 8-4.2.2 Frequency Response of Practical Filters8-4.2.3 Passband Defined for the Frequency Response8-4.2.4 Stopband Defined for the Frequency Response8-4.2.5 Transition Zone of the LPF8-4.2.6 Summary of Filter Specifications8-4.3 GUI for Filter DesignTable of Fourier Transform Properties and PairsSummary and LinksProblemsThe Discrete Fourier Transform Discrete Fourier Transform (DFT)9-1.1 The Inverse DFT9-1.2 DFT Pairs from the DTFT9-1.2.1 DFT of Shifted Impulse9-1.2.2 DFT of Complex Exponential9-1.3 Computing the DFT9-1.4 Matrix Form of the DFT and IDFTProperties of the DFT9-2.1 DFT Periodicity for XOEk]9-2.2 Negative Frequencies and the DFT9-2.3 Conjugate Symmetry of the DFT9-2.3.1 Ambiguity at XOEN=2]9-2.4 Frequency Domain Sampling and Interpolation9-2.5 DFT of a Real Cosine SignalInherent Periodicity of xOEn] in the DFT9-3.1 DFT Periodicity for xOEn]9-3.2 The Time Delay Property for the DFT9-3.2.1 Zero Padding9-3.3 The Convolution Property for the DFTTable of Discrete Fourier Transform Properties and PairsSpectrum Analysis of Discrete Periodic Signals9-5.1 Periodic Discrete-time Signal: Fourier Series9-5.2 Sampling Bandlimited Periodic Signals9-5.3 Spectrum Analysis of Periodic SignalsWindows9-6.0.1 DTFT of WindowsThe Spectrogram9-7.1 An Illustrative Example9-7.2 Time-Dependent DFT9-7.3 The Spectrogram Display9-7.4 Interpretation of the Spectrogram9-7.4.1 Frequency Resolution9-7.5 Spectrograms in MATLABThe Fast Fourier Transform (FFT)9-8.1 Derivation of the FFT9-8.1.1 FFT Operation CountSummary and LinksProblemsz-Transforms Definition of the z-TransformBasic z-Transform Properties10-2.1 Linearity Property of the z-Transform10-2.2 Time-Delay Property of the z-Transform10-2.3 A General z-Transform FormulaThe z-Transform and Linear Systems10-3.1 Unit-Delay System10-3.2 z-1 Notation in Block Diagrams10-3.3 The z-Transform of an FIR Filter10-3.4 z-Transform of the Impulse Response10-3.5 Roots of a z-transform PolynomialConvolution and the z-Transform10-4.1 Cascading Systems10-4.2 Factoring z-Polynomials10-4.3 DeconvolutionRelationship Between the z-Domain and the !O -Domain10-5.1 The z-Plane and the Unit CircleThe Zeros and Poles of H.z/10-6.1 Pole-Zero Plot10-6.2 Significance of the Zeros of H.z/10-6.3 Nulling Filters10-6.4 Graphical Relation Between z and !O10-6.5 Three-Domain MoviesSimple Filters10-7.1 Generalize the L-Point Running-Sum Filter10-7.2 A Complex Bandpass Filter10-7.3 A Bandpass Filter with Real CoefficientsPractical Bandpass Filter DesignProperties of Linear-Phase Filters10-9.1 The Linear-Phase Condition10-9.2 Locations of the Zeros of FIR Linear-Phase SystemsSummary and LinksProblemsIIR Filters The General IIR Difference EquationTime-Domain Response11-2.1 Linearity and Time Invariance of IIR Filters11-2.2 Impulse Response of a First-Order IIR System11-2.3 Response to Finite-Length Inputs11-2.4 Step Response of a First-Order Recursive SystemSystem Function of an IIR Filter11-3.1 The General First-Order Case11-3.2 H.z/ from the Impulse Response11-3.3 The z-Transform MethodThe System Function and Block-Diagram Structures11-4.1 Direct Form I Structure11-4.2 Direct Form II Structure11-4.3 The Transposed Form StructurePoles and Zeros11-5.1 Roots in MATLAB11-5.2 Poles or Zeros at z D 0 or 111-5.3 Output Response from Pole LocationStability of IIR Systems11-6.1 The Region of Convergence and StabilityFrequency Response of an IIR Filter11-7.1 Frequency Response using MATLAB11-7.2 Three-Dimensional Plot of a System FunctionThree DomainsThe Inverse z-Transform and Some Applications11-9.1 Revisiting the Step Response of a First-Order System11-9.2 A General Procedure for Inverse z-TransformationSteady-State Response and StabilitySecond-Order Filters11-11.1 z-Transform of Second-Order Filters11-11.2 Structures for Second-Order IIR Systems11-11.3 Poles and Zeros11-11.4 Impulse Response of a Second-Order IIR System11-11.4.1 Distinct Real Poles11-11.5 Complex PolesFrequency Response of Second-Order IIR Filter11-12.1 Frequency Response via MATLAB11-12.23-dB Bandwidth11-12.3 Three-Dimensional Plot of System FunctionsExample of an IIR Lowpass FilterSummary and LinksProblems