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