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 XŒk]
9-2.2 Negative Frequencies and the DFT
9-2.3 Conjugate Symmetry of the DFT
9-2.3.1 Ambiguity at XŒN=2]
9-2.4 Frequency Domain Sampling and Interpolation
9-2.5 DFT of a Real Cosine Signal
Inherent Periodicity of xŒn] in the DFT
9-3.1 DFT Periodicity for xŒn]
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
