[eBook] Digital Signal Processing First, Global Edition > eBook

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