## Table of Contents

**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) System**s

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