Signals and systems

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Introduction to signals

Signal classifications and properties

0.1 Introduction

0.2 Classifications of signals

0.3 Continuous-time vs. discrete-time

0.4 Analog vs. digital

0.5 Periodic vs. aperiodic

0.6 Finite vs. infinite length

0.7 Causal vs. anticausal vs. noncausal

0.8 Even vs. odd

0.9 Deterministic vs. random

0.10 Signal classifications summary

1 Signal size and norms

2 Signal operations

3 Common continuous time signals

4 Common discrete time signals

5 Continuous time impulse function

6 Discrete time impulse function

7 Continuous time complex exponential

8 Discrete time complex exponential

Introduction to systems

System classifications and properties

1 Linear time invariant systems

Time domain analysis of continuous time systems

Continuous time systems

1 Continuous time impulse response

2 Continuous time convolution

2.1 Introduction

2.2 Convolution and circular convolution

2.3 Convolution

2.4 Operation definition

2.5 Definition motivation

2.6 Graphical intuition

2.7 Circular convolution

2.8 Definition motivation

2.9 Graphical intuition

2.10 Convolution demonstration

2.11 Convolution summary

3 Properties of continuous time convolution

3.1 Introduction

3.2 Continuous time convolution properties

3.3 Associativity

3.4 Commutativity

3.5 Distributivity

3.6 Multilinearity

3.7 Conjugation

3.9 Differentiation

3.10 Impulse convolution

3.12 Convolution properties summary

4 Eigenfunctions of continuous time lti systems

5 Bibo stability of continuous time systems

6 Linear constant coefficient differential equations

7 Solving linear constant coefficient differential equations

Time domain analysis of discrete time systems

Discrete time systems

1 Discrete time impulse response

2 Discrete time convolution

3 Properties of discrete time convolution

4 Eigenfunctions of discrete time lti systems

5 Bibo stability of discrete time systems

6 Linear constant coefficient difference equations

7 Solving linear constant coefficient difference equations

Introduction to fourier analysis

Introduction to fourier analysis

Continuous time fourier series (ctfs)

Continuous time periodic signals

0.1 Introduction

0.2 Relevant spaces

0.3 Periodic signals

0.4 Demonstration

1 Continuous time fourier series (ctfs)

2 Common fourier series

3 Properties of the ctfs

4 Continuous time circular convolution and the ctfs

5 Convergence of fourier series

6 Gibbs phenomena

6.1 Introduction

6.3 Explanation

6.4 Square wave

6.5 Redefine equality

6.6 Visualizing gibb's phenomena

Discrete time fourier series (dtfs)

Discrete time periodic signals

1 Discrete time fourier series (dtfs)

1.1 Introduction

1.3 Eigenfunction analysis

1.4 Dtfs synthesis

1.5 Dft synthesis demonstration

1.6 Dtfs analysis

1.7 Complex sinusoids

1.8 In the complex plane

1.9 Periodic extension to dtfs

1.11 Dtfs conclusion

2 Common discrete fourier series

3 Properties of the dtfs

4 Discrete time circular convolution and the dtfs

Continuous time fourier transform (ctft)

Continuous time aperiodic signals

1 Continuous time fourier transform (ctft)

2 Common fourier transforms

3 Properties of the ctft

3.1 Introduction

3.2 Discussion of fourier transform properties

3.5 Time scaling

3.6 Time shifting

3.7 Convolution

3.8 Time differentiation

3.9 Parseval's relation

3.10 Modulation (frequency shift)

3.11 Properties demonstration

3.12 Summary table of ctft properties

4 Continuous time convolution and the ctft

Discrete time fourier transform (dtft)

Discrete time aperiodic signals

1 Discrete time fourier transform (dtft)

2 Common discrete time fourier transforms

3 Properties of the dtft

4 Discrete time convolution and the dtft

Sampling and reconstruction

Signal sampling

1 Sampling theorem

1.1 Introduction

1.2 Nyquist-shannon sampling theorem

1.3 Statement of the sampling theorem

1.4 Proof of the sampling theorem

1.5 Perfect reconstruction

1.6 Practical implications

1.7 Discrete time processing of continuous time signals

1.8 Psychoacoustics

1.9 Sampling theorem summary

2 Signal reconstruction

3 Perfect reconstruction

4 Aliasing phenomena

5 Anti-aliasing filters

5.1 Introduction

5.2 Anti-aliasing filters

5.3 Anti-aliasing filters summary

6 Discrete time processing of continuous time signals

6.1 Introduction

6.2 Discrete time processing of continuous time signals

6.3 Process structure

6.4 Discrete time filter

6.5 Practical considerations

6.6 Anti-aliasing filter

6.7 Signal quantization

6.8 Filter implementability

6.9 Anti-imaging filter

6.10 Discrete time processing of continuous time signals summary

Laplace transform and continuous time system design

Laplace transform

0.1 Introduction

0.2 The laplace transform

0.3 Bilateral laplace transform pair

0.4 Relation between laplace and ctft

0.5 Visualizing the laplace transform

0.6 Using a computer to find the laplace transform

0.7 Laplace transform definition demonstration

0.8 Interactive demonstrations

1 Common laplace transforms

2 Properties of the laplace transform

3 Inverse laplace transform

4 Poles and zeros in the s-plane

4.1 Introduction to poles and zeros of the laplace-transform

4.2 The s-plane

4.3 Examples of pole/zero plots

4.4 Interactive demonstration of poles and zeros

4.5 Applications for pole-zero plots

4.6 Stability and control theory

4.7 Pole/zero plots and the region of convergence

4.8 Frequency response and pole/zero plots

5 Region of convergence for the laplace transform

6 Rational functions and the laplace transform

7 Differential equations

8 Continuous time filter design

8.1 Introduction

8.2 Estimating frequency response from z-plane

8.3 Drawing frequency response from pole/zero plot

8.4 Types of filters

8.5 Butterworth filters

8.6 Chebyshev filters

8.7 Bessel filters

8.8 Elliptic filters

8.9 Filter design demonstration

8.10 Conclusion

Z-transform and discrete time system design

0.1 Introduction

0.2 The z-transform

0.3 Bilateral z-transform pair

0.4 Relation between z-transform and dtft

0.5 Visualizing the z-transform

0.6 Application to discrete time filters

0.7 Interactive z-transform demonstration

1 Common z-transforms

2 Properties of the z-transform

3 Inverse z-transform

4 Poles and zeros in the z-plane

5 Region of convergence for the z-transform

6 Rational functions and the z-transform

7 Difference equations

8 Discrete time filter design

8.1 Estimating frequency response from z-plane

8.2 Drawing frequency response from pole/zero plot

8.3 Interactive filter design illustration

Capstone signal processing topics

Dft: fast fourier transform

1 The fast fourier transform (fft)

1.1 Introduction

1.2 Deriving the fft

1.3 Fft and the dft

1.4 Speed comparison

2 Deriving the fast fourier transform

3 Matched filter detector

3.1 Introduction

3.2 Matched filter detector theory

3.3 Signal comparison

3.4 Pattern detection

3.5 Practical applications

3.6 Image detection

3.7 Communications systems

3.9 Matched filter demonstration

3.10 Matched filter summary

Appendix a: linear algebra overview

Basic linear algebra

0.1 Linear independence

1 Eigenvectors and eigenvalues

2 Matrix diagonalization

3 Eigen-stuff in a nutshell

4 Eigenfunctions of lti systems

Appendix b: hilbert spaces overview

Fields and complex numbers

1 Vector spaces

3 Inner products

4 Hilbert spaces

5 Cauchy-schwarz inequality

6 Common hilbert spaces

7 Types of bases

8 Orthonormal basis expansions

9 Function space

10 Haar wavelet basis

11 Orthonormal bases in real and complex spaces

12 Plancharel and parseval's theorems

12.1 Parseval's theorem

12.2 Prove: plancherel theorem

12.3 Periodic signals power

12.4 Plancharel theorem

12.5 Plancharel theorem

12.6 Plancharels theorem demonstration

12.7 Parseval's theorem: a different approach

12.8 Parseval's theorem

13 Approximation and projections in hilbert space

Appendix c: analysis topics overview

Introduction to metric spaces

0.1 Introduction

0.2 Metric spaces

0.3 A notion of distance

0.4 Relationship with norms

0.5 Metric spaces summary

1 Convergence of sequences

2 Convergence of sequences of vectors

3 Uniform convergence of function sequences

Appendix d: viewing interactive content

Viewing embedded labview content in connexions

1 Getting started with mathematica

Source: OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15

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