Signals and Systems
Outline

First course

I. Introduction

II. Notion of signals and systems
1. What is a signal?
2. What is a system?
3. Signal classification with examples
4. Dirac impulse, Dirac impulse train and properties
5. Discrete impulse, discrete impulse train and properties

III. Fourier representations for signals
1. Introduction
2. Fourier transform (FT)
3. Discrete-time Fourier transform (DTFT)
4. Fourier representation of Dirac impulse and discrete impulse
5. Link between TF and DTFT
6. Fourier series (FS)
7. Discrete-time Fourier series (DFS)
8. Link between FS and FT and between DFS and DTFT
9. FT of a Dirac impulse train, Poisson formula
10. DTFT of a discrete impulse train, discrete Poisson formula

IV. Convolution product
1. Introduction
2. Continuous-time linear convolution
3. Discrete-time linear convolution
4. Dirac impulse and convolution, Generalized Poisson formula
5. Discrete impulse and convolution, Generalized  discrete Poisson formula

V. Sampling and reconstruction
1. Introduction
2. Periodic sampling, Frequency correspondence
3. Reconstruction of band limited signals
4. Practical (causal) reconstruction

VI. Linear and time-invariant (LTI) systems
1. Introduction
2. Time-domain representation of continuous-time LTI system
3. Time-domain representation of discrete-time LTI system
4. Frequency response
5. Four important additional properties of LTI systems

VII. Correlation products
1. Introduction
2. Continuous-time linear autocorrelation and cross-correlation
3. Discrete-time linear autocorrelation and cross-correlation
4. Energy spectral density
5. Cross-correlation of the outputs of two LTI systems

Second course

I. Discrete Fourier series
1. Introduction
2. Definition
3. Important properties
4. Discrete-time circular convolution
5. Link with discrete-time linear convolution of time-limited signals
6. Discrete cosine transforms

II. The Laplace transform
1. Introduction
2. Definition
3. Important properties
4. Transfert function of LTI systems and properties
5. Rational systems and properties
6. Cross-correlation of the continuous-time outputs of two LTI systems

III. The z transform
1. Introduction
2. Definition
3. Important properties
4. Transfert function of LTI systems and properties
5. Rational systems and properties
6. Cross-correlation of the discrete-time outputs of two LTI systems

IV. Continuous-time pass-band signal representation
1. Introduction
2. Fourier transform of the unit step function
3. Hilbert transform
4. Analytical signal of a real signal
5. Pass-band signal and complex envelope

V. Decimation and expansion
1. Introduction
2. Definitions
3. Frequency-domain representations
4. Decimation and expansion filters
5. Application to sampling-rate conversion