PLD Autumn 2016 Signals and Linear Systems Lecture 1

EE2 Signals and Linear Systems

Pier Luigi Dragotti Electrical & Electronic Engineering Department

Imperial College London

URL: http://www.commsp.ee.ic.ac.uk/~pld/Teaching/ E-mail: p.dragotti@imperial.ac.uk

PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 2

Aims and Objectives

‣  “The concepts of signals and systems arise in a variety of fields and the techniques associated with these notions play a central role in many areas of science and technology such as, for example, communications, aeronautics, bio-engineering, energy, circuit design, ect. Although the physical nature of the signals and systems involved in these various disciplines are different, they all have basic features in common. The aim of the course is to provide the fundamental and universal tools for the analysis of signals, and for the analysis and design of basic systems and this independently of the domain of application.”

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Aims and Objectives

By the end of the course, you will have understood:

-  Basic signal analysis (mostly continuous-time)

-  Basic system analysis (also mostly continuous systems)

-  Time-domain system analysis (including convolution)

-  Laplace and Fourier Transform

-  System Analysis in Laplace and Fourier Domains

-  Filter Design

-  Sampling Theorem and signal reconstructions

-  Basics on z-transform

PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 4

About the course

• Lectures - 15 hours over 8-9 weeks • Problem Classes – 7-8 hours over 8-9 weeks • Assessment – 100% examination in June • Handouts in the form of pdf slides also available at http://www.commsp.ee.ic.ac.uk/~pld/Teaching/ and on BlackBoard • Discussion board available on Blackboard • Text Book: -  Recommended Text Book: B.P. Lathi, “Linear Systems and Signals”, 2nd Ed., Oxford University Press -  Also useful: A.V. Oppenheim & A.S. Willsky “Signals and Systems”, Prentice Hall

PLD Autumn 2016 Signals and Linear Systems Lecture 1

Lecture 1 Basics about Signals

(Lathi 1.1-1.5)

Pier Luigi Dragotti Electrical & Electronic Engineering Department

Imperial College London

URL: http://www.commsp.ee.ic.ac.uk/~pld/Teaching/ E-mail: p.dragotti@imperial.ac.uk

PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 6

Outline

‣  Examples of Signals

‣  Some useful Signal Operations

‣  Classification of Signals

‣  Some useful Signals

-  Unit Step Function

-  Unit Impulse Function

-  The Exponential Function

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Examples of Signals (1)

‣  Electroencephalogram (EEG) Signal

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Examples of Signals (2)

‣  Stock Market Data as a discrete-time signal

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Examples of Signals (3)

‣  Magnetic Resonance Image (MRI) as a 2-dimensional signal

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Useful Signal Operations: Time Shifting

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Useful Signal Operations: Time Scaling

φ(t/2)=x(t)

φ(2t)=x(t)

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Signals Classification

‣  Signals may be classified into:

-  Energy and power signals

-  Discrete-time and continuous-time signals

-  Analogue and digital signals

-  Deterministic and probabilistic signals

-  Periodic and aperiodic signals

-  Even and odd signals

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Signals Classification: Energy vs Power

Size of a signal x(t)

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Signals Classification: Energy vs Power (2)

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Signals Classification: Energy vs Power (3)

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Signals Classification: Continuous-time vs Discrete-time

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Signals Classification: Analogue vs Digital

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Signals Classification: Deterministic vs Random

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Signals Classification: Periodic vs Aperiodic

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Signals Classification: Even vs Odd

fe(t)=fe(-t)

fo(t)=-fo(-t)

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Signals Classification: Even vs Odd (2)

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Useful Signals: Unit Step Function u(t)

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Useful Signals: Pulse Signal

A pulse signal can be represented using two step functions. E.g:

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Useful Signals: Unit Impulse Function δ(t)

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Sampling Property of the Unit Impulse Function

It follows that:

If we want to sample φ(t) and t=T, we multiply by δ(t-T):

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The Exponential Function est (1)

‣ The exponential function is very important in signals & systems. ‣ The parameter s is a complex variable given by:

s=σ+jω

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The Exponential Function est (2)

‣ The exponential function est can be used to model a large class of signals. ‣ Here are a few examples:

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The Exponential Function est (3)

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The Complex Frequency Plane s=σ+jω

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Discrete-Time Exponential γn

‣ A continuous-time exponential est can be expressed in alternate form as with

‣ Similarly we have for discrete-time exponentials

‣ The form γn is preferred for discrete-time exponentials

‣ When Re{λ}<0, then and the exponential decays

‣ When Re{λ}>0, then and the exponential grows

‣ When λ is purely imaginary then and the exponential is constant-amplitude and oscillates

L3.3-3 p254

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est = γ t

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γ = es

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eλn = γ n

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γ <1

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γ >1

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γ =1

PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 31

Discrete-Time Exponential γn (cont’d)

L3.3-3 p255