EE2 Signals and Linear Systems
Transcript of EE2 Signals and Linear Systems
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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: [email protected]
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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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 3
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
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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
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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: [email protected]
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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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 7
Examples of Signals (1)
‣ Electroencephalogram (EEG) Signal
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 8
Examples of Signals (2)
‣ Stock Market Data as a discrete-time signal
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 9
Examples of Signals (3)
‣ Magnetic Resonance Image (MRI) as a 2-dimensional signal
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 10
Useful Signal Operations: Time Shifting
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 11
Useful Signal Operations: Time Scaling
φ(t/2)=x(t)
φ(2t)=x(t)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 12
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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 13
Signals Classification: Energy vs Power
Size of a signal x(t)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 14
Signals Classification: Energy vs Power (2)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 15
Signals Classification: Energy vs Power (3)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 16
Signals Classification: Continuous-time vs Discrete-time
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 17
Signals Classification: Analogue vs Digital
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 18
Signals Classification: Deterministic vs Random
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 19
Signals Classification: Periodic vs Aperiodic
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 20
Signals Classification: Even vs Odd
fe(t)=fe(-t)
fo(t)=-fo(-t)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 21
Signals Classification: Even vs Odd (2)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 22
Useful Signals: Unit Step Function u(t)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 23
Useful Signals: Pulse Signal
A pulse signal can be represented using two step functions. E.g:
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 24
Useful Signals: Unit Impulse Function δ(t)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 25
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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 26
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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 27
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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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 28
The Exponential Function est (3)
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 29
The Complex Frequency Plane s=σ+jω
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 30
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
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PLD Autumn 2016 Signals and Linear Systems Lecture 1 Slide 31
Discrete-Time Exponential γn (cont’d)
L3.3-3 p255