continuous time analog systems
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107.07.2015OVGU Präsentation
Continuous Time Analog Systems
Aranya Sarkar
M.Sc- Electrical Engineering and Information Technology07.07.2015
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207.07.2015OVGU Präsentation
Introduction- Signals
Analog and Digital signals
Classification of signals and elementary continuous-time signals
Systems, classifications and properties
Time invariance and linearity
Analog Signal Processing and tools used
Convolution
Conclusion
References
Contents
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Fig Variation of the Earth’s electric field recorded, for a period of twelve (12) months by ATH monitoring site.
An electric current or electromagnetic field
Varies with time, space or any other independent variable
Mainly used to convey data from one place to another
Signals- An Introduction
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Analog: Continuous function V of continuous variable t (time, space etc.) : V(t)
Example: Human voice in air,analog devices.
Digital: Discrete function Vk of discrete sampling variable tk, with k = integer: Vk = V(tk)
Example: Computer, CDs, DVDs,digital devices.
Analog and Digital Signals
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507.07.2015OVGU Präsentation
Continuous:
defined for every instant of time
denoted by x(t)
Discrete:
defined at the discrete-instant of
time
denoted by x(n)
Continuous and Discrete Time Signals
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607.07.2015OVGU Präsentation
Unit Step Function: u(t)=1, for t≥ 0
=0, for t< 0
Unit Ramp Function: r(t)=t, for t≥ 0
=0, for t<0
Unit Parabolic Function: p(t)= 𝑡2
2, for t≥ 0
= 0, for t<0
Unit Impulse Function: −∞∞𝛿 𝑡 = 1
𝛿 𝑡 = 0, 𝑓𝑜𝑟 t≠ 0
Elementary Continuous Time Signals
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Physical device that generates a response or output signal, for a
given input signal
Continuous-time system:
x(t) y(t)
input output
Discrete-time system:
x[n] y[n]
input output
Systems
Continuous-time Systems
Discrete-time Systems
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Memoryless Systems
Invertible
Causality
Stablity
Time invariance
Linearity
Properties of Continuous Time Systems
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time-shift of the input signal results in the same time-shift of the
output signal
𝑥(𝑡 − 𝑡0) 𝑦(𝑡 − 𝑡0)
Example: y(t) = sin[x(t)]
Time Invariance
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1007.07.2015OVGU Präsentation
Systems obeying superposition principle is linear
If linear, systems are both additive and scalable
For continuous systems: 𝑇 𝑎𝑥1 𝑡 + 𝑏𝑥2(𝑡) = 𝑎𝑇 𝑥1(𝑡) + 𝑏𝑇 𝑥2(𝑡)
Example: y(n) = nx(n)
Linearity
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1107.07.2015OVGU Präsentation
Any type of signal processing conducted on analog signals by
analog means
Examples – crossover filters in loudspeakers
‘volume’ control in stereos
‘tint’ control on TVs
Common elements- capacitors, resistors, inductors, transistors
Analog Signal Processing
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Convolution
Fourier Transform
Laplace Transform
Bode Plots
Tools Used
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1307.07.2015OVGU Präsentation
Given a signal x(t) and impulse response h(t), the convolution between them is defined as:
𝑦 𝑡 = −∞∞𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏
Denoted as y(t)=x(t) * h(t)
Convolution
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Commutative property: 𝑥1(t) ∗ 𝑥2 (t) = 𝑥2(𝑡) ∗ 𝑥1(t)
Distributive property: 𝑥1(𝑡) ∗ (𝑥2(𝑡) + 𝑥3(𝑡)) = (𝑥1(𝑡) ∗ 𝑥2(𝑡)) +(𝑥1(𝑡) ∗ 𝑥3(𝑡))
Associative property: 𝑥(𝑡) ∗ (𝑥2(𝑡) ∗ 𝑥3(𝑡)) = (𝑥(𝑡) ∗ 𝑥2(𝑡)) ∗ 𝑥3(𝑡)
Shift property
Convolution with an impulse : 𝑥 𝑡 ∗ 𝛿 𝑡 = 𝑥(𝑡)
Width property
Properties of Convolution
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1507.07.2015OVGU Präsentation
Example of a Convolution
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1607.07.2015OVGU Präsentation
Analog signal is a continuous signal which represents physical measurements whereas , digital signals are discrete time signals generated by digital modulation
Continuous signals are defined for every instant of time, whereas discrete signals are defined for discrete. Instant of time
Systems that take in continuous time input and provides a continuous time output are known as continuous time systems
Any type of processing that is done on an analog signal by some analog means is known as analogue signal processing
The various tools used for analog signal processing include convolution, fourier transformation, laplace transformation and bode plot.
Conclusion
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P.Ramesh Babu, 2007, ‘Signals and Sytems’, 3rd edition, Scitech Publications, Ch.1-4
Stanley Chan, 2011, ‘Classnotes for Signals and Systems’, 2nd
Edition, Ch. 1-4http://scholar.harvard.edu/stanleychan/files/note_0.pdf
Mauricio, 2011, ‘Analog System Properties’ 2nd notes, http://control.ucsd.edu/mauricio/courses/mae143a/lectures/2analogsystemsproperties.pdf
Sparkfun, 2012, ‘Analog v/s digital’, e-book
http://www.google.de/imgres?imgurl=https://cdn.sparkfun.com/assets/3/7/6/6/0/51c48875ce395f745a000000.png&imgrefurl=https://learn.sparkfun.com/tutorials/analog-vs-digital/analog-signal.html
References
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1807.07.2015OVGU Präsentation
QUESTIONS ?
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1907.07.2015OVGU Präsentation
Thank You For Your Attention