Gives a basic introduction on Continuous Time Analog Systems

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Continuous Time Analog Systems
Aranya Sarkar
M.Sc- Electrical Engineering and Information Technology
07.07.2015

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 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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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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 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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 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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 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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 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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Example of a Convolution

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 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. 1-4
http://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/3/7/6/6/0/51c48875ce395f74
5a000000.png&imgrefurl=https://learn.sparkfun.com/tutorials/analog-vs-digital/analog-signal.html
References

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QUESTIONS ?

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Thank You For Your Attention