Continuous Time Analog Systems

107.07.2015OVGU Präsentation

Continuous Time Analog Systems

Aranya Sarkar

M.Sc- Electrical Engineering and Information Technology07.07.2015


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


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


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


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


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


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


Memoryless Systems

Invertible

Causality

Stablity

Time invariance

Linearity

Properties of Continuous Time Systems


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


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


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


Convolution

Fourier Transform

Laplace Transform

Bode Plots

Tools Used


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


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


Example of a Convolution


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


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

http://scholar.harvard.edu/stanleychan/files/note_0.pdf

http://control.ucsd.edu/mauricio/courses/mae143a/lectures/2analogsystemsproperties.pdf

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


QUESTIONS ?


Thank You For Your Attention