3. Introduction
The
concepts of signals and systems
arise in a wide variety of areas:
communications,
circuit design,
biomedical engineering,
power systems,
speech processing,
etc.
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4. What is a Signal?
SIGNAL
A set of information or data.
Function of one or more
independent variables.
Contains information about the
behavior or nature of some
phenomenon.
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7. What is a System?
SYSTEM
Signals
may be processed further
by systems, which may modify
them or extract additional from
them.
A
system is an entity that
processes a set of signals
(inputs) to yield another set of
signals (outputs).
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8. What is a System? (2)
A
system may be made up of
physical components, as in
electrical or mechanical systems
(hardware realization).
A system may be an algorithm
that computes an outputs from
an inputs signal (software
realization).
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9. Examples of signals and systems
Voltage (x1) and current (x2) as functions of
time in an electrical circuit are examples of
signals.
A circuit is itself an example of a system (T),
which responds to applied voltages and
currents.
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11. Signal Models: Unit Step Function
Continuous-Time
unit step function, u(t):
u(t)
is used to start a signal, f(t) at t=0
f(t) has a value of ZERO for t <0
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12. Signal Models: Unit Impulse Function
A
possible approximation
to
a
unit
impulse:
An overall area that has
been
maintained
at
unity.
Graphically, it is
represented by an arrow
"pointing to infinity" at
t=0 with its length equal
to its area.
Multiplication
of
a
function by an Impulse?
bδ(t) = 0; for all t≠0
is an impulse function
which the area is b.
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13. Signal Models: Unit Impulse Function
(3)
May
use functions other than a rectangular
pulse. Here are three example functions:
Note that the area under the pulse function
must be unity.
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14. Signal Models: Unit Ramp Function
Unit
ramp function is defined by:
r(t) = t∗u(t)
Where
can it be used?
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15. Signal Models: Exponential Function
est
Most
important function in SNS where s is
complex in general, s = σ+jϖ
Therefore,
est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt)
(Euler’s formula: ejϖt = cosϖt + jsinϖt)
s∗ = σ-jϖ,
es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt)
If
From
the above, e cosϖt = ½(e +e )
σt
st
-st
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16. Signal Models: Exponential Function
est (2)
Variable s is complex frequency.
est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt)
es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt)
eσtcosϖt = ½(est +e-st )
There are special cases of est :
1.
2.
3.
4.
A constant k = ke0t (s=0 σ=0,ϖ=0)
A monotonic exponential eσt (ϖ=0, s=σ)
A sinusoid cosϖt (σ=0, s=±jϖ)
An exponentially varying sinusoid eσtcosϖt
(s= σ ±jϖ)
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17. Signals Classification
Signals
may be classified into:
1. Continuous-time and Discrete-time signals
2. Deterministic and Stochastic Signal
3. Periodic and Aperiodic signals
4. Even and Odd signals
5. Energy and Power signals
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18. Continuous v/S Discrete Signals
Continuous-time
A signal that is
specified for every
value of time t.
Discrete-time
A signal that is
specified only at
discrete values
of time t.
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19. Deterministic v/s Stochastic
Signal
Signals
that can be written in any
mathematical expression are called
deterministic signal.
(sine,cosine..etc)
Signals that cann’t be written in mathematical
expression are called stochastic signals.
(impulse,noise..etc)
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20. Periodic v/s Aperiodic Signals
Signals
that repeat itself at a proper interval
of time are called periodic signals.
Continuous-time signals are said to be
periodic.
Signals that will never repeat themselves,and
get over in limited time are called aperiodic or
non-periodic signals.
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22. Even v/s Odd Signals
A
signal x(t) or x[n] is referred to as an even
signal if
CT:
DT:
A
signal x(t) or x[n] is referred to as an odd
signal if
CT:
DT:
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23. Even and Odd Functions: Properties
Property:
Area:
Even signal:
Odd signal:
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24. Even and Odd Components of a
Signal (1)
Every
signal f(t) can be expressed as a sum
of even and odd components because
Example,
f(t) = e-atu(t)
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25. Energy v/s Power Signals
Signal with finite energy (zero power)
Signal with finite power (infinite energy)
Signals that satisfy neither property are referred
as neither energy nor power signals
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26. Size of a Signal, Energy (Joules)
Measured
by signal energy Ex:
Generalize
CT:
Energy
for a complex valued signal to:
DT:
must be finite, which means
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27. Size of a Signal, Power (Watts)
If
amplitude of x(t) does not → 0 when t → ∞,
need to measure power Px instead:
Again,
generalize for a complex valued signal
to:
CT:
DT:
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28. OPERATIONS ON SIGNALS
It
includes the transformation of independent
variables.
It is performed in both continuous and
discrete time signals.
Operations that are performed are-
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29. 1.ADDITION &SUBSTRACTION
Let two signals x(t) and y(t) are given,
Their addition will be,
z(t) = x(t) + y(t)
Their substraction will be,
z(t) = x(t) – y(t)
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30. 2.MULTIPLICATION OF
SIGNAL BY A CONSTANT
If a constant ‘A’ is given with a signal x(t)
z(t) = A.x(t)
If A>1,it is an amplified signal.
If A<1,it is an attenuated signal.
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32. 4.SHIFTING IN TIME
Let a signal x(t),than the signal x(t-T)
represented a delayed version of x(t),which is
delayed by T sec.
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33. Signal Operations: Time Shifting
Shifting
of a signal in time
adding or subtracting the amount of the
shift to the time variable in the function.
x(t) x(t–t )
o
to > 0 (to is positive value),
signal is shifted to the right (delay).
to < 0 (to is negative value),
signal is shifted to the left (advance).
x(t–2)?
x(t) is delayed by 2 seconds.
x(t+2)? x(t) is advanced by 2 seconds.
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34. Signal Operations: Time Shifting (2)
Subtracting
a fixed amount from the time
variable will shift the signal to the right that
amount.
Adding
to the time variable will shift the signal
to the left.
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36. 5.COMPRESSION/EXPANSION
OF SIGNALS
This is also known as ‘Time Scaling’ process.
Let a signal x(t) is given,we will examine as
x(at)
where a =real number
and how it is related to x(t) ?
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