Transcript Signal Properties

SiSy 2012 Dqtm Chapter 1,Introduction, 1
Signals and Systems
SiSy
Chapter 1
Signals and Systems: Introduction
Marina de Queiroz Tavares
[email protected]
TB511
Content, Calendar, and «Rules of the Game»
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Inhalt
Signals, Systems and Transformations
(base for several engineering areas)
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Chapter 1
Signals and Systems: Introduction
Contents
1. SIGNALS AND SYSTEMS: EXAMPLES
2. SIGNAL PROPERTIES
3. SYSTEM PROPERTIES
4. TEST SIGNALS
5. DISCRETE SIGNALS
6. OPERATIONS WITH THE TIME-VARIABLE
7.
VOCABULARY
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Input
Signals
Process modifying
signal, can be described
in many ways:
- diff.equation
- block diagram
- step response
- transfer function
- …
Measurements
Measurements
Time Functions
Output
Signals
Systems
Time Functions
can also be represented
in frequency domain (spectrum)
Example: Radio Receiver
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Antenna
el. magn. Waves
Signal
Loud speaker
acoustic waves
System
Signal
Question 1-1
Next steps
Identify the input and output signals, plus the system and
some sub-systems for the following applications:
How can we describe signals?
Internet Radio; Cable TV ; Music Synthesiser
How can we model systems?
Examples
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Robot Arm
Source: http://www.technovelgy.com/graphics/
content07/kuka-bible-2.jpg
Electronic Stability Control
for vehicles
Ultrasound
Medical Instrument
Source: http://www.divrad.com/index.php/imagingprocedures/services/category/ultrasound/
Source:
http://www.caradvice.com.au/wp-content/uploads/2007/11/esccomponents.jpg
Sensors used to measure : position, speed, acceleration, force, pressure,…
Actuators used to generate: voltage, current, …
Transducers transforming voltage and current in other physical signals
(e.g. ultrasound waves, traction in a motor, …)
Signal Properties: Analog vs. Digital
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Signals can be described as mathematical
functions of one or several variables
we will deal with time signals
x(t)
analog signal
(continuous range for time and values)
t
Sampling
-Ts
Ts = 1/fs
fs : sample frequency or rate [Samples/s]
t
Quantisation
digital signals
(time and value are discrete)
ADC
t
Time discrete
(continuous values) signals
Signal Properties
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Periodic
Regular pattern over time, with period length T
T
Question 1-2
How can you mathematically describe a periodic
sinus signal y(t) with period of 0,5 seconds, an
amplitude within the range [-2 ; 2], and by t=0
(initial condition) of y(0)=2 ?
T
Beispiele periodisch
Netzspannung (sinus)
Beispiele aperiodisch
Blitzschlag (Licht)
Temperaturverlauf (Wetter)
Signal Properties
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Energy
Signal
vs.
Comes and disappears
after some time
totally integratable
Question 1-3
If you are calculating the power of a periodic signal, it
is interesting to make the integral interval equals to
the signal period. Can you explain why?
Power
Signal
can last for a while
calculate energy over time
periodische Signale sind
normalerweise Leistungssignale
Burst-Signale sind meist
Energiesignale
Signal Properties: Symmetry
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Even
vs.
Odd
Question 1-4
The signal you described in Question 1-2, is it an odd or an even signal? Can you change its
symmetry property by varying the phase of the sinusoidal function?
Signal Properties
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• Deterministic
Pattern or form is known, for example periodic signals
• Random
Unknown form, description with statistical values;
Very important/useful (represent noise or information signals)
Sample pattern (every
sample will show a
different form; but same
statistical values)
Question 1-5
The input signal coming into a TV set via cable, is it a deterministic or a random signal? Can
a deterministic signal carry out information?
System Properties
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Linearity
Superposition
Principle
Time Invariance
Delayed input
 delayed output
Causality
Cause => consequence
«Output starts after input starts»
System Properties
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Example 1-1
Passive RC-circuit
Question 1-7
The response of the R-C
circuit for a staircase
input signals u(t) is given
besides.
Use this result to discuss
and justify if the system
is or is not time invariant.
Test Signals
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Step or Heaviside Step Function
«switch-on» function
ε(t)
1
t
Question 1-8
Question 1-9
Use the definition of the step signal σ(t)
to describe mathematically the staircase
input signal from Question 1-7.
Use the information of Figure 1-1 to define
the step response of the passive RC circuit.
Can you imagine a mathematical function
that describes this step response?
Test Signals
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Impulse or
Dirac Delta Function
Ideal sampler (unity energy in dt)
Question 1-10
Calculate the following integral and compare
the result to the step function.
What do you think is the relationship between the step and the impulse functions?
Test Signals
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Sine/Cosine waves
Complex exponential
Euler’s Formula
Questions 1-11 ;
1-12;
1-13
Discrete Signals
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via sampling continuous signal
Special Discrete Signals
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Unit Impulse
Kronecker Delta
Obs: can express any discrete function as
Unit Step
Relationship
Operations with the time-variable
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Time Shift
Time scaling
a>1 : compressed
a<1 : stretched
Mirroring
For example
check shift with one point