0
SIGNALS AND SYSTEM
SURAJ MISHRA
SUMIT SINGH
AMIT GUPTA
PRATYUSH SINGH
(E.C 2ND YEAR ,MCSCET)
1
Topics





Introduction
Classification of Signals
Some Useful Signal Operations
Some useful signal models

2
Introduction
 The

concepts of signals and systems
arise in a wide variety of areas:
 communications,
 circuit design,
...
What is a Signal?
SIGNAL





A set of information or data.
Function of one or more
independent variables.
Contains inf...
Examples of Signals
 BRAIN

WAVE

5
Examples of Signals
 Stock

Market data as signal (time series)

6
What is a System?
SYSTEM
Signals

may be processed further
by systems, which may modify
them or extract additional from
t...
What is a System? (2)
A

system may be made up of
physical components, as in
electrical or mechanical systems
(hardware r...
Examples of signals and systems




Voltage (x1) and current (x2) as functions of
time in an electrical circuit are exam...
Some Useful Signal
Models

10
Signal Models: Unit Step Function
 Continuous-Time

unit step function, u(t):

 u(t)

is used to start a signal, f(t) at...
Signal Models: Unit Impulse Function
A

possible approximation
to
a
unit
impulse:
An overall area that has
been
maintaine...
Signal Models: Unit Impulse Function
(3)
 May

use functions other than a rectangular
pulse. Here are three example funct...
Signal Models: Unit Ramp Function
 Unit


ramp function is defined by:

r(t) = t∗u(t)

 Where

can it be used?

14
Signal Models: Exponential Function
est
 Most

important function in SNS where s is
complex in general, s = σ+jϖ
 Theref...
Signal Models: Exponential Function
est (2)





Variable s is complex frequency.
est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt ...
Signals Classification
 Signals






may be classified into:

1. Continuous-time and Discrete-time signals
2. Deter...
Continuous v/S Discrete Signals
 Continuous-time

A signal that is
specified for every
value of time t.

 Discrete-time
...
Deterministic v/s Stochastic
Signal
 Signals

that can be written in any
mathematical expression are called
deterministic...
Periodic v/s Aperiodic Signals
 Signals

that repeat itself at a proper interval
of time are called periodic signals.
 C...
Even v/s Odd Signals

21
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 ...
Even and Odd Functions: Properties
 Property:

 Area:


Even signal:



Odd signal:
23
Even and Odd Components of a
Signal (1)
 Every

signal f(t) can be expressed as a sum
of even and odd components because
...
Energy v/s Power Signals


Signal with finite energy (zero power)



Signal with finite power (infinite energy)



Sign...
Size of a Signal, Energy (Joules)
 Measured

by signal energy Ex:

 Generalize


CT:

 Energy

for a complex valued si...
Size of a Signal, Power (Watts)
 If

amplitude of x(t) does not → 0 when t → ∞,
need to measure power Px instead:

 Agai...
OPERATIONS ON SIGNALS
 It

includes the transformation of independent
variables.
 It is performed in both continuous and...
1.ADDITION &SUBSTRACTION





Let two signals x(t) and y(t) are given,
Their addition will be,
z(t) = x(t) + y(t)

Thei...
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 a...
3.MULTIPLICATION OF TWO
SIGNALS


If two signals x(t) and y(t) are given,than their
multiplication will be
z(t) = x(t).y(...
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...
Signal Operations: Time Shifting
 Shifting

of a signal in time
  adding or subtracting the amount of the
shift to the ...
Signal Operations: Time Shifting (2)
 Subtracting

a fixed amount from the time
variable will shift the signal to the rig...
Signal Operations: Time Shifting
 Shifting

of a signal in time

35
5.COMPRESSION/EXPANSION
OF SIGNALS





This is also known as ‘Time Scaling’ process.
Let a signal x(t) is given,we wil...
Time Scaling

37
Signal Operations: Time Inversion
 Reversal

of the time axis, or folding/flipping
the signal (mirror image) over the y-a...
THANKS....................... FOR
YOUR

ATTENTION !
39
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Transcript of "Signals and classification"

  1. 1. SIGNALS AND SYSTEM SURAJ MISHRA SUMIT SINGH AMIT GUPTA PRATYUSH SINGH (E.C 2ND YEAR ,MCSCET) 1
  2. 2. Topics     Introduction Classification of Signals Some Useful Signal Operations Some useful signal models 2
  3. 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. 3
  4. 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. 4
  5. 5. Examples of Signals  BRAIN WAVE 5
  6. 6. Examples of Signals  Stock Market data as signal (time series) 6
  7. 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). 7
  8. 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). 8
  9. 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. 9
  10. 10. Some Useful Signal Models 10
  11. 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 11
  12. 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. 12
  13. 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. 13
  14. 14. Signal Models: Unit Ramp Function  Unit  ramp function is defined by: r(t) = t∗u(t)  Where can it be used? 14
  15. 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 15
  16. 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) 16
  17. 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 17
  18. 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. 18
  19. 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) 19
  20. 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. 20
  21. 21. Even v/s Odd Signals 21
  22. 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: 22
  23. 23. Even and Odd Functions: Properties  Property:  Area:  Even signal:  Odd signal: 23
  24. 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) 24
  25. 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 25
  26. 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 26
  27. 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: 27
  28. 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- 28
  29. 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) 29
  30. 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.  30
  31. 31. 3.MULTIPLICATION OF TWO SIGNALS  If two signals x(t) and y(t) are given,than their multiplication will be z(t) = x(t).y(t) 31
  32. 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. 32
  33. 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. 33
  34. 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. 34
  35. 35. Signal Operations: Time Shifting  Shifting of a signal in time 35
  36. 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) ? 36
  37. 37. Time Scaling 37
  38. 38. Signal Operations: Time Inversion  Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis. 38
  39. 39. THANKS....................... FOR YOUR ATTENTION ! 39
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