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Introduction Signal & System

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Introduction Signal & System Powered By Docstoc
					                                                                              Forward
                                                                               Concepts of signal and systems arise in wide variety of
   Signal and Systems                                                          fields.
                                                                               Ideas and techniques associated with these concepts play
                                                                               important role in diverse areas of science and technology:
                                              Introduction                       Communication
                                                                                 Aeronautics and astronautics
                                                                                 Circuit design
                                                                                 Seismology
                                                                                 Biomedical engineering
                                                                                 Control
                                                                                 Speech processing
                                                                                 Image processing

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Forward                                                                       Forward
 Although the different physical natures of the signals that
 arise in different disciplines, they have two basic features in                 Automobile itself is a system, the pressure on accelerator
 common:                                                                         pedal is the input and the automobile speed is the response
                                                                                 (output).
   Signal are function of one or more independent variable.
   Contain information about the behavior or nature of some                      Computer program for electrocardiogram can be viewed as
   phenomena.                                                                    the system that has its input from sensors and produce an
                                                                                 estimate of heart rate (output).
 System process particular signal to produce other signal or
 to follow a desired behavior.                                                   Camera receives light and produce photograph.
   Voltages and currents as a function of time in electrical circuits are        Robot arm whose movement are a response to control
   example of signals.                                                           inputs.
   Circuit is itself an example of a system
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Forward                                                                       Forward
 In the many contexts in which signal and systems arise,                       Another very important class of application in which
 there are a variety of problem:                                               concepts of signal and systems arise is control.
   System characterization                                                       Control system to regulate chemical processing plant
      Circuit response, aircraft response, … etc                                 Aircraft autopilot system.
   Designing systems to process signals in particular way
      Pilot communicating with air traffic control tower (signal
                                                                               Importance of concepts of signals and systems stems not
      enhancement).                                                            only from the diversity of application, but from the rich
      Image from deep space or earth-observing satellite (image restoration    collection of ideas, analytical tools, and methodologies,
      or enhancement).
                                                                               developed over the years.
      Transmitter in communication link.
                                                                                 Fourier analysis, Laplace trnasformation, z-trnasformation, …
   Designing system to extract specific peace of information
                                                                                 etc.
   form the signal.
      Electrocardiogram, Radar, economic forecast, communication link, …
      etc                 Prepared by Dr Nidal Kamel                   5                               Prepared by Dr Nidal Kamel                6




                                                                                                                                                     1
 Forward                                                               Forward
   In some phenomena, signal vary continuously with time,                Over the last several decades, disciplines of CT and DT
   whereas with others, evolution is described only at discrete          signals and systems have increasingly entwined.
   points of time.                                                       The major impetus for this has come from the dramatic
     Signals in electrical circuits and mechanical systems are           advance in technology.
     continuous.                                                           High-speed digital computer
     Daily closing stock market average is discrete signal.                Integrated circuits
   Concepts and techniques associated with continuous-time and           This made it advantageous to process CT signals by
   discrete-time signal and systems are conceptually related.            representing them by time samples.



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Continuous-Time and Discrete-Time Signals                             Continuous-Time and Discrete-Time Signals
Examples                                                              Examples




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Continuous-Time and Discrete-Time Signals                             Continuous-Time and Discrete-Time Signals
Examples                                                              Signal Energy and Power

                                                                         In a broad range of phenomena, signals are related to
For continuous-time we                                                   power and energy in the physical system.
use symbol t to denote                                                   If v(t) and i(t) are the voltage and current across resistor
the independent                                                          R, then the instantaneous power is
variable.
                                                                                                                             1 2
                                                                                          p(t ) = v (t )i (t ) =               v (t )
For discrete time we use                                                                                                     R
symbol n to denote the
                                                                         Total energy expended over time interval t1 ≤ t ≤ t2 is
independent variable
                                                                                               t2                   t2   1 2
                                                                                           ∫t1
                                                                                                    p(t )dt = ∫
                                                                                                                   t1    R
                                                                                                                           v (t )dt

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                                                                                                                                             2
Continuous-Time and Discrete-Time Signals                                   Continuous-Time and Discrete-Time Signals
Signal Energy and Power                                                     Signal Energy and Power

   The average power over the time interval t1 ≤ t ≤ t2           is           In may systems, we are interested in examining power
                   1      t2               1 t2 1 2                            and energy in signals over infinite time interval.
                t2 − t1 ∫t1             t2 − t1 ∫t1 R
                             p (t )dt =               v ( t )dt
                                                                               In this case, we define the energy of x(t) as
   Energy of arbitrary complex - continuous signal over                                             ∆          T                          ∞
                                                                                                 E∞ = lim ∫ x(t ) dt = ∫ x( t ) dt
                                                                                                                           2                       2

   t1 ≤ t ≤ t2            t2    2                                                                       T → ∞ −T                          −∞

                         ∫ x(t ) dt
                                 t1
                                                                                                 E∞ = lim
                                                                                                                +N

                                                                                                                ∑ x( t )
                                                                                                                               2
                                                                                                                                   dt =
                                                                                                                                           +∞

                                                                                                                                           ∑ x (t )
                                                                                                                                                       2
                                                                                                                                                           dt
                                                                                                        N →∞
   Energy of discrete-time signal over time interval                                                           n =− N                     n = −∞

   n1 ≤ n ≤ n2 is                 2
                                 n2                                            E∞ = ∞ signal has infinite energy
                                ∑ x[n]
                               n =n1
                                                                               E∞ < ∞ signal has finite energy
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Continuous-Time and Discrete-Time Signals                                   Continuous-Time and Discrete-Time Signals
Signal Energy and Power                                                     Signal Energy and Power

   Time-averaged power of x(t) over infinite interval                               With these definitions we have three classes of signals:
                         ∆
                                                                               1.    Energy signal: Signal that has finite energy E∞ < ∞ and
                                   1           T
                                           ∫
                                                           2
                    P∞ = lim                       x(t ) dt                          thus zero averaged power
                             T →∞ 2T        −T
                                                                                                                     ∆
                         ∆             +N                                                                                          E∞
                                 1                                                                           P∞ = lim                 =0
                                       ∑N x (t ) dt
                                                2
                    P∞ = lim                                                                                             T →∞      2T
                         N →∞ 2 N + 1
                                      n=−

                                                                               2.    Power signal: Signal that has finite average power P∞ and
   P∞ = ∞ signal has infinite averaged power                                         thus infinite energy E∞ = ∞
   P∞ < ∞ signal has finite averaged power                                     3.    Signal for which neither E∞ nor P∞ are finite.


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Transformation of the Independent Variable                                  Transformation of the Independent Variable
Time-shift                                                                  Time-reversal

   Transformation of the independent variable is a central
   concept in signal and systems analysis.
   A simple and very important transformation of the
   independent variable is the time-shift.




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                                                                                                                                                                     3
Transformation of Independent Variable                           Transformation of Independent Variable
Time-scaling                                                     Example




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Transformation of Independent Variable                           Transformation of Independent Variable
Periodic Signals                                                 Periodic Signals

 Continuous-time signal x(t) has the property that there is a     Discrete-time signal x[n] is periodic with period N if
 positive value of T for which                                                          x[n] = x[ n ± N ]
                           x(t ) = x(t ± T )
                                                                  If signal is periodic with a fundamental period N (or N0),
 Periodic signal remains unchanged with time-shift of T.          then it is also periodic with periods 2N, 3N, 4N,…
 Periodic signal with T is also periodic with 2T, 3T, 4T
 T (or T0) is called the fundamental period the signal.




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Transformation of Independent Variable                           Transformation of Independent Variable
Even and Odd Signals                                             Even and Odd Signals

 In continuous time signal is even if
                         x (−t ) = x (t )
 Discrete-time signal is even if
                               x[ − n ] = x[ n ]
 The signal is odd if
                          x ( −t ) = − x ( t )
                          x[− n ] = − x[n ]



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                                                                                                                                    4
                                                                                                               Exponential and Sinusoidal Signals
 Transformation of Independent Variable                                                                        Continuous-Time Complex Exponential and Sinusoidal
 Even and Odd Signals                                                                                          Signals

   Any signal can be broken into the                                                                              Continuous-time complex exponential signal is of the form
   sum of two signals one is odd and
   one is even.                                                    Ev{x( t )} =
                                                                                  1
                                                                                  2
                                                                                    [x (t ) + x (− t )]                                   x (t ) = Ce at
   The even signal is
                                                                                                                  where C and a, in general, complex numbers.
                    1
     Ev{x (t )} =     [x (t ) + x (− t )]                                                                         If C and a are real, x(t) is called real exponential signal.
                    2
   The odd signal is                                                                                              For real exponential signal there are two types of behavior.
                    1                                                                                                If a is positive then x(t) is a growing exponential.
    Od {x (t )} =     [x (t ) − x (− t )]                                                                            If a is negative then x(t) is a decaying exponential.
                    2


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Exponential and Sinusoidal Signals                                                                              Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and Sinusoidal                                                              Continuous-Time Complex Exponential and
Signals                                                                                                         Sinusoidal Signals

                                                                                                                  Important class is obtained when a is purely imaginary.

                                                                                                                                              x(t ) = e jωot
                                                                                                                  To verify that this signal is periodic

                                                                                                                                     e jωot = e jωo (t +T )
                                                                                                                                             = e jωo t e jωoT = e jωo t
                                                                                                                  The fundamental period of x(t) is
                                                                                                                                                         2π
                                                                                                                                                 To =
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                                                                                                                                                          ωo                 28




 Exponential and Sinusoidal Signals                                                                             Exponential and Sinusoidal Signals
 Continuous-Time Complex Exponential and                                                                        Continuous-Time Complex Exponential and
 Sinusoidal Signals                                                                                             Sinusoidal Signals

   Signal closely related to periodic complex exponential is                                                      If we decrease the
   sinusoidal signal.                                                                                             magnitude of ωo we
                                                                                                                  slow       down       the
  x(t ) = A cos (ωot + φ )                                                                                        oscillation and therefore
  e jω0t = cos (ω0 t ) + j sin (ω0t )                                                                             increase the period.
                          A j Φ jω0 t A − jΦ − jω0t
  A cos (ω0t + Φ ) =        e e + e e
                          2           2
                               {
  A cos (ω0t + Φ ) = Aℜe e j (ω0 t +Φ )     }
                               {
  A sin (ω0t + Φ ) = A Im e j (ω0 t + Φ)    }
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                                                                                                                                                                                  5
Exponential and Sinusoidal Signals                                                   Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and                                              Continuous-Time Complex Exponential and
Sinusoidal Signals                                                                   Sinusoidal Signals

   General             Complex                   Exponential                             Discrete time complex
   Signals                                                                               exponential signal is
                                                                                         defined by
                          Ceat
                                                                                      x[ n] = Cα n = Ce β n where α=e β
where C = C e jθ and a = r + jw0 then
                                                                                         If C and α are real,
Ce at = C e jθ e(
                    r + jω0 ) t                j (ω0t +θ )
                                  = C e rt e                                             signal      is real
     = C e rt cos (ω0t + θ ) + j C e rt sin (ω0 t + θ )                                  exponential.


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Exponential and Sinusoidal Signals                                                   Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and                                              Continuous-Time Complex Exponential and
Sinusoidal Signals                                                                   Sinusoidal Signals

   Important       complex                                                                General complex signal                                       α >1
   exponential signal is
                                                                                  C = C e jθ and α = α e jωo then
   obtained by constraining                                                                   n                       n
                                                                                  Cα n = C α cos (ωo n + θ ) + j C α sin (ωo n + θ )
   β to imaginary.
                                                                                                                                                        α <1
            x[ n] = e jω0 n

   The signal is closely
   related sinusoidal signal.
   By Euler’s relation
e jωo n = cos ω0 n + j sin ω0 n and
                          A jΦ jωo n A − j Φ − jωo n
 Acos ( ωo n + Φ ) =        e e     + e e                                    33                                           Prepared by Dr Nidal Kamel           34

                          2          2




Exponential and Sinusoidal Signals                                                   Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and                                              Continuous-Time Complex Exponential and
Sinusoidal Signals                                                                   Sinusoidal Signals

        Two basic properties of CT exponential signal ej ωot
        from its DT counterpart ej ωon :

   1.     The larger the magnitude of ω0 the higher the frequency is.
   2.     ej ωot is periodic for any value of ω0 .

        First property doesn’t hold true for the DT exponential
        signal, because
                         e j (ω0 + 2π ) n = e j 2π n e jω0 n = e jω0 n
        Signal doesn’t have a continually increasing rate of
        oscillation as ω0 increases.
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                                                                                                                                                                    6
Exponential and Sinusoidal Signals                                                  Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and                                             Continuous-Time Complex Exponential and
Sinusoidal Signals                                                                  Sinusoidal Signals

  Next, is ejωon is periodic for any value of ω0?                                     If x[n] is periodic with fundamental period N, its fundamental
    In order ejωon is periodic with period N > 0, we must have                        frequency is 2π/N.
                   jω0 ( n + N ) )
                                                                                      Since the discrete-time exponential signal is periodic if
              e                      = e jω0 n or equivelently e jω0 N = 1 ⇒                                         2π ω0
                        ω0 m                                                                                           =
              ω0 N = 2π m, or=                                                                                       N   m
                        2π N                                                          the fundamental frequency of discrete-time exponential signal is
  Accordingly signal ejωon is periodic if ω0/2π is rational                                                               ωo
  number and not periodic otherwise.
                                                                                                                           m


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Exponential and Sinusoidal Signals
Continuous-Time Complex Exponential and                                             Unit Impulse and Step Functions
Sinusoidal Signals                                                                  Discrete-Time Unit Impulse and Unit Step

                                                                                      One of the simplest
                                                                                      discrete-time is the unit
                                                                                      impulse (or unit sample),
                                                                                      defined as
                                                                                                    0 , n ≠ 0
                                                                                           δ [n ] = 
                                                                                                    1, n = 0
                                                                                      The second basic discrete-
                                                                                      time signal is unit step,
                                                                                                    0, n < 0
                                                                                             u[n] = 
                                                                                                    1, n ≥ 0
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Unit Impulse and Step Functions                                                     Unit Impulse and Step Functions
Discrete-Time Unit Impulse and Unit Step                                            Discrete-Time Unit Impulse and Unit Step

  Relationship between unit                                                           Relationship between unit
  impulse and unit step.                                                              impulse and unit step.
    δ [n ] = u[n ] − u[n − 1]                                                                      ∞
               n                                                                         u[n] = ∑ δ [n − k ]
    u[n] =   ∑ δ [m ]
             m = −∞
                                                                                                  k =0




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                                                                                                                                                              7
Unit Impulse and Step Functions                                   Unit Impulse and Step Functions
Continuous-Time Unit Impulse and Unit Step                        Continuous-Time Unit Impulse and Unit Step

  Continuous-time step function is defined in similar way           Continuous-time unit impulse function is related to unit
  to discrete-time counterpart                                      step in a manner analogous to discrete time.
                                    0, t < 0                                                 t
                            u(t) =                                               u (t ) = ∫ δ (τ )dτ                         δ [ n] = u[n ] − u[ n − 1]
                                   1, t > 0                                                 −∞

                                                                                  Thus                                                    n

                                                                                           du(t )
                                                                                                                               u[n ] =   ∑ δ [ m]
                                                                                  δ (t ) =                                               m =−∞
                                                                                            dt



  Unit step is discontinuous at t = 0.
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Unit Impulse and Step Functions                                   Unit Impulse and Step Functions
Continuous-Time Unit Impulse and Unit Step                        Continuous-Time Unit Impulse and Unit Step

  u(t) is discontinuous at                                          δ∆(t) is a short pulse of duration
  t = 0, thus not                                                   ∆ and unit area.
  differentiable.                                                   As ∆→0, δ∆(t) becomes
  Consider u∆(t), as shown                                          narrower and higher, maintaining
  in the figure                                                     its unit area.
  Formally u(t) is the limit                                        Its limiting form is
  of u∆(t) as ∆ → 0, and
                                                                          δ(t ) = lim δ∆ (t )
                                                                                       ∆→ 0
                  du (t )
        δ ∆ (t ) = ∆                                                Since δ(t) has, no duration but
                    dt
                                                                    unit area, we adopt the graphical
                                Prepared by Dr Nidal Kamel   45
                                                                    notation, shown the Figure.                                                            46




Unit Impulse and Step Functions                                   Unit Impulse and Step Functions
Continuous-Time Unit Impulse and Unit Step                        Continuous-Time Unit Impulse and Unit Step

  More generally, a scaled                                          CT-impulse has very
  impulse kδ(t) has an area                                         important sampling
  of k, thus                                                        property.
                                                                           x(t ) = x (t )δ ∆ (t )
        t                                                                        ≈ x (0)δ ∆ (t )
    ∫−∞
            kδ(τ )dτ = kδ(t )
                                                                    Since δ(t) is the limit as
                                                                    ∆→0 of δ∆(t), it follows that
  The height of the arrow is
                                                                       x(t )δ (t ) = x(0)δ (t )
  proportional to the area of
                                                                    By the same argument, we
  impulse.                                                          have
                                                                      x(t )δ (t − t 0 ) = x(t 0 )δ (t − t 0 )
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                                                                                                                                                                8
                                                                                        Continuous-Time and Discrete-Time Systems
Continuous-Time and Discrete-Time Systems                                               Simple Examples of Systems

   Continuous-time       (CT)                                   x (t ) → y ( t )
   systems,         transform                                                                      v s ( t ) − vc ( t )
                                                                                          i (t ) =
   continuous input signals                                                                                  R
   into continuous outputs.                                                                            dvc (t )
                                                                                          i (t ) = C
                                                                  x[ n] → y[ n ]                           dt
       x (t ) → y (t )                                                                    dvc (t )         1               1
                                                                                                    +          vc ( t ) =    v s (t )
                                                                                              dt         RC               RC
   Discrete-time          (DT)
   systems,          transform
   discrete    inputs      into
   discrete outputs.
      x[ n] → y[ n]                Prepared by Dr Nidal Kamel                      49                                          Prepared by Dr Nidal Kamel   50




Continuous-Time and Discrete-Time Systems                                               Continuous-Time and Discrete-Time Systems
Interconnection of Systems                                                              Interconnection of Systems




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 Basic System Properties                                                                 Basic System Properties
 System with and without Memory                                                          System with and without Memory

   The system is memoryless if its output for each value of                                Voltage across capacitor in RC circuit is an example of CT
   the independent variable at any given time is dependent                                 system with memory:
   on the input at only the same time.                                                                                                  1
                                                                                                                                            t

                              y [n ] = 2 x[ n ] + x 2 [n ]
                                                                                                                           v( t ) =
                                                                                                                                        C   ∫ i(τ )dτ
                                                                                                                                            −∞
                              y ( t ) = Rx ( t )
   Examples of DT system with memory:                                                      Concept of memory corresponds to the presence of a
                                                                                           mechanism in the system that stores information about
                                      n
                                                                                           input at times other than the current time.
                         y[ n] =    ∑ x[k]
                                   k = −∞                                                      capacitor in RC circuit
                         y[ n] = x[ n − 1] + x[ n]                                             kinetic energy in automobile
                         y[ n] = x[ n + 1] + x[n ] + x[ n − 1]                                 memory and registers in digital computers.
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                                                                                                                                                                 9
Basic System Properties                                                   Basic System Properties
Invertibility and Inverse Systems                                         Causality

 A system is invertible                                                    System is causal if the output at any time depends on
 if distinct inputs lead to                                                values of the input at only the present and past times.
 distinct output.                                                            Voltage across capacitor in RC circuit ( running integral).
 The        system        is                                                 Velocity of automobile as a function of fuel flow (driver action).
 invertible if the inverse                                                 Causal system is also called nonanticipative system.
 system exists.                                                                                         n

 Cascading the inverse                                                                       y[n] =   ∑ x[k ]
                                                                                                      k = −∞
                                                                                                                                            (Causal)

 system        with     the                                                                  y[n] = x[ n − 1] + x[ n]                       (Causal)
 original system, yields                                                                     y[n] = x[ n + 1] + x[n] + x[ n − 1]            (Noncausal)
 output equal to the                                                                         y[n] = x[-n]                                    (?)
 input.                                                                                             1 t
                                                                                             y(t) =
                                                                                                    C −∞∫  x(τ ) dτ                           (Causal)
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                                                                                             y(t) = 2.6 x(t Prepared by Dr Nidal Kamel
                                                                                                            + 1)                            (Noncausal)   56




Basic System Properties                                                   Basic System Properties
Stability                                                                 Stability

  Informally, a stable system                                              Consider applying force
  is the one in which small                                                f(t)=F to the automobile,
  inputs lead to responses                                                 with the vehicle initially
  that do not diverge.                                                     at rest.
  In     different     words,                                              The velocity will increase,
  unstable system is the one                                               but not without bound.
  in which responses grow                                                  The bound when the
  without bound in response                                                frictional forces exactly
  to small inputs.                                                         balance the applied force,
                                                                           i.e.
                                                                                 ρ         1        F
                                                                                     V =     F ⇒V =
                            Prepared by Dr Nidal Kamel               57          m         m        ρPrepared by Dr Nidal Kamel                           58




Basic System Properties                                                   Basic System Properties
Stability                                                                 Time Invariance

  A useful strategy to test the stability of a system is to                A system is time invariant if the behavior and
  look for a specific bounded input that leads to                          characteristics are fixed over time.
  unbounded output.                                                          RC circuit with constant values for R and C over time.
  To Check the stability of the system S1 : y (t ) = tx(t ) we               constantly loaded auto’s trunk.
  consider a simple bounded input like unit step ( x(t) = u(t) ).          In systems language, system is time invariant if time shift in
  Accordingly this system is unstable
                                                                           input signal results in identical time shift in the output.
  To check the stability of the system S 2 : y (t ) = e x (t ) we
  consider abounded input by B. Thus –B < x(t) < B for all t. Then           In DT systems, we express this as, if y[n] is the output when x[n] is
  y(t) must satisfy                                                          the input then y[n-n0] is the output when x[n-n0] is applied.
                         e − B < y (t ) < e B                                In CT systems, we express this as, if y(t) is the output when x(t) is
                                                                             the input then y(t-t0) is the output when x(t-t0) is applied.
  Accordingly, this system is stable.

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                                                                                                                                                               10
Basic System Properties                                                   Basic System Properties
Time Invariance/Example                                                   Time Invariance

 Check whether the system y(t) = sin[x(t)] is time invariant or             Check whether the system y[n] = n x[n] is time invariant
 not.                                                                       or not.
 We must determine whether the time-invariance property holds
 for any input and any time shift t0.                                       Consider input signal x1[n] = δ[n] which yields output y1[n]=0
                                                                            (since nδ[n] = 0) .
 Let x1(t) be an arbitrary input, and let y1(t) = sin[x1(t)].
                                                                            However the input x2[n] = δ[n-1] yields output
 Then consider second input obtained by shifting x1(t) in time:
                                                                            y2[n] = nδ[n-1]= δ[n-1], which is not a shifted version of y1(t).
 x2(t) = x1(t-t0).
                                                                            Thus, the system is time-variant.
 The output corresponding to this input is y2(t) = sin[x1(t-t0)].
 Since y2(t)= y1(t-t0) the system is time invariant.


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Basic System Properties                                                   Basic System Properties
Linearity                                                                 Linearity/Example

 Linear system, in continuous or discrete time, is a system                Consider a system S whose input and output are related as
 that possesses the property of superposition.                                                          y (t ) = tx(t )
 More precisely, let y1(t) be the response of CT system to                 determine whether the system is linear or not.
 input x1(t) and y2(t) to input x2(t) Then the system is
                                                                           let x1(t) and x2(t) be a two arbitrary inputs with their outputs
 linear if:
                                                                           y1(t) = t x1(t) and y2(t) = t x2(t), respectively.
   The response to x1(t) + x2(t) is y1(t) + y2(t) [additive property].
   The response to ax1(t) is ay1(t), where a is any complex constant.      Let x3(t) a linear combination of the two inputs
   [scaling property]                                                                              x3 (t ) = ax1 (t ) + bx2 (t )
 The two properties can be combined into single statement                  If x3(t) is the input of S, then the output is
   CT: ax1(t) + bx2(t)+… → ay1(t) + by2(t)+….                                                      y3 (t ) = tx3 (t )
   DT: ax1[n]+ bx2[n]+… → ay1[n] + by2[n]+….                                                              = t ( ax1 (t ) + bx2 (t ) )
                                                                                                          = atx1 (t ) + btx2 (t )
                          Prepared by Dr Nidal Kamel                 63                              Prepared by Dr Nidal Kamel               64
                                                                                                          = ay1 (t ) + by2 (t )




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posted:4/15/2012
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Description: This article show the introduction of signal and system in our life. How the concept of Signal and System will be applied in our life such as communication, circuit design, control, speech processing, image processing and others.