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					                               Linear Systems
 A system is a relation between input and output signal.

                      r(t)                               s(t)
                                      System



                                  r (t )  s(t )

   Linearity property:

                    ar1 (t )  br2 (t )  as1 (t )  bs2 (t )
   Time invariant
                             r (t  t0 )  s (t  t0 )


KMUTT: S. Srakaew
                             Linear Systems
        A linear time-invariant system is a system that has linearity
 and time-invariant properties.
             Let h(t) be the impulse response

                     (t )          Linear              h(t)
                                    System


                                (t )  h(t )
  We can show that:
                               s(t )  r (t )  h(t )
                             S ( f )  R( f ) H ( f )
                                       S( f )
   Transfer function:        H( f ) 
                                       R( f )

KMUTT: S. Srakaew
                              Linear Systems
   Time Response of a First-Order System:
                                  R
                      +      + v (t ) -                    +
                                R
                    v (t )                C               vC (t )
                      -                                    -


                             dv c (t )
                       RC               v c (t )  v (t )
                               dt
   Step Response:
                                                t
                     g (t )  (1  e                 RC
                                                          )u (t )

   Impulse Response:
                                 1        t
                       h (t )     e           RC
                                                    u (t )
                                RC
KMUTT: S. Srakaew
                        Linear Systems
   Time Response of a First-Order System:




KMUTT: S. Srakaew
                        Linear Systems
   Time Response of a First-Order System:




KMUTT: S. Srakaew
                                 Linear Systems
   Complex Transfer Function:
                                     R
                          +      + v (t ) -        +
                                    R
                        v (t )                C   vC (t )
                          -                        -


                                  VC ( f )         XC
                    H( f )                     
                             VR ( f )  VC ( f ) R  X C

                                   1
                                   j 2fC            1
                                            
                                R 1           1  j 2fRC
                                      j 2fC


KMUTT: S. Srakaew
                        Linear Systems
   Frequency Response of a First-Order System:




                                                1
                                           B
                                              2RC




KMUTT: S. Srakaew
                       Linear Systems
   Time Operations and Transfer Functions:




KMUTT: S. Srakaew
                       Linear Systems
   Block-Diagram Analysis:




KMUTT: S. Srakaew
                    Distorting Transmission
   Distortionless Transmission

                    x(t)           Comm.             y(t)
                                   Channel



                           y (t )  Kx (t  t d )

   Transfer Function
                                         jt d
                       H ( f )  Ke
                       H( f )  K                 constant amplitude response

                    arg H ( f )  2t d f  m180           linear phase shift


KMUTT: S. Srakaew
                     Distorting Transmission
   Linear Distortion

                •Amplitude Distortion

                                    H( f )  K

                •Delay Distortion

                          arg H ( f )  2t d f  m180 

   Nonlinear Distortion




KMUTT: S. Srakaew
                    Distorting Transmission
   Linear Distortion: Example

                                1          1
               x(t )  cos 0t  cos3 0t  cos5 0t
                                3          5




KMUTT: S. Srakaew
                    Distorting Transmission
   Linear Distortion: Example (continued)

   Test signal with amplitude distortion (a) low frequency
   attenuated; (b) high frequency attenuated




         1          1                                1
x(t )   cos3 0t  cos5 0t       x(t )  cos 0t  cos3 0t
         3          5                                3
KMUTT: S. Srakaew
                    Distorting Transmission
   Linear Distortion: Example (continued)
   Test signal with constant phase shift   90




KMUTT: S. Srakaew
                      Distorting Transmission
   Nonlinear Distortion:
   Polynomial approximation of y(t)

                    y (t )  a1 x(t )  a2 x 2 (t )  a3 x 3 (t )  ...

      Y ( f )  a1 X ( f )  a2 X  X ( f )  a3 X  X  X ( f )  ...




                     Transfer Characteristic of Nonlinear device
KMUTT: S. Srakaew
              Transmission Loss and Decibels
   Power Gain:
                          Pout
                       g
                          Pin
   Power Gain in dB:

                                        Pout   
                       g dB    10 log 
                                       P       
                                                
                                        in     


                    PoutdB  g dB  PindB


KMUTT: S. Srakaew
              Transmission Loss and Decibels
   Typical values of transmission loss:




KMUTT: S. Srakaew
                              Linear Systems
   Filters:
                                                                        H( f )
   Ideal Distortionless Filter:                                K


                                                                                         f
                                                                       arg H ( f )
                       j 2ft 0
     H ( f )  Ke                      Slope  2f 0
                                                                                         f



                                                                         H( f )
   Ideal Lowpass Filter (LPF):                                     K


                                                                                     f
                                                         fm
               Ke  j 2ft 0
                                  , f  fm
                                                                              fm

     H( f )  
               0,
                                   f  fm                             arg H ( f )


                                                                                     f
                                                         fm                  fm


KMUTT: S. Srakaew
                                         Linear Systems
   Ideal Bandpass Filter (BPF):

          Ke  j 2ft 0                 , f L  f  fU
 H( f )  
           0,                              elsewhere

                       H( f )                                                  H lp ( f )
                                                                        K
                  K

                                           f                        f L  fU    f L  fU    f
    fU     fL             fL      fU                                  2           2




                      arg H ( f )
                                                                   f L  fU              f L  fU 
                                               H ( f )  H lp  f             H lp  f           
    fU     fL              fL     fU
                                           f                           2                     2 




KMUTT: S. Srakaew
                               Linear Systems
   First-Order Lowpass Filter (LPF):
                           R
                  +                     +
                    vin (t )      C    vout (t )
                      -                 -

                      1 / j 2fC            1
           H( f )                  
                     R 1             1  j 2fRC
                             j 2fC
                             1
          H( f )                                 Cutoff frequency, fc
                      1  (2fRC)   2

                                                             1
         arg H ( f )   tan (2fRC )
                              1                     fc 
                                                           2RC

KMUTT: S. Srakaew
                          Linear Systems
If we set RC  1/ 2 , cutoff frequency = 1 Hz. The magnitude and phase

of H(f) are shown below




KMUTT: S. Srakaew
                                          Linear Systems
   Butterworth Lowpass Filter:

                                                1                    1
                    Hn ( f )                              
                                          1  (2f ) 2 n       1  ( f / B) 2 n

   Cutoff frequency, B
                                            1
                                      B
                                           2

                         H lp ( f )
                     1



                                                    f
          -B                          B




KMUTT: S. Srakaew
                       Linear Systems
   Second-Order Butterworth Lowpass Filter:




KMUTT: S. Srakaew
                               Linear Systems
   Bandpass Filter (BPF):
                                R
                      +                                 +
                    vin (t )          L        C       vout (t )
                      -                                 -


                                       j 2fL
                    H( f ) 
                             R  (2f ) 2 RLC  j 2fL
                                                   1
                    H( f ) 
                                    R 2 [1 /(2fL)  2fC]2  1
   Resonance frequency, f0
                                1          1
                          f0 
                               2         LC
KMUTT: S. Srakaew
                                    Linear Systems
                                              vin(t)
   Example:
                                             1




                     -2T       -T            0         T          2T
                                                                            t



                                            R
                               +                            +
                            vin (t )              C        vout (t )
                               -                            -
                                            1
                                        j 2fRC            j 2fRC
                    H( f )                           
                               R 1                     1  j 2fRC
                                           j 2fC
                                    1
           H( f )                                         arg H ( f )   tan 1 (2fRC )
                           1  (2fRC) 2
KMUTT: S. Srakaew
                                  Linear Systems
                                  1
                            cn        sin(n / 2)
                                 nf 0
                            c0 = 1/2
Let f = 1, and RC = 1

       n            0       1        2     3         4     5      6      7

      |cn|          0.5    1/       0   1 / 3      0   1 / 5   0   1 / 7

     arg cn                 0             180            0           180

     H (nf )        1     0.1572         0.0530          0.0318       0.0227

  arg H (nf )              81           87            88         89


KMUTT: S. Srakaew
                                  Linear Systems
   Frequency Spectrum: (Vin)




                                        Vin(f)
                                        0.5
                                   1              1
                                                 


    1                                                           1
   5                                                          5
                                        0
                                                                        f
     -5                      -3    -1         1       3             5
                         1                                 1
                                                     
                        3                                3




KMUTT: S. Srakaew
                                            Linear Systems
                                       Fourier Series of Input Signal Vin(t) : n=15
                    1.2


                      1


                    0.8
    Input Vin (V)




                    0.6


                    0.4


                    0.2


                      0


                    -0.2
                        -1   -0.8   -0.6   -0.4   -0.2    0     0.2        0.4        0.6   0.8   1
                                                    Frequency (Hz)

KMUTT: S. Srakaew
                                               Linear Systems
                                         Fourier Series of Output Signal Vout(t) : n=15
                           2



                         1.5



                           1
        Input Vout (V)




                         0.5



                           0



                         -0.5



                          -1
                            -1   -0.8   -0.6   -0.4   -0.2    0     0.2      0.4     0.6   0.8   1
                                                        Frequency (Hz)

KMUTT: S. Srakaew
              Correlation and Spectral Density
   Average Power:

                    Pv  v(t )        v(t )v * (t )  0
                                 2




   Schwarz’s Inequality:
                                          2
                         v(t ) w * (t )        Pv Pw

   where v(t) and w(t) are power signals




KMUTT: S. Srakaew
              Correlation and Spectral Density
   Crosscorrelation of two power signals:

             Rvw ( )  v(t )w * (t   )  v(t   )w * (t )

                           Rvw ( )  Pv Pw
                                    2



                          Rwv ( )  Rvw (  )
                                      



                           Rwv ( )  Rvw ( )




KMUTT: S. Srakaew
              Correlation and Spectral Density
   Autocorrelation:

       Rv ( )  Rvv ( )  v(t )v * (t   )  v(t   )v * (t )

                            Rv (0)  Pv
                            Rv ( )  Rv (0)
                           Rv (  )  R ( )
                                          
                                          v




KMUTT: S. Srakaew
              Correlation and Spectral Density
   Total Energy:
                              
                       Ev   v(t )v  (t )dt  0
                             


   Correlation of energy signals:
                                  
                    Rvw ( )   v(t ) w (t   )dt
                                         
                               


                    Rv ( )  Rvv ( )




KMUTT: S. Srakaew
              Correlation and Spectral Density
   Spectral Density Function:

        Gv ( f ) is the distribution of power or energy in frequency domain
                                
                     Rv (0)   Gv ( f )df
                                


                    Gy ( f )  H ( f ) Gx ( f )
                                       2




                      x(t)                        y(t)
                                     LTI




KMUTT: S. Srakaew
              Correlation and Spectral Density
   Wiener-Kinchine Theorem:


                Gv ( f )  F Rv ( )   Rv ( )e  j 2f d
                                            

                                            


   Energy Spectral Density of Energy Signal v(t):

                             Gv ( f )  V ( f )
                                                      2



   Power Spectral Density of Periodic Signal v(t):
                                     
                         v(t )     c(nf )e
                                   n  
                                            0
                                                     j 2nf 0t



                                     
                     Gv ( f )      c(nf )           ( f  nf 0 )
                                                 2
                                            0
                                   n  
KMUTT: S. Srakaew
                    Assignment 2
                               Due: June 26, 2003




KMUTT: S. Srakaew

				
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