# Fourier Series

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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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