Lecture 2
Signals and Systems (II)
Principles of Communications
Fall 2008
NCTU EE Tzu-Hsien Sang
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Outlines
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Signal Models & Classifications
Signal Space & Orthogonal Basis
Fourier Series &Transform
Power Spectral Density & Correlation
Signals & Linear Systems
Sampling Theory
DFT & FFT
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Examples
• Symmetry Properties of x(t) and Its Fourier
Function
*
X

X
For real periodic x(t), n
n
For real aperiodic x(t), X ( f )  X * ( f )
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• Fourier Transform of Singular Functions
 (t ) is not an energy signal (hence doesn’t
satisfy Dirichlet condition).
However, its FT can be obtained by formal
definition.
FT
FT
 (t ) 1, 1  ( f ),
 j 2f0
A (t  t0 )  Ae
FT
• Example: The FT of
, Ae
jf0t
FT

A ( f  f 0 ),

  (t  nT ) ?
n  
0
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• Fourier Transform of Periodic Signals—
Periodic signals are not energy signals (don’t
satisfy Dirichlet’s conditions). But we are
doing it anyway (at least formally)…
• Given a periodic signal


jn 0t  X ( f ) 
 X n ( f  nf0 )
x(t )   X n e
n
n
2f 0t
• Example-1: cos

• Example-2:   (t  nT0 )
n  
(A pulse train! What good are they for?)
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Note: This table uses “” instead of “f”. But it doesn’t hurt the fundamental facts.
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Transform Pairs (There is something nice
to know in life…)
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• Let FT of an aperiodic pulse signal p(t) be
{ p(t )}  P( f )
• We can generate a periodic signal x(t) by
duplicating p(t) at every interval Ts, then

x(t )  [

 (t  nT )] * p(t )   p(t  nT )
s
s
n
n
• From convolution
theorem,

X ( f )  {[
 (t  nT )]} P( f )
s
n

 fs

 ( f  nf )  P( f )   f P(nf ) ( f  nf )
s
n
s
n
s
s
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Taking inverse FT of the eq. on previous page.
1
 { X ( f )}  x(t ) 

s
n  


1
f
P
(
nf
)

 s s { ( f  nfs )} 
n 

 p(t  nTs ) 
n  

 p(t  nT )   {  f P(nf ) ( f  nf )}
1
n 
s
s
s

j 2nf st
f
P
(
nf
)
e
 s s
n 

j 2nf s t
f
P
(
nf
)
e
 s s
n  
Poisson sum formula
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Power Spectral Density & Correlation
• Why should we care about the “frequency
components” of a signal?
• For energy signals:
 ( )  1{G( f )}  1[ X ( f ) X * ( f )]  1[ X ( f )]  1[ X * ( f )]

T
 x( )  x( )   x( ) x(   )d  lim  x( ) x(   )d

T  T
• The time-averaged autocorrelation function
• The squared magnitude of the FT represents
the “energy” distributed on the frequency axis.
 (0)  E  signal energy.
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• For power signals:
R( )  x( ) x* (t   )
1 T

*
lim
x(t ) x (t   )dt, if aperiodicpowersignal

T  2T T
 1
*
if periodicpowersignal
x(t ) x (t   )dt,


 T0 T0



R(0)   S ( f )df
S ( f )  {R( )}

“Power spectral density function”
• For periodic power signals:
S ( f )  {R( )} 


| X n |2  ( f  nf0 )
n
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• The functions () and R() measure the
similarity between the signal at time t and t+.
• G(f) and S(f) represents the signal energy or
power per unit frequency at freq. f.
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R
(
0
)

power

x
(t )  R( ) ,  , max{R( )}  R(0).
•
• R() is even for real x(t): R( )  x(t ) x* (t   )  R( ).
• If x(t) does not contain a periodic component:
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lim R( )  x(t ) .
| |
• If x(t) is periodic with period T0, then R() is
periodic in  with the same period.
• S(f) is non-negative. S ( f )  {R( )}  0, f
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• Cross-correlation of two power signals:
Rxy ( )  x(t ) y * (t   )  x(t   ) y * (t )
1
 lim
T  2T
T

T
x(t ) y * (t   )dt
• Cross-correlation of two energy signals:
 xy ( ) 



x(t ) y* (t   )dt
• Remarks: Rxy ( )  R ( ),  xy ( )
*
yx
*
  yx ( )
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Signals & Linear Systems
x (t )
Η
y (t )
y (t )  H {x(t )}
• The standard input/output black box model for
linear systems. Q: Why does it work?
• Linear: Satisfies superposition principle
y(t )  H{1 x1 (t )   2 x2 (t )}
 H{1 x1 (t )} H{ 2 x2 (t )}  y1 (t )  y2 (t )
• Time-invariant: Delayed input produces an
output with the same delay.
H {x(t  t0 )}  y(t  t0 )
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Describing LTI Systems with Impulse Responses
• Let h(t) be the impulse response: h(t )  H { (t )}.
x(t ) 



x( ) (t   )d

y(t )  H {x(t )}  H { x( ) (t   )d}


  x( )H { (t   )}d

If time-invariant,

y(t )  H {x(t )}   x( )h(t   )d  x(t ) * h(t )


H {x(t  t0 )}   x( )h(t  t0   )d  y(t  t0 )

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Note: This example is a linear, but not time-invariant system.
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• The convolution form holds iff LTI.
• Duality of signal x(t) & system h(t):
y(t ) 



x( )h(t   )d 



h( )x(t   )d
• The Convolution Theorem:


{ y(t )}  Y ( f ){

h( )x(t   )d}  H ( f ) X ( f )
Key application: generally H ( f ) X ( f ) is easier
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than x(t )  h(t ) …