Transcript Signals and Systems - National Chiao Tung University

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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More on LTI Systems
• A system is BIBO if output is bounded, given
any bounded input. 
max{| y (t ) |}  max{|  h( )x(t   )d |}





 max{| x(t ) |} | h( ) |d     | h( ) |d  
 main elementof Dirichletcondition
• A system is causal if: current output does not
depend on future input; or current input does
not contribute to the output in the past.
y (t ) 



h( )x(t   )d 
 h(t )  0, for t  0

 h( )x(t   )d
0
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• Paley-Wiener Condition:

If  H ( f ) df  , and h(t )  0 for t  ,
2


ln H(f)

1 f

2
df  .
• Remarks: (1) |H(f)| cannot grow too fast.
(2) |H(f)| cannot be exactly zero over a finite
band of frequency.

 ln H(f)
2
nd
• 2 ver.: If  H ( f ) df   and 
df  ,
2

 1  f
 H such thatH ( f ) is causal (h(t )  0 for t  ).
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Eigenfunctions of LTI Systems
• Another way of taking complicated things part.
• Instead of trying to find a set of orthogonal
basis functions, let’s look for signals that will
not be changed “fundamentally” when passing
them through an LTI system.
• Why?
• Consider the key words: analysis/synthesis.
• Note: Eigen-analysis is not necessarily
consistent with orthogonal basis analysis.
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• If Η{g (t )}  ag (t ), where a is a constant, then a
is the eigenvalue for the eigenfunction g(t).
• Let x(t )  Aest , s : an arbitrarycomplexnumber

y(t )   h( )Ae
s ( t  )


d  [  h( )e
 s

d ] Ae
st

 ax(t ), wherea   h( )Ae s d.
Let s  j 2f i t ,


then y(t )  [  h( )e

 j 2f i 
d ] Ae
j 2f i t
 ax(t ).
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• (Cross)correlation functions related by LTI
systems:

1. R yx ( )  h( )  Rx ( )   h( ) Rx (   )d

2. R y ( )  h ( )  h( )  Rx ( )
*
3. S yx ( f )  H ( f )  S x ( f )
4. S y ( f ) | H ( f ) | S x ( f )
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Filters!!!
• Note: In proving them, we use:
*
*
{h( )}  H ( f )
{h ( )}  H ( f )
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• Since almost any input x(t) can be represented
by a linear combination of orthogonal
sinusoidal basis functions e j 2ft , we only need
to input Ae j 2ft to the system to characterize
the system’s properties, and the eigenvalue
a



h(t )e  j 2ft dt  H ( f )
carries all the system information responding
to Ae j 2ft . (Frequency response!!!)
• In communications, signal distortion is of
primary concern in high-quality transmission
of data. Hence, the transmission channel is the
key investigation target.
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• Three major types of distortion caused by a
transmission channel:
1. Amplitude distortion: linear system but the
amplitude response is not constant.
2. Phase (delay) distortion: linear system but the
phase shift is not a linear function of
frequency. (Q: What good is linear phase?)
3. Nonlinear distortion: nonlinear system
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• Example: Group Delay
1 d ( f )
Definition: Tg ( f )  
,  ( f )  H ( f ).
2 df
For a linear system, ( f )  H 0  2ft0
 Tg ( f )  t0 , which is a constant.
If thesystemis not linear phase  Tg ( f ) is not
a constantdifferentfrequencycomponentshave
differentdelays.
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( f )
Phasedelay : Tp ( f )  
.
2f
It shows therelativephase relationsbetween the input
and theoutput frequencycomponents
.
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• Example: Ideal general filters
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• Realizable filters approximating ideal filters
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The Uncertainty Principle
• It can be argued that a narrow time signal has a
wide (frequency) bandwidth, and vice versa:
(duration)  (bandwidth)  constant
By theequal - area argument(thisis not a proof),




Tx(0)   | x(t ) | dt   x(t )dt  X ( f ) | f 0 ,




and 2WX (0)   | X ( f ) | df   X ( f )df  x(0)
 2W 
x(0) 1
1
  TW 
X (0) T
2
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Sampling Theory
• You’ve probably heard of “signal processing.” But
how to process a signal?
• For instance, the rectifier – max{x(t), 0}.
• But, how to do Fourier transform of an arbitrary
signal x(t)?
• Computers seem a good idea. But computers can only
work on numbers.
• We need to “transform” the signal first into numbers.
• Q: Tell discrete signals from digital signals.
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Hopefully the math becomes easier in ideal case. The concept actually is harder.
• Ideal sampling signal: impulse train (an analog

signal) st     t  nT  , T: the sampling period
n  
• Analog (continuous-time) signal: x(t )
• Sampled (continuous-time) signal: x (t )

x t   xt st   xt    t  nTs 
n  



 xt  t  nT    xnT  t  nT 
n  
s
n  
X   f   X ( f )  S  f   X  f [ f s
 fs
s
s

   f  kf ]
s
k  


 X  f    f  kf   f  X  f  kf 
k  
s
s
k  
s
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• Aliasing: If f s  2W . The replicas of X(f)
overlap in frequency domain. That is, the
higher frequency components of overlap with
the lower frequency components of X(f-fs).
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– Nyquist Sampling Theorem:
• Let x(t) be a bandlimited signal with X(f) = 0
for | f | W . (i.e., no components at frequencies
greater than W.) Then x(t) is uniquely
determined by its samples x[n]  x(nTs ), n  0,1,2,
1
if f s   2W.
Ts
Undersampling: f s  2W
Oversampling: f s  2W
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• In other words, oversampling preserves all the
information that x(t) contains. It is possible to
reconstruct x(t) purely by its samples.
• Ideal reconstruction filter (interpretation in
frequency domain:
f  j 2ft0
H ( f )  H 0  ( )e
,
W  B  fs W
2B
 j 2ft0
 Y ( f )  f s H 0 X ( f )e
 y (t )  f s H 0 x(t  t 0 )
• In time domain:
y(t ) 


 x(nT )h(t  nT )  2BH  x(nT )sinc[2B(t  t
n 
s
s
0
n 
s
0
 nTs )]
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• Two types of reconstruction errors
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DFT & FFT
• You can view DFT as a totally new definition
for a totally different set of signals. Or you can
try to connect it to the Fourier Transform.
1
xn 
N
N 1
X
k 0
N 1
X k   xn e
k
j
e
j
2nk
N
2nk
N
,
,
n  0,1,  , N  1
k  0,1,  , N  1
n 0
H (e j )  H ( z ) | z e j
H (k )  H (e j ) |

N 1
2k 
N

n 0
h(n)e  j 2nk / N 
N 1

n 0
h(n)WNnk
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• Fast Fourier Transform (FFT) is not a new
transform, it is simply a fast way to compute
DFT. So, don’t use FFT to denote the object
that you want to compute; only use it to denote
the tool that you use to compute it. (Gauss
knew the method already!)
• Application example: Fast convolution via
N 1
N 1
FFT:
1
1
nk
nk
y(n) 
Y (k )W

N
N
k 0

H (k ) X (k )W

N
N
k 0
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