Transcript Chapter 2

Chapter 2 Discrete-time signals
and systems
2.1
Discrete-time signals:sequences
2.2
Discrete-time system
2.3
Frequency-domain representation of
discrete-time signal and system
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2.1 Discrete-time signals:sequences
2.1.1 Definition
2.1.2 Classification of sequence
2.1.3 Basic sequences
2.1.4 Period of sequence
2.1.5 Symmetry of sequence
2.1.6 Energy of sequence
2.1.7 The basic operations of sequences
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2.1.1 Definition
x  x[n]
EXAMPLE
   n  
Enumerative
representation
x[n]  {1,2,1.2,0,1,2,2.5},1  n  5
x[n]  0.9 cos(0.2n   / 2),0  n  10
n
Function
representation
3
2
0.5
1
0
0
-1
-0.5
-2
-3
-2
-1
0
2
4
6
0
5
10
Graphical
representation
4
Generate and plot the
sequence in MATLAB
n=-1:5
x=[1,2,1.2,0,-1,-2,-2.5]
stem(n,x, '.')
n=0:9
y=0.9.^n.*cos(0.2*pi*n+pi/2)
stem(n,y,'.')
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x[n]  xa (t ) |t nT  xa (nT )
EXAMPLE
Sampling the
analog
waveform
Figure 2.2
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Display the wav speech signal in ULTRAEDIT
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Display the wav speech signal in COOLEDIT
The whole
waveform
Display the wav speech signal in
local Blowup
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2.1.2 Classification of sequence
Right-side
x[n]  0, for    n  N
Left-side
x[n]  0, for N  n  
Two-side
x[n]  0, for    n  
Finite-length
x[n]  0, outside of N1  n  N2
Causal
x[n]  0, for
n0
Noncausal
x[n]  0, for n  0
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2.1.3 Basic sequences
1. Unit sample sequence
1
 [ n]  
0
n0
1
n0
n
0
2．The unit step sequence
1
u[n]  
0
δ[n]
n0
u[n]
1
n0
n
0
3．The rectangular sequence
1
RN [ n ]  
0
0  n  N 1
other
R[n]
1
10
0
N-1
n
4. Exponential sequence
x[n]  a n
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x[n]  a n  (re j ) n
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5. Sinusoidal sequence
x[n]  A cos(n  )
 : frequency, radians / sample

 cos(0.9n)
 cos(1.8n)
x[n]  A cos(n  )  A cos(2n  (n  ))  A cos((2   )n  )
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x[n]  A cos(n  )  A cos((  2k )n  )
For convenience, sinusoidal signals are usually expressed by exponential sequences.




A j (n   )
A sin n    
e
 e  j (n   )
2j
A j (n   )
A cosn    
e
 e  j (n   )
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The relationship between ω and Ω：
 x[n]  A sin( n  )
 xc (t ) |t  nT  A sin( t  ) |t  nT  A sin( nT  )
  T   / f s
f   /( 2 )
unit :
f : Hz( period / sec ond )
 : radians / sec ond
 : radians / sample
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2.1.4 Period of sequence
if x[n]  x[n  N ],  n  , then period is N
x(t )  A sin(t  )  A sin(t    2 )
 A sin((t  2 / ） )
 T  2 / 
x[n]  A sin(n  
) A sin(n    2l )
 A sin((n  2l /  )  )
 x[n  2 l /  ]
integer N，period  N

2 /   rational number P / Q，period  P
irrational number ，period  

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Three kinds of period of sequence
2 /   N
2 /   P / Q，N  P
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2.1.5 Symmetry of sequence
x[n]  x[n], even sequence
x[n]   x[n], odd sequence
x[n]  x*[n] Conjugate-symmetric sequence
x[n]   x*[n] Conjugate-antisymmetric sequence
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x[n]  xe [n]  xo [n]
xe [n]  xe [n]
x e [ n] 
x[n]  x[n]
2
xo [n]   xo [n]
x o [ n] 
x[n]  x[n]
2
x[n]  xe [n]  xo [n]
xe[n]  xe*[n]
x[n]  x * [n]
x e [ n] 
2
xo[n]   xo*[n]
x[n]  x * [n]
x o [ n] 
2
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EXAMPLE
x[n]  (n  1) R6[n]
n=[-5:5];
x=[0,0,0,0,0,1,2,3,4,5,6];
xe=(x+fliplr(x))/2;
xo=(x-fliplr(x))/2;
subplot(3,1,1)
stem(n,x)
subplot(3,1,2)
stem(n,xe)
subplot(3,1,3)
stem(n,xo)
Real sequences can be
decomposed into two
symmetrical sequences.
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EXAMPLE
x[n]  (1  j ) R6[n]
n
Complex sequences can be decomposed into two
symmetrical sequences.
n=[-5:5];
x=zeros(1,11);
2
x((n>=0)&(n<=5))=(1+j).^[0:5]0
xe=(x+conj(fliplr(x)))/2;
-2
xo=(x-conj(fliplr(x)))/2
-4
subplot(3,2,1);
-5
stem(n,real(x))
2
subplot(3,2,2);
stem(n,imag(x))
0
subplot(3,2,3);
-2
stem(n,real(xe))
-5
subplot(3,2,4);
2
stem(n,imag(xe))
subplot(3,2,5);
0
stem(n,real(xo))
subplot(3,2,6);
-2
-5
stem(n,imag(xo))
2
0
-2
0
5
-4
-5
2
0
5
0
5
0
5
0
0
5
-2
-5
1
0
-1
0
5
-2
-5
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2.1.6 Energy of sequence

E


| x[n] |2 
n  

x[n]x*[n]
n  
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2.1.7 The basic operations of sequences
1. y[n]  x[n  n0 ]
2. y[n]  x[n]
3. y[n]  a  x[n]
4. y[n]  x[n]  w[n]
5. y[n]  a  x[n]
6. y[n]  x[n]  w[n]
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Basic operations of sequences
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Original speech
sequences
sequences after
vector addition
sequences after vector
multiplication
Original music
sequence
sequences after scalar
multiplication
echo
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EXAMPLE
The matlab codes on the
addition of two sequences
x[n]  [1,2,4,6,5,8,10]
y[n]  3x[n  2]  x[n  4]
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20
0
-2 0
-1 0
-5
0
5
10
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n=[-4:2] ;
x=[1,-2,4,6,-5,8,10] ;
%x1[n]=x[n+2]
n1=n-2;
x1=x;
%x2[n]=x[n-4]
n2=n+4;
x2=x;
%y[n]
m=[min(min(n1),min(n2)): max(max(n1),max(n2))] ;
y1=zeros(1,length(m)) ;
y2=y1;
y1((m>=min(n1))&(m<=max(n1)))=x1;y2((m>=min(n2))&(m<=max(n2)))=x2;
y=3*y1+y2;
stem(m,y)
Output:y =3
-6 12 18 -15 24 31 -2
4 6 -5 8 10
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7.convolution sum:
y[n]  x[n]  h[n] 


k  
k  
 x[k ]h[n  k ]   x[n  k ]h[k ]
(1) x[n] * h[n]  h[n] * x[n]
(2) x[n] * (h1[n]  h2 [n])  x[n] * h1[n]  x[n] * h2 [n]
(3)( x[n] * h1[n]) * h2 [n]  x[n] * (h1[n] * h2 [n])
(4) x[n] *  [n] 

 x[k ] [n  k ]  x[n], x[n] * [n  n ]  x[n  n ]
k  
0
0
(5)if x[n]  0, outside N 0  n  N1 , length  L1
h[n]  0, outside N 2  n  N 3 , length  L2
then, y[n]  0, outside N 0  N 2  n  N1  N 3 ,
length  L1  L2  1
steps：turnover, shift, vector multiplication, addition
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EXAMPLE
1, n  1,...4
x[n]  0.5 u[n], h[n]  
, y[n]  x[n] * h[n]
0, other
n
nx=0:10;
nh=-1:4;
y=conv(x,h);
x=0.5.^nx;
h=ones(1,length(nh))
stem([min(nx)+min(nh):max(nx)+max(nh)],y)
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8.crosscorrelation:

 x[k ] y[k  n]  x[n]  y[n]  r
rxy [n] 
yx [  n]
k  
or,

rxy [n] 
 x[k ] y[k  n]
k  
aotocorrelation:

rxx [n] 
 x[k ]x[k  n]  x[n]  x[n]  r
xx [  n]
k  
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example：correlation detection in digital audio watermark
pseudo random sequences : w[n]  {1,1}, n  0 : 1023
x[n] : original audio sequences, n  0 : 1023
embed the watermark ：xw [n]  x(n)  0.1 | x[n] | w(n), n  0 : 1023
detect the watermark ：xw _ o[n]  high pass filter ( xw [n])
w _ o[n]  high pass filter ( w[n])
rxw _ o , w _ o , compare the maximum with the threshold
to confirm the presence of the watermark
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33
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2.1 summary
•2.1.1 Definition
•2.1.2 Classification of sequence
•2.1.3 Basic sequences
•2.1.4 Period of sequence
•2.1.5 Symmetry of sequence
•2.1.6 Energy of sequence
•2.1.7 The basic operations of sequences
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requirements：judge the period of sequence ;
calculate convolution with graphical
and analytical evaluation .
key： convolution
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2.2
Discrete-time system
2.2.1 Definition：input-output description of systems
2.2.2 Classification of discrete-time system
2.2.3 Linear time-invariant system（LTI）
2.2.4 Linear constant-coefficient difference equation
2.2.5. Direct implementation of discrete-time system
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2.2.1 definition：input-output description of systems
y[n]  T [ x[n]]
x[n]
h[n]  T [ [n]]
T[ ]
y[n]
the impulse response
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EXAMPLE
y[n]  2 x[n]
y[n]  max{ x[n  1], x[n], x[n  1]}
echo system : y[n]  x[n]  ax[n  nd ]
0
accumulato r : y[n] 
 x[n  k ]
k 
ideal delay : y[n]  x[n  5]
M2
1
moving average : y[n] 
x[n  k ]

M 1  M 2  1 k  M1
backward difference : y[n]  x[n]  x[n  1]
forward difference : y[n]  x[n  1]  x[n]
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2.2.2 classification of discrete-time system
1．Memoryless (static) system
the output depends only on the current input.
2．Linear system
T [ax1[n]  bx2 [n]]  aT [ x1[n]]  bT [ x2 [n]]
3．Time-invariant system：
if T [ x[n]]  y[n], then T [ x[n  n0 ]]  y[n  n0 ]
4．Causal system：
the output does not depend on the latter input.
5．Stable system：
if | x[n] | , then | T [ x[n]] | 
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2.2.3 linear time-invariant system（LTI）
characteri zed
by
y[n]  x[n] * h[n] 
h[n]


k  
k  
 x[k ]h[n  k ]   x[n  k ]h[k ]
How to get h[n] from the input and output：
let
x[n]   [n], then
y[n]  h[n]
41
the impulse response in LTI
EXAMPLE
(1) y[n]  2 x[n], h[n]  2 [n]
(2) y[n]  x[n]  0.5 x[n  50]
h[n]   [n]  0.5 [n  50]  h[0]  1, h[50]  0.5
0
(3) y[n] 
 x[n  k ]
k 
0
h[n] 
 [n  k ]  u[n]
k 
(4) y[n] 
1
M1  M 2  1
1
h[n] 
M1  M 2  1
M2
 x[n  k ]
k   M1
1

,  M1  n  M 2

 [ n  k ]   M1  M 2  1

0, other
k   M1

M2

42
Properties of LTI
h[n]
h[n]
h1[n]
h2[n]
h2[n]
Figure 2.12
x[n]
h1[n]
h1[n]  h2[n]
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classification of linear time-invariant system
(1) FIR : h[n]' s length is
IIR:
finite
h[n]’s length is infinite
(2)causal : h[n]  0, for n  0
causal

 causal
FIR
IIR
prove :

y[ n]  x[ n]  h[ n] 
k  

1

 h[k ]x[n  k ]
 h[k ]x[n  k ]   h[k ]x[n  k ]
k  
k 0
the latter input
the former input

(3) stable :
 | h[n] | 
n  
FIR must be stable。
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2.2.4 linear constant-coefficient difference equation
N
a
k 0
M
k
y[n  k ]   bk x[n  k ]
k 0
1.relation with input-output description and convolution
EXAMPLE
For IIR，the latter two are consistent.
accumulator system( IIR )

input-output description
y[n]   x[n  k ] 
k 0

 u[k ]x[n  k ]
k  


convolution description
 y[n]  x[n] * h[n] 
h[k ]x[n  k ], while h[n]  u[n]
infinite items，unrealizable
k  
difference equation description  y[n  1] 
Finite items, realizable

 x[n  1  k ] y[n]  x[n]  y[n  1]
k 0
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EXAMPLE
For FIR，the followings are consistent
M
2
1
input-output description and
y[n] 
x[n  k ]
difference equation description
M 2  1 k 0
（non-recursion）

 1
,0  n  M 2

 y[n]  x[n] * h[n]，while h[n]   M 2  1

0, other
Convolution description
Another difference equation description，recursion，lower rank
y[n]  y[n  1] 

1
M2 1
1
M2 1
M2
M2
 x[n  k ]  M  1  x[n  1  k ]
k 0
1
2
k 0
( x[n]  x[n  M 2  1])
For FIR and IIR，difference equations are not exclusive.
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2.Recursive computation of difference equations：
For IIR, there needs N initial conditions , then ,the solution is unique.
For FIR, there needs no initial conditions.
With initial-rest conditions (linear, time invariant, and causal), the solution is unique.
EXAMPLE
y[n]  ay[n  1]  x[n]
y[1]  1, x[n]   [n], det er min y[n]
y[0]  a  1
y[1]  a ( a  1)  0
y[ 2]  a  a ( a  1)  0

y[ n]  a n ( a  1), n  0
y[ 2]  1 / a ( y[ 1]  x[ 1])  a 1
y[ 3]  a  2

y[ n]  a n 1, n  0
y[ n]  a n 1  a nu[ n]
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3.computation of difference equations with homogeneous
and particular solution
with the input x[n] and initial conditions ，the output can be written as
y[n]  y p [n]  yh [n]
the particular solution y p [n] is the response to the input x[n] under some
initial condition.
ususlly, take the casual LTI as initial condition, then get y p [n]，
which
is called zero - state response and can be solved by the z - transform ；
the homogeneou s solution yh [n] is the response when the input is zero ，
N
viz. the solution of
a
m 1
k
yh [n  k ]  0，
which is called zero - input response
N
y h [ n]   A z ,
m 1
n
m m
N
z m is the root of
a z
m 1
k
k
 0, Am can be solved from
the initial conditions .
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2.2.5. Direct implementation of discrete-time system
EXAMPLE
FIR
h[n]   [n]   [n  1]  2 [n  2]
2
y[n]  x[n] * h[n] 
 h[k ]x[n  k ]  x[n]  x[n  1]  2 x[n  2]
k 0
x[n]
z-1
z-1
2
-1
+
+
y[n]
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EXAMPLE
IIR
h[n]  u[n]

y[n]  x[n] * h[n] 
 x[n  k ]
k 0
y[n]  x[n]  y[n  1]
x[n]
+
y[n]
z-1
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x[ n]  u[ n]
EXAMPLE
y[ n]  x[ n]  y[ n  1]
H ( z) 
1
1 z
1
A=[1,-1]
n=[0:100];
y=filter(B,A,x);
stem(n,y);
The matlab codes on the
direct realization of LTI
B=1;
x=[n>=0];
axis([0,20,0,20])
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keys：
judge the type of a system（from the relationship between
the input and output, and from h[n] for LTI).
the physics meaning of convolution representation for LTI：
the output signals are the weighted combination of the input signals，
h[n] is the weight。
the similarities and differences between linear constant-coefficient
difference equations and convolution representation，recursive computation。
the difference between IIR and FIR：
FIR
IIR
h[n]
finite length
infinite length
y[n]是x[n]的加权
finite items
infinite items
realization
stability
convolution or difference
stable
difference , recursion
maybe stable
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2.3 frequency-domain representation of
discrete-time signal and system
2.3.1 definition of fourier transform
2.3.2 frequency response of system
2.3.3 properties of fourier transform
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EXAMPLE
The intuitionistic meaning of frequencydomain representation of signals
x(t )  cos(2 100 t )  0.5 cos(2 200 t ), f1  100 Hz , f 2  200 Hz
2
10
1
5
0
-1
0
0.01
0.02
0
0 100 200 300 400 500
55
The intuitionistic meaning of frequencydomain representation of systems
y(t )  T {x(t )}  cos(2 100t ), f  100Hz
1
频响
10
0.5
0
5
-0.5
-1
0
0.01
0.02
0
0 100 200 300 400 500
150Hz
f
56
EXAMPLE
The effect of lowpass and highpass
filters to image signals
57
Frequency-domain analysis of de-noise
process through bandstop filter
58
Derivation of Fourier transform
3
x[n] 
| X ( ) | cos(n   ( ))

 

3


 

e j (n  ( ))  e j (n  ( ))
| X ( ) |
2
0
3


 

3

0
| X ' ( ) | e j (n  ( ))
3

 

X ( )e jn
3
X ( ) | X ' ( ) | e j ( )
59
2.3.1 definition of fourier transform
1 
j
jn
x[n] 
X
(
e
)
e
d



2 
X e j 
x[n]e  jn
  
n  
X (e j )  X R (e j )  jX I (e j ) | X (e j ) | e jX (e
j )
)
 
| X e j |, magnitude


   ARGX e    : principal
 X e j , phase arbitrary phase
j
value phase
  
arg X e j , continuous phase

X e
j (  2 )
   x[n]e


 j (  2 ) n
 
 X e j
60
x[n]  0.2n u[n], X (e j ) 
EXAMPLE
1
1  0.2e j
subplot(2,2,1); fplot('real(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);
subplot(2,2,2); fplot('imag(1/(1-0.2*exp(-1*j*w)))',[-2*pi ,2*pi]);
subplot(2,2,3); fplot('abs(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);
subplot(2,2,4); fplot('angle(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);
ʵ²¿
Ð鲿
1.5
0.5
1
0
0.5
-0.5
-5
0
·ù¶È
5
1.5
0.5
1
0
0.5
-0.5
-5
0
title('实部')
title('虚部')
title('幅度')
title('相位')
5
Matlab codes to draw the
frequency chart of signals
-5
0
Ïàλ
5
-5
0
5
61
Fourier transforms of nonabsolutely summable or nonsquare summable signals
EXAMPLE
sin(c n)
x[n] 
,  n  
n
1, |  | c
j
X (e )  
0, |  | c
sin( c n)
1 c jn
j
IFT [ X e ] 
e d 
2 c
n
 

EXAMPLE x[n]  1
    2   2r 
X e
j

r  
 
1  
jn


 IFT [ X e ] 
[
2



2

r
]
e
d




2
r  
1 
jn



2



e
d  1



2
j
62
2.3.2 frequency response of system
    h[n]e
He
j

 jn

1 
h[n] 
H (e j )e jn d
2 

 
| H e j |: amplitude response
 
 H e j : phase response
63
EXAMPLE
Ideal filter in frequency
and time domain
|  | c
1
H lp (e )  
c |  | 
0
sin(  c n)
1 c jn
hlp [n] 
e d 



c
2
n
j
64
EXAMPLE
Matlab codes to draw the
frequency response of a system
echo system : y[n]  x[n]  0.5x[n  10]
h=[1,0,0,0,0,0,0,0,0,0.5]
freqz(h,1)
65
Eigenfunction and steady-state response：
(1) x[n]  e j 0 n ,  n  

y[n]  x[n] * h[n] 

e j 0 k h[n  k ]
k  
let , n  k  k ' 


e j 0 ( n  k ') h[k ' ]  H (e j 0 )e j 0 n
k  
| H (e
j 0
)|e
j  0 n  j  H ( e j 0 )
(2) x[n]   ak e j k n ,  n  
k
y[n]  x[n] * h[n] 

a k H (e j k )e j k n
k
(3) x[n]  e j0 n u[n]
y[n]  H (e
j 0
)e
j 0 n

(

h[k ]e  jk )e jn
k  n 1
Steady-state response
transient response
66

y[n] 
 h[k ]x[n  k ]
k 0
 h[0]x[n]  h[1]x[n  1]   h[]x[n  ]
causal FIR system acts
on causal signal
h[N-1]
h[0]
Causal and stable IIR system
acts on causal signal
h[0]
Figure 2.20
67
Sin(0.1*pi*n)
example of steady-state response
h[n]=[1,1,1,1,1,1,1,1,1,1]/4
B=[1,0,1,0,1];A=[1,0.81,0.81,0.81]
68
2.3.3 properties of fourier transform
F
1.linearity : ax[n]  by[n] 
aX (e j )  bY (e j )
F
2.time shifting : x[n  n0 ] 
e  jn0 X (e j )
F
3. frequency shifting : e j0n x[n] 
X (e j ( 0 ) )
69
5.x[n]  y[n]  X (e j )Y (e j )
F
1
1
6.x[n] y[n] 
X (e j )  Y (e j ) 
2
2
F

7. parseval :

n  



X (e j )Y (e j (  ) )d
1 
x[n] y [n] 
X (e j )Y * (e j )d
2 

*

1 
j
2
E   | x[n] | 
|
X
(
e
)
|
d



2
n  
2
70
8.x*[n]  X * (e j ), x[n]  X (e j ), x*[n]  X * (e j )
F
F
F
x[n]  x*[n] F
X (e j )  X * (e j )
j
Re( x[n]) 
 X e (e ) 
2
2
x[n]  x*[n] F
X (e j )  X * (e j )
j
j Im( x[n]) 
 X o (e ) 
2
2
X (e j )  X * (e j )
Im( x[n]) 
2j
F
x[n]  x*[n] F
X (e j )  X * (e j )
j
xe[n] 
 Re[ X (e )] 
2
2
x[n]  x*[n] F
X (e j )  X * (e j )
j
xo [n] 
 j Im[ X (e )] 
2
2
71
9.x[n]  x * [n]  X (e j )  X * (e j ), for
real
sequences
Re[ X (e j )]  Re[ X (e j )]
Im[ X (e j )]   Im[ X (e j )]
| X (e j ) || X (e j ) |,  X (e j )    X (e j )
X (e j )  X * (e j (  2 ) )  X * (e j (2  ) )
72
2.3
summary
2.3.1 definition of fourier transform
2.3.2 frequency response of system
2.3.3 properties of fourier transform
requirements：calculation of fourier transforms
steady-state response
linearity
time shifting
frequency shifting
the convolution theorem
windowing theorem
Parseval’s theorem
symmetry properties
73
Keys and difficulties：
the convolution theorem;
the frequency spectrum of a real sequence is conjugate
symmetric;
the frequency spectrum of a conjugate symmetric
sequence is a real function.
exercises：
2.35 2.45 2.57
74