Transcript Introduction to Digital Logic

Digital Signal Processing
Prof. Nizamettin AYDIN
[email protected]
http://www.yildiz.edu.tr/~naydin
1
Digital Signal Processing
Lecture 20
Fourier Transform
Properties
2
READING ASSIGNMENTS
• This Lecture:
– Chapter 11, Sects. 11-5 to 11-9
– Tables in Section 11-9
• Other Reading:
– Recitation: Chapter 11, Sects. 11-1 to 11-9
– Next Lectures: Chapter 12 (Applications)
LECTURE OBJECTIVES
• The Fourier transform
X ( j ) 

x
(
t
)
e

 j t
dt

• More examples of Fourier transform pairs
• Basic properties of Fourier transforms
– Convolution property
– Multiplication property
Fourier Transform
1
x (t ) 
2
X ( j ) 

 X ( j )e
j t
d
Fourier Synthesis
(Inverse Transform)


 x (t )e
 j t
dt
Fourier Analysis
(Forward Transform)

T ime- Domain Frequency- Domain
x(t )  X ( j )
WHY use the Fourier transform?
• Manipulate the “Frequency Spectrum”
• Analog Communication Systems
– AM: Amplitude Modulation; FM
– What are the “Building Blocks” ?
• Abstract Layer, not implementation
• Ideal Filters: mostly BPFs
• Frequency Shifters
– aka Modulators, Mixers or Multipliers: x(t)p(t)
Frequency Response
• Fourier Transform of h(t) is the Frequency
Response
t
h(t )  e u(t )
1
h(t )  e u(t )  H ( j ) 
1  j
t

1

x (t )  

0
t T /2
sin(T / 2)
 X ( j ) 
 /2
t T /2
sin(bt )
x (t ) 
t

1



b

 X ( j )  

0   b
 j t0
x(t )   (t  t0 )  X ( j )  e
t0  0
Table of Fourier Transforms
t
x (t )  e u (t ) 

1
x (t )  

0
1
X ( j ) 
1  j
t T /2
sin(T / 2)
 X ( j ) 
 /2
t T /2
sin(bt )
x (t ) 
t

1



b

 X ( j )  

0   b
 j t0
x(t )   (t  t0 )  X ( j )  e
x(t )  e
jct
 X ( j )  2 (  c )
x(t )  cos(ct )  X ( j )   (  c )   (  c )
Fourier Transform of a General
Periodic Signal
• If x(t) is periodic with period T0 ,
x (t ) 

 ak e
jk0 t
k  
1
ak 
T0
Therefore,since e
X ( j ) 
jk0t

T0
 x (t )e
 jk0 t
0
 2 (  k0 )
 2 ak (  k0 )
k 
dt
Square Wave Signal
x(t)  x(t  T0 )
2T0
1
ak 
T0
T0
0
T0 / 2
 (1)e
 j0 kt
e
ak 
 j 0 kT0
T0 / 2
0
T0
1
 j 0 kt
dt 
(1)e
dt

T0 T0 / 2
 j0 kt
0
2T0 t
T0
 j 0 kt
T0
e

 j  0 kT0 T
0
 j k
/2
1 e

j k
Square Wave Fourier Transform
x(t)  x(t  T0 )
2T0
T0
0
T0
X( j ) 
2T0
t

 2 a  (  k 
k 
k
0
)
Table of Easy FT Properties
Linearity Property
ax1(t)  bx2 (t)  aX1( j )  bX2 ( j )
Delay Property
x(t  td )  e
 j t d
X( j )
Frequency Shifting
x(t)e
j 0 t
 X( j(   0 ))
Scaling
x(at) 
1
|a|

X( j( a ))
Scaling Property
x(at ) 

x
(
at
)
e

 j t
1
a
X(

j )
a

dt 


x(2t ) shrinks;
 j (  / a ) d
x (  )e
a


1
a
X ( j a )
1
2

X ( j ) expands
2
Scaling Property
x(at ) 
1
a
X ( j a )
x2 (t )  x1(2t )
Uncertainty Principle
• Try to make x(t) shorter
– Then X(j) will get wider
– Narrow pulses have wide bandwidth
• Try to make X(j) narrower
– Then x(t) will have longer duration
• Cannot simultaneously reduce time
duration and bandwidth
Significant FT Properties
x(t) h(t)  H( j )X( j )
x(t)p(t) 
x(t)e
j 0 t
1
2
X( j ) P( j )
 X( j(   0 ))
Differentiation Property
dx(t)
 ( j )X( j )
dt
Convolution Property
y(t)  h(t)  x(t)
x(t)
X( j )
Y( j )  H( j )X( j )
• Convolution in the time-domain

y(t)  h(t)  x(t)   h( )x(t   )d

corresponds to MULTIPLICATION in the frequencydomain
Y( j )  H( j )X( j )
Convolution Example
• Bandlimited Input Signal
– “sinc” function
• Ideal LPF (Lowpass Filter)
– h(t) is a “sinc”
• Output is Bandlimited
– Convolve “sincs”
Ideally Bandlimited Signal
sin(100 t )
x (t ) 
t
b  100

1   100
 X ( j )  

0   100
Convolution Example
x(t) h(t)  H( j )X( j )
sin(100  t) sin(200 t) sin(100  t)


t
t
t
Cosine Input to LTI System
Y (j  )  H( j )X(j  )
 H( j )[ (   0 )   (   0 )]
 H( j 0 ) (   0 )  H( j 0 ) (   0 )
y(t) 


j0 t
 j 0t
H(j  0 ) e  H( j  0 ) e
*
1 j0 t
1  j0 t
H( j 0 ) 2 e  H ( j 0 ) 2 e
H( j 0 ) cos( 0t H( j 0 ))
1
2
1
2
Ideal Lowpass Filter
Hlp ( j )
 co
y(t)  x(t)
y(t)  0
 co
if  0   co
if  0   co
Ideal Lowpass Filter
1
H( j )  
0
   co
   co
f co "cutoff freq."
y(t) 
4

sin50t  
4
sin 150t 
3
Signal Multiplier (Modulator)
y(t)  p(t)x(t)
x(t)
X( j )
Y( j ) 
p(t)
1
2
X( j )  P( j )
• Multiplication in the time-domain corresponds
to convolution in the frequency-domain.
1 
Y( j ) 
 X( j )P( j(   ))d
2 
Frequency Shifting Property
x(t)e

e


j 0t
j 0 t
x (t )e
 X( j(   0 ))
 j t

dt 
x
(
t
)
e

 j (  0 ) t
dt

 X ( j (   0 ))
sin 7t j 0 t
1  0 7     0 7
y(t) 
e  Y( j )  
elsewhere
t
0
y(t)  x(t)cos( 0 t) 
Y( j ) 
1
X( j(
2
  0 )) 
1
X( j(
2
  0 ))
x(t)
Differentiation Property

dx(t)
d  1 
j t

 X( j )e d 

dt
dt 2 

1

2


 ( j ) X( j )e
j t
Multiply by j
d


d at
at
at
e u(t)  ae u(t)  e  (t)
dt
  (t)  ae
at
u(t)
j

a  j