digital signal processing
DESCRIPTION
Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. Digital Signal Processing. Lecture 20 Fourier Transform Properties. READING ASSIGNMENTS. This Lecture: Chapter 11, Sects. 11-5 to 11-9 Tables in Section 11-9 - PowerPoint PPT PresentationTRANSCRIPT
Prof. Nizamettin AYDIN
http://www.yildiz.edu.tr/~naydin
Digital Signal Processing
1
Lecture 20
Fourier TransformFourier Transform
PropertiesProperties
Digital Signal Processing
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
• More examples of Fourier transform pairs• Basic properties of Fourier transforms
– Convolution property
– Multiplication property
dtetxjX tj )()(
Fourier Transform
Fourier Analysis(Forward Transform)
dtetxjX tj )()(
Fourier Synthesis(Inverse Transform)
dejXtx tj)(2
1)(
)()(
Domain-FrequencyDomain-Time
jXtx
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
jjHtueth t
1
1)()()(
)()( tueth t
2/
)2/sin()(
2/0
2/1)(
T
jXTt
Tttx
b
bb jX
t
ttx
0
1)(
)sin()(
0)()()( 0tjejXtttx
00 t
Table of Fourier Transforms
)(2)()( ctj jXetx c
0)()()( 0tjejXtttx
b
bb jX
t
ttx
0
1)(
)sin()(
2/
)2/sin()(
2/0
2/1)(
T
jXTt
Tttx
jjXtuetx t
1
1)()()(
)()()()cos()( ccc jXttx
Fourier Transform of a General Periodic Signal
• If x(t) is periodic with period T0 ,
0
00
00
)(1
)(T
tjkk
k
tjkk dtetx
Taeatx
)(2 since Therefore, 00 ke tjk
k
k kajX )(2)( 0
Square Wave Signal
x(t) x(t T0 )
T0 2T0 T0 2T00 t
ak e j0kt
j0kT0 0
T0 / 2
e j 0kt
j0kT0 T0 /2
T0
1 e jk
jk
ak 1
T0
(1)e j0 ktdt 1
T0
( 1)e j 0ktdtT0 / 2
T0
0
T0 / 2
Square Wave Fourier Transform
X( j ) 2 ak( k0 )k
x(t) x(t T0 )
T0 2T0 T0 2T00 t
Table of Easy FT Properties
ax1(t) bx2 (t) aX1( j ) bX2 ( j )
x(t td ) e jtd X( j )
x(t)e j0t X( j( 0 ))
Delay Property
Frequency Shifting
Linearity Property
x(at) 1|a | X( j(
a ))Scaling
Scaling Property
expands)(shrinks;)2( 221 jXtx
)(
)()(
1
)/(
aa
adajtj
jX
exdteatx
)()( 1aa
jXatx
Scaling Property
)()( 1aa
jXatx
)2()( 12 txtx
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 Cannot simultaneously reduce time duration and bandwidthduration and bandwidth
Significant FT Properties
x(t)h(t) H( j )X( j )
x(t)e j0t X( j( 0 ))
x(t)p(t) 1
2X( j ) P( j )
dx(t)
dt ( j)X( j)
Differentiation Property
Convolution Property
• Convolution in the time-domain
corresponds to MULTIPLICATIONMULTIPLICATION in the frequency-
domain
y(t) h(t) x(t) h( )
x(t )d
Y( j ) H( j )X( j )
y(t) h(t) x(t)x(t)
Y( j ) H( j )X( 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
1000
1001)(
)100sin()( jX
t
ttx
100b
Convolution Example
sin(100 t)
t
sin(200t)
t
x(t)h(t) H( j )X( j )
sin(100 t)
t
Cosine Input to LTI System
Y ( j) H( j )X( j)
H( j )[( 0 ) ( 0)]
H( j0 ) ( 0 ) H( j0 ) ( 0 )
y(t) H (j0 ) 12 e j0t H( j0 ) 1
2 e j 0t
H( j0 ) 12 e j0t H *( j 0)
12 e j0t
H( j0 ) cos( 0t H( j0 ))
Ideal Lowpass Filter
Hlp( j )
co co
y(t) x(t) if 0 co
y(t) 0 if 0 co
Ideal Lowpass Filter
y(t) 4
sin 50t 4
3sin 150t
fco "cutoff freq."
H( j ) 1 co
0 co
Signal Multiplier (Modulator)
• Multiplication in the time-domain corresponds to convolution in the frequency-domain.
Y( j ) 1
2X( j ) P( j )
y(t) p(t)x(t)
X( j)
x(t)
p(t)
Y( j ) 1
2X( j )
P( j( ))d
Frequency Shifting Property
x(t)e j0t X( j( 0 ))
y(t) sin 7t
te j 0 t Y ( j )
1 0 7 07
0 elsewhere
))((
)()(
0
)( 00
jX
dtetxdtetxe tjtjtj
y(t) x(t)cos(0t)
Y( j ) 12
X( j( 0 )) 12
X( j( 0 ))
x(t)
Differentiation Property
dx(t)
dt
d
dt1
2X( j )e j td
1
2( j )X( j )e j td
d
dte atu(t) ae atu(t) e at (t)
(t) ae atu(t)
ja j
Multiply by j