Transcript Fourier - Gupta Lab

Review on Fourier …
Slides edited from:
• Prof. Brian L. Evans and Mr. Dogu
Arifler Dept. of Electrical and
Computer Engineering The
University of Texas at Austin course:
EE 313 Linear Systems and Signals
Fall 2003
Fourier Series
Spectrogram Demo (DSP First)
• Sound clips
–
–
–
–
Sinusoid with frequency of 660 Hz (no harmonics)
Square wave with fundamental frequency of 660 Hz
Sawtooth wave with fundamental frequency of 660 Hz
Beat frequencies at 660 Hz +/- 12 Hz
• Spectrogram representation
– Time on the horizontal axis
– Frequency on the vertical axis
Frequency Content Matters
• FM radio
– Single carrier at radio station frequency (e.g. 94.7 MHz)
– Bandwidth of 165 kHz: left audio channel, left – right
audio channels, pilot tone, and 1200 baud modem
– Station spacing of 200 kHz
• Modulator/Demodulator (Modem)
Transmitter
Receiver
Home
Channel
upstream
downstream
Receiver
Transmitter
Service
Provider
Residential Application
Downstream Upstream Willing to pay
rate (kb/s)
rate (kb/s)
384
9
High
Database Access
384
9
Low
On-line directory; yellow pages
1,500
1,500
High
Video Phone
1,500
64
Low
Home Shopping
1,500
1,500
Medium
Video Games
3,000
384
High
Internet
6,000
0
Low
Broadcast Video
24,000
0
High
High definition TV
Demand
Potential
Medium
High
Medium
Medium
Medium
Medium
High
Medium
Business Application
Demand
Potential
High
Low
Low
High
Low
Medium
Medium
Low
Downstream Upstream Willing to pay
rate (kb/s)
rate (kb/s)
384
9
Medium
On-line directory; yellow pages
1,500
9
Medium
Financial news
1,500
1,500
High
Video phone
3,000
384
High
Internet
3,000
3,000
High
Video conference
6,000
1,500
High
Remote office
10,000
10,000
Medium
LAN interconnection
45,000
45,000
High
Supercomputing, CAD
Courtesy of Milos Milosevic (Schlumberger)
Demands for Broadband Access
T1
HDSL
SHDSL
Splitterless
ADSL
Full-Rate
ADSL
VDSL
T-Carrier One
(requires two pairs)
High-Speed Digital
Subscriber Line
(requires two pairs)
Single Line HDSL
Splitterless
Asymmetric DSL
(G.Lite)
Asymmetric DSL
(G.DMT)
Very High-Speed
Digital Subscriber
Line (proposed)
1.544 Mbps Symmetric Business, Internet
Service
1.544 Mbps Symmetric Pair Gain (12 channels),
Internet Access, T1/E1
replacement
1.544 Mbps Symmetric Same as HDSL except
pair gain is 24 channels
Up to 1.5 Mbps Downstream Internet Access, Video
Up to 512 kbps Upstream Phone
Up to 10 Mbps Downstream Internet Access, Video
Up to 1 Mbps Upstream Conferencing, Remote
LAN Access
Up to 22 Mbps Downstream Internet Access, VideoUp to 3 Mbps Upstream on-demand, ATM,
Up to 6 Mbps Symmetric Fiber to the Hood
Courtesy of Shawn McCaslin (Cicada Semiconductor, Austin, TX)
xDSL
ISDN
DSL Broadband Access
Meaning
Data Rate
Mode
Applications
Integrated Services Standards
144 kbps Symmetric Internet Access, Voice,
Digital Network
Pair Gain (2 channels)
• Discrete Multitone (DMT) modulation
ADSL (ANSI 1.413) and proposed for VDSL
• Orthogonal Freq. Division Multiplexing (OFDM)
magnitude
Digital audio/video broadcasting (ETSI DAB-T/DVB-T)
channel frequency
response
a carrier
a subchannel
Courtesy of Güner Arslan (Cicada Semiconductor)
Multicarrier Modulation
frequency
Harmonically related carriers
Periodic Signals
• f(t) is periodic if, for some positive constant T0
For all values of t, f(t) = f(t + T0)
Smallest value of T0 is the period of f(t).
sin(2pfot) = sin(2pf0t + 2p) = sin(2pf0t + 4p): period 2p.
• A periodic signal f(t)
Unchanged when time-shifted by one period
Two-sided: extent is t  (-, )
May be generated by periodically extending one period
Area under f(t) over any interval of duration equal to the
period is the same; e.g., integrating from 0 to T0 would
give the same value as integrating from –T0/2 to T0 /2
Sinusoids
• f0(t) = C0 cos(2 p f0 t + q0)
• fn(t) = Cn cos(2 p n f0 t + qn)
• The frequency, n f0, is the nth harmonic of f0
• Fundamental frequency in Hertz is f0
• Fundamental frequency in rad/s is w = 2 p f0
Cn cos(n w0 t + qn) =
Cn cos(qn) cos(n w0 t) - Cn sin(qn) sin(n w0 t) =
an cos(n w0 t) + bn sin(n w0 t)
Fourier Series
• General representation
of a periodic signal
• Fourier series
coefficients
• Compact Fourier
series

f t   a0   an cosnw0t   bn sin nw0t 
n 1
1
a0 
T0
 f t dt
T0
0
an 
2
T0
 f t cosnw t dt
bn 
2
T0
 f t sin nw t dt
T0
0
0
T0
0
0

f t   c0   cn cosnw 0t  q n 
n 1
where c0  a0 , cn  an2  bn2 , and
  bn 

q n  tan 
 an 
1
Existence of the Fourier Series
• Existence
T0

0
• Convergence for all t
f t  dt  
f t    t
• Finite number of maxima and minima in one
period of f(t)
Example #1

f t   a0   an cos2nt  bn sin 2nt 
f(t)
n 1
1
e-t/2
p
0
a0 
p
• Fundamental period
T0 = p
• Fundamental frequency
f0 = 1/T0 = 1/p Hz
w0 = 2p/T0 = 2 rad/s
an 
1
p
2
p
p

0
p

0
p
2  2 
e dt    e  1  0.504
p

e

t
2

t
2
 2 
cos2nt  dt  0.504
2 
 1  16n 
 8n 


e
sin
2
nt
dt

0
.
504

2 
p 0
1

16
n


an and bn decreasein amplit udeas n  .
bn 
2
p

t
2

2








f t   0.5041  
cos
2
nt

4
n
sin
2
nt
2

 n1 1  16n

Example #2

f t   a0   an cos(π n t )  bn sin π n t 
f(t)
n 1
A
1
0
-A
1
a0  0
(by inspectionof theplot)
an  0
(because it is odd symmetric)
bn 
2
p
2
• Fundamental period
T0 = 2
• Fundamental frequency
f0 = 1/T0 = 1/2 Hz
w0 = 2p/T0 = p rad/s
p
1/ 2
 2 A t sin(π n t ) dt 
1 / 2
3/ 2
 (2 A  2 A t ) sin(π n t ) dt
1/ 2

 0
 8 A
bn   2 2
 np
 8 A
 n 2p 2
n is even
n  1,5,9,13,
n  3,7,11,15,
Example #3
f(t)
1
2p
p p/2
• Fundamental period
T0 = 2p
• Fundamental frequency
f0 = 1/T0 = 1/2p Hz
w0 = 2p/T0 = 1 rad/s
p/2
p
2p
1
2
 0 n even
Cn   2
n odd
p n
 0 for all n  3,7,11,15,
qn  
n  3,7,11,15, 
 p
C0 
Fourier Analysis
Periodic Signals
• For all t, x(t + T) = x(t)
x(t) is a period signal
• Periodic signals have
a Fourier series
representation
xt  

C
m  
T
2
1
Cn 
T
n
e
 xt  e

m
j 2p   t
T 
m
 j 2p   t
T 
dt
T
2
• Cn computes the projection (components) of
x(t) having a frequency that is a multiple of the
fundamental frequency 1/T.
Fourier Integral

G  f    g t  e

X w    xt  e  j w t dt



1 
j 2p f t
jw t


g t    G  f  e
df xt  
X
w
e
dw




2p
Communication Systems
Signal Processing
 j 2p f t
dt
• Conditions for the Fourier transform of g(t) to
exist (Dirichlet conditions):
x(t) is single-valued with finite maxima and minima in
any finite time interval
x(t) is piecewise continuous; i.e., it has a finite number of
discontinuities in any finite time interval

x(t) is absolutely integrable
 g t  dt  
Laplace Transform
• Generalized frequency variable s = s + j w

F s    f t  e  s t dt

1
f t  
2p


F s  e s t ds

• Laplace transform consists of an algebraic
expression and a region of convergence (ROC)
• For the substitution s = j w or s = j 2 p f to be
valid, the ROC must contain the imaginary axis
Fourier Transform
• What system properties does it possess?
 Memoryless
 Causal
 Linear
 Time-invariant
•
•
•
•
What does it tell you about a signal?
Answer: Measures frequency content
What doesn’t it tell you about a signal?
Answer: When those frequencies occurred in time
Useful Functions
• Unit gate function (a.k.a. unit pulse function)
rect(x)
1
-1/2
0
1/2
x

0
 1
rect x   
2
1

• What does rect(x / a) look like?
• Unit triangle function
(x)
1
-1/2
0
1/2
x

 0
 x   
1  2 x

1
2
1
x
2
1
x
2
x
1
2
1
x
2
x
Useful Functions
• Sinc function
x
3p 2p
p
0 p
sin  x 
x
How to comput esinc(0)?
As x  0, numeratorand
denominator are bot h going
to 0. How to handleit?
sinc x  
sinc(x) 1
2p
3p
– Even function
– Zero crossings at x  p ,  2p ,  3p , ...
– Amplitude decreases proportionally to 1/x
Fourier Transform Pairs
t / 2  jwt
 t   jwt
F w    rect e dt   e dt

t / 2
t 
 wt 
 wt 
2 sin 
sin



1  jwt / 2
2
2



  t sinc wt 

e
 e jwt / 2 
t


jw
w
2
 wt 




 2 



F(w)
f(t)
F
1
-t/2
0
t
t/2
t
w
6p
t
4p
t
2p
t
0 2p
t
4p
t
6p
t
Fourier Transform Pairs
From thesamplingpropertyof theimpulse,

F  t     t  e  j w t dt  e  j 0t  1

F(w) = 2 p (w)
f(t) = 1
F
1
0
t
(2p)
0
w
(2p) means that the area under the spike is (2p)
Fourier Transform Pairs
1 
1 jw 0 t
jwt


F  w  w 0  

w

w
e
d
w

e
0



2p
2p
1 jw 0 t
e
  w  w 0  or e jw 0t   w  w 0 
2p
1 jw 0 t
Since cosw 0t   e  e  jw0t
2
cosw 0t   p  w  w 0    w  w 0 
1


F(w)
f(t)
(p)
(p)
F
0
t
w0
0
w0
w
Fourier Transform Pairs
1 t 0
 lim e  at u t   e at u  t 
sgn t   
 1 t  0 a  0



 

F sgn t   lim F e  at u t   F e at u  t 
a 0
 1
1 

 lim

a  0 a  jw
w
j

a


  2 jw  2

 lim 2
a 0 a  w 2 
 jw

sgn(t)
1
t
-1
Fourier Transform Properties
Fourier vs. Laplace Transform Pairs
f(t)
e-at u(t)
e-a|t|
(t)
1
u(t)
cos(w0t)
sin(w0t)
eat u(t)
F(s)
1
s+a
2a
2
a – s2
1
Region of Convergence
F(w)
Re{s} > -Re{a}
1
jw + a
2a
w2 + a2
1
-Re{a} < Re{s} < Re{a}
complex plane
2p(s) complex plane
Re{s} > 0
1
s
2p(w)
p(w) + 1/(jw
p[(w + w0) + (w – w0)]
jp[(w + w0) - (w – w0)]
1
s-a
Re{s} > Re{a}
Assuming that Re{a} > 0
Duality
• Forward/inverse transforms are similar

1 
jw t
 jw t


f t  
F
w
e
dw
F w    f t  e
dt




2p
F t   2p f  w 
f t   F w 
rect(t/t)  t sinc(w t / 2)
• Example:
t sinc(t t/2)  2 p rect(-w/t)
t sinc(t t /2)  2 p rect(w/t)
– Apply duality
– rect(·) is even
f(t)
t
F(w)
1
w
-t/2
0
t/2
t
6p
t
4p
t
2p
t
0 2p
t
4p
t
6p
t
Scaling
• Same as Laplace
transform scaling property
f t   F w 
f at 
1 w 
F 
a a
|a| > 1: compress time axis, expand frequency axis
|a| < 1: expand time axis, compress frequency axis
• Effective extent in the time domain is inversely
proportional to extent in the frequency domain
(a.k.a bandwidth).
f(t) is wider  spectrum is narrower
f(t) is narrower  spectrum is wider
Time-shifting Property
• Shift in time
f t  t0   e j w t0 F w 
– Does not change magnitude of the Fourier transform
– Does shift the phase of the Fourier transform by -wt0
(so t0 is the slope of the linear phase)
Frequency-shifting Property
e j w0 t f t   2p F w  w0 
e  j w0 t f t   2p F w  w0 
1
pF w  w0   pF w  w0 
cosw0t  f t  
2p
1
1
cosw0t  f t   F w  w0   F w  w0 
2
2
1
 jpF w  w0   jpF w  w0 
sin w0t  f t  
2p
sin w0t  f t   
p
j
F w  w0  
p
j
F w  w0 
Modulation
y t   f t  cosw 0t 
Mult iplicat ionin t he t imedomain is
convolut ion in t hefrequency domain:
1
Y w  
F w   p w  w 0   p w  w 0 
2p
Recall t hat

x t    t     t x t  t dt  x t 


x t    t  t0     t  t0 x t  t dt  x t  t0 

So,
1
1
Y w   F w  w 0   F w  w 0 
2
2
Modulation
F(w)
• Example: y(t) = f(t) cos(w0 t)
f(t) is an ideal lowpass signal
Assume w1 << w0
Y(w)
1/2 Fww0
1/2
1
-w1
w1
0
w
1/2 Fww0
w
-w0 - w1
w0
-w0 + w1
0
w0 - w1
w0
w0 + w1
• Demodulation is modulation followed by
lowpass filtering
• Similar derivation for modulation with sin(w0 t)
Time Differentiation Property
• Conditions
f(t)  0, when |t|  
f(t) is differentiable
• Derivation of property:
Given f(t)  F(w)
d

Let B w   F  f (t ) 
 dt

 df t 
B w   
e  j w t dt
  dt

  e  j w t df t 

From t hechain rule, recall t hat
 u dv  u v -  v du
Let u  e  j w t and dv  df (t ),
so du   jw e  j w t
B w   e  j w t f t 
 jw 



t  



f t d e  j w t 
f t e  j w t dt  jw F w 
df t 
 jw F w 
dt
df n t 
n



j
w
F w 
n
dt
Time Integration Property
Find  f  x dx  ?
t
-
From thepropertyof timeconvolution :

t
-

f  x dx   f  x u t  x dx
-
 f t   u t 

1 
 F w  p  w  

jw 

F w 
 p F 0  w  
jw
T herefore,
t
F w 






f
x
dx

p
F
0

w

-
jw
Summary
• Definition of Fourier Transform

F w    f t  e j w t dt

• Two ways to find Fourier Transform
– Use definitions
– Use properties