Signal & Systems - KOCWcontents.kocw.net/KOCW/document/2015/chungnam/kimjin... · 2016. 9. 9. ·...
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Kim, J. Y. IC & DSP Research Group Signal & Systems Chonnam National University Dept. of Electronics Engineering Kim, Jinyoung
Transcript of Signal & Systems - KOCWcontents.kocw.net/KOCW/document/2015/chungnam/kimjin... · 2016. 9. 9. ·...
Kim, J. Y.
IC & DSP
Research
Group
Signal & Systems
Chonnam National University
Dept. of Electronics Engineering
Kim, Jinyoung
Kim, J. Y.
IC & DSP
Research
Group
4. Applications of Fourier
Representation To Mixed
Classes
A/D Conversion by Flash/Parallel Technique
Kim, J. Y.
IC & DSP
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Time. Periodic Nonperiodic
Co
ntin
uo
us
(t)
Fourier Series Fourier Transform
No
np
eriod
ic
Discrete
[n]
Discrete-Time Fourier
Series
Discrete-Time Fourier
Transform
Perio
dic
Discrete
[k]
Continuous
1[ ] ( )
2
( ) ( )
j t
j t
x t X j e d
X j x t e dt
0
0
0
( ) [ ]
1[ ] ( )
2( ) has period ,
jk t
k
jk t
T
x t X k e
X k x t e dtT
x t TT
0
0
0
[ ] [ ]
1[ ] [ ]
[ ] and [ ] have period
2
jk n
k N
jk n
n N
x n X k e
X k x n eN
x n X k N
N
1[ ] ( )
2
( ) [ ]
( ) has period 2
j j n
j j n
n
j
x n X e e d
X e x n e
X e
( , )
( , )k
( , )k
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4.1 Introduction
Mixing of the following classes of signals
– Periodic and nonperiodic singals
– Continuous and discrete time signals
Sampling and reconstruction
DTFS approximation of FT
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4.2 Fourier Transform Representations of Periodic Singals
Relating the FT to the FS
Relating the DTFT to the DTFS
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4.2.1 Relating the FT to the FS 1
FS representation for a periodic signal x(t)
FT representation
0( ) [ ] : FS .jk t
k
x t X k e repr
0
0
1( )
2
FTjk t
e k
0
0( ) [ ] ( ) 2 [ ] ( )FT
jk t
k k
x t X k e X j X k k
' 1( )
2
( ) 2 ( ')
j t j te X j e d
X j
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Relating the FT to the FS 2
Graphic explanation
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Relating the FT to the FS 3
Example4.1 x(t)=cos(0t)
0;
0
1/ 2, 1cos( ) [ ]
0, 1
FS kt X k
k
0 0 0cos( ) ( ) ( ) ( )FT
t X j
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Relating the FT to the FS 4
Example4.2 Unit impulse train
( ) ( )n
p t t nT
0
0
/ 2
/ 2
0
2 /
1 1[ ] ( )
2( ) ( )
T
jk t
T
k
T
P k t e dtT T
P j kT
… …
… …
t
ω
T
ω0
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4.2.2 Relating the DTFT to the
DTFS
DTFS representation for a periodic signal
x[n] DTFT representation 1
0[ ] [ ] : DTFS .
jk t
k N
x n X k e repr
0
0
1( 2 )
2
DTFTjk n
m
e k m
0
0[ ] [ ] 2 [ ] ( 2 )DTFT
jk n
k N k N m
x n X k e X k k m
' 1( )
2
( ) 2 ( ')
j n j j n
j
e X e e d
X e
2 periodic
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Relating the DTFT to the DTFS 2
2
0
0
0
Since [ ] is N periodic and 2
[ ] [ ] 2 [ ] ( )DTFT
jk n
k N k
X k N
x n X k e X k k
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Relating the DTFT to the
DTFS 2
Example4.3 Determine the inverse DTFT
of the frequency-domain representation.
1 1
1 1
1
1 1( ) ( ) ( ),
2 2
1 1 1[ ] sin( )
2 2 2
j
j n j n
X ej j
x n e e nj
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4.3 Convolution and Multiplication with Mixtures of Periodic and Nonperiodic Signals
Convolution of Periodic and Nonperiodic
Signals
Modulation of Periodic and Nonperiodic
Signals
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4.3.1 Convolution of Periodic
and Nonperiodic Signals
Basic Theory
x(t) is periodic signal
( ) ( )* ( ) ( ) ( ) ( )FT
y t x t h t Y j X j H j
FT
0( ) 2 [ ] ( )k
x(t) X j X k k
0
0 0
( ) ( )* ( )
( ) 2 [ ] ( ) ( )
2 ( ) [ ] ( )
FT
k
k
y t x t h t
Y j X k k H j
H jk X k k
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Convolution of Periodic and
Nonperiodic Signals 2
Graphic Explanation
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Convolution of Periodic and
Nonperiodic Signals 3
Example 4.4
(sol)
x(t) is periodic with period 4
( ) (1/ )sin( )h t t t
1,| | 1( ) , | | 2
0,
tx t for t
otherwise
1,| |( )
0,
2sin( / 2)( ) ( )
2k
H jotherwise
kX j k
k
( ) 2 ( ) ( ) 2 ( )2 2
1 1( ) cos( )
2 2
Y j
y t t
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Convolution of Periodic and
Nonperiodic Signals 4
Discrete time case
– Basic convolution property
– Convolution of periodic and nonperiodic signals
[ ] [ ]* [ ] ( ) ( ) ( )j j jy n x n h n Y e X e H e
0
0[ ] [ ]* [ ] ( ) 2 ( ) [ ] ( )jkj
k
y n x n h n Y e H e X k k
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4.3.2 Modulation of Periodic
and Nonperiodic Signals 1
Basic Theory
x(t) is periodic signal
1( ) ( ) ( ) ( ) ( )* ( )
2
FT
y t g t x t Y j G j X j
FT
0( ) 2 [ ] ( )k
x(t) X j X k k
0( ) ( ) ( ) ( ) [ ] ( ( ))FT
k
y t g t x t Y j X k G j k
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Modulation of Periodic and
Nonperiodic Signals 2
Graphic Explanation
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Modulation of Periodic and
Nonperiodic Signals 3
Example4.10 y(t)=g(t)x(t), g(t)=cos(t/2)
(sol)
x(t) is periodic with period 4
1,| | 1( ) , | | 2
0,
tx t for t
otherwise
2sin( / 2)( ) ( )
2k
kY j G j k
k
1 1
( ) ( )2 2
G j
2sin( / 2) 1 1( ) ( ) ( )
2 2 2 2k
kY j k k
k
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Modulation of Periodic and
Nonperiodic Signals 4
Example4.10 continued
2sin( / 2) 1 1( ) ( ) ( )
2 2 2 2k
kY j k k
k
2sin(2*pi/2)=0
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Modulation of Periodic and
Nonperiodic Signals 5
Example 4.6 AM Radio
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( ) ( )cos( )
( ) (1/ 2) ( ( )) (1/ 2) ( ( ))
c
c c
r t m t t
R j M j M j
( ) ( )cos( )
( ) (1/ 2) ( ( )) (1/ 2) ( ( ))
( ) (1/ 4) ( ( 2 )) (1/ 2) ( ) (1/ 4) ( ( 2 ))
c
c c
c c
q t r t t
Q j R j R j
Q j M j M M j
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Modulation of Periodic and
Nonperiodic Signals 6
The discrete-time modulation property
x[n] is periodic signal
1[ ] [ ] [ ] ( ) ( ) ( )
2
DTFTjy n x n g n Y e G j X j
DTFT
0[ ] ( ) 2 [ ] ( )j
k
x n X e X k k
( )
0
2
( )
0
2
( ) [ ] ( ) ( )
[ ] ( ) ( )
j j
k
j
k N
Y e X k k G e d
X k k G e d
0( )[ ] [ ] [ ] ( ) [ ] ( )
DTFTj kj
k N
y n g n x n Y e X k G e
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Example 4.7 : Windowing data
Effect of windowing a data record. Y(ej) for different
values of M, assuming that 1 = 7/16 and 2 = 9/16. (a)
M = 80, (b) M = 12, (c) M = 8.
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4.4 Fourier Transform Representation for Discrete-Time Signals
Relating the FT to the DTFT
Relating the FT to the DTFS
g[n] to be equal to the samples of x(t)
taken at intervals of T.
- g[n]=x(nT);
- ejn=ejTn
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Preliminary
Sample the x(t) with impulse sequences
x(t)
x(nT)=x[n]
x(t)
( )n
t nT
j n j Tne e
T
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4.4.1 Relating the FT to the DTFT 1
Basic theory
( ) [ ]j j n
k
X e x n e
( ) ( ) [ ]j j Tn
Tk
X j X e x n e
( )
( ) [ ] ( )
FTj Tn
n
t nT e
x t x n t nT
( ) [ ] ( ) ( ) [ ]FT
j Tn
n n
x t x n t nT X j x n e
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Relating the FT to the DTFT 2
Graphic explanation
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x(t)
x(nT)=x[n]
x(t)
( ) ( )n
p t t nT
x[n]
( ) ( ) ( ) ( ) ( )n
x t x t p t x nT t nT
( ) [ ]j j n
n
X e x n e
( ) ( ) ( )
( ) ( )
( ) ( )
j t
n
j t
n
j nT j
Tn
X j x nT t nT e dt
x nT t nT e dt
x nT e X e
1( ) ( )* ( )
2
1 2( ( ))
k
X j X j P j
X j kT T
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Relating the FT to the DTFT 3
Example 4.8 Determine the FT pair
associated with a signal whose DTFT is
(sol)
1( )
1
j
jX e
ae
[ ] [ ]
( ) ( )
n
n
n
x n a u n
x t a t nT
1( ) ( )
1
FT
j Tx t X j
ae
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IC & DSP
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4.4.2 Relating the FT to the DTFS
1
Basic Theory
Using the scaling property of the impulse function, (av)=a(v)
0( ) 2 [ ] ( )j
k
X e X k k
0
0
( ) ( )
2 [ ] ( )
2 [ ] ( )
j T
k
k
X j X e
X k T k
X k T kT
02( ) [ ]
k
X j X k kT T
DTFT of N periodic signal x[n]
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IC & DSP
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4.5 Sampling
Sampling continuous-time signals
Subsampling: Sampling discrete-time
signals
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4.5.1 Sampling continuous-time
signals
Sampling Process
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Sampling continuous-time signals
2
Basic Theory
( ) [ ] ( )
( ) ( )
( ) ( ) ( ), where ( ) ( )
n
n
n
x t x n t nT
x nT t nT
x t x t p t p t t nT
s
1( ) ( )* ( )
2
1 2( )* ( )
2
1( ( ))
, where 2 /
s
k
s
k
X j X j P j
X j kT
X j kT
T
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IC & DSP
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Graphic Explanation
The FT of a sampled
signal for different
sampling frequencies.
(a) Spectrum of
continuous-time signal.
(b) Spectrum of
sampled signal when s
= 3W.
(c) Spectrum of sampled
signal when s = 2W.
(d) Spectrum of
sampled signal when s
= 1.5W.
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Sampling continuous-time signals
4
DTFT of sampled signal
/[ ] ( ) ( )
DTFTj
Tx n X e X j
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Sampling continuous-time signals
5
Example4.13 x(t)=cos(t)
(a) T=1/4 (b) T=1 (c) T=3/2
(sol)
( ) ( ) ( ) ( )
( ) ( ) ( )
FT
s s
k
x t X j
X j k kT
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IC & DSP
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The effect of sampling a sinusoid at different rates (Example 4.9).
(a) Original signal and FT. (b) Original signal, impulse sampled
representation and FT for Ts = ¼ .
(c) Original signal, impulse sampled representation and FT for cT = 1.
(d) Original signal, impulse sampled representation and FT for Ts = 3/2.
A cosine of frequency /3 is shown as the dashed line.
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IC & DSP
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Example4.10 Aliasing in Movies Aliasing in a movie.
(a) Wheel rotating at
radians per second and
moving from right to left
at meters per second.
b) Sequence of movie
frames, assuming that the
wheel rotates less than
one-half turn between
frames.
(c) Sequences of movie
frames, assuming that the
wheel rotates between
one-half and one turn
between frames.
(d) Sequence of movie
frames, assuming that the
wheel rotates one turn
between frames.
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IC & DSP
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Example 4.11 : Multipath Communication
Channel : Discrete Time Model - 생략
Two-path
communication model
( ) ( ) ( )
[ ] [ ] [ 1]
diffy t x t x t T
t nT
y n x n ax n
( ) 1
( ) 1
( ) 1
diff
s
j T
j
s
j T
H j e
H j ae
T
H j ae
2( ) ( )
2
s
s
T
sT
TMSE H j H j d
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22
min | | (1 )MSE
( )
2
s
diff s
s
Tj T Ts
T
diff s
s
Te d
T Tsinc
T
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IC & DSP
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4.5.2 Sampling : Discrete-Time
Signals - 생략
Subsampling : y[n]=x[qn]
( ) [ ] ( )n
x t x n t nT
1
( ) ( ( ))s
k
X j X j kT
[ ] [ ] ( )y n x qn x nqT
( ) ( ) ( )
1 ( ) ( ( ))
1 ( ( ))
FT
n
s
k
s
k
y t x t t nT
Y j X j kT
kX j
qT q
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IC & DSP
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Sampling Discrete-Time Signals 2
Represent Y(j) as a function of X(j)
1
0
( ( ))
1
0
1 1( ) ( ( ))
1( ( ))
s
q
s s
m l
mX j
q
q
s
m
k ml
q q
mY j X j l
q T q
mX j
q q
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IC & DSP
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Sampling Discrete-Time Signals 3
Represent Y(ej) as a function of X(ej) : DTFT
version
1
/0
1
0
1
0
1( / 2 / )
0
1 1( 2 ) / ( 2 )
0 0
1( ) ( )
1
12
1, ( ) /
1 1
qj
sTm
q
s
m
q
m
qj q m q j
m
q qj m q j m
q
m m
mY e Y j X j
q T q
mX j
q qT q
j mX
q T q q
X e X e X j Tq
X e X eq q
Kim, J. Y.
IC & DSP
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Effect of subsampling on
the DTFT.
(a) Original signal
spectrum.
(b) m = 0 term, Xq(ej), in
Eq. (4.27)
(c) m = 1 term in Eq.
(4.27).
(d m = q – 1 term in Eq.
(4.27).
(e) Y(ej), assuming that
W < /q.
(f) Y(ej), assuming that
W > /q.
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IC & DSP
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4.6 Reconstruction of Continuous-Time Signals from Samples
Conversion of a discrete-time signal to a
continuous signal
Reconstruction
System x(t) x[n]
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Sampling Theorem 1
The samples do not tell us anything about the
behavior of the signal in between the sample
times.
x[n]=x1(nT)=x2(nT)
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Sampling Theorem 2
Sampling Theorem : Let x(t)FTX(j)
represent a bandlimited singal so that
X(j)=0 for ||>m, where s=2/T is the
sampling frequency, then x(t) is uniquely
determined by its samples x(nT), n=0, 1,
2, 3,…
Nyquist sampling rate=2m
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Sampling Theorem 3
Example 4.12 x(t)=sin(10t)/t, Determine the
condition on the sampling interval.
(sol) 1,| | 10
( )0,| | 10
10m
X j
220
1
10
T
or
T
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Sampling Theorem 4
Anti-aliasing filter
Suppress any frequency components
above s/2
Anti-aliasing
filter
Sampling
s x(t)
bandlimited signal
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4.6.2 Ideal Reconstruction 1
Fourier transform of sampled signal 1
( ) ( )s
k
X j X j jkT
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Ideal Reconstruction 2
Ideal lowpass filtering ,| | / 2
( )0,| | / 2
s
r
s
TH j
( ) ( )* ( )
( )* [ ] ( )
[ ] ( )
r
r
n
r
n
x t x t h t
h t x n t nT
x n h t nT
( ) [ ]sinc2
sin2
( ( ) )
s
n
s
r
x t x n t nT
T t
h tt
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4.6.3 Practical Reconstruction -
The Zero-Order Hold 1
Zero-order hold
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Practical Reconstruction -
The Zero-Order Hold 2
Mathematical analysis
( ) ( )* ( )
( )* [ ] ( )
( ) ( ) ( )
o o
o
n
o o
x t x t h t
h t x n t nT
X j H j X j
/ 2
sin2
( ) ( ) 2FT
j T
o o
where
T
h t H j e
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Practical Reconstruction -
The Zero-Order Hold 3
Influences of zero-order hold
1. A linear phase shift corresponding to a time delay of T/2 seconds.
2.The portion of X(j) between -m and m is distorted by the curvature of the main lobe of Ho(j).
3.Distorted and attenuated version of X(j) remain centered at nonzero multiples of .
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Effect of the zero-order hold in the frequency
domain
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Practical Reconstruction -
The Zero-Order Hold 5
Practical reconstruction system
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4.7 Discrete-Time Processing of Continuous Signals - 생략
Block diagram
1( ) ( ) ( ) ( ) ( )sj T
o C a
s
Y j H j H j H e H jT
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Mathematical Description ( ) ( ) ( )
1( ) ( ( ))
1( ( )) ( ( ))
a a
a s
ks
a s s
ks
X j H j X j
X j X j kT
H j k X j kT
1( ) ( ) ( ) ( ) ( )sj T
o C a
s
Y j H j H j H e H jT
/ 2
1( ) ( ) ( ( )) ( ( ))
sin( / 2)( ) 2
1( ) ( ) ( ) ( ) ( ( )) ( ( ))
1( ) ( ) ( ) ( ) ( ) ( )
s
s
s
s
j T
a s s
ks
j T so
j T
o C a s s
ks
j T
o C a
s
Y j H e H j k X j kT
TH j e
Y j H j H j H e H j k X j kT
Y j H j H j H e H j X jT
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IC & DSP
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4.8 Fourier-Series Representation of Finite-Duration Nonperiodic Signals - 생략 Use of DTFS and FS for representing
finite-duration nonperiodic signals
Relating the DTFS to the DTFT
Relating the FS to the FT
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IC & DSP
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4.8.1 Relating the DTFS to the
DTFT 1
DTFS of a finite-duration nonperiodic
signal :
-x[n]=0, n<0 or nM
DTFT of x[n]
- NM DTFS coeffs using x[n], 0nN-1
1
0
( ) [ ]M
j j n
n
X e x n e
0
0
0
1
0
0
1
0
1( ) [ ] , 2 /
1 1[ ] ( )
Nj kn
n
Mj kn j
n k
X k x n e NN
x n e X eN N
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IC & DSP
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Relating the DTFS to the
DTFT 2
Convert the DTFS coeffs back to a time-
domain signal
The effect of sampling the DTFT of a finite-
duration nonperiodic signal is to periodically
extend the signal in the time domain.
0[ ] [ ] (N periodic signal)
[ ] [ ],0 1
jk n
k N
x n X k e
x n x n n N
0
0
; 1[ ] [ ] [ ] ( )
DTFSjk
m
x n x n mN X k X eN
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Relating the DTFS to the DTFT 3
Graphic illustration
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Relating the DTFS to the
DTFT 4
Example4.14
(sol)
cos((3 8) ),0 31[ ]
0,
n nx n
otherwise
[ ] [ ] [ ],
1,0 313where [ ] cos( ) and [ ]
0,8
x n g n w n
ng n n w n
otherwise
(31/ 2)
3 3( ) ,
8 8
as one 2π period of ( )
sin(16 )( )
sin( / 2)
j
j
j
G e
G e
W e e
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Relating the DTFS to the
DTFT 5
Example 4.14 : continued
(31/ 2)( 3 /8)
(31/ 2)( 3 /8)
( ) 1/ 2 ( ) ( )
3sin 16
8
2 1 3sin
2 8
3sin 16
8
2 1 3sin
2 8
j j j
j
j
X e G e W e
e
e
Kim, J. Y.
IC & DSP
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The DTFT and length-N DTFS of a
32-point cosine. The dashed line
denotes |X(ej)|, while the stems
represent N|X[k]|. (a) N = 32, (b) N
= 60, (c) N = 120.
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IC & DSP
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4.8.2 Relating the FS to the FT 1
The relationship between the FS coeffs
and the FT of a finite-duration
nonperiodic signal
0
0 0
0
0
0 0
( ) 0, 0 or
( ) ( )
1 1[ ] ( ) ( )
( ) ( )
1[ ] ( )
m
TT
jk t jk t
j t
k
x t t t T
x t x t mT
X k x t e dt x t e dtT T
X j x t e dt
X k X jT
Kim, J. Y.
IC & DSP
Research
Group
4.9 The Discrete-Time Fourier Series Approximation to the Fourier Transform - 생략
Approximating the FT
Requirement
– aliasing error
– resolution
Kim, J. Y.
IC & DSP
Research
Group
Requirement 1 : Aliasing Error
Sampling induces aliasing error
Approximate X(jω) on the interval
–ωa< ω< ωa (x(t) is bandlimitted with ωm > ωa
( ) ( )
1( ) ( ( ))
FT
s
ks
x t X j
X j X j kT
2s
m a
T
Kim, J. Y.
IC & DSP
Research
Group
Requirement 2 : Resolution
1( ) ( ) ( )
2
1( ) ( ) ( )
2
j j jY e X e W e
Y j X j W j
The windowing operation of M -> periodic
convolution
( 1) / 2
1,0 1[ ]
0,
sin( / 2)( )
sin( )2
sj T M s
s
n Mw n
otherwise
M TW j e
T
Kim, J. Y.
IC & DSP
Research
Group
Requirement 3 : Resolution
Resolution
– We cannot resolve details in the spectrum that
are closer than a mainlobe width apart.
– Resolution = mainlobe width ωs/M
– Requirement of Resolution ωr : 2s
s
r r
M MT
Total time over which
we sample x(t)
Kim, J. Y.
IC & DSP
Research
Group
Requirement 4 : Resolution
DTFS sample the DTFT at intervals of 2π/N
If desired sampling interval is at least Δω, then we
require that
DTFS approximation
;2 /
2 /
/
[ ] [ ] ( )
[ ] (1/ ) ( )
1[ ] ( ) ( 2 /( ) / )s
sampleDTFS Nj
j k N
jk N
s s
y n Y k Y e
Y k N Y e
Y k Y e NT NN
sN
1[ ] ( / )s
s
Y k X jk NNT
Kim, J. Y.
IC & DSP
Research
Group
Example 4.15 DTFS Approximation
of the FT for Damped Sinusoid
/10
/10
( ) ( )(cos(10 ) cos(12 ))
( ) ( )
( ) ( )
( ) cos(10 ) cos(12 )
t
t
x t e u t t t
f t g t
f t e u t
g t t t
2 2 2 2
1( )
1
10
( ) ( 10) ( 10) ( 12) ( 12)
1 1
10 10( )1 1
( ) 10 ( ) 1210 10
F j
j
G j
j j
X j
j j
Kim, J. Y.
IC & DSP
Research
Group
The DTFS approximation to the FT of x(t) = e-1/10 u(t)(cos(10t) + cos(12t).
The solid line is the FT |X(j)|, and the stems denote the DTFS
approximation NTs|Y[k]|. Both |X(j) and NTs|Y[k]| have even symmetry, so
only 0 < < 20 is displayed. (a) M = 100, N = 4000. (b) M = 500, N = 4000. (c)
M = 2500, N = 4000. (d) M = 2500, N = 16,0000 for 9 < < 13.
Kim, J. Y.
IC & DSP
Research
Group
DTFS Approximation to the FT of
a periodic signals
A complex sinusoid case
0
0
0
0
0
0
( )
( ) ( ) 2 ( )
2( ) ( )
( ) ( ( ))
( ) ( ( ))
[ ] ( ( ))
j t
FT
s
ks
s
k
s
x t ae
x t X j a
X j kT
Y j a W j k
Simplify
Y j aW j
aY k W j k
N N
? , 0[ ]
0,
a kY k
otherwise
Kim, J. Y.
IC & DSP
Research
Group
Example 4.16 DTFS Approximation of Sinusoids
The DTFS approximation to the FT of x(t) = cos(2(0.4)t) + cos(2(0.45)t). The
stems denote |Y[k]|, while the solid lines denote (1/M|Y (j)|. The frequency
axis is displayed in units of Hz for convenience, and only positive frequencies
are illustrated. (a) M = 40. (b) M = 2000. Only the stems with nonzero
amplitude are depicted. (c) Behavior in the vicinity of the sinusoidal frequencies
for M = 2000. (d) Behavior in the vicinity of the sinusoidal frequencies for M =
2010.
Kim, J. Y.
IC & DSP
Research
Group
4.10 Efficient Algorithm for
Evaluating the DTFS - 생략
Fast DTFS algorithm : FFT
