signal & systems -...
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![Page 1: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/1.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Signal & Systems
Chonnam National University
Dept. of Electronics Engineering
IC&DSP Research Group
Kim, Jin Young
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Kim, J. Y.
IC & DSP
Research
Group
3. Fourier Representations
for Signals and Linear Time-
Invariant Systems
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Kim, J. Y.
IC & DSP
Research
Group
3.1 Introductions
Represent a signal as a weighted
superposition of complex sinusoids.
The study of signals and systems using
sinusoidal representation is termed
Fourier analysis after Joseph Fourier
(1768-1830) for his contributions to the
theory of representing functions as
weighted superposition of sinusoids.
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Kim, J. Y.
IC & DSP
Research
Group
3.2 Complex Sinusoids and
Frequency Response of LTI Systems
Frequency response : the response of an
LTI system to a sinusoidal input
h(t)
A
-A
A|H(j)|
-A|H(j)|
( ) j tx t e
( ) ( ) j tx t H j e
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Kim, J. Y.
IC & DSP
Research
Group
Complex Sinusoids and LTI
Systems 1
A complex sinusoid input to a LTI system
generates an output equal to the
sinusoidal input multiplied by the system
frequency response
[ ] ( ) , ( ) [ ]j j n j j k
k
y n H e e where H e h k e
( ) ( ) , ( ) ( )j t jy t H j e where H j h e d
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Kim, J. Y.
IC & DSP
Research
Group
Complex Sinusoids and LTI
Systems 2
Eigenfunction and eigenvalue of a system
H (t)
[n]
(t)
[n]
H
H nj
e njj
eeH
)(
j te ( ) j tH j e
1 1
( ) ( ) ( )k k
M Mi t i t
k k k
k k
x t a e y t a H j e
Signal decomposition Convolution is not necessary!
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Kim, J. Y.
IC & DSP
Research
Group
3.3 Fourier Representation for
Four Classes of Signals
Time Periodic Nonperiodic
Continuous Fourier Series
Fourier
Transform
Discrete
Discrete-Time
Fourier Series
Discrete-Time
Fourier
Transform
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Kim, J. Y.
IC & DSP
Research
Group
A function in the
functional space
Othogonal basis functions
of the functional space
Saw tooth wave
1
cosnx
sinnx
Coordinate system
ak=<x(t), coskx>
bk=<x(t), sinkx>
*,k m k m
T
dt
….
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Kim, J. Y.
IC & DSP
Research
Group
http://www.nst.ing.tu-
bs.de/schaukasten/fourier/en_idx.html#DIRI
With sound!!
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Kim, J. Y.
IC & DSP
Research
Group
Orthogonality of Complex
Sinusoids 1
The orthogonality of complex sinusoids plays a
key role in Fourier representations.
Orthogonal if their inner product is zero.
Orthogonality of periodic signals
- Discrete time signal
- Continuous time signal
)(],[][ ,
*
, mkInnI mkm
Nn
kmk
*
, ,, ( )k m k m k m
T
I dt I k m
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Kim, J. Y.
IC & DSP
Research
Group
Walsh function is not a eigen
function of LTI system
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Kim, J. Y.
IC & DSP
Research
Group
Orthogonality of Complex
Sinusoids 2
Complex sinusoid with frequency k0
- Discrete time case
- Continuous time case
0
0
1( ) 2
,
0
,,
10,,
1
Nj k m n jk n
k m
n jk
N k mN k m
I e ek mk m
e
0
0
( )
, ( )
0
0 0
,,
10,,
( )
TTj k m t
k m j k m t
T k mT k m
I e dtk me k m
j k m
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Kim, J. Y.
IC & DSP
Research
Group
3.4 Discrete-Time Periodic
Singals : Discrete-Time
Fourier Series
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Kim, J. Y.
IC & DSP
Research
Group
N Complex Sinusoids in N-
periodic Functional Space
0 0 0 0
0 0 0
( )
2
2
j kn j k N n j kn j Nn
j Nnj kn j kn j knj nN
e e e e
e e e e e
0
0
1( ) 2
,
0
,,
10,,
1
Nj k m n jk n
k m
n jk
N k mN k m
I e ek mk m
e
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Kim, J. Y.
IC & DSP
Research
Group
The DTFS Representation
The DTFS representation for x[n] is given
by
where x[n] has fundamental period N and
0=2/N.
0
0
[ ] [ ]
1[ ] [ ]
jk n
k N
jk n
n N
x n X k e
X k x n eN
0;
[ ] [ ]DTFS
x n X k
e(jΩ0ln)
e(jΩ0kn)
….
e(jΩ0mn) x(n)=x(n+N)
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Kim, J. Y.
IC & DSP
Research
Group
The DTFS Representation
X[k] is N periodic in k
0
0 0
0
0
( )
2
1[ ] [ ]
1[ ]
( )
1[ ] [ ]
j k N n
k N
jk n jN n
k N
jN n j n
jk n
k N
X N k x n eN
x n e eN
e e
x n e X kN
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3.3, pp205
x[n]=cos(/3n+) (0=2/6)
/ 2, 1
[ ] / 2, 1
0, 2 3
j
j
e k
X k e k
k
3 3
3 3
3
3
2
[ ]2
1 1
2 2
[ ]
j n j n
j n j nj j
j kn
k
e ex n
e e e e
X k e
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3.6 (1)
DTFS for the N periodic square wave
M -M
N
0
0 0
0 0
0
2
0
(2 1)
1[ ] 1
11
1, 0, , 2
1
Mjk n
n M
Mjk M jk n
n
jk M jk M
jk
X k eN
e eN
e ek N N
N e
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Kim, J. Y.
IC & DSP
Research
Group
0 0
0 0
0 0
0 0
(2 1) / 2 (2 1)
/ 2
(2 1) / 2 (2 1) / 2
/ 2 / 2
0
0
1 1[ ]
1
1
sin( (2 1) / 2)1
sin( / 2)
sin( (2 1))1
, 0, , 2
sin( )
jk M jk M
jk jk
jk M jk M
jk jk
e eX k
N e e
e e
N e e
k M
N k
k MN k N N
Nk
N
sin( (2 1))1
, 0, , 2
sin( )[ ]
2 1, , 0, , 2
k MN k N N
NkX k
N
Mk N N
N
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3.6 (3)
See figure 3.12(pp. 211)
(a) M=4
(b) M=12
The DTFS coefficients have even symmetry, X[k]=X[-k], and we my rewrite the DTFS as a series involving harmonically related cosines.
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3.6 (4) :
Harmonically Related Cosines
0
0 0
0
0 0
/ 2
/ 2 1
/ 2 1
1
( / 2)
0
/ 2 1
1
/ 2 1
0
1
[ ] [ ]
[0] ( [ ] [ ] )
[ / 2]
( [ ] [ ] and 2 )
[0] 2 [ ]( ) [ / 2]2
[0] 2 [ ]cos( ) [ / 2]cos( )
Njk n
k N
Njm n jm n
m
j N n
jm n jm nNj n
m
N
m
x n X k e
X X m e X m e
X N e
X m X m N
e eX X m X N e
X X m m n X N n
/ 2
0
0
[ ], 0, / 2[ ]
2 [ ], 1,2,..., / 2 1
[ ] [ ]cos( )N
k
X k k NB k
X k k N
x n B k k n
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3.7
Define a partial sum approximation to x[n]
as
where JN/2. N=50 and M=12
J=1,3,5,23, and 25.
(sol)
0
0
[ ], 0, / 2[ ]
2 [ ], 1,2,..., / 2 1
ˆ [ ] [ ]cos( )J
J
k
X k k NB k
X k k N
x n B k k n
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Kim, J. Y.
IC & DSP
Research
Group
Example 3.8 Numerical analysis of
the ECG
Electrocardiogram
waveform
- normal
- ventricular complexes
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Kim, J. Y.
IC & DSP
Research
Group
3.5 Continuous-Time Periodic
Signals : The Fourier Series
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Kim, J. Y.
IC & DSP
Research
Group
0
0
( )
, ( )
0
0 0
,,
10,,
( )
TTj k m t
k m j k m t
T k mT k m
I e dtk me k m
j k m
Infinite Complex Sinusoids in
T-periodic Functional Space
0 0 if j kt j mt
e e k m
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Kim, J. Y.
IC & DSP
Research
Group
The FS Representation
Fourier Series
0
0
( ) [ ]
1[ ] ( )
jk t
k
jk t
T
x t X k e
X k x t e dtT
0;
( ) [ ]FS
x t X k
e(jω0lt)
e(jω0kt)
….
e(jω0mt)
x(t)=x(t+T)
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Kim, J. Y.
IC & DSP
Research
Group
The FS Representation
Truncated approximation
Under what conditions does the infinite
series actually converge to x(t)?
0
0ˆ( ) [ ] , 2 /
Jjk t
k J
x t A k e where T
T
dttxT
2|)(|
1
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Kim, J. Y.
IC & DSP
Research
Group
Demo Program
FourierSeries
http://users.ece.gatech.edu/mcclella/matlabGUIs/
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Kim, J. Y.
IC & DSP
Research
Group
Example 3.9 Direct Calculation of FS
Coef.
Examples 1
2 42
0
1 1[ ]
2 4 2
t jk t eX k e e dt
jk
※만약 k가 크면, 크기는 k에 반비례
위상은 분모가 허수로 보이므로 π/2(90도)
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Kim, J. Y.
IC & DSP
Research
Group
Figure 3.17 (p. 217)
Magnitude and phase spectra for Example 3.9.
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Kim, J. Y.
IC & DSP
Research
Group
Examples 2
(Example3.11) by inspection
(sol) ( ) 3cos2 4
x t t
/ 4 ( / 2) / 4 ( / 2)
0
/ 4
/ 4
( ) 3cos2 4
3 3( / 2)
2 2
(3/ 2) , 1
[ ] (3 / 2) , 1
0, otherwise
j j t j j t
j
j
x t t
e e e e
e k
X k e k
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Kim, J. Y.
IC & DSP
Research
Group
Figure 3.18 (p. 219)
Magnitude and phase spectra for Example 3.11.
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Kim, J. Y.
IC & DSP
Research
Group
Examples 2
(Example3.13) a square wave
0
0
/ 2
/ 2
0
0
1[ ] ( )
1( )
2sin( ), 0
2, 0
s
s
T
jk t
T
T
jk t
T
s
X k x t e dtT
x t e dtT
k Tsk
Tk
Tk
T
※X(k)는 k에
관한 우함수
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Kim, J. Y.
IC & DSP
Research
Group
Figure 3.22a&b (p. 222)
The FS coefficients, X[k], –50 k 50, for three square
waves. (see Fig. 3.21.) (a) Ts/T = 1/4 . (b) Ts/T = 1/16.
(c) Ts/T = 1/64.
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Kim, J. Y.
IC & DSP
Research
Group
0
0 0
0 0
1
1
0
1
( ) [ ]
[0] ( [ ] [ ] )
[0] 2 [ ]( )2
[0] 2 [ ]cos( )
jk t
k
jm t jm t
m
jm t jm t
m
m
x t X k e
X X m e X m e
e eX X m
X X m m t
0
0
[0], 0[ ]
2 [ ], 0
[ ] [ ]cos( )k
X kB k
X k k
x n B k k t
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Kim, J. Y.
IC & DSP
Research
Group
Examples 4
(Example3.14) Partial sum approximation
( 1) / 2
0
0
1/ 2, 0
[ ] 2( 1) /( ),
0,
ˆ [ ] [ ]cos( )
k
J
J
k
k
B k k k odd
k even
x t B k k t
![Page 40: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/40.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Figure 3.25 Individual terms (left panel) in the FS expansion of a square wave and the corresponding partial-sum approximations J(t) (right panel). The square wave has period T = 1 and Ts/T = ¼ . The J = 0 term is 0(t) = ½ and is not shown.
(a) J = 1. (b) J = 3. (c) J = 7. (d) J = 29. (e) J = 99.
x
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Kim, J. Y.
IC & DSP
Research
Group
Example 5
(Example3.15) 0 / 1/ 4, 1 , 0.1T T T s RC s
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Kim, J. Y.
IC & DSP
Research
Group
0 0
0( ) [ ] ( ) ( ) [ ]
1/( )
1/
jk t jk t
k k
x t X k e y t H jk X k e
RCH j
j RC
0
2
100
100
10 sin( / 2)[ ]
2 10
[ ] 1/
( ) [ ]jk t
k
kY k
j k k
Y k k
y t Y k e
The FS coefficients Y[k], –25
k 25, (a) Magnitude
spectrum. (b) Phase
spectrum. c) One period of
the input signal x(t) dashed
line) and output signal y(t)
(solid line). The output
signal y(t) is computed from
the partial-sum
approximation
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Kim, J. Y.
IC & DSP
Research
Group
3.6 Discrete-Time Nonperiodic
Signals : The Discrete-Time
Fourier Transform
![Page 44: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/44.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Derivation 1
Develop the DTFT from the DTFS by
describing a non-periodic signal as the
limit of a periodic signal whose period N,
approaches infinity.
Approximate x[n] with periodic signal.
1) [ ] [ ],
2) [ 2 1] [ ] : periodic DT Fourier series
3) [ ] lim [ ]M
x n x n M n M
x n M x n
x n x n
![Page 45: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/45.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Derivation 2 0
0
[ ] [ ]
1[ ] [ ]
2 1
Mjk n
k M
Mjk n
k M
x n X k e
X k x n eM
0
continuous function of frequency ( )
( ) [ ]
[ ] ( ) /(2 1)
j
Mj j n
k M
jk
X e
X e x n e
X k X e M
0 0
0 0
0
0
1[ ] ( )
2 1
using the relation 2 /(2 1)
1[ ] ( )
2
Mjk jk n
k M
Mjk jk n
k M
x n X e eM
M
x n X e e
-M M
-M M
x(n)
~x(n)
![Page 46: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/46.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Derivation 3
0 0
0
0
0
[ ] is the limiting value of [ ] as
1[ ] lim ( )
2
1 lim ( ) Rieman Integral
2
1[ ] ( )
2
Mjk jk n
Mk M
Mj j n
kMk M
j j n
x n x n M
x n X e e
X e e
x n X e e d
-π π
( )j j nX e e
….
0
2
2 1M
0
2
2 1k k k
M
![Page 47: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/47.jpg)
Kim, J. Y.
IC & DSP
Research
Group
The DTFT is expressed as
DTFT Representation 1
1[ ] ( )
2
( ) [ ] inner product ( ),
j j n
j j n j n
n
x n X e e d
X e x n e x n e
[ ] [ ]DTFT
jx n X e
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Kim, J. Y.
IC & DSP
Research
Group
DTFT Representation 2
If x[n] is absolutely summable, that is,
then the sum of DTFT converges uniformly a continuous function of .
If x[t] is not absolutely summable, but does have finite energy that is,
then it can be shown that sum of DTFT converges in a mean-square error sense
| [ ] |n
n
x n
2| [ ] |n
n
x n
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Kim, J. Y.
IC & DSP
Research
Group
Examples 1
Example3.17 x[n]=nu[n]
0
0
( ) [ ]
( )
1,| | 1
1
j n j n
n
n j n
n
j n
n
j
X e u n e
e
e
e
2 2 2 1/ 2
1( )
((1 cos ) sin )
sinarg{ ( )} arctan
1 cos
j
j
X e
X e
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Kim, J. Y.
IC & DSP
Research
Group
Figure 3.29 (p.232)
The DTFT of an exponential signal x[n] = ()nu[n]. (a) Magnitude
spectrum for = 0.5. (b) Phase spectrum for = 0.5. (c) Magnitude
spectrum for = 0.9. (d) Phase spectrum for = 0.9.
![Page 51: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/51.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 2
Example3.18 : Rectangular pulse
(sol)
1,| |[ ]
0,| |
n Mx n
n M
2 2( )
0 0
(2 1)
( ) 1 1
1, 0, 2 , 4 ,...
1
2 1, 0, 2 , 4 ,..
sin( (2 1) / 2)
sin( / 2)
M M Mj j n j m M j M j m
n M m m
j Mj M
j
X e e e e e
ee
e
M
M
![Page 52: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/52.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Figure 3.30 (p. 233)
Example 3.18. (a) Rectangular pulse in the time domain. (b)
DTFT in the frequency domain.
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Kim, J. Y.
IC & DSP
Research
Group
Example 3
Inverse DTFT
1[ ]
2
1sin( )
W
j n
W
x n e d
Wnn
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Kim, J. Y.
IC & DSP
Research
Group
Examples 4
Example3.20 : impulse x[n]=[n]
(sol)
[ ] 1DTFT
n
( ) [ ] 1j j n
n
X e n e
※임펄스 응답의 의미??
![Page 55: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/55.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 5
Example3.21 : Find the inverse DTFT of
(sol)
( ) ( ),jX e
1 1[ ] ( )
2 2
j nx n e d
1( )
2
DTFT
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Kim, J. Y.
IC & DSP
Research
Group
Examples 6
Example 3.22 : Moving-average system
1
1
1
/ 2
1[ ] [ ] [ 1]
2
1 1[ ] [ ] [ 1]
2 2
1 1
2 2
cos2
j j
j
y n x n x n
h n n n
H e e
e
2
2
2
/ 2
1[ ] [ ] [ 1]
2
1 1[ ] [ ] [ 1]
2 2
1 1
2 2
sin2
j j
j
y n x n x n
h n n n
H e e
je
![Page 57: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/57.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 7
Multipath communication channel
[ ] [ ] [ 1]
1j j
y n x n ax n
H e ae
Magnitude response of the system in Example 3.23 describing
multipath propagation. (a) Echo coefficient a = 0.5ej/3. (b)
Echo coefficient a = 0.9ej2/3.
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Kim, J. Y.
IC & DSP
Research
Group
Figure 3.38 (p. 241)
Magnitude response of the inverse system for multipath
propagation in Example 3.23. (a) Echo coefficient a =
0.5ej/3. (b) Echo coefficient a = 0.9ej/3
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Kim, J. Y.
IC & DSP
Research
Group
3.7 Continuous-Time Nonperiodic Singals : The Fourier Transform
![Page 62: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/62.jpg)
Kim, J. Y.
IC & DSP
Research
Group
FT Representation 1
The FT is expressed as
1( ) ( )
2
( ) ( )
j t
j t
x t X j e d
X j x t e dt
( ) [ ]FT
x t X j
![Page 63: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/63.jpg)
Kim, J. Y.
IC & DSP
Research
Group
0
0
0
0
0
0
00
0 0
( ) ( ),
( 2 ) ( )
1[ ] ( )
2
( ) [ ]
( ) ( )
1[ ] ( )
2
1( ) ( )
2
2 1
2 2 2
1( ) ( )
2
( )
Tjk t
T
jk t
k
Tj t
T
jk t
k
jk t
k
x t x t T t T
x t T x t
X k x t e dtT
x t X k e
X j x t e dt
X k X jkT
x t X jk eT
T T
x t X jk e
x t
1( )
2
( ) ( )
j t
j t
X j e d
X j x t e dt
-T T
-T T
x(t)
~x(t)
![Page 64: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/64.jpg)
Kim, J. Y.
IC & DSP
Research
Group
FT Representation 2
About convergence - Square integrable
: MSE between x(t) and x’(t), where
*Zero MSE does not imply pointwise convergence
2| ( ) |x t dt
1'( ) ( )
2
j tx t X j e d
![Page 65: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/65.jpg)
Kim, J. Y.
IC & DSP
Research
Group
FT Representation 3
- Dirichlet condition
o Absolute integrable
o A finite number of local maxima, minma, and discontinuities in any finite interval o The size of each discontinuity is finite
Pointwise convergence at all values of t except those corresponding to discontinuities.
| ( ) |x t dt
![Page 66: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/66.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 1
Example 3.24 x(t)=e-atu(t)
(sol)
0
, 0, FT dose not converge for a 0ate dt a
( )
0
( )
0
For a 0
) ( )
1 1
at j t a j t
a j t
X(jω e u t e d e d
ea j a j
![Page 67: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/67.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Figure 3.39 (p.
243)
Example 3.24. (a)
Real time-domain
exponential signal.
(b) Magnitude
spectrum.
(c) Phase spectrum.
![Page 68: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/68.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 2
Example 3.25
(sol)
1,[ ]
0,| |
T t Tx n
t T
) ( )
1 2sin( )
T
j t j t
T
T
j t
T
X(jω x t e d e d
e Tj
![Page 69: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/69.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Normalized sinc function
Unnormalized sinc function
![Page 70: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/70.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Examples 3
Example 3.26
1,)
0,| |
1( ) sin( )
( )
W WX(jω
W
x t Wtt
W Wtx t sinc
t
![Page 71: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/71.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Example 4
Example 3.27
Example 3.28
( ) 1FT
t
( ) 1FT
t
1 2 ( )FT
![Page 75: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/75.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.8 Properties of Fourier
Representation
Periodicity properties
Linearity Symmetry properties Time-shift properties Scaling properties Differentiation and integration Convolution and modulation properties Parseval relationships Duality Time-bandwidth product
![Page 76: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/76.jpg)
Kim, J. Y.
IC & DSP
Research
Group
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
![Page 77: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/77.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Periodicity Properties
Time-Domain Properties Frequency-Domain
Properties Properties Continuous Nonperiodic Discrete Periodic Periodic Discrete Nonperiodic Continuous
![Page 78: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/78.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.9 Linearity and Symmetric
Properties
All the four Fourier representations: linear
operator
Symmetric properties : real and Imaginary
signals, even and odd signals
![Page 79: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/79.jpg)
Kim, J. Y.
IC & DSP
Research
Group
0
0
;
;
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) [ ] [ ]
[ ] [ ] [ ] ( ) [ ] [ ]
[ ] [ ] [ ] ( ) [ ] [ ]
FT
FS
DTFTj j j
DTFS
z t ax t by t Z j aX j bY j
z t ax t by t Z j aX k bY k
z n ax n by n Z e aX e bY e
z n ax n by n Z j aX k bY k
Linearity 1
![Page 80: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/80.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Linearity 2
Example3.30 : Find the FS coefficients of z(t)
3 1( ) ( ) ( )
2 2z t x t y t
2
( ) [ ] (1/( ))sin( / 4)x t X k k k
2
( ) [ ] (1/( ))sin( / 2)y t Y k k k
3 1[ ] [ ] [ ]
2 2Z k X k Y k
![Page 81: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/81.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Symmetry Properties - Real
and Imaginary Signals
x(t) is real , table3.4
x(t) is imaginary, table3.5
*
*
* ( )
( ) ( )
( ) ( )
( )
j t
j t j t
X j x t e dt
x t e dt x t e dt
X j
*
*
* ( )
( ) ( )
( ) ( )
( )
j t
j t j t
X j x t e dt
x t e dt x t e dt
X j
![Page 82: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/82.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Symmetry Properties – Even and
Odd Signals
x(t) is real and even
x(t) is real and odd
( ) is realX j
( ) is imaginaryX j
* *( ) ( ) ( )
( ) ( )
( )
Im[ ( )] 0
j t j t
j
X j x t e dt x t e dt
x e d t
X j
X j
![Page 84: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/84.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.10 Convolution Property
T *
Convolution Multiplication
Analyze the input-output behavior of a linear
system in the frequency domain by multiplying
transforms instead of convolving time signals!
[ ] [ ]* [ ]DTFT
j j jy n x n h n Y e X e H e
![Page 85: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/85.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Nonperiodic Convolution
( )
( )
( )
( )
( )
1( ) ( )
2
1( ) ( ) ( )
2
1( ) ( )
2
1( ) ( )
2
1( ) ( )
2
1( ) ( )
2
j t
j t
j t
j t
j t
j j t
x t X j e d
y t h X j e d d
h X j e d d
h X j e d d
h X j e d d
h e d X j e
1( ) ( ) ( )
2
j t
d
y t H j X j e d
( ) ( )* ( )
( ) ( )
y t h t x t
h x t d
[ ] [ ]* [ ]DTFT
j j jy n x n h n Y e X e H e
( ) ( )* ( ) ( ) ( ) ( )FT
y t h t x t Y j X j H j
![Page 86: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/86.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Example 3.31 : A convolution
problem in the frequency domain
y(t)=x(t)*h(t) ( ) (1/( ))sin( )
( ) (1/( ))sin(2 )
x t t t
h t t t
1,( ) ( )
0,
FT
x t X j
1, 2( ) ( )
0, 2
FT
h t H j
1,( ) ( ) (1/( ))sin( )
0,Y j y t t t
![Page 87: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/87.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Example 3.32 : Inverse FT
Find x(t)
2
2
4( ) ( ) sin ( )
FT
x t X j
( ) ( ) ( )
2( ) sin( )
( ) ( )* ( )
X j Z j Z j
Z j
x t z t z t
![Page 88: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/88.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Filtering
Filtering : multiplication that occurs in the
frequency-domain representation
A system performs filtering on the signal by
presenting a different response to components
of the input that are at different frequencies.
The term, filtering, implies that some frequency
components of the input are eliminated while
others are passed by the system unchanged
![Page 89: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/89.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Frequency response of ideal continuous- (left panel)
and discrete-time (right panel) filters. (a) Low-pass
characteristic. (b) High-pass characteristic. (c)
Band-pass characteristic
![Page 90: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/90.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Example 3.3 RC Circuit : filtering
/( )1( ) ( )
1( )
1
t RC
C
C
h t e u tRC
H jj RC
![Page 91: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/91.jpg)
Kim, J. Y.
IC & DSP
Research
Group
System Identification
If the spectrum in nonzero at all frequencies,
the frequency response of a system be
determined from the knowledge of the input
and output spectra.
( )( )
( )
( )( )
( )
jj
j
Y jH j
X j
Y eH e
X e
H y(t) x(t) 2( ) ( )tx t e u t
1( )
2
1( )
1
X jj
Y jj
1( ) 1
1
( ) ( ) ( )t
H jj
h t t e u t
( ) ( )ty t e u t
Example 3.4
![Page 92: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/92.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Inverse System : Recover the input
of the system from the output
1( ) ( ) ( )
( )
invX j H Y j Y jH j
H y(t) x(t) Hinv x(t)
[ ] [ ] [ 1],| | 1y n x n ax n a
1, 0
[ ] , 1
0,
n
h n a n
ohterwise
( ) 1
1( )
1
j j
inv j
j
H e ae
H eae
Example 3.5 : equalization
[ ] ( ) [ ]inv nh n a u n
![Page 93: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/93.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Convolution of Periodic Signals
Periodic convolution and (DT)FS
representations
0
2;
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) [ ] [ ] [ ]
T
FST
y t x t z t x z t d
y t x t z t Y k TX k Z k
1
0
2;
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]
N
k
DTFSN
y n x n z n x k z n k
y n x n z n Y k NX k Z k
![Page 94: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/94.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Example3.36 : Convolution of Two
Periodic Signals
Periodic convolution of two signals
( ) 2cos(2 ) sin(4 )z t t t
1, 1
1/(2 ), 2[ ]
1/(2 ), 2
0,
2sin( / 2)[ ]
2
k
j kz k
j k
otherwise
kX k
k
1, 1[ ]
0,
( ) (2 / )cos(2 )
ky k
otherwise
y t t
![Page 95: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/95.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.11 Differentiation and
Integration Properties
Integration circuit
Differentiation
circuit
![Page 96: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/96.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Differentiation in Time 1
Differentiation of (non)periodic signal in time
1( ) ( )
2
1( ) ( )
2
j t
j t
x t X j e d
dx t X j j e d
dt
( ) ( )FTd
x t j X jdt
0:
0( ) [ ]FSd
x t jk X kdt
0
0
0
( ) [ ]
( ) [ ]
jk t
k
jk t
k
x t X k e
dx t X k jk e
dt
nonperiodic periodic
![Page 97: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/97.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Differentiation in Time 2
Example3.37 : verify the following result.
(sol)
( ( ))FT
atd je u t
dt a j
( ( )) ( ) ( )
( ) ( )
at at at
at
de u t ae u t e t
dt
ae u t t
( ( )) 1FT
atd a je u t
dt a j a j
![Page 98: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/98.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Differentiation in Frequency 1
( ) ( )
( ) ( )
j t
j t
X j x t e dt
dX j jtx t e dt
d
( ) ( )FT d
jtx t X jd
( ) [ ]j j n
n
X e x n e
[ ] ( )FT
jdjnx n X e
d
nonperiodic periodic
Differentiation of (non)periodic signal in
frequency
( ) [ ]j j n
n
dX e jnx n e
d
![Page 99: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/99.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Differentiation in Frequency 2
Exampe 3.40 ; FT of a Gaussian Pulse
pp.275-276
2 2/ 2 / 2(1/ 2 )FT
te e
2 / 2( ) (1/ 2 ) tg t e
???
![Page 100: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/100.jpg)
Kim, J. Y.
IC & DSP
Research
Group
2
2
2
/ 2
/ 2
/ 2
( ) (1/ 2 ) ( )
( ) ( )
( ) ( )
1( ) ( )
( ) ( )
( )
( 0) 1
( )
tdg t e tg t
dt
dg t j G j
dt
tg t j G j
dtg t G j
j d
dG j G j
d
G j ce
G j
G j e
Differentiation in time domain
Differentiation in frequency domain
( ) 1g t dt
![Page 101: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/101.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Integration 1
Integration (smooth signal in time) : FT,
( ) ( ) , ( ) ( )
1( ) ( )( 0)
1( ) ( ) ( )
td
y t x d y t x tdt
Y j X jj
Y j X j cj
1( ) ( ) ( 0) ( )
t FT
x d X j X jj
![Page 102: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/102.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Integration 2
Step function as the sum of a constant and a signum
function : pp.278
( ) ( )
( ) 1
1( ) ( ) ( )
t
FT
FT
u t d
u t U jj
sgn( ) 2 ( )
( ) 2
2, 0
( )
0, 0
dt t
dt
j S j
jS j
1/ 2 ( )
1[ ] ( )
2
j tx t X j e d
![Page 103: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/103.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Integration 3 : Proof
Integration is the convolution with
unit step function
![Page 104: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/104.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.12 Time- and Frequency-Shift
Properties
The effect of time and frequency shifts on the
Fourier representation
![Page 105: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/105.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Time-Shift Properties 1
z(t)=x(t-t0) : time shifted version of x(t)
0
0
0
0
( )
0
( ) ( ) ( )
( ) ( )
( )
( ) ( )
j t j t
j t
j t j
j t
Z j z t e dt x t t e dt
x e d t t
e x e d
Z j e X j
![Page 106: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/106.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Time-Shift Properties 2
Time-shift properties of Fourier representation
0
0
0 0
0
0
0 0
0
;
0
0
;
0
( ) ( )
( ) [ ]
[ ] ( )
[ ] [ ]
FTj t
FSjk t
DTFTj n j
DTFSjk n
x t t e X j
x t t e X k
x n n e X e
x n n e X k
![Page 107: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/107.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Time-Shift Properties 3
Example3.41
0
0
2( ) sin( )
2( ) sin( )j T
X j T
Z j e T
![Page 108: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/108.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Frequency response of a system by
a difference equation
Discrete time system case
0 0
0 0
0
0
[ ] [ ]
( [ ] )
N M
k k
k k
DTFTjk j
N Mk k
j j j j
k k
k k
Mk
jj k
j k
Nj kj
k
k
a y n k b x n k
z n k e Z e
a e Y e b e X e
b eY e
H eX e
a e
![Page 109: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/109.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Frequency-Shift Properties 1
Z(j)=X(j(-))
( )
1( ) ( )
2
1( ( ))
2
( )
1( )
2
1( )
2
( )
j t
j t
j t
j t j t
j t
z t Z j e d
X j e d
X j e d
e X j e d
e x t
![Page 110: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/110.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Frequency-Shift Properties 2
Frequency-shift properties of Fourier
representation
0
0 0
0
0 0
;
0
( )
;
0
( ) ( ( ))
( ) [ ]
[ ] ( )
[ ] [ ]
FTj t
FSjk t
DTFTj n j
DTFSjk n
e x t X j
e x t X k k
e x n X e
e x n X k k
![Page 111: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/111.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Frequency-Shift Properties 3
Example3.42
(sol)
10 ,| |( )
0,
j te tz t
otherwise
10
10
1,| |( ) , ( ) ( )
0,
2( ) ( ) sin( )
( ) ( ( 10))
2( ) sin(( 10) )
10
j t
FT
FTj t
FT
tx t z t e x t
otherwise
x t X j
e x t X j
z t
![Page 112: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/112.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Find the FT of the signal
(sol)
Example 3.43 Using multiple
properties to find an FT
3( ) ( ( ))*( ( 2))t tdx t e u t e u t
dt
22 ( 2) 2
( ) ( )* ( ) ( ) ( )
1( ) ( )
3
( ) ( 2)1
FT
FTat
jFTt
dx t w t v t j W j V j
dt
w t e u tj
ev t e e u t e
j
![Page 113: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/113.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.13 Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions
Partial fraction expansion of rational
function : ???
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Kim, J. Y.
IC & DSP
Research
Group
Inverse Fourier Transform
1 0
1
1 1 0
0
( ) ( ) ( )( )
( ) ( ) ( ) ( )
( )( )
( )
M
M
N N
N
M Nk
k
k
b j b j b B jX j
j a j a j a A j
B jf j
A j
IFTs are obtained from
the pair δ(t) ↔1 and the
differentiation property
1
1
( )( )
( ) ( )k
Nk
k k
FT
Nd t
k
k
CX j
j d
x t C e u t
X(jω) expressed of polynomial in jω
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Kim, J. Y.
IC & DSP
Research
Group
Inverse Discrete-Time Fourier
Transform
1 0
( 1)
1 1 0
1
1
( )( )
( )
1
( ) ( ) [ ]
j M j jj M
j N j N j j
N
Nk
jk k
FT
Nn
k k
k
b e b e b B eX e
e a e a e a A e
C
d e
x t C d u n
X(ejΩ) expressed of polynomial in ejΩ
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Kim, J. Y.
IC & DSP
Research
Group
Example 3.40 : MEMS
Accelerometer : impulse response
Find the impulse response
2 2
1( )
( ) 25,000( ) (10,000)
1/15,000 1/15,000
20,000 5,000
H jj j
j j
5,000 20,000( ) (1/15,000)( ) ( )t th t e e u t
MEMS accelerometer
(Analog Device)
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Kim, J. Y.
IC & DSP
Research
Group
3.14 Multiplication Property
Fourier representation of a product of time-
domain signals : non-periodic, continuous
time 1
( ) ( )2
1( ) ( )
2
j t
j t
x t X j e d
z t X j e d
( )
2
1( ) ( ) ( )
(2 )
( )
1 1( ) ( ( ))
2 2
j t
j t
y t X j Z j e d d
X j Z j d e d
1( ) ( ) ( ) ( )* ( )
2
FT
y t x t z t X j Z j
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Kim, J. Y.
IC & DSP
Research
Group
Multiplication Property 2
( )
2
1[ ] [ ] [ ] ( ) ( ) ( )
2
,
( ) ( ) ( ) ( )
DTFTj j j
j j j j
y n x n z n Y e X e Z e
where
X e Z e X e Z e d
Fourier representation of a product of time-
domain signals : non-periodic, discrete time
Periodic convolution
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Kim, J. Y.
IC & DSP
Research
Group
Windowing
Windowing=Truncating w(t)
x(t)
y(t)
( ) ( ) ( )
1( ) ( ) ( )* ( )
2
,
2( ) sin( )
FT
y t x t w t
y t Y j X j W j
where
W j T
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Kim, J. Y.
IC & DSP
Research
Group
Example 3.46 : Truncating the
Impulse Response
Ideal lowpass filter
IDTFT
DTFT
Ideal lowpass filter
H(ej)
Ideal lowpass filter
h(n) :sinc function
Truncate ILF h[n]
ht(n)
Truncate ILF
Ht(ej)
truncate
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Kim, J. Y.
IC & DSP
Research
Group
The effect of truncating the impulse
response of a discrete time system
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Kim, J. Y.
IC & DSP
Research
Group
3.15 Scaling Property
Effect of scaling the time variable :
Fourier transform z(t)=x(at)
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Kim, J. Y.
IC & DSP
Research
Group
Scaling Properties 1
FT : z(t)=x(at)
( / )
( ) ( )
( )
( )
1( ) ( ) ( , 0)
1( )
| |
j t
j t
j a
Z j z t e dt
x at e dt
at
sign a x e d aa
X ja a
단
( ) ( ) (1/ | |) ( / )FT
z t x at a X j a
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Kim, J. Y.
IC & DSP
Research
Group
Scaling Properties 2
The FT scaling property. The figure assumes that
0 < a < 1.
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Kim, J. Y.
IC & DSP
Research
Group
Scaling Properties 3
Example 3.49 : Using multiple properties to
find an inverse FT
2
( )1 ( / 3)
jd eX j j
d j
2( ) ( / 3)
1, ( ) ( ) ( )
1
j
IFTt
dX j j e S j
d
where S j s t e u tj
3
3
2
3( 2)
3( 2)
( ) ( / 3)
( ) 3 (3 )
3 (3 )
3 ( )
( ) ( )
( ) ( 2)
3 ( 2)
( ) ( )
( ) ( )
3 ( 2)
t
t
j
t
t
Y j S j
y t s t
e u t
e u t
W j e Y j
w t y t
e u t
dX j j W j
d
x t tw t
te u t
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Kim, J. Y.
IC & DSP
Research
Group
Scaling Properties 4
FS : Continuous periodic signal x(t) with period
Tx(at) is also periodic with period T/a
0
/
[ ] ( ) [ ]jka t
T a
aZ k z t e dt X k
T
0;
( ) ( ) [ ] [ ], 0FS a
x at z t Z k X k a
( ( )) ( ) ( )T
x a t x at T x ata
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Kim, J. Y.
IC & DSP
Research
Group
3.16 Parseval Relationships
Energy or power in the time-domain representation of
a signal = energy or power in frequency-domain
representation
2| ( ) |xE x t dt
* *1( ) ( )
2
j tx t X j e d
*
*
*
1( ) ( )
2
1 1( ) ( )
2 2
1( ) ( )
2
j t
x
j t
E x t X j e d dt
X j x t e dt d
X j X j d
2 21| ( ) | | ( ) |
2xE x t dt X j d
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Kim, J. Y.
IC & DSP
Research
Group
Representaion Parseval Relation
FT
FS
DTFT
DTFS
Parseval Relationships 2
PR for the 4 Fourier Representation
2 21| ( ) | | ( ) |
2x t dt X j d
2 21| ( ) | | [ ] |
kT
x t dt X kT
2 2
2
1| [ ] | | ( ) |
2
j
n
x n X e d
2 21| [ ] | | [ ] |
n N k N
x n X kN
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Kim, J. Y.
IC & DSP
Research
Group
Parseval Relationships 3
Example3.50
(sol) By Parseval’s relations
2
2 2
sin ( )
n
Wn
n
2sin( )[ ] , | [ ] |
n
Wnx n x n
n
2
2
1| ( ) |
2
jX e d
1,| |[ ] ( )
0, | |
DTFTj
Wx n X e
W
11
2
W
W
Wd
![Page 131: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/131.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.17 Time-Bandwidth Product
An inverse relationship between the
time and frequency extent of a signal
0 00
0
1,| |( ) ( ) 2 sin
0,| |
FTt T Tx t X j T c
t T
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Kim, J. Y.
IC & DSP
Research
Group
Uncertainty principle
Effect duration and bandwidth
(cf)
Uncertainty principle
1/ 2 1/ 2
2 2 2 2
2 2
| ( ) | | ( ) |
,
| ( ) | | ( ) |
d W
t x t dt X j d
T B
x t dt X j d
1
2d WT B
2 2
: random variable
( ) : probability density function
( ) 0
( ) ( ) : variance
x
p x
E x
E x x p x dx
![Page 133: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/133.jpg)
Kim, J. Y.
IC & DSP
Research
Group
3.18 Duality
Duality of rectangular pulses and sinc functions.
![Page 134: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/134.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Duality 2
Duality : interchangeability property
1( ) ( )
2
2 ( ) ( )
( , )
2 ( ) ( )
j t
j t
j t
x t X j e d
x t X j e d
t t
x X jt e dt
( ) ( )
( ) 2 ( )
FT
FT
f t F j
F jt f
![Page 136: Signal & Systems - contents.kocw.netcontents.kocw.net/KOCW/document/2015/chungnam/kimjinyoung/3.… · IC & DSP Research Group Signal & Systems Chonnam National University Dept. of](https://reader033.vdocuments.net/reader033/viewer/2022060606/605c433e48e6ec40955dc2d8/html5/thumbnails/136.jpg)
Kim, J. Y.
IC & DSP
Research
Group
Duality 4
Example 3.52 : find the FT of
(solution)
1( )
1x t
jt
1( ) ( ) ( )
1
1( )
1
( ) 2 ( ) implies that
( ) 2 ( ) 2 ( )
FTt
FT
f t e u t F jj
F jtjt
F jt f
X j f e u