Post on 18-Oct-2020
Kim, J. Y.
IC & DSP
Research
Group
Signal & Systems
Chonnam National University
Dept. of Electronics Engineering
IC&DSP Research Group
Kim, Jin Young
Kim, J. Y.
IC & DSP
Research
Group
3. Fourier Representations
for Signals and Linear Time-
Invariant Systems
Kim, J. Y.
IC & DSP
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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.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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!
Kim, J. Y.
IC & DSP
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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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IC & DSP
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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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http://www.nst.ing.tu-
bs.de/schaukasten/fourier/en_idx.html#DIRI
With sound!!
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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Walsh function is not a eigen
function of LTI system
Kim, J. Y.
IC & DSP
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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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3.4 Discrete-Time Periodic
Singals : Discrete-Time
Fourier Series
Kim, J. Y.
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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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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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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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IC & DSP
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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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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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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
Kim, J. Y.
IC & DSP
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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.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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Example 3.8 Numerical analysis of
the ECG
Electrocardiogram
waveform
- normal
- ventricular complexes
Kim, J. Y.
IC & DSP
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3.5 Continuous-Time Periodic
Signals : The Fourier Series
Kim, J. Y.
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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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IC & DSP
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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)
Kim, J. Y.
IC & DSP
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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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Demo Program
FourierSeries
http://users.ece.gatech.edu/mcclella/matlabGUIs/
Kim, J. Y.
IC & DSP
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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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IC & DSP
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Figure 3.17 (p. 217)
Magnitude and phase spectra for Example 3.9.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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Figure 3.18 (p. 219)
Magnitude and phase spectra for Example 3.11.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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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IC & DSP
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Example 5
(Example3.15) 0 / 1/ 4, 1 , 0.1T T T s RC s
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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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3.6 Discrete-Time Nonperiodic
Signals : The Discrete-Time
Fourier Transform
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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
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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)
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IC & DSP
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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
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IC & DSP
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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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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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IC & DSP
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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
Kim, J. Y.
IC & DSP
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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.
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IC & DSP
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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
Kim, J. Y.
IC & DSP
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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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Example 3
Inverse DTFT
1[ ]
2
1sin( )
W
j n
W
x n e d
Wnn
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Examples 4
Example3.20 : impulse x[n]=[n]
(sol)
[ ] 1DTFT
n
( ) [ ] 1j j n
n
X e n e
※임펄스 응답의 의미??
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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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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
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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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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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3.7 Continuous-Time Nonperiodic Singals : The Fourier Transform
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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
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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)
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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
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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
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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
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Figure 3.39 (p.
243)
Example 3.24. (a)
Real time-domain
exponential signal.
(b) Magnitude
spectrum.
(c) Phase spectrum.
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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
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Normalized sinc function
Unnormalized sinc function
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Examples 3
Example 3.26
1,)
0,| |
1( ) sin( )
( )
W WX(jω
W
x t Wtt
W Wtx t sinc
t
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Example 4
Example 3.27
Example 3.28
( ) 1FT
t
( ) 1FT
t
1 2 ( )FT
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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
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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
Kim, J. Y.
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Periodicity Properties
Time-Domain Properties Frequency-Domain
Properties Properties Continuous Nonperiodic Discrete Periodic Periodic Discrete Nonperiodic Continuous
Kim, J. Y.
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3.9 Linearity and Symmetric
Properties
All the four Fourier representations: linear
operator
Symmetric properties : real and Imaginary
signals, even and odd signals
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
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Example 3.3 RC Circuit : filtering
/( )1( ) ( )
1( )
1
t RC
C
C
h t e u tRC
H jj RC
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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3.11 Differentiation and
Integration Properties
Integration circuit
Differentiation
circuit
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
???
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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Integration 3 : Proof
Integration is the convolution with
unit step function
Kim, J. Y.
IC & DSP
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3.12 Time- and Frequency-Shift
Properties
The effect of time and frequency shifts on the
Fourier representation
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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Time-Shift Properties 3
Example3.41
0
0
2( ) sin( )
2( ) sin( )j T
X j T
Z j e T
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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3.13 Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions
Partial fraction expansion of rational
function : ???
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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)
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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The effect of truncating the impulse
response of a discrete time system
Kim, J. Y.
IC & DSP
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3.15 Scaling Property
Effect of scaling the time variable :
Fourier transform z(t)=x(at)
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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Scaling Properties 2
The FT scaling property. The figure assumes that
0 < a < 1.
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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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
Kim, J. Y.
IC & DSP
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3.18 Duality
Duality of rectangular pulses and sinc functions.
Kim, J. Y.
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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
Kim, J. Y.
IC & DSP
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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