Discrete Time Rect Function(4B)
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Discrete Time Rect Functions
Young Won Lim
4/20/13
Copyright (c) 2009 - 2013 Young W. Lim.
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Young Won Lim
4/20/13
Fourier Transform Types
Discrete Time Fourier Transform
∞
j

X e  =
∑
1
x [n] =
2π
−j
n
x [n ] e
n = −∞
+π
∫−π
j ω̂
̂n
+ jω
X (e ) e
Discrete Fourier Transform
N−1
X [k ] =
∑
x [n] e
1
x [n ] =
N
− j2 / N  k n
n=0
DT Rect (4B)
3
N −1
∑
 j 2 / N  k n
X [k ] e
k =0
Young Won Lim
4/20/13
DTFT and DTFS
L = 2 N +1
1
L
2π
( L−1) zero
crossings
0
−N
+N
jω
̂
X (e ) =
DTFT (Discrete Time Fourier Transform)
sin( ω
̂ L/2)
jω
̂
= L D L (e )
̂ 2)
sin( ω/
= L⋅ diric( ω
̂ , L)
L = 2 N +1
N0
1
L/ N 0
N0
( L−1) zero
crossings
−N
0
+N
1 sin (π k L / N 0 )
N 0 sin (π k / N 0 )
L
=
⋅drcl (k / N 0 , L)
N0
X [k ] =
DTFS (Discrete Time Fourier Series)
DT Rect (4B)
4
Young Won Lim
4/20/13
DT Rect (4B)
5
Young Won Lim
4/20/13
RectN[n] DTFT
Discrete Time Fourier Transform
DTFT
∞
∑
j

X e  =
1
x [n] =
2π
−j
n
x [n ] e
n = −∞
+N
∑
̂
jω
X (e ) =
̂
− j ωn
e
x [n]
+ j ω̂ N
+⋯+ e
= e
+jω
̂N
e
− j ω(2
̂ N +1)/2
= e
+j ω
̂N
− j ω̂ N
{1 + ⋯ + e
}
=
− jω
̂ 2N
e
+ j ω(2
̂ N +1)/ 2
}
jω
̂
= e
− jω
̂ (2 N +1)
1−e
1 − e− j ω̂
+ j ω(
̂ 2 N +1)/ 2
e
− j ω(
̂ 2 N +1)/ 2
=
DT Rect (4B)
− j ω(2
̂ N +1)/ 2
sin( ω(2
̂ N +1)/ 2)
̂
sin( ω/2)
sin( ω
̂ L/2)
jω
̂
= L D L (e )
̂ 2)
sin( ω/
Dirichlet Function
jω
̂
0
̂
−e
̂ 2
̂ 2
e+ j ω/
− e− j ω/
D L (e ) =
−N
̂
= L⋅ diric( ω
̂ , L)
L = 2 N +1
1
X (e j ω ) e+ j ω n
−e
̂ 2
̂ 2
e+ j ω/
− e− j ω/
X (e ) =
+j ω
̂N
∫−π
̂ 2
e− j ω/
n=−N
= {e
+π
sin( ω
̂ L/2)
̂
Lsin( ω/2)
+N
6
Young Won Lim
4/20/13
Dirichlet Functions
jω
̂
D9 (e )
jω
̂
D9(e ) =
2π
jω
̂
D11 (e ) =
jω
̂
D13 (e ) =
8 zero crossings
jω
̂
D10 (e )
jω
̂
D 10 (e ) =
2π
jω
̂
D12 (e ) =
9 zero crossings
jω
̂
D14 (e ) =
DT Rect (4B)
7
sin( ω
̂ 9/2)
̂
9 sin ( ω/2)
8 zero crossings
sin ( ω
̂ 11/ 2)
̂ 2)
11 sin( ω/
10 zero crossings
sin ( ω
̂ 13/ 2)
̂ 2)
13 sin( ω/
12 zero crossings
sin( ω
̂ 10 / 2)
̂ / 2)
10 sin ( ω
9 zero crossings
sin ( ω
̂ 12/ 2)
̂ 2)
12 sin( ω/
11 zero crossings
sin( ω
̂ 14 /2)
̂ /2)
14 sin ( ω
13 zero crossings
Young Won Lim
4/20/13
Magnitude Response
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-15
DT Rect (4B)
-10
-5
0
5
10
8
15
20
Young Won Lim
4/20/13
Phase Response
3.5
3
2.5
2
1.5
1
0.5
0
-20
-15
DT Rect (4B)
-10
-5
0
5
10
9
15
20
Young Won Lim
4/20/13
DT Rect (4B)
10
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS (1)
Discrete Time Fourier Series
1
X [k] =
N
1
X [k ] =
N0
N−1
N−1
∑
x [ n] e
x [n]e
− j (2 π/ N 0 )k n
− j (2 π / N ) k n
x [n] =
= e
+ j (m) N k
n=0
+N
1
− j (2 π/ N
=
x [n ]e
∑
N 0 n=−N
+ j (2 π N / N 0) k
N 0 X [k ] = e
= e
+ j (2 π/ N 0) N k
L = 2 N +1
∑
+ j(2 π/ N )k n
X [k] e
k=0
n=0
N 0−1
∑
DTFS
0
)k n
= e
− j (2 π N / N 0) k
+⋯+ e
=
− j (m)(2 N +1) k
1−e
⋅
1 − e− j ( m)k
e
⋅
+ j (m) N k
− j (m)( 2 N +1) k / 2
e− j (m) k / 2
+ j (m)(2 N +1) k / 2
e
⋅
sin((m)(2 N +1)k / 2)
sin((m)k / 2)
0
)k
X [k ] =
1 sin ((2 π/ N 0 )(2 N +1) k / 2)
N0
sin ((2 π / N 0 ) k /2)
N0
Dirichlet Function
drcl (t , L) =
DT Rect (4B)
− j (m)(2 N +1) k / 2
−e
e+ j (m) k / 2 − e− j (m) k / 2
− j (2 π/ N 0 )( 2 N +1) k
1−e
⋅
− j (2 π/ N
1−e
1
−N
m = (2 π / N 0)
0
sin(π L t )
Lsin(π t)
+N
11
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS (2)
Discrete Time Fourier Series
1
X [k] =
N
X [k ] =
=
X [k ] =
DTFS
N−1
N−1
∑
x [ n] e
− j (2 π / N ) k n
∑
x [n] =
+ j(2 π/ N )k n
X [k] e
k=0
n=0
1 sin ((2 π/ N 0 )(2 N +1) k / 2)
N0
sin ((2 π / N 0 ) k /2)
drcl (k / N 0 , (2 N +1)) =
1 sin (π k (2 N +1)/ N 0 )
N0
sin (π k / N 0 )
X [k ] =
1 sin (π k L/ N 0 )
N 0 sin (π k / N 0 )
sin (π k (2 N +1)/ N 0 )
(2 N +1)sin (π k / N 0 )
(2 N +1)
⋅drcl (k / N 0 , (2 N +1))
N0
X [k ] =
L
⋅drcl (k / N 0 , L)
N0
Dirichlet Function
L = 2 N +1
N0
1
drcl (t , L) =
jω
̂
−N
DT Rect (4B)
0
D L (e ) =
+N
12
sin(π Lt )
Lsin (π t)
sin( ω
̂ L/2)
̂
Lsin( ω/2)
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS (3)
Discrete Time Fourier Series
1
X [k] =
N
X [k ] =
X [k ] =
DTFS
N−1
N−1
∑
x [ n] e
− j (2 π / N ) k n
x [n] =
∑
+ j(2 π/ N )k n
X [k] e
k=0
n=0
Period : N0 (odd L), 2N0 (even L)
1 sin (π k L/ N 0 )
N 0 sin (π k / N 0 )
L
⋅drcl (k / N 0 , L)
N0
(L-1) zero crossings
Dirichlet Function
L = 2 N +1
N0
1
drcl (t , L) =
jω
̂
−N
DT Rect (4B)
0
D L (e ) =
+N
13
sin(π Lt )
Lsin (π t)
sin( ω
̂ L/2)
̂
Lsin( ω/2)
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS (4)
t = −2
t = −1
t =0
t = +1
t = +2
t = −2
t = −1
t =0
odd L=9
t = +1
t = +2
even L=10
9 zero crossings
8 zero crossings
k=−32
k=−16 k=0
k=+16
k =+32
k=−32
(L-1) zero crossings
k=−16 k=0
k=+16
k=+32
(L-1) zero crossings
L
Dirichlet Function
drcl (t , L) =
DT Rect (4B)
X [k ] =
sin(π L t )
L sin(π t )
9
⋅drcl (k /16 , 9)
16
⋯ −3, −2, −1, 0, +1, +2, +3, ⋯
14
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS (5)
Period : N0 (odd L), 2N0 (even L)
k=−32
k=−16
k=0
k=+16
k=+32
(L-1) zero crossings
DT Rect (4B)
X [k ] =
9
⋅drcl (k /16
15, 9)
16
Young Won Lim
4/20/13
Rect2[n] * δ8[n] DTFS Example
Discrete Time Fourier Series
1
X [k] =
N
DTFS
N−1
N−1
∑
x [ n] e
− j (2 π / N ) k n
x [n] =
∑
+ j(2 π/ N )k n
X [k] e
k=0
n=0
X [k ] =
1 sin (π k (2 N +1)/ N 0 )
N0
sin (π k / N 0 )
X [k ] =
L
⋅drcl (k / N 0 , L)
N0
1 sin(π k 5 /8)
8 sin (π k /8)
5
X [k ] = ⋅drcl (k /8 , 5)
8
X [k ] =
Period : N0 = 8 (odd L = 5)
(L – 1) = 4 zero crossings
N 0=8
L = 5 ( N = 2)
Dirichlet Function
N 0=8
L = 2 N +1
drcl (t , L) =
1
jω
̂
D L (e ) =
−N
DT Rect (4B)
0
sin(π Lt )
Lsin (π t)
sin( ω
̂ L/2)
̂
Lsin( ω/2)
+N
16
Young Won Lim
4/20/13
Magnitude Response
0.45
( )
7
k
drcl
,7
16
16
1 sin(π k 7/16)
=
16 7 sin(π k /16)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-15
DT Rect (4B)
-10
-5
0
5
10
17
15
20
Young Won Lim
4/20/13
Phase Response
3.5
( )
7
k
drcl
,7
16
16
1 sin(π k 7/16)
=
16 7 sin(π k /16)
3
2.5
2
1.5
1
0.5
0
-20
-15
DT Rect (4B)
-10
-5
0
5
10
18
15
20
Young Won Lim
4/20/13
RectN[n] * δN0[n] DFT
Discrete Fourier Transform
N−1
X [k ] =
∑
x [n] e
− j2 / N  k n
x [n ] =
n=0
X [k ] =
sin ((2 π / N 0 )(2 N +1)k / 2)
sin ((2 π / N 0 )k / 2)
=
sin (π k / N 0 (2 N +1))
sin (π k / N 0 )
=
sin (π k / N 0 L)
sin (π k / N 0 )
1
N
N −1
∑
 j 2 / N  k n
X [k ] e
k =0
drcl (k / N 0 , (2 N +1)) =
sin (π k / N 0 (2 N +1))
(2 N +1)sin (π k / N 0 )
X [k ] = (2 N +1)⋅drcl (k / N 0 , (2 N +1))
= L⋅ drcl (k / N 0 , L)
Dirichlet Function
L = 2 N +1
N0
1
drcl (t , L) =
jω
̂
D L (e ) =
−N
DT Rect (4B)
0
sin(π Lt )
Lsin (π t)
sin( ω
̂ L/2)
̂
Lsin( ω/2)
+N
19
Young Won Lim
4/20/13
RectN[n] * δN0[n] DTFS & DFT
Discrete Time Fourier Series
1
X [k] =
N
X [k ] =
DTFS
N−1
N−1
∑
x [ n] e
− j (2 π / N ) k n
x [n] =
∑
+ j(2 π/ N )k n
X [k] e
k=0
n=0
1 sin (π k L/ N 0 )
N 0 sin (π k / N 0 )
X [k ] =
L
⋅drcl (k / N 0 , L)
N0
Discrete Fourier Transform
N−1
X [k ] =
∑
x [n] e
− j2 / N  k n
x [n ] =
n=0
X [k ] =
sin (π k / N 0 L)
sin (π k / N 0 )
DT Rect (4B)
1
N
N −1
∑
 j 2 / N  k n
X [k ] e
k =0
X [k ] = L ⋅drcl (k / N 0 , L)
20
Young Won Lim
4/20/13
References
[1]
[2]
[3]
[4]
http://en.wikipedia.org/
J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003
G. Beale, http://teal.gmu.edu/~gbeale/ece_220/fourier_series_02.html
C. Langton, http://www.complextoreal.com/chapters/fft1.pdf
Young Won Lim
4/20/13