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Young Won Lim4/29/13
● Discrete Time Rect Functions
Discrete Time Rect Function(4B)
Young Won Lim4/29/13
Copyright (c) 2009 - 2013 Young W. Lim.
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3 Young Won Lim4/29/13DT Rect (4B)
Fourier Transform Types
Discrete Time Fourier Transform
X e j = ∑n=−∞
∞
x [n ] e− j n x [n] = 12π ∫−π
+πX (e j ω̂) e+ j ω̂n
Discrete Fourier Transform
X [k ] = ∑n= 0
N − 1
x [n] e− j2 /N k n x [n ] = 1N ∑
k = 0
N − 1
X [k ] e j 2/N k n
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
DFT
DTFT
4 Young Won Lim4/29/13DT Rect (4B)
DTFS and DTFT
DTFS (Discrete Time Fourier Series)
DTFT (Discrete Time Fourier Transform)X (e j ω̂) = sin(ω̂ L/2)
sin(ω̂/2) = L DL(ej ω̂)
X [k ] = 1N 0
sin (π Lk /N 0)sin(π k /N 0)
= LN 0
⋅drcl (k /N 0 , L)
N 0
(L−1) zerocrossings
L/N 0
0
1
+N−N
L = 2 N+1 N 0
2πL
= L⋅diric(ω̂ , L)
(L−1) zerocrossings
0
1
+N−N
L = 2 N+1
1
k
ω̂
2πL
5 Young Won Lim4/29/13DT Rect (4B)
6 Young Won Lim4/29/13DT Rect (4B)
DTFS
DTFS (Discrete Time Fourier Series)X [k ] = 1
N 0
sin (π Lk /N 0)sin(π k /N 0)
= LN 0
⋅drcl (k /N 0 , L)
N 0
(L−1) zerocrossings
L/N 0
0
1
+N−N
L = 2 N+1 N 01
k
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
7 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS (1)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
X [k ] = 1N 0
∑n=0
N 0−1
x [n]e− j (2π/N 0)k n
= 1N 0
∑n=−N
+N
x [n ]e− j (2π/N 0)k n
N 0 X [k ] = e+ j (2π N /N 0)k +⋯+ e− j (2π N /N 0)k
= e+ j (m)N k⋅e− j (m)(2N+1)k /2
e− j (m) k /2 ⋅e+ j (m)(2 N+1)k /2 − e− j (m)(2 N+1)k /2
e+ j (m)k /2 − e− j (m) k /2
= e+ j (m)N k⋅1 − e− j (m)(2 N+1) k
1− e− j (m)km = (2π/N 0)
= sin((m)(2 N+1)k /2)sin((m)k /2)
X [k ] = 1N 0
sin ((2π/N 0)(2 N+1)k /2)sin ((2π/N 0)k /2)
drcl (t , L) = sin(π Lt )Lsin(π t)
= e+ j (2π/N 0) N k⋅ 1− e− j (2π/N 0)(2N+1)k
1 − e− j (2π/N 0)k
0
1
+N−N
L = 2 N+1 N 0
8 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS (2)
0
1
+N−N
L = 2 N+1
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
N 0
drcl (t , L) = sin(π Lt )Lsin(π t)
drcl (k /N 0 , (2 N+1)) =sin (π k (2 N+1)/N 0)(2 N+1)sin (π k /N 0)
X [k ] = (2 N+1)N 0
⋅drcl (k /N 0 , (2 N+1))
X [k ] = 1N 0
sin((2π/N 0)(2 N+1)k /2)sin ((2π/N 0)k /2)
= 1N 0
sin (π k (2N+1)/N 0)sin (π k /N 0)
X [k ] = LN 0
⋅drcl (k /N 0 , L)X [k ] = 1N 0
sin(π k L /N 0)sin(π k /N 0)
9 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS (3)
0
1
+N−N
L = 2 N+1
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
N 0
drcl (t , L) = sin(π Lt )Lsin(π t)
X [k ] = LN 0
⋅drcl (k /N 0 , L)
X [k ] = 1N 0
sin(π k L /N 0)sin(π k /N 0)
Period : N0 (odd L), 2N0 (even L)
(L-1) zero crossings
10 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS (4)
odd L=9 even L=10
t = 0 t =+1 t =+2t =−1t =−2 t = 0 t =+1 t =+2t =−1t =−2
Dirichlet Function
drcl (t , L) =sin(π L t)L sin(π t)
X [k ] = 916 ⋅drcl (k /16 , 9)
k=0 k=+16 k=+32k=−16k=−32 k=0 k=+16 k=+32k=−16k=−32
8 zero crossings
9 zero crossings
LL
⋯ −3, −2, −1, 0, +1, +2, +3,⋯
(L-1) zero crossings (L-1) zero crossings
11 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS (5)
X [k ] = 916 ⋅drcl (k /16 , 9)
Period : N0 (odd L), 2N0 (even L)
k=0 k=+16 k=+32k=−16k=−32
(L-1) zero crossings
12 Young Won Lim4/29/13DT Rect (4B)
Rect2[n] * δ8[n] DTFS Example
0
1
+N−N
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
N 0=8 drcl (t , L) = sin(π Lt )Lsin(π t)
X [k ] = 1N 0
sin (π k (2 N+1)/N 0)sin (π k /N 0)
X [k ] = 58⋅drcl (k /8 , 5)
L = 2N+1
X [k ] = LN 0
⋅drcl (k /N 0 , L)
L = 5 (N = 2)N 0=8
X [k ] = 18sin(π k 5 /8)sin (π k /8)
Period : N0 = 8 (odd L = 5)(L – 1) = 4 zero crossings
13 Young Won Lim4/29/13DT Rect (4B)
Rect3[n] * δ16[n] DTFS Example
0
1
+N−N
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
N 0=16 drcl (t , L) = sin(π Lt )Lsin(π t)
X [k ] = 1N 0
sin (π k (2 N+1)/N 0)sin (π k /N 0)
X [k ] = 716
⋅drcl (k /16 , 7)
L = 2N+1
X [k ] = LN 0
⋅drcl (k /N 0 , L)
L = 7 (N = 3)N 0=16
X [k ] = 116sin (π k 7 /16)sin (π k /16)
Period : N0 = 16 (odd L = 7)(L – 1) = 6 zero crossings
Rect3 [n ]∗ δ16 [n]
14 Young Won Lim4/29/13DT Rect (4B)
7/16 drcl(k/16, 7)
= 716drcl ( k16 ,7)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-20 -15 -10 -5 0 5 10 15 20
k16
= 17m
6 zero crossings
167⋅1 16
7⋅3
167⋅2
167⋅4
167⋅5
167⋅6
k = 167m
L = 7
N 0 = 16
k
LN 0
= 716
Zeros
= 716sin(π7 k /16)7sin(π k /16)
116sin(π7 k /16)sin(π k /16)
N 0
L =167
15 Young Won Lim4/29/13DT Rect (4B)
Phase of 7/16 drcl(k/16, 7)
π
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
-20 -15 -10 -5 0 5 10 15 20
π = −π
= 716drcl ( k16 ,7)
= 716sin(π7 k /16)7sin(π k /16)
116sin(π7 k /16)sin(π k /16)
16 Young Won Lim4/29/13DT Rect (4B)
Magnitude Response
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-20 -15 -10 -5 0 5 10 15 20
= 716drcl ( k16 ,7)
= 716sin(π7 k /16)7sin(π k /16)
116sin(π7 k /16)sin(π k /16)
k
N 0 = 16
17 Young Won Lim4/29/13DT Rect (4B)
Phase Response (1)
0
0.5
1
1.5
2
2.5
3
3.5
-20 -15 -10 -5 0 5 10 15 20
= 716drcl ( k16 ,7)
= 716sin(π7 k /16)7sin(π k /16)
116sin(π7 k /16)sin(π k /16)
k
N 0 = 16
18 Young Won Lim4/29/13DT Rect (4B)
Phase Response (2)
-4
-3
-2
-1
0
1
2
3
4
-20 -15 -10 -5 0 5 10 15 20
k
= 716drcl ( k16 ,7)
= 716sin(π7 k /16)7sin(π k /16)
116sin(π7 k /16)sin(π k /16)
N 0 = 16
19 Young Won Lim4/29/13DT Rect (4B)
20 Young Won Lim4/29/13DT Rect (4B)
RectN[n-N] * δN0[n] DTFS (1)
Dirichlet Function
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
drcl (t , L) = sin(π Lt )Lsin(π t)
0
1
+N−N
L = 2 N+1 N 0
21 Young Won Lim4/29/13DT Rect (4B)
22 Young Won Lim4/29/13DT Rect (4B)
DFT
DFT (Discrete Fourier Transform)X [k ] =
sin (π L k /N 0)sin (π k /N 0)
= L⋅drcl (k /N 0 , L)
N 0
(L−1) zerocrossings
L
0
1
+N−N
L = 2 N+1 N 01
k
Discrete Fourier Transform
X [k ] = ∑n= 0
N − 1
x [n] e− j2 /N k n x [n ] = 1N ∑
k = 0
N − 1
X [k ] e j 2/N k n
DFT
23 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DFT
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
drcl (t , L) = sin(π Lt )Lsin(π t)
drcl (k /N 0 , (2 N+1)) =sin (π k /N 0(2N+1))(2 N+1)sin (π k /N 0)
X [k ] = (2 N+1)⋅drcl (k /N 0 , (2N+1))
X [k ] =sin((2π/N 0)(2N+1)k /2)
sin ((2π/N 0)k /2)
=sin (π k /N 0(2 N+1))
sin (π k /N 0)
= L⋅drcl (k /N 0 , L)=sin (π k /N 0 L)sin (π k /N 0)
Discrete Fourier Transform
X [k ] = ∑n= 0
N − 1
x [n] e− j2 /N k n x [n ] = 1N ∑
k = 0
N − 1
X [k ] e j 2/N k n
0
1
+N−N
L = 2 N+1 N 0
24 Young Won Lim4/29/13DT Rect (4B)
RectN[n] * δN0[n] DTFS & DFT
Discrete Time Fourier Series DTFS
X [k ] = 1N ∑
n= 0
N − 1
x [n] e− j (2π /N )k n x [n] = ∑k = 0
N − 1
X [k ] e+ j(2π/N )k n
Discrete Fourier Transform
X [k ] = ∑n= 0
N − 1
x [n] e− j2 /N k n x [n ] = 1N ∑
k = 0
N − 1
X [k ] e j 2/N k n
X [k ] = L⋅drcl (k /N 0 , L)X [k ] =sin (π k L /N 0)sin (π k /N 0)
DFT
X [k ] = LN 0
⋅drcl (k /N 0 , L)X [k ] = 1N 0
sin(π k L /N 0)sin(π k /N 0)
25 Young Won Lim4/29/13DT Rect (4B)
26 Young Won Lim4/29/13DT Rect (4B)
DTFT
DTFT (Discrete Time Fourier Transform)X (e j ω̂) = sin(ω̂ L/2)
sin(ω̂/2) = L DL(ej ω̂)
2πL
= L⋅diric(ω̂ , L)
(L−1) zerocrossings
0
1
+N−N
L = 2 N+1
ω̂
2πL
Discrete Time Fourier Transform
X e j = ∑n=−∞
∞
x [n ] e− j n x [n] = 12π ∫−π
+πX (e j ω̂) e+ j ω̂n
DTFT
27 Young Won Lim4/29/13DT Rect (4B)
RectN[n] DTFT
0
1
+N−N
Discrete Time Fourier Transform DTFT
X e j = ∑n=−∞
∞
x [n ] e− j n x [n] = 12π ∫−π
+πX (e j ω̂) e+ j ω̂n
= {e+ j ω̂ N +⋯+ e− j ω̂ N}
X (e j ω̂) = ∑n=−N
+N
e− j ω̂n x [n]
= e+ j ω̂N {1 +⋯+ e− j ω̂ 2N }
= e+ j ω̂N 1− e− j ω̂ (2N+1)
1 − e− j ω̂
= e+ j ω̂N e− j ω̂(2 N+1)/2
e− j ω̂ /2e+ j ω̂(2N+1)/2 − e− j ω̂(2 N+1)/2
e+ j ω̂ /2 − e− j ω̂ /2
= e+ j ω̂(2N+1)/2 − e− j ω̂(2N+1)/2
e+ j ω̂ /2 − e− j ω̂ /2 = sin(ω̂(2N+1)/2)sin(ω̂/2)
X (e j ω̂) = sin(ω̂ L/2)sin(ω̂/2)
L = 2 N+1
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
= L DL(ej ω̂)
= L⋅diric(ω̂ , L)
28 Young Won Lim4/29/13DT Rect (4B)
D9(ej ω̂) = sin(ω̂9/2)
9sin (ω̂/2)
D11(ej ω̂) = sin (ω̂11/2)
11sin(ω̂/2)
D13(ej ω̂) = sin (ω̂13/2)
13sin(ω̂/2)
D9(ej ω̂)
8 zero crossings
8 zero crossings
10 zero crossings
12 zero crossings
Dirichlet Functions
2π
D10(ej ω̂) = sin(ω̂10 /2)
10sin (ω̂ /2)
D12(ej ω̂) = sin (ω̂12/2)
12sin(ω̂/2)
D14(ejω̂) = sin(ω̂14 /2)
14sin (ω̂ /2)
D10(ej ω̂)
9 zero crossings
9 zero crossings
11 zero crossings
13 zero crossings
2π
29 Young Won Lim4/29/13DT Rect (4B)
L DL(ejω) (1)
-2
-1
0
1
2
3
4
5
6
7
-10 -8 -6 -4 -2 0 2 4 6 8 10
X (e j ω̂) = sin(ω̂ L/2)sin(ω̂/2)
= L DL(ej ω̂)
= L⋅diric(ω̂ , L)
6 zero crossings
2π7
⋅1 2π7
⋅3
2π7
⋅22π7
⋅4
2π7
⋅5
2π7
⋅6
L = 7
2πL = 7
2πL = 2π
7
30 Young Won Lim4/29/13DT Rect (4B)
Magnitude Response of L DL(ejω)
0
1
2
3
4
5
6
7
-10 -8 -6 -4 -2 0 2 4 6 8 10
X (e j ω̂) = sin(ω̂ L/2)sin(ω̂/2)
= L DL(ej ω̂)
= L⋅diric(ω̂ , L)
31 Young Won Lim4/29/13DT Rect (4B)
Phase Response of L DL(ejω)
0
0.5
1
1.5
2
2.5
3
3.5
-10 -8 -6 -4 -2 0 2 4 6 8 10
X (e j ω̂) = sin(ω̂ L/2)sin(ω̂/2)
= L DL(ej ω̂)
= L⋅diric(ω̂ , L)
32 Young Won Lim4/29/13DT Rect (4B)
33 Young Won Lim4/29/13DT Rect (4B)
RectN[n-N] DTFT
0
1
+N−N
Discrete Time Fourier Transform DTFT
X e j = ∑n=−∞
∞
x [n ] e− j n x [n] = 12π ∫−π
+πX (e j ω̂) e+ j ω̂n
L = 2 N+1
DL(ej ω̂) = sin(ω̂L /2)
Lsin(ω̂/2)
Dirichlet Function
34 Young Won Lim4/29/13DT Rect (4B)
Young Won Lim4/29/13
References
[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] G. Beale, http://teal.gmu.edu/~gbeale/ece_220/fourier_series_02.html[4] C. Langton, http://www.complextoreal.com/chapters/fft1.pdf[5] M. J. Roberts, Fundamentals of Signals and Systems