(2πn/10)
N
(2πn/10)
N
H
0, 1, ..., 29
2 3
5
cq (n)
q = 6 x(n) =
[5, −1, 5, −1, 5, −7]
zi (n)
3−
30
∈
∈
/
⊆
∀
∃
∩
∪
ℜ
ℑ
∗
!
⋆
F{f }
M{f }
f
f
i
R
Rn
Rm×n
R+
R++
C
Cn
Cm×n
Z
Z+
N(= Z++ )
[a, b] {x | a ≤ x ≤ b}
(a, b) {x | a < x < b}
ei
In
XT
XH
X†
C = AB
D =A◦B
√
n−
m×n
n−
m×n
−1
n×1
n×1
a
a
i−
n×n
b
b
X
X
X
!
Cjk = N
l=1 Ajl Blk
Djk = Ajk Bjk
10
{0, 1, ..., 9} = {n | n ∈ Z, n < 10}.
[a, b] ∩ Z
[0, 10) ∩ Z
[0, 9] ∩ Z
x[n]
n
f (t)
t
a 1 | a2
a2
a1
a1
a1 ! a2
a2
a1
a2
p
q
(p, q)
p
q
(p, q)
k
[a1 , ..., ak ] := [a1 , ..., ak , a1 , ..., ak , a1 , ..., ak , ...].
r=
n
(n, m)
r
r ∈ [0, m − 1] ∩ Z
m
(26, 11) = 4
26
11
4
≡
26 ≡ 48 (
26
48
11)
11
x[n] = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
[3, 7, 11]
[x[n]]↓d,l
d
l
[x[n]]↓4,3 = [3, 7, 11].
q
ζq =
ζq k
(2πi/q)
xq − 1 = 0
1
k = 0, 1, 2, ..., q − 1
(ζq k )q =
(2πik) = 1.
xq − 1 = 0
{1 = ζq 0 , ζq , ζq 2 , ..., ζq q−1 }
q
q
q
q
q
ζq
ζq n = 1
n
x q0 − 1 = 0
q
n=q
xq − 1 = 0
ζq
q0 < q
ζq
ζq p
(p, q) = 1
ζq
(ζq p )q/k = (ζq q )p/k = 1
(p, q) = k
ζq p
q/k
p/k
q = 12
ζ12 ζ12 5 ζ12 7 ζ12 11
ζ12 2 ζ12 10
ζ12 3 = i ζ12 9 = −i
ζ12 4 = ω ζ12 8 = ω 2
ζ12 6 = −1
ζ12 0 = 1
ω = (−1 +
√
△
3i)/2
q
q
k
ζq k
k
q
n1 n2
k
nk
a1 a2
ak
x
x ≡ a1
(
n1 )
x ≡ a2
..
.
(
n2 )
x ≡ ak
(
nk )
N = n1 n2 ...nk
x
[0, N )
[0, N )
x ≡ ai (
ai
x ≡ ai + kni (
ni )
ni
ai − 1
ni )
ni
n1 n2 ...nk = N
Nk
x ∈ [0, N )∩Z
N
x
N
x1 , x2 ∈ [0, N ) ∩ Z
(x1 , ni ) =
(x2 , ni ),
(x1 − x2 , ni ) = 0,
x 1 − x2
ni
x1 ̸= x2
∀i = 1, ..., k
∀i = 1, ..., k.
N = n1 n2 ...nk |(x1 − x2 )
[0, N ) ∩ Z ⊆ [0, N − 1] |x1 − x2 | ≤ N − 1
x1 , x 2 ∈
x1 = x2
n1 = 2 n2 = 3 n3 = 5 a1 = 3 a2 = 9 a3 = −3
N = 2 × 3 × 5 = 30
x≡3≡1 (
x≡9≡0 (
x ≡ −3 ≡ 2 (
2)
3)
5)
x = 0, 1, ..., 29
(
(x, 2),
(x, 3),
3
(x, 5))
3
x
27 △
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
0, 1, ..., 29
2 3
N
(n)
n∈N
1
id(n)
idk (n)
id(n) = n,
k
k
id (n) = n ,
ε(n)
∀n ∈ N.
∀n, k ∈ N.
5
ε(n)
" # $
0,
1
ε(n) =
=
n
1,
n > 1.
n = 1.
φ(n)
φ(n)
n
1 3 5
7
8
φ(8) = 4
σ(n) σk (n)
σ(n)
σk (n)
n
n σ(n) = σ1 (n)
k
σk (n) =
%
dk
d|n
τ (n)
τ (n) = σ0 (n)
n
µ(n)
µ(n)
n
n=1
(−1)k
n
k
⎧
⎪
⎨ 1,
µ(n) =
(−1)k ,
⎪
⎩
0,
n = 1.
n = p1 p2 ...pk
.
χ(n)
(
pi
.
r)
r
χ(n) =
$
1,
0,
(n, r) = 1.
(n, r) ̸= 1.
ω(n)
cq (n)
Ω(n)
λ(n) = (−1)Ω(n)
π(n) =
p(n)
Λ(n)
Λ(n) =
$
p,
0,
n = pα , p
.
.
!
p≤n,p∈P
1
a(n)
p∈N q∈N
a(pq) = a(p) + a(q)
(p, q) = 1
ω(n)
a(n)
a(pq) = a(p) + a(q)
p∈N q∈N
Ω(n)
a(n)
p∈N q∈N
a(pq) = a(p)a(q)
(p, q) = 1
σk (n)
σk (n)
σ(n) τ (n)
φ(n)
µ(n)
cq (n)
a(n)
a(pq) = a(p)a(q)
p∈N q∈N
λ(n)
q1
n∈N
q2
χ(n)
(q1 q2 , n) =
(q1 , n)
(q2 , n)
f (n)
ε(n)
(n)
id(n)
idk (n)
µ(n)
τ (n)
σ(n)
σk (n)
φ(n)
χ(n)
ω(n)
Ω(n)
cq (n)
λ(n)
π(n)
Λ(n)
p(n)
q1
q2
q1
q1 = p1 α1 p2 α2 ...pN αN
q2 = pN +1 αN +1 pN +2 αN +2 ...pN +M αN +M ,
q1 q2 = p1 α1 p2 α2 ...pN +M αN +M .
(q1 , n) = p1 β1 p2 β2 ...pN βN
(q2 , n) = pN +1 βN +1 pN +2 βN +2 ...pN +M βN +M ,
(q1 q2 , n) = p1 β1 p2 β2 ...pN +M βN +M ,
(q1 q2 , n) =
(q1 , n)
(q2 , n)
q2
n
Φ(n) = {x ∈ [0, n) ∩ Z |
n1 , n 2 ∈ N
φ(n) = |Φ(n)|
φ(n1 )φ(n2 )
(x, n) = 1}
g1 ∈ Φ(n1 )
(n1 , n2 ) = 1
(g1 , g2 )
g2 ∈ Φ(n2 )
g
g ≡ g1
(
g1 ∈ Φ(n1 )
g ≡ g2
n1 ),
(
g2 ∈ Φ(n2 )
n2 ).
g ∈ Φ(n1 n2 )
φ(n1 )φ(n2 ) ≤ φ(n1 n2 )
n1 n2 − φ(n1 )φ(n2 )
(h1 , h2 )
h1 ∈
/ Φ(n1 )
h ∈ Φ(n1 n2 )
h ≡ h1
(
h ≡ h2
n1 ),
n1 n2 −φ(n1 )φ(n2 ) ≤ n1 n2 −φ(n1 n2 )
n = pk
(
n2 ).
φ(n1 )φ(n2 ) ≥ φ(n1 n2 )
φ(n) = pk − pk−1 = pk−1 (p − 1)
p
n∈N
φ(n) = n
*+
p∈P
p|n
1
1−
p
,
.
h2 ∈
/ Φ(n2 )
pk
pk
pk /p =
p
pk−1
n = pα1 1 ...pαk k
n
φ(n) =
φ(pα1 1 )...φ(pαk k )
=
k
*
pi
k−1
i=1
(pi − 1) =
k
*
i=1
pi
k
+
1
1−
pi
,
=n
*+
p∈P
p|n
1
1−
p
∗
⋆
f (n)
(f ∗ g)(n) :=
n
%
k=0
f (k)g(n − k).
g(n)
,
F{g ∗ h} = F{g}F{h}
p(x) =
!n
k=0
ak xk
q(x) =
!m
l=0 bl x
l
p(x)q(x) = an bm xn+m + (an bm−1 + an−1 bm )xn+m−1 + ... + (a0 b1 + a1 b0 )x + a0 b0
- k
.
n+m
n+m
% %
%
k
=
aj bk−j x =
( ∗ ) k xk ,
k=0
j=0
= [a0 , a1 , ..., an ]
k=0
= [b0 , b1 , ..., bm ]
p(x)q(x)
f (n)
(f ⋆ g)(n) :=
%
d|n
f (d)g
/n0
d
.
(f ⋆ g)(n) = (g ⋆ f )(n)
d
n/d
((f ⋆ g) ⋆ h)(n) = (f ⋆ (g ⋆ h))(n)
g(n)
%
((f ⋆ g) ⋆ h)(n) =
(f ⋆ g)(a)h(b) =
ab=n
=
%
%
ab=n
-
%
.
f (c)g(d) h(b)
cd=a
f (c)g(d)h(b)
bcd=n
b c
d
ε(n)
f (n)
(f ⋆ ε)(n) = (ε ⋆ f )(n) = f (n).
(f ⋆ ε)(n) =
%
d|n
" #
d
= f (n)
f (d)
n
d | n [d/n] = 1
f
d=n
g
f ⋆g
d | mn
(m, n) = 1
d = d1 d 2
d1 | m
d2 | n
(d1 , d2 ) = 1
f ⋆ g(mn) =
%
d|mn
=
%
d1 |m
f (d)g
/ mn 0
f (d1 )g
d
+
m
d1
=
%%
d1 |m d2 |n
,%
d2 |n
f (d2 )g
f (d1 d2 )g
+
n
d2
,
+
mn
d1 d2
,
= [(f ⋆ g)(m)][(f ⋆ g)(n)]
f
g
h(n) =
h = f ⋆g
% /n0
g(d).
f
d
d|n
c∈N
h(cn) =
% / cn 0
g(d) = (fc ⋆ g)(n),
f
d
d|n
fc (n) := f (cn)
τ=
h(cn) = (f ⋆gc )(n)
⋆
⋆ (n) =
%
1(d)1
d|n
σν =
gc (n) := g(cn)
/n0
d
=
%
△
1 = τ (n)
d|n
⋆ idν
⋆ idν (n) =
% /n0
%
dν =
1
dν = σν (n),
d
d|n
d|n
φ⋆
φ ⋆ (n) =
%
∀ν ∈ N
△
= id
△
φ(d) = n
d|n
F (n) = φ ⋆ (n) =
!
d|n
φ(d)
φ
F
n
n = p1 α1 ...pk αk
p i αi
F (pi αi )
F (n)
1, pi , pi 2 , ..., pi αi
F (pi αi ) = φ(1) + φ(pi ) + φ(pi 2 ) + ... + φ(pi αi )
= 1 + (pi − 1) + (pi 2 − pi ) + ... + (pi αi − pi αi −1 ) = pi αi .
F (n)
F (n) = F (p1 α1 )F (p2 α2 )...F (pi αi ) = p1 α1 p2 α2 ...pk αk = n
f (n)
g(n)
f ⋆g = ε
(n)
µ ⋆ (n) =
n=1
=
!
d|n
⋆ µ(n) = ε(n)
µ(d) = 1 = ε(1)
n>1
n
n = p1 α1 p2 α2 ...pk αk
%
µ(d) =1 + [µ(p1 ) + ... + µ(pk )] + [µ(p1 p2 ) + ... + µ(pk−1 pk )]
d|n
+ ... + [µ(p1 p2 ...pk−1 ) + ... + µ(p2 p3 ...pk )] + µ(p1 p2 ...pk )
=1 + C1k (−1) + C2k (−1)2 + ... + Ckk (−1)k
=(1 + (−1))k = 0 = ε(n).
τ=
τ=
⋆
⋆
⇒τ ⋆µ=
1=
%
⋆ ⋆µ=
τ (d)µ
d|n
σν =
/n0
d
⋆ε= .
.
⋆ idν
△
idν =
σµ ⋆ µ
ν
n =
%
σν (d)µ
d|n
/n0
d
,
ν ∈ N.
id = φ ⋆
φ(n) =
%
d|n
µ(d)
△
φ = id ⋆ µ
% µ(d)
n
=n
.
d
d
d|n
△
τ ⋆φ = σ
d =
⋆
φ = id ⋆ µ
τ ⋆φ=
⋆ ⋆ id ⋆ µ = ( ⋆ id) ⋆ ( ⋆ µ).
⋆ id = σ
⋆µ=ε
△
τ ⋆ φ = σ ⋆ ε = σ.
f (n)
g(n)
f (a)g(b)
f (a)g(b)
ab = n
a+b = n
!∞
ai
n=1
an /ns
a(n)
s
Fa (s) :=
∞
%
a(n)
n=1
ns
xs−1
1
FM (s) = M{f (x)} = √
2π
1
∞
f (x)xs−1 dx,
0
x ∈ R++ , s ∈ C.
x=0
√
1/ 2π
s = 0, −1, −2, ...
Ff ⋆g (s) = Ff (s)Fg (s)
Ff (s)Fg (s) =
∞
∞ %
%
f (k)g(l)
k=1 l=1
k s ls
∞
∞
%
%
1 %
(f ⋆ g)(n)
=
f (k)g(l) =
s
n kl=n
ns
n=1
n=1
ε(n)
Fε (s) = 1
s∈R
f (n)
ε(n)
(n)
id(n)
idk (n)
µ(n)
τ (n)
σ(n)
σk (n)
φ(n)
Ff (s)
ζ(s)
ζ(s − 1)
ζ(s − k)
1
ζ(s)
ζ 2 (s)
ζ(s)ζ(s − 1)
ζ(s)ζ(s − k)
ζ(s − 1)
ζ(s)
ℜ{s} > 1
ℜ{s} > 1
ℜ{s} > 2
ℜ{s} > k + 1
ℜ{s} > 1
ℜ{s} > 1
ℜ{s} > 2
ℜ{s} > k + 1
τ= ⋆
σ = ⋆ id
σk = ⋆ idk
ℜ {s} > 2
φ = id ⋆ µ
∞
%
1
.
ζ(s) :=
ns
n=1
ζ(n)
ζ(s)
∞
%
k=−∞
f (n − k)g(k)
%
f (d)g
d|n
/n0
d
f ∗g =g∗f
f ⋆g =g⋆f
f ∗ (g ∗ h) = (f ∗ g) ∗ h
f ⋆ (g ⋆ h) = (f ⋆ g) ⋆ h
h=f ∗g
h(n − m) = f (n − m) ∗ g(n)
h=f ⋆g
h(cn) = f (cn) ⋆ g(n)
δ(n) =
$
X(z) =
1, n = 0
0,
!∞
n=0
x(n)z −n
ε(n) =
$
FM (s) =
1, n = 1
0,
!∞
n=1
x(n)n−s
cq (n)
n−
q
cq (n) =
+
q
%
p=1
(p,q)=1
q∈N
n∈N
2iπpn
q
q
,
.
n
q
cq (n)
q
n
q
cq (n)
k∈N
cq (n + kq) =
q
%
p=1
(p,q)=1
=
+
(2iπpk)
2iπp(n + kq)
q
q
%
p=1
(p,q)=1
+
,
2iπpn
q
,
= 1 × cq (n).
q
N
Z
cq (n)
cq (n) =
+
q
%
p=1
(p,q)=1
q∈N
2iπpn
q
,
n∈Z
q ∈ N,
cq (n) = cq (−n),
r =q−p
cq (−n) =
(p, q) = 1
q
%
p=1
(p,q)=1
=
q
%
p=1
(q−p,q)=1
=
n∈Z
q
%
r=1
(r,q)=1
+
2iπp(−n)
q
+
+
,
=
2iπ(q − p)n
q
2iπrn
q
,
(r, q) = 1
+
q
%
p=1
(p,q)=1
,
q−1
%
=
r=0
(r,q)=1
= cq (n)
cq (n)
q
n
cq (n) ∈ Z ∀q ∈ N, n ∈ Z
cq (n) ∈ Z ∀q ∈ N
∀n ∈ Z
2iπ(−p)n
q
+
2iπrn
q
,
,
cq (n)
ζq =
ηq (n) :=
q−1
%
ζq
kn
(2iπ/q)
=
k=0
q−1
%
(ζq k )n
k=0
ζq
{ζq k | k = 0, 1, ..., q − 1},
φ(q)
(k, q)
(k, q) = q/d
q
%
k=1
(k,q)=q/d
k
ζq =
q
%
k=1
(k,q)=q/d
+
2iπk
q
,
.
k = lq/d
q
%
+
d
%
k
ζq =
k=1
(k,q)=q/d
l=1
(l,d)=1
ηq (n) =
%
2iπl
d
,
= cd (n).
cd (n).
d|q
q an (q) := cq (n)
n
bn (q) := ηq (n)
bn = an ⋆
an = µ ⋆ bn
% /q0
bn (q).
µ
d
an (q) =
d|q
ηq (n)
cq (n)
% /q 0
ηd (n).
µ
d
cq (n) =
d|q
g(x) = xq−1 + xq−2 + ... + x + 1
ηq (n) = g(ζq n )
(x − 1)g(x)
ηq (n) =
!q
k=1
1=q
q!n
f (x) = xq − 1 =
ζq n
f (x)
g(x)
ηq (n) =
ζq n = 1
q | n
$
0,
q,
ηq (n) = g(ζq n ) = 0
q ! n.
q | n.
cq (n) =
% /q0
d=
µ
d
d|q
d|n
d|
%
µ
(q,n)
/q0
d
d.
cq (n)
q
(q1 , q2 ) = 1
∀n ∈ Z.
cq1 q2 (n) = cq1 (n)cq2 (n),
ηq (n)
q
(q1 , q2 ) = 1
q !n
ηq (n) = 0
ηq1 (n) = 0
q = q1 q2
q1 ! n
q2 ! n
ηq2 (n) = 0
ηq1 (n)ηq2 (n) = 0 = ηq1 q2 (n).
q ! n
q = q1 q2 | n
ηq (n) = q
q2 | n
ηq1 (n) = q1
q1 | n
ηq2 (n) = q2
ηq1 (n)ηq2 (n) = q1 q2 = q = ηq1 q2 (n).
cq (n)
q
n
µ(n)
n
q
ηq (n)
cq (n)
cp (p) = p−1
q
µ(n)
q
cq (n)
p ̸= q
ηq (n)
n
cp (q) = −1
cp (pq) = p−1 ̸= cp (p)cp (q)
p
⎧
s−1
s
⎪
⎨ p (p − 1), p | n
cps (n) =
ps−1 | n, ps ! n
−ps−1 ,
⎪
⎩
0,
.
(k, ps ) = 1
(k, p) = 1
(k, p) ̸= 1
p
+
s
cps (n) =
p
%
k=1
(k,p)=1
s
=
p
%
k=1
+
p
2iπkn
ps
2iπkn
ps
,
,
s
−
p
%
k=1
(k,p)̸=1
+
2iπkn
ps
,
p
k = lp
p
k
+
s
cps (n) =
p
%
k=1
2iπkn
ps
,
ps | n
s
cps (n) =
p
%
k=1
ps−1
−
%
l=1
n = n1 ps
+
2iπln
ps−1
l
,
n1 ∈ Z
ps−1
(2iπkn1 ) −
%
l=1
(2iπlpn1 ) = ps − ps−1 = ps−1 (p − 1).
n
n2 ∈ Z
.
p
ps ! n
ps−1 | n
ps ! n
n = n2 ps−1
(n2 , p) = 1
+
s
cps (n) =
p
%
k=1
=
+
2iπkn2
p
2iπn2
p
,
,1−
1−
= 0 − ps−1 = −ps−1 ,
!N
j=1
ps−1
−
%
(2iπln2 )
l=1
/
2iπn2 ps
p
/
2iπn2
p
0
0 − ps−1
z∈C
az j = a(1 − z N )/(1 − z)
ps−1 ! n
s−1
, p%
,
+
2iπkn
2iπln
−
cps (n) =
s
p
ps−1
k=1
l=1
,
,
+
+
(2iπn)
2iπn 1 −
2iπn 1 −
/
0−
=
2iπn
ps
ps−1 1 −
1−
s
p
%
+
ps
= 0 − 0 = 0.
q1
q2
(2iπn)
/
0
2iπn
ps−1
cq (n) = µ(m)
m = q/
(q, n)
φ(q)
,
φ(m)
φ(n)
q = ps
ps | n
µ(1)
m = ps /ps = 1
φ(ps )
= φ(ps ) = ps−1 (p − 1).
φ(1)
ps ! n
ps−1 | n
µ(p)
m = ps /ps−1 = p
ps−1 (p − 1)
φ(ps )
= −1
= −ps−1 .
φ(p)
p−1
ps−1 ! n
m
p2
m
µ(m) = 0
q
m
id(q)
(q, n)
q
φ(n) µ(n)
m
(q1 ,q2 )
%
q1 ̸= q2
cq1 (n)cq2 (n) = 0,
n=1
q
%
cq 2 (n) = qφ(q)
n=1
(q1 ,q2 )
%
(q1 ,q2 )
cq1 (n)cq2 (n) =
n=1
⎡
% ⎢
⎣
n=1
=
q1
%
p1 =1
(p1 ,q1 )=1
!N
j=1
(q1 ,q2 )
%
n=1
=
=
q1
%
p1 =1
(p1 ,q1 )=1
q2
%
p2 =1
(p2 ,q2 )=1
⎡
⎣
+
2iπp1 n
q1
(q1 ,q2 )
%
n=1
,
+
q2
%
p2 =1
(p2 ,q2 )=1
+
2iπp2 n ⎥
⎦
q2
⎤
,
2iπp1 n 2iπp2 n ⎦
.
+
q1
q2
az j = a(1 − z N )/(1 − z)
,
2iπp1 n 2iπp2 n
+
q1
q2
+
,
(q1 ,q2 )
%
p 1 q2 + p 2 q1
2iπn
q
q
1
2
n=1
,
+
p1 q2 + p2 q1
,1−
+
2iπ[
(q1 , q2 )]
p1 q2 + p2 q1
q1 q2
+
,
2iπ
p1 q2 + p2 q1
q 1 q2
1−
2iπ
q1 q2
+
⎤
,
(p1 q2 + p2 q1 )
q
%
1−1 = 0
(q1 , q2 )/q1 q2
q
cq 2 (n) =
n=1
+
q
%⎢ %
⎣
n=1
=
⎡
p1 =1
(p1 ,q)=1
q
%
q
%
p1 =1
(p1 ,q)=1
p2 =1
(p2 ,q)=1
2iπp1 n
q
+
q
%
n=1
,
+
q
%
p2 =1
(p2 ,q)=1
2iπ(p1 + p2 )n
q
,
⎤
,
2iπp2 n ⎥
⎦
q
.
!q
q | p1 + p2
q
%
n=1
+
2iπ(p1 + p2 )n
q
,
+
=
2iπ(p1 + p2 )
q
,
1−
n=1
(2iπ(p1 + p2 ))
, = 0.
+
2iπ(p1 + p2 )
q
1−
q ! p 1 + p2
p1
q | p1 + p2
p1 + p2 = q
q = p1 + p2
q
%
%
cq 2 (n) =
n=1
(p1 , q) = 1
q
p1 ,p2 ∈{1,2,...,q}
(p1 ,q)=1
(p2 ,q)=1
p1 +p2 =q
p2 = q−p1
p2
(q − p1 , q) = 1
q
%
n=1
2
cq (n) =
q
%
p1 =1
(p1 ,q)=1
1 = q
q = qφ(q).
p2
cq1 (n)
q1
cq2 (n)
q2
k1 k2 ∈ Z
N
−1
%
n=0
+
2iπk1 n
N
,
+
2iπk2 n
N
,
=
$
k1 =
̸ k2
k 1 = k2
0,
N,
N
N
x(n)
N
x(n) =
∞
%
xq cq (n)
q=1
xq =
xq
N
1
1 %
x(n)cq (n)
φ(q) N →∞ N n=1
x(n)
1/φ(q)
x(n)
xq
σ(n)
n
φ(n)
n
φ(n)Λ(n)
b(n) :=
n
C(n)
π2 1
6 q2
6 µ(q)
π 2 φ2 (q)
µ(q)
+ φ(q),2
µ(q)
φ(q)
Λ(n)
C(n)
C2 ≈ 0.66
⎧
*p−1
⎪
⎪
,
⎨ 2C2
p−2
p|n
C(n) =
p∈P
⎪
⎪
⎩ 0,
φ2 (n) = q
2
*+
p∈P
p|q
h
.
,
h
1
1− 2
p
.
φ2 (n)
,
.
b(n)
µ(n)/φ(n)
a(n) = a0
a0
(2πn/n0 + δ)
n0
(δ)/φ(n0 )
r
r
x(n) = x(
(n, r))
r
r
x(n)
X(k) = x(0) +
% /r0
cd (k).
x
d
d|r
d>1
r
r
r
r
1/N
0.25
0.2
xq
0.15
0.1
0.05
0
−0.05
0
5
10
15
20
25
15
20
25
q
0.25
0.2
xq
0.15
0.1
0.05
0
−0.05
0
5
10
q
x(n) =
(2πn/11)
x(n) =
x(n)
(2πn/12)
cq (n)
N
c1 (n), ..., cN (n)
!N
n=1 ck1 (n)ck2 (n)
̸= 0
k1 , k2 ! N
N
xq
q = 11
q = 12
q = 10 14
30
P = 10 14
x(n)
30
0.25
period P=10
period P=14
period P=30
0.2
0.15
x
q
0.1
0.05
0
−0.05
−0.1
0
10
20
30
40
50
q
60
70
80
0.25
90
100
period P=10
period P=14
period P=30
0.2
0.15
x
q
0.1
0.05
0
−0.05
−0.1
x(n) =
0
(2πn/P )
10
20
30
40
50
q
x(n) =
60
70
80
90
(2π(n + 37)/P )
100
f (t)
T = aab abc
T = abbd
abda
f (t + T ) = f (t)
acb
abaa
aba
abcd
a
T >0
ab
p=4
T = cbacc cbacc
cbacc
P
...
P
(2πn/5)
0.2
P
5
P−
P
x(t)
T
Ts
x(t) = x(t+T )
xs [n] = x(nT s)
T
x(t) =
(ωt)
x[n] = x(nTs )
Ts
P = 2π/ω
Ts1 = P /10 = π/5ω
x1 [n] =
P
Ts
(πn/5)
10
10Ts = P
Ts2 = 2P /5 = 4π/5ω
P
Ts
x2 [n] =
(4πn/5)
6T s = 2P
Ts3 = 7P /10π = 7/5ω
x3 [n] =
(n/5)
9Ts = 63P /10π ≈ 2.0054P ≈ 2P
P
Ts
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
P
2P
t (P=π/ω.)
3P
4P
0
P
2P
t (P=π/ω.)
3P
4P
0
P
2P
t (P=π/ω.)
3P
4P
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
Ts1 =
P /10 = π/5ω
Ts2 = 2P /5 = 4π/5ω
Ts3 = 7P /10π = 7/5ω
n
x(n) = [1, −1, 1, −1, 1, −0.9].
x(n)
x(n)
x(n)
x[n] = [1, −1] + [0, 0, 0, 0, 0, 0.1].
1
x(n)
−1
2
6−
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
a[n] = [−0.8391, 0.9346, −0.7455, 0.6900, −0.8163, 0.5552]
[0.9829, 0.9325, −0.0593, −0.6822, −1.0793, −0.5002]
a[n] = [−0.8391, 0.9346, −0.7455, 0.6900, −0.8163, 0.5552],
b[n]
=
b[n] = [0.9829, 0.9325, −0.0593, −0.6822, −1.0793, −0.5002].
6−
2−
a[n]
2
6
b[n]
2
2
a[n]
[−0.8391, −0.7455, −0.8163]
2
[0.9346, 0.6900, 0.5552]
−0.8033
(−1, 1)
0.7266
[0.9829, −0.0593, −1.0793]
b[n]
2
[0.9325, −0.6822, −0.5002]
−0.0519
−0.0833
x(n)
q
q−
x(n)
q
x(n)
l∈N
r|q (r ̸= q)
q/r−1
%
x(kr + l) = 0.
k=0
l
l
%
n
[x(n)]↓r,l
[x(n)]↓r,l = 0,
∀r|q, l ∈ Z.
x(n) = [0.32, 0.27, −0.05, −0.32, −0.27, 0.05]
6−
r = 3
l = 0, 1, 2
x(0) + x(3) = 0 x(1) + x(4) = 0
x(2) + x(5) = 0
x(0)+x(2)+x(4) = 0
r=2
l=0
x(1)+x(3)+x(5) = 0
1
r = 1 x(0)+x(1)+x(2)+x(3)+x(4)+x(5) =
0
x(n)
△
6−
q
q
q
1/q
x(n)
r ∈ Z x(n − r)
q−
q
x(n − r)
x(n)
x(n)
k = 0, ..., p1 − 1
q
E[[x(n)]↓p2 ,l = 0]
q/p1 −1
%
l = 0, ..., p2 − 1
x(kp1 + k) = 0,
k = 0, ..., p1 − 1
x(kp2 + l) = 0,
l = 0, ..., p2 − 1.
k=0
q/p2 −1
%
k=0
p2 | p1
E[[x(n)]↓p1 ,k ] = 0
p1 = 2
x(2) + x(4) = 0
x(1) + x(3) = 0
x(1) + x(2) + x(3) + x(4) = 0
p2 = 1
x(n)
y(n) = [x(n)]↓r,l
y(n)
y(n)
x(n)
y(n)
q 1 q2
x(n)
q1 q2
l1 l2 ∈ Z [x(n)]↓q1 ,l1
x(n)
q2
[x(n)]↓q2 ,l2
q1 q2
d
d1 |q1
l∈Z
d1 = 1
d2 |q2
[x(n)]↓d,l = [x(n)]↓d2 ,l
q1
q1
q2
d = d1 d 2
[x(n)]↓q2 ,l
d2 = 1
d1 > 1 d2 > 1
q1 q2
−1
d
d %
E[[x(n)]↓d,l ] =
x(kd + l)
q1 q2 k=0
q1 q2
−1
d d2
d1 d2 1%
x(kd1 d2 + l).
=
q1 q2 k=0
k
q2 /d2
k = (q2 /d2 )k1 + k2
k
k1
k2
q1
q
−1 d2 −1
d1
2
,
++
,
d 1 d2 % %
q2
x
k 1 + k2 d 1 d 2 + l
E[[x(n)]↓d,l ] =
q1 q2 k =0 k =0
d2
1
2
q2
q1
−1
−1
d2
d1
d 1 d2 % %
x(d1 q2 k1 + d1 d2 k2 + l)
q1 q2 k =0 k =0
2
⎡ 1 q
⎤
q2
1 −1
−1
d2
d1
d 1 % ⎢ d2 %
⎥
=
x(d1 q2 k1 + (d1 d2 k2 + l))⎦ .
⎣
q1 k =0 q2 k =0
=
2
1
[[x(n)]↓q2 ,0 ]↓d1 ,d1 d2 k2 +l
[x(n)]↓q2 ,0
E[[x(n)]↓q1 q2 ,l ] = 0
x1 (n)
q1 ̸= q2
(q1 ,q2 )
%
q1 −
q2 −
x2 (n)
x1 (n)x2 (n) = 0.
n=1
T =
q1′
q2′
x1 (kq1 + l) = x1 (l)
(q1 , q2 )
L=
T = q1′ q2′ L
k ∈Z
q1 = Lq1′
(q1 , q2 )
x1 (n)
n
n = n 1 q1 + n 2
q2 = Lq2′
q1
n1 = 0, ..., L − 1
n2 = 1, ..., q1
T
%
x1 (n)x2 (n) =
n=1
q2′ −1
q1
% %
x1 (n1 q1 + n2 )x2 (n1 q1 + n2 )
n1 =0 n2 =1
=
q1
%
x1 (n2 )
n2 =1
q2′ −1
%
x2 (n1 q1 + n2 ).
n1 =0
n1 q1 = n1 Lq1′
(q1′ q2′ ) = 1
(n1 q1′ , q2′ ) ,
n1 = 1, ..., q2′ − 1,
1, 2, ..., q2′ − 1
((n1 q1′ )L, q2 ) =
((n1 q1′ )L, q2′ L) ,
n1 = 1, ..., q2′ − 1,
L, 2L, ..., (q2′ − 1)L
q2′ −1
%
x2 (n1 q1 + n2 ) =
n1 =0
q2′ −1
%
x2 (kL + n2 ),
k=0
x2 (n)
L
q1 = 4 q2 = 6 T = 12 L = 2 x1 (n)
12
%
n=1
x1 (n)x2 (n) =
4
%
n2 =1
x1 (n2 )
2
%
n1 =0
x2 (4n1 + n2 )
x2 (n)
= x1 (1) [x2 (1) + x2 (5) + x2 (9)] + x1 (2) [x2 (2) + x2 (6) + x2 (10)]
+ x1 (3) [x2 (3) + x2 (7) + x2 (11)] + x1 (4) [x2 (4) + x2 (8) + x2 (12)] .
x2 (n)
6
[x1 (1) + x1 (3)] [x2 (1) + x2 (3) + x2 (5)] + [x1 (2) + x1 (4)] [x2 (2) + x2 (4) + x2 (6)] .
△
P
x(n) =
P ∈N
q
P−
(2πn/P +θ)
θ∈R
x(n)
P−
q|P
l∈Z
y(m) = x(mq+l)
y(m)
m = P /q − 1
P /q−1
%
m=0
,
2π
(mq + l) + θ
x(mq + l) =
P
m=0
,9
+
P /q−1
% 8
2iπ
(mq + l) + iθ
ℜ
=
P
m=0
⎫
⎧
,⎬
+
/q−1
⎨P%
2iπ
(mq + l) + iθ
=ℜ
⎭
⎩ m=0
P
0⎫
/
⎧
2iπq P ⎬
,1−
+
⎨
P q
2iπl
= 2iπq >
=ℜ
iθ +
⎭
⎩
P
1−
P
P /q−1
%
+
= ℜ{0} = 0.
m=0
cq (n)
q−
q
cq (n)
q
1
!q
n=1 cq (n)
=
q = ps
?!q−1
n=1 (−1)
p
@
+ (q − 1) = 0
s∈N
cq (n)
ps
cq (n)
ps−1
p
q
ps−1
cq (n)
q = p1 s1 p2 s2 ...pk sk
qi = q/pi
i
x(n)
i = 1, ..., k
qi
q
qi
p1
l∈N
p1
%
cq (kq1 + l) = 0.
k=1
q2 , ..., qk
q1
q1 = q/p1 = p1 s1 −1 p2 s2 ...pk sk .
pi
p1
%
cq (kq1 + l) =
k=1
p1
%
cp1 s1 (kq1 + l)...cpk sk (kq1 + l).
k=1
s
pj j
q1
j = 2, ..., k
cpsj (kq1 +l)
j
cpj sj (l)
p1
%
cq (kq1 + l) = cp2 s2 (l)...cpk sk (l)
k=1
p1
%
cp1 s1 (kq1 + l).
k=1
p1 s1 −1
q1
kq1
l ̸= 0
kq1 +l
p1 s1 −1
l=0
p1
%
cq (kq1 ) = cp2 s2 (0)...cpk sk (0)
k=1
= cp2 s2 (0)...cpk sk (0)
p1
%
cp1 s1 (kq1 )
k=1
Ap −1
1
%
B
cp1 s1 (kq1 ) + cp1 s1 (p1 q1 ) .
k=1
kq1 = (kp2 s2 ...pk sk ) ps11 −1 .
k
1
p1 − 1
p1
p1 s1 −1
k
=
(kp2 s2 ...pk sk ) p1 s1 −1 , p1 s1
(kq1 , p1 s1 ) =
1 × p1 s1 −1 2 × p1 s1 −1
k
>
(p1 − 1) × p1 s1 −1
cp1 s1 (.)
−p1 s1 −1
p1
%
cq (kq1 ) = cp2 s2 (0)...cpk sk (0)
k=1
Ap −1
1
%
k=1
−p1 s1 −1 + cp1 s1 (p1 q1 )
B
?
@
= cp2 s2 (0)...cpk sk (0) −(p1 − 1)p1 s1 −1 + (p1 − 1)p1 s1 −1 = 0,
x(n)
q
τ (q)
n
x(n)
di
τ (n)
n
di
q−
x(n)
q
τ (q)
1 = d1 < d2 < ... < dk = q k = τ (q)
xd1 (n), xd2 (n), ..., xdk (n)
i xdi (n)
di −
x(n) q
τ (q)
x1 (n), ..., xτ (q) (n)
j
j ← 1 x (n) ← x(n)
j = 1, · · · τ (q)
k = 1, · · · , dj
d
yk j (n) ← [xj (n)]↓dj ,k
d
xdj (m) ← E[ymj (n)]
dj
x(n)
xj+1 (n) ← xj (n) − xdj (n)
x(n) =
!k
i=1
xdi (n)
q = 12
d5 = 6
d6 = 12
q
d 1 = 1 d 2 = 2 d 3 = 3 d4 = 4
12−
6
q
1 = d1 < d2 < ... <
dk = q
[5, −1, 5, −1, 5, −7]
x2 (n) x3 (n)
x1 (n)
x6 (n)
xdj (n)
xj+1 (n) = xj (n) − xdj (n)
xj+1 (n)
dj
xdj (n)
xdj (n)
dj
j
dj
dj
s | dj
xdj (n)
s | q
x(n)
q
j=1
x1 (n) = x(n) = [5, −1, 5, −1, 5, −7]
→ y11 (n) = [x1 (n)]↓1,1 = [5, −1, 5, −1, 5, −7]
→ x1 (n) = [E[y11 ]] = [1]
j=2
x2 (n) = x1 (n) − x1 (n) = [4, −2, 4, −2, 4, −8]
→ y12 (n) = [x2 (n)]↓2,1 = [4, 4, 4]
→ y22 (n) = [x2 (n)]↓2,2 = [−2, −2, −8]
→ x2 (n) = [E[y12 ], E[y22 ]] = [4, −4]
j=3
x3 (n) = x2 (n) − x2 (n) = [0, 2, 0, 2, 0, −4]
→ y13 (n) = [x3 (n)]↓3,1 = [0, 2]
→ y23 (n) = [x3 (n)]↓3,2 = [2, 0]
→ y33 (n) = [x3 (n)]↓3,3 = [0, −4]
→ x3 (n) = [E[y13 ], E[y23 ], E[y33 ]] = [1, 1, −2]
j=4
x4 (n) = x3 (n) − x3 (n) = [−1, 1, 2, 1, −1, −2]
→ x6 (n) = x4 (n) = [−1, 1, 2, 1, −1, −2]
q = 6 x(n) =
[5, −1, 5, −1, 5, −7]
xj (n)
xdj (n)
x(n)
q
xj (n)
s
1 = d1 < ... < dk = q
q
k = τ (q)
x(n)
x(n) = x1 (n) + ... + xk (n) = u1 (n) + ... + uk (n)
xi (n)
ui (n)
i = 1, ..., k
xdj (n) ̸= udj (n)
n=1
x(n)
dj
xdj (1) ̸= udj (1)
"
+
[x(n)]↓di ,1 = x(1), x(1 + di ), x(1 + 2di ), ..., x 1 +
[x(n)]↓dj ,1
xdi (n)
+
, ,#
q
− 1 di .
di
E[[x(n)]↓dj ,1 ] = δ
udi (n) i = 1, ..., k
ydi (n)
vdi (n)
[x(n)]↓dj ,1 = y1 (n) + ... + yk (n)
= v1 (n) + ... + vk (n),
ys (n) := [xs (n)]↓dj ,1 ,
s = 1, ..., k.
vs (n) := [us (n)]↓dj ,1 ,
s = 1, ..., k.
i ̸= j ydi (n)
xdi (n)
udi (n)
dj ̸= di
i ̸= j.
E [ydi (n)] = E [vdi (n)] = 0,
ydj (n, 1)
vdj (n)
dj
?
@
ydj (n) = xdj (1), xdj (1), ..., xdj (1)
?
@
vdj (n) = udj (1), udj (1), ..., udj (1) .
vdi (n)
E[ydj (n)] = xdj (1)
E[vdj (n)] = udj (1)
E[[x(n)]↓dj ,1 ] = E [y1 (n, 1) + ... + yk (n, 1)]
= E [v1 (n, 1) + ... + vk (n, 1)] .
@
?
@
?
δ = E ydj (n, 1) = E vdj (n, 1) ,
δ = xdj (1) = udj (1)
x(n)
[1, −1, 1, −1, 1, −0.9] =[0.017] + [0.983, −0.983] + [−0.017, −0.017, 0.033]
+ [0.017, −0.017, −0.033, −0.017, 0.017, 0.033].
x(n)
[0.983, −0.983]
△
x(n)
1 = d1 , d2 , ..., dk = P
P
x(n)
x(n) =
%
xdi (n),
%
∥xdi (n)∥2 ,
di |P
∥x(n)∥2 =
di |P
P
∥x(n)∥ :=
∥x(n)∥2 =
P
%
i=1
x(n)2 =
-
P
1 %
x(i)2
P i=1
P
%
i=1
⎡
⎛
⎝
%
di |P
.1/2
.
⎞2
xdi (n)⎠
⎤
P
%
%
⎥
⎢%
2
⎥.
⎢
=
x
(n)
+
2
x
(n)x
(n)
d
d
d
i
i
j
⎦
⎣
i=1
di
di |P
di <dj
di |P,dj |P
dj
P
(di , dj )
P
%
i=1
xdi (n)xdj (n) = 0,
di < dj ,
di , dj |P,
P
2
∥x(n)∥ =
P %
%
xdi (n)2 =
i=1 di |P
%
di |P
∥xdi (n)∥2 .
q
φ(q)
q
x(n)
x(0), x(1), ..., x(q − 1)
q
q
x(n)
q
q = ps
q
q
φ(q)
1, p, p2 , ..., ps
1 p p2
x(n)
ps−1
⎧ ps −1
%
⎪
⎪
⎪
x(k) = 0
⎪
⎪
⎪
⎪
k=0
⎪
⎪
⎪
ps−1 −1
⎪
%
⎪
⎪
⎪
x(pk + l) = 0, l = 0, 1, ...p
⎪
⎪
⎪
⎪
k=0
⎨ s−2
p
%−1
⎪
x(p2 k + l) = 0, l = 0, 1, ..., p2
⎪
⎪
⎪
⎪
k=0
⎪
⎪
..
⎪
⎪
⎪
.
⎪
⎪
⎪
p−1
⎪
%
⎪
⎪
⎪
⎪
x(ps−1 k + l) = 0, l = 0, 1, ..., ps−1
⎩
k=0
p∈P
s∈N
p−1
%
x(ps−1 k + l) = 0,
k=0
l = 0, 1, ..., ps−1 − 1.
p−1
l
p
ps−1
x(n)
l
ps−1 (p − 1) = φ(ps ) = φ(q)
q = p1 s1 p2 s2 ...pk sk
q
y0 (n) = [x(n)]↓p1 s1 ,0
y1 (n) = [x(n)]↓p1 s1 ,1
y2 (n) = [x(n)]↓p1 s1 ,2
..
.
s
s
yps−1
−1 (n) = [x(n)]↓p1 1 ,p1 1 −1 ;
1
p2 s2 ...pk sk
p1 s 1
x(0), x(p2 s2 ...pk sk ), x(2p2 s2 ...pk sk ), ..., x((p1 s1 − 1)p2 s2 ...pk sk ).
(p1 s1 , p2 s2 ...pk sk ) = 1
y0 (n)
yps−1
−1 (n)
1
y0 (n)
yps−1
−s (n)
1
..
.
x(0)
x(β)
x(2β)
..
.
x(α)
x(β + α)
x(2β + α)
..
.
x(2α)
x(β + 2α)
x(2β + 2α)
..
.
α−1
x((α−1)β)
x((α−1)β+α)
x((α−1)β+2α)
..
.
x((β − 1)α)
x(β + (β − 1)α)
x(2β + (β − 1)α)
..
.
x((α−1)β+(β−1)α)
β−1
α = p1 s1 β = p2 s2 ...pk sk
zi (n)
z0 (n) z1 (n)
zps−1
−1 (n)
1
p2 s2 ...pk sk
zi (n)
α = p1 s1
x(n)
β = p2 s2 ...pk sk
αβ
α
β
α
q
p1 s1
p2 s2 ...pk sk
φ(p1 s1 )
x(n)
φ(p1 s1 )φ(p2 s2 )...φ(pk sk ) = φ(q)
q = 36 = 22 32
β
q = 36 = 22 32
y0 (n) = [x(0), x(4), x(8), x(12), x(16), x(20), x(24), x(28), x(32)]
y1 (n) = [x(1), x(5), x(9), x(13), x(17), x(21), x(25), x(29), x(33)]
y2 (n) = [x(2), x(6), x(10), x(14), x(18), x(22), x(26), x(30), x(34)]
y3 (n) = [x(3), x(7), x(11), x(15), x(19), x(23), x(27), x(31), x(35)].
x(n)
x(18)
x(0) x(9)
yi (n)
x(27)
y1 (n) y2 (n)
y3 (n)
(4, 9) = 1
x(9) x(18)
x(27)
z0 (n) = [x(0), x(4), x(8), x(12), x(16), x(20), x(24), x(28), x(32)]
z1 (n) = [x(9), x(13), x(17), x(21), x(25), x(29), x(33), x(1), x(5)]
z2 (n) = [x(18), x(22), x(26), x(30), x(34), x(2), x(6), x(10), x(14)]
z3 (n) = [x(27), x(31), x(35), x(3), x(7), x(11), x(15), x(19), x(23)].
z1 (n)
x(1) = x(37)
x(5) = x(41)
zi (n)
z2 (n)
x(33)
z3 (n)
x(n)
x(0)
x(4)
x(8)
x(12) x(16) x(20) x(24) x(28) x(32)
x(9)
x(13) x(17) x(21) x(25) x(29) x(33) x(37) x(41)
x(18) x(22) x(26) x(30) x(34) x(38) x(42) x(46) x(50)
x(27) x(31) x(35) x(39) x(43) x(47) x(51) x(55) x(59).
φ(9)
φ(4)
φ(9)φ(4) = φ(36)
P
P
x(n) =
%
xdi (n),
di |P
xdi (n)
φ(di )
φ ⋆ (n) =
%
φ(d) = n,
d|n
P
x(n)
△
6−
a[n] = [−0.8391, 0.9346, −0.7455, 0.6900, −0.8163, 0.5552].
a1 [n] = [−0.0368]
a2 n] = [−0.7634, 0.7634]
a3 [n] = [−0.0377, 0.0960, −0.0583]
a6 [n] = [−0.0011, 0.1120, 0.1131, 0.0011, −0.1120, −0.1131].
△
a2 [n]
2
6−
b[n] = [0.9829, 0.9325, −0.0593, −0.6822, −1.0793, −0.5002].
1
0.8
a1[n]
a [n]
2
0.8
0.6
a3[n]
a [n]
6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−1
0
2
4
6
8
10
12
14
16
18
20
−0.8
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
a[n] =
[−0.8391, 0.9346, −0.7455, 0.6900, −0.8163, 0.5552]
b1 [n] = [−0.0676]
b2 [n] = [0.0157, −0.0157]
b3 [n] = [0.2179, −0.0058, −0.2121]
b6 [n] = [0.8168, 1.0216, 0.2047, −0.8168, −1.0216, −0.2047].
△
b6 [n]
6
a[n]
6−
b[n]
x[n] = a[n] + b[n]
x1 [n] = [−0.1044]
x2 [n] = [−0.7477, 0.7477]
1
1.5
b1[n]
b [n]
2
0.8
b3[n]
1
b [n]
6
0.6
0.4
0.5
0.2
0
0
−0.2
−0.5
−0.4
−0.6
−1
−0.8
−1
0
2
4
6
8
10
12
14
16
18
20
−1.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
b[n] =
[0.9829, 0.9325, −0.0593, −0.6822, −1.0793, −0.5002]
x3 [n] = [0.1802, 0.0902, −0.2704]
x6 [n] = [0.8157, 1.1336, 0.3178, −0.8157, −1.1336, −0.3178].
△
ak [n] + bk [n] = xk [n]
6
x[n]
x[n]
1 2 3
4
k = 1, 2, 3, 6
1
1.5
x1[n]
x [n]
2
0.8
x3[n]
1
x [n]
6
0.6
0.4
0.5
0.2
0
0
−0.2
−0.5
−0.4
−0.6
−1
−0.8
−1
0
2
4
6
x[n] = a[n] + b[n]
8
10
12
14
16
18
20
−1.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
x(n + r)
r = 1, 2, ...
r−
x(n)
r
N
X(q, r)
x(n)
N
1 %
x(n + r − 1)cq (n).
X(q, r) =
N n=1
1/φ(q)
x(n) =
(2πn/10)
N = 100
r−
N = 500
x(n − r + 1)
N = 100
N = 500
(2πn/10)
N
φ(q)
X(q, r) = φ(q)yq ,
y(n) = x(n − r + 1).
q = 10
1/φ(q)
x(n)
φ(q)
q = 10
10−
N
q
r
x(n)
x(1), ..., x(r − 1)
R
cq (n)
x(n)
r−
X (q, r)
N
x(n)
N
1 %
x(n)cq (n − r + 1),
X (q, r) =
N n=1
cq (n)
x(n) =
(2πn/10)
q
X(q, .)
x(n)
cq (n)
⎡
⎤
⎡
cq (R)
⎢ X(q, 1) ⎥ ⎢cq (1) cq (2) · · ·
⎢
⎥ ⎢
⎢ X(q, 2) ⎥ ⎢ 0
cq (1) · · · cq (R − 1)
⎢
⎥ ⎢
=
⎢
⎥
⎢
..
..
..
..
⎢
⎥ ⎢ ..
.
.
.
.
⎢
⎥ ⎢ .
⎣
⎦ ⎣
0
0
···
cq (1)
X(q, R)
···
cq (N )
⎤⎡
⎤
⎥ ⎢ x(1) ⎥
⎥⎢
⎥
⎢ x(2) ⎥
···
cq (N − 1) ⎥
⎥⎢
⎥
⎥⎢ . ⎥.
..
..
⎥⎢ . ⎥
.
.
⎥⎢ . ⎥
⎦⎣
⎦
· · · cq (N − R + 1)
x(N )
X (q, .)
⎡
⎤
⎡
⎢ X (q, 1) ⎥ ⎢ cq (1)
⎢
⎥ ⎢
⎢ X (q, 2) ⎥ ⎢ cq (0)
⎢
⎥ ⎢
⎢
⎥=⎢
..
.
⎢
⎥ ⎢
..
.
⎢
⎥ ⎢
⎣
⎦ ⎣
cq (−R+2)
X (q, R)
cq (2)
cq (1)
..
.
cq (−R+3)
···
cq (R)
⎤⎡
···
⎤
cq (N ) ⎥ ⎢ x(1) ⎥
⎥⎢
⎥
⎢ x(2) ⎥
· · · cq (R − 1) · · · cq (N − 1)⎥
⎥⎢
⎥
⎥⎢ . ⎥.
..
..
..
..
⎥⎢ . ⎥
.
.
.
.
⎥⎢ . ⎥
⎦⎣
⎦
cq (1)
···
· · · cq (N −R+1)
x(N )
X (q, r)
N = 100
N = 500
(2πn/10)
N
x(n)
cq (n)
N = 100
500
10−
x(n)
X (q, r)
P
N
X (q, r)
q−
q|P
q|N
P |N
q−
xq (n)
q!P
x(n)
x(n)
q!N
P !N
q−
N
q|N
P !N
P |N
q−
q!N
x(n)
N
q ! N
q | N
N
q ! 100
10−
10−
N
x(n)
x(n) =
k
%
xdi (n).
i=1
xd (n)
d
xd (n)
d−
x(n)
xd (n)
P |N
x(n)
d|N
xd (n)
q ̸= d
N
Xd (., .)
D=
Xd (., .)
d−
q | N
q−
N
(q, d)
cq (n − r + 1)
D
%
n=1
r∈N
r ∈ N
xd (n)cq (n − r + 1) = 0
N = CD + l
xd (n)
C∈Z
l ∈ {0, 1, 2, ..., Di − 1}
q−
r
CD+l
1 %
xd (n)cq (n − r + 1).
Xd (q, r) =
N n=1
q|N
d | N
d|N
l=0
q!N
d!N
l>0
B
A CD
CD+l
%
1 %
Xd (q, r) =
xd (n)cq (n − r + 1) +
xd (n)cq (n − r + 1)
N n=1
n=CD+1
B
A
l
%
1
=
xd (n)cq (n − r + 1) .
0+
N
n=1
Xd (q, r)
N
d
x(n)
d−
d|N
xd (n)
N
Xd (., .)
d−
Xd (d, r) = xd (r − 1)
q|N
M = p1 p2 ...pk
n1 , n 2 ∈ Z
pi
(n1 , pi ) =
⇔
d!N
(n2 , pi ),
(n1 , M ) =
∀i = 1, ..., k
(n2 , M ).
N
M = p1 p2 ...pk
pi
m1 , n1 ∈ [0, p1 − 1] ∩ Z
m2 , n2 ∈ [0, p2 − 1] ∩ Z
..
.
mk , nk ∈ [0, pk − 1] ∩ Z,
mi ni
mj ̸= nj
j
A = m1 (p2 p3 ...pk ) + m2 (p1 p3 ...pk ) + ... + mk (p1 p2 ...pk−1 )
B = n1 (p2 p3 ...pk ) + n2 (p1 p3 ...pk ) + ... + nk (p1 p2 ...pk−1 ),
(A, M ) ̸=
(B, M )
(A, M ) =
(B, pi )
(A, pi ) =
(B, M )
i = 1, ..., k
j
mj ̸= nj
(A, pj ) =
(mj (p1 p2 ...pj−1 pj+1 ...pk ), pj )
(B, pj ) =
(nj (p1 p2 ...pj−1 pj+1 ...pk ), pj ).
((mj − nj )p1 p2 ...pj−1 pj+1 ...pk , pj ) = 0.
pi
mj − nj
i ̸= j
pj
pj
(p1 p2 ...pj−1 pj+1 ...pk , pj ) = 1
mj , nj ∈ [0, pj − 1] ∩ Z
mj = nj
k = 3 p1 = 2 p2 = 3 p3 = 5
(n1 , n2 , n3 )
n1 ∈ {0, 1} n2 ∈ {0, 1, 2}
3−
n3 ∈ {0, 1, 2, 3, 4}
n1 (p2 p3 ) + n2 (p1 p3 ) + n3 (p1 p2 )
=
3−
M = 30
[0, M )
△
k−
M = p1 p2 ...pk
i = 1, ..., k
S1 = {0, 1, ..., M − 1}
$
- k
.
G
%
mj (p1 p2 ...pj−1 pj+1 ...pk ), M | mi ∈ [0, pi − 1] ∩ Z
S2 =
j=1
j ∈ [0, M − 1] ∩ Z
[0, pi − 1] ∩ Z
j =
m1 , ..., mk
(
!k
j=1
x(n)
{x(i)|i ∈ S1 }
m1 , ..., mk
mi ∈
mj (p1 p2 ...pj−1 pj+1 ...pk ), M )
M
S1x =
S2x = {x(j)|j ∈ S2 }
x(n)
30−
x(0(3 × 5) + 0(1 × 5) + 3(1 × 3)) = x(3)
x(0(3 × 5) + 0(1 × 5) + 0(1 × 3)) = x(0)
x(0), x(1), ..., x(29)
= (0, 0, 0) :
= (0, 0, 1) :
= (0, 0, 2) :
= (0, 0, 3) :
= (0, 0, 4) :
= (0, 1, 0) :
= (0, 1, 1) :
= (0, 1, 2) :
= (0, 1, 3) :
= (0, 1, 4) :
= (0, 2, 0) :
= (0, 2, 1) :
= (0, 2, 2) :
= (0, 2, 3) :
= (0, 2, 4) :
= (1, 0, 0) :
= (1, 0, 1) :
= (1, 0, 2) :
= (1, 0, 3) :
= (1, 0, 4) :
= (1, 1, 0) :
= (1, 1, 1) :
= (1, 1, 2) :
= (1, 1, 3) :
= (1, 1, 4) :
= (1, 2, 0) :
= (1, 2, 1) :
= (1, 2, 2) :
= (1, 2, 3) :
= (1, 2, 4) :
(0(3 × 5) + 0(1 × 5) + 0(1 × 3), 30) = 0
(0(3 × 5) + 0(1 × 5) + 1(1 × 3), 30) = 3
(0(3 × 5) + 0(1 × 5) + 2(1 × 3), 30) = 6
(0(3 × 5) + 0(1 × 5) + 3(1 × 3), 30) = 9
(0(3 × 5) + 0(1 × 5) + 4(1 × 3), 30) = 12
(0(3 × 5) + 1(1 × 5) + 0(1 × 3), 30) = 5
(0(3 × 5) + 1(1 × 5) + 1(1 × 3), 30) = 8
(0(3 × 5) + 1(1 × 5) + 2(1 × 3), 30) = 11
(0(3 × 5) + 1(1 × 5) + 3(1 × 3), 30) = 14
(0(3 × 5) + 1(1 × 5) + 4(1 × 3), 30) = 17
(0(3 × 5) + 2(1 × 5) + 0(1 × 3), 30) = 10
(0(3 × 5) + 2(1 × 5) + 1(1 × 3), 30) = 13
(0(3 × 5) + 2(1 × 5) + 2(1 × 3), 30) = 16
(0(3 × 5) + 2(1 × 5) + 3(1 × 3), 30) = 19
(0(3 × 5) + 2(1 × 5) + 4(1 × 3), 30) = 22
(1(3 × 5) + 0(1 × 5) + 0(1 × 3), 30) = 15
(1(3 × 5) + 0(1 × 5) + 1(1 × 3), 30) = 18
(1(3 × 5) + 0(1 × 5) + 2(1 × 3), 30) = 21
(1(3 × 5) + 0(1 × 5) + 3(1 × 3), 30) = 24
(1(3 × 5) + 0(1 × 5) + 4(1 × 3), 30) = 27
(1(3 × 5) + 1(1 × 5) + 0(1 × 3), 30) = 20
(1(3 × 5) + 1(1 × 5) + 1(1 × 3), 30) = 23
(1(3 × 5) + 1(1 × 5) + 2(1 × 3), 30) = 26
(1(3 × 5) + 1(1 × 5) + 3(1 × 3), 30) = 29
(1(3 × 5) + 1(1 × 5) + 4(1 × 3), 30) = 2
(1(3 × 5) + 2(1 × 5) + 0(1 × 3), 30) = 25
(1(3 × 5) + 2(1 × 5) + 1(1 × 3), 30) = 28
(1(3 × 5) + 2(1 × 5) + 2(1 × 3), 30) = 1
(1(3 × 5) + 2(1 × 5) + 3(1 × 3), 30) = 4
(1(3 × 5) + 2(1 × 5) + 4(1 × 3), 30) = 7
3−
30
d′ = d/M = p1 s1 −1 p2 s2 −1 ...pk sk −1
d = p1 s1 p2 s2 ...pk sk M = p1 p2 ...pk
r∈Z
T1 = {
$
T2 =
i = 1, ..., k
(id′ + r, d) | i = 0, ..., M − 1}
-A k
B
.
G
%
mj (p1 p2 ...pj−1 pj+1 ...pk ) d′ + r, d | mi ∈ [0, pi − 1] ∩ Z
j=1
(kd′ + r, d)
k = 0, ..., M − 1
q′
S1 = S2
M
d
T1′ = {
$
T2′ =
(id′ , d) |i = 0, ..., M − 1}
-A k
B
.
G
%
mj (p1 p2 ...pj−1 pj+1 ...pk ) d′ , d | mi ∈ [0, pi − 1] ∩ Z
j=1
r
xd (n)
d
i ∈ T1
j ∈ T2
d
xd (i) = xd (j)
d
d = p1 s1 p2 s2 ...pk sk .
pi si
pj sj
cd (n) = cp1 s1 (n)cp2 s2 (n)...cpk sk (n)
N
d
H ∈ Z+
N = Hd + l
l ∈ [0, d) ∩ Z
d−
N
1 %
Xd (d, r) =
xd (n)cd (n − r + 1)
N n=1
A Hd
B
Hd+l
%
1 %
=
xd (n)cd (n − r + 1) +
xd (n)cd (n − r + 1)
N n=1
n=Hd+1
B
A
d
l
%
%
1
=
xd (n)cd (n − r + 1) +
xd (n)cd (n − r + 1)
H
N
n=1
n=1
d
H%
=
xd (n)cd (n − r + 1) + Rd (r)
N n=1
d
H%
=
xd (n)cp1 s1 (n − r + 1)cp2 s2 (n − r + 1)...cpk sk (n − r + 1) + Rd (r),
N n=1
l
1 %
Rd (r) :=
xd (n)cd (n − r + 1).
N n=1
cpi si (m)
pi si −1
m
n = 1, ..., d
p1 s1 −1 p2 s2 −1
n−r+1
n
p1 s1 −1 p2 s2 −1 ...pk sk −1 | n − r + 1
d′ = p1 s1 −1 p2 s2 −1 ...pk sk −1 = d/p1 p2 ...pk
n = md′ + r − 1
H
Xd (d, r) =
N
pk sk −1
p1 p2 ...pk −1
%
n − r + 1 = md′
m = 0, ..., p1 ...pk − 1
xd (md′ +r−1)cp1 s1 (md′ )cp2 s2 (md′ )...cpk sk (md′ )+Rd (r).
m=0
m ∈ [0, p1 ...pk − 1] ∩ Z
k−
mi ∈ [0, pi − 1] ∩ Z
(m1 , m2 , ..., mk )
md′ ≡ [m1 (p2 p3 ...pk ) + m2 (p1 p3 ...pk ) + ... + mk (p1 p2 ...pk−1 )] d′
,
+
mk
m1 m2
d (
d),
+
+ ... +
≡
p1
p2
pk
psi i
p i d′
mi d′
cpi si (md′ ) = cpi si ([m1 (p2 p3 ...pk ) + ... + mk (p1 p2 ...pk−1 )] d′ )
= cpi si (mi (p1 p2 ...pi−1 pi+1 ...pk )d′ )
+
,
mi
d .
= cpi si
pi
pi si
cpi si
d
d
k−
m=0
xd (md′ + r − 1)cd (md′ )
m
md′
p1 p2 ...pk − 1
m1
0
p1 − 1 m2
0
p2 − 1
mk
0
Xd (d, r)
+"
#
,
p1 −1
pk −1 "
%
H %
mk
m1
xd
d+r−1 ×
...
+ ... +
Xd (d, r) =
N m =0 m =0
p1
pk
1
k
+
+
,
,#
m1
mk
cp1 s1
d ...cpk sk
d + Rd (r).
p1
pk
!pk −1
mk =0
xd (.)cpk sk (.)
pk − 1
mk = 0
#
,
+
,
mk
m1
mk
d + r − 1 cpk sk
xd
+ ... +
d
p1
pk
pk
mk =0
+
,
pk −1
%
mk
sk −1
r)(pk − 1)(pk
)+
xd
d + rH (−pk sk −1 )
= xd (H
pk
mk =1
B
A
pk −1
%
r) −
xd (mk (p1 p2 ...pk−1 )d′ + rH) ,
= pk sk −1 (pk − 1)xd (H
pk −1
%
+"
mk =1
rH = [m1 /p1 + ... + mk /pk ] d + r − 1
p1 p2 ...pk−1 d′ = d/pk
xd (H
r)
r)
−xd (H
0
pk −1
%
mk =0
xd
+"
xd (n)
#
,
+
,
mk
m1
mk
d + r − 1 cpk sk
+ ... +
d
p1
pk
pk
r) + xd (H
r)]
= pk sk −1 [(pk − 1)xd (H
r).
= pk sk xd (H
rI
+"
#
,
pk−1 −1 "
p1 −1
%
Hpk sk %
mk−1
m1
xd
d+r−1 ×
...
+ ... +
Xd (d, r) =
N m =0 m =0
p1
pk−1
1
k
+
+
,
,#
m1
mk−1
c p1 s 1
d ...cpk−1 sk−1
d + Rd (r).
p1
pk−1
pk
pk sk
pk−1 , pk−2 , ..., p1
[m1 /p1 + ... + mk−1 /pk−1 ]
Hd
Hp1 s1 p2 s2 ...pk sk
=
.
N
N
Rd (r)
l
Hd
1 %
Xd (d, r) =
xd (r − 1) +
xd (n)cd (n − r + 1).
N
N n=1
d | N Hd/N = 1
Xd (d, r) = xd (r − 1)
l=0
N = Hd + l
Rd (r)
Hd/N
N
d!N
N
0≤l<d
cq (n)
q
q
cq (n)
q
q
q = p1 s1 p2 s2 ...pk sk
cq (n) = cp1 s1 (n)cp2 s2 (n)...cpk sk (n).
cq (n)
cq (n)
q
x(n)
Q×R
Q×R
x(n)
O(Q2 R)
cq (n)
q−
x(n)
O(Q( Q)R)
xq
Q
O(QR)
R
Q
N
N
N
N
q
q
x(n)
q
X (., .)
q
Y(q)
yq (r)
q
X (., .)
q
cq (n)
1/N
yq (r) := X (q, .) =
1
(x ! cq )(r).
N
N
Yq (k) =
1
X(k)Cq (k),
N
x(n)
Yq (k) X(k)
Cq (k)
yq (n) x(n)
∥x(n)∥ :=
-
P
1 %
x(i)2
P i=1
Y(q)
.1/2
.
yq (n)
N
1 %
yq (n)2
Y(q) := ∥yq (n)∥ =
N n=1
2
2
N
%
yq (n)2 =
n=1
N −1
1 %
∥Yq (k)∥2 .
N k=0
N −1
1 %
∥Yq (k)∥2
Y(q) = 2
N k=0
2
N −1
1 % 1
= 2
∥X(k)Cq (k)∥2
N k=0 N 2
N −1
1 %
= 4
∥X(k)Cq (k)∥2 .
N k=0
φ(q)
N
q
q
q−1
1 N%
∥X(k)Cq (k)∥2
Y(q) = 4
N q k=0
2
1
=
qN 3
q−1
%
k=0
(k,q)=1
q 2 ∥X(k)∥2
Cq (k)
cq (n)
q
= 3
N
q−1
%
k=0
(k,q)=1
J
K
K q
Y(q) = K
LN3
X(k)
N
∥X(k)∥2 .
q−1
%
k=0
(k,q)=1
∥X(k)∥2 ,
x(n)
q
q
N
Y(q)
q
N
q|N
q
a[n] = [−0.8391, 0.9346, −0.7455, 0.6900, −0.8163, 0.5552],
b[n] = [0.9829, 0.9325, −0.0593, −0.6822, −1.0793, −0.5002].
a(n)
b(n)
a6 (n)
a3 (n)
a2 (n)
a1 (n)
IIPF components
RS map
a(n)
Xa (q, r)
a(n)
a(n)
b6 (n)
b3 (n)
b2 (n)
b1 (n)
IIPF components
RS map
b(n)
Xb (q, r)
b(n)
b(n)
5
4
4
3
3
2
2
1
1
f(n)
f(n)
5
0
0
−1
−1
−2
−2
−3
−3
−4
−4
−5
20
40
60
80
100
120
−5
140
20
40
60
n
80
100
120
140
n
0.06
0.04
0.04
0.02
0.02
xq
0.1
0.08
0.06
xq
0.1
0.08
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
0
5
10
15
20
25
q
30
35
40
45
50
−0.1
1
1
0.9
0.9
0.8
0.8
Norm of q−th column of X(q,r)
Norm of q−th column of X(q,r)
−0.1
0.7
0.6
0.5
0.4
0.3
0.2
15
20
25
q
30
35
40
45
50
0
5
10
15
20
25
q
30
35
40
45
50
0
5
10
15
20
25
k
30
35
40
45
50
0.6
0.5
0.4
0.3
0.1
0
5
10
15
20
25
q
30
35
40
45
0
50
30
25
25
Fourier spectrum Xk
35
30
k
Fourier spectrum X
10
0.7
35
20
15
20
15
10
10
5
5
0
5
0.2
0.1
0
0
0
5
10
15
20
25
k
30
35
40
45
50
0
x(n)
(2πn/10)
20
(2πn/20)
N = 500
14
x(n) =
(2πn/10) +
x(n)
(2πn/14) +
q−
q−
0.71 1.00
0.58
q
q=2 5
12
q = 12 14
20
q = 10 14
20
7
14
4
3
3
2
2
1
1
f(n)
f(n)
4
0
0
−1
−1
−2
−2
−3
−4
−3
0
5
10
15
20
25
n
30
35
40
45
50
f1(n)
1
0
5
10
15
20
25
n
30
35
40
45
1
g (n)
1
0.8
0.6
0.6
0.4
0.4
0.2
50
f1(n)
1
g (n)
0.8
−4
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
0
5
10
15
20
25
n
30
35
40
45
50
f2(n)
1
5
10
15
20
25
n
30
35
40
45
g2(n)
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
50
f2(n)
1
g2(n)
0.8
0
−1
0
5
10
15
20
25
n
30
35
40
45
50
f3(n)
1
5
10
15
20
25
n
30
35
40
45
3
g (n)
3
0.8
0.6
0.6
0.4
0.4
0.2
50
f3(n)
1
g (n)
0.8
0
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
0
5
10
15
20
25
n
30
35
40
45
x(n)
50
0
5
6 12
fi (n)
0.5
10
15
18
x(n)
20
25
n
30
35
40
45
50
gi (n)
6 12
18
6−
R
12−
18−
Q
R×Q
q
N
N
x[n]
X(z) =
∞
%
n=0
x[n]z −n .
z=
(iω)
N
%
k=0
ak y[n − k] =
H(z) =
M
%
k=0
M
%
k=0
N
%
bk x[n − k],
bk z −k
.
ak z −k
k=0
H(z)
x[1] x[2]
z −1 )
X(z)
x[n]
X(z)
[M /N ] M
X(z)
N
R[M /N ] (z)
1.5
1
0.8
1
0.6
0.4
0.5
0.2
0
0
−0.2
−0.5
−0.4
−0.6
−1
−0.8
−1.5
0
50
100
150
200
250
300
50
100
−1
0
50
100
200
250
300
150
200
250
0.25
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
0
150
M
*
K
(z − zj )
P (z)
j=1
.
= N
R[M /N ] (z) =
Q(z)
*
(z − pj )
j=1
[m/m + 1]
m
m+1
300
pj
zj
[M /N ]
0.905 ± 0.403i
M
0.905
M
[0, 2π]
N
r = 1
N
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1.5
−1
−0.5
0
m=7
0.5
1
1.5
2
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
m = 11
m = 7
m = 11
2
f (x) ∈ C m m
a ∈ Df
f
Pm (x) =
m
%
k=0
ck (x − a)k .
Pm (a) = f (a),
Pm ′ (a) = f ′ (a), ...,
ck =
Pm (m) (a) = f (m) (a).
f (k) (a)
.
k!
a=0
m+1
M (n+1) M
Mf
(x)M ≤ M
Rm (x) := |f (x) − Pm (x)|
|Rm (x)| ≤
M
|x − a|n+1 ,
(n + 1)!
|x − a| ≤ d
|x − a| ≤ d.
n
P (x)/Q(x) P (x)
Q(x)
f (x)
m
Q(x)
Rf,[m/n]
b0
O(xn )
n
1
P (x)
,
:=
Q(x)
1
P (x) =
m
%
j=0
j
aj x ,
Q(x) =
n
%
j=0
b j xj .
m+1
P (x)
Q(x)
n
O(xm+n+1 )
Rf,[m/n]
O(xm+n+1 )
f (x) − Rf,[m/n] (x)
O(xm+n+1 )
Q(x)f (x) − P (x)
O(xm+n+1 )
Q(x)f (x) − P (x)
f (x) =
∞
%
f (x)
c j xj ,
j=0
Q(x)f (x) − P (x)
Q(x)f (x) − P (x) =
=
-
n
%
bk x
k
k=0
-
∞
n
%
%
k=0
.-
i<0
bi = 0
j=0
bj ck−j = ak ,
j=0
n
%
bj ck−j = 0,
j=0
m+n+1
.
i<0
j
.
xk −
Q(x)
−
m
%
m
%
a l xl
l=0
al xl
l=0
n
i>n
1 x x2
n
%
cj x
j=0
bj ck−j
n+1
ci = 0
∞
%
xm+n
k = 0, 1, · · · , m
k = m + 1, m + 2, · · · , m + n.
m+n+1
O(xm+n+1 )
f (x)
⎡
⎤
⎡
0
0
c
⎢ 0
a
0
⎢
⎢ ⎥ ⎢c
c0
0
⎢ ⎥ ⎢ 1
⎢ a1 ⎥ ⎢
⎢ ⎥ ⎢
⎢ . ⎥ = ⎢ c2
c1
c0
⎢ . ⎥ ⎢
⎢ . ⎥ ⎢ ..
..
..
⎣ ⎦ ⎢ .
.
.
⎣
am
cm cm−1 cm−2
a
G
⎡ ⎤ ⎡
cm
⎢0⎥ ⎢ cm+1
⎢ ⎥ ⎢
⎢0⎥ ⎢ cm+2 cm+1
⎢ ⎥ ⎢
⎢.⎥ = ⎢ .
..
⎢.⎥ ⎢ .
.
⎢.⎥ ⎢ .
⎣ ⎦ ⎣
0
cm+n cm+n−1
0
···
c0
0
···
···
..
.
c1
..
.
c0
..
.
···
..
.
· · · cn−1 cn−2 · · ·
H′
⎡ ⎤ ⎡
cm
⎢0⎥ ⎢ cm+1
⎢ ⎥ ⎢
⎢0⎥ ⎢ cm+2 cm+1
⎢ ⎥ ⎢
⎢.⎥ = ⎢ .
..
⎢.⎥ ⎢ .
.
⎢.⎥ ⎢ .
⎣ ⎦ ⎣
cm+n cm+n−1
0
H′
0
···
···
..
.
···
⎤
··· 0 ⎡ ⎤
⎥ b
⎥⎢ 0 ⎥
··· 0⎥
⎥
⎥⎢
⎥
b
⎥⎢
1
⎢ ⎥
⎥
⎢
· · · 0 ⎥ . ⎥,
⎥
⎥⎢
⎢ .. ⎥
..
⎥
. 0 ⎥⎣ ⎦
⎦ bm
· · · c0
b′
⎤⎡ ⎤
0 ⎥⎢ b 0 ⎥
⎥⎢ ⎥
⎢ ⎥
0⎥
⎥⎢ b1 ⎥
⎥
⎥,
.. ⎥⎢
⎢ .. ⎥
. ⎥⎢ . ⎥
⎦⎣ ⎦
cm bn
m<n−1
b
⎤⎡ ⎤
cm−n+1 ⎥⎢ b0 ⎥
⎥⎢ ⎥
⎢ ⎥
cm−n+2 ⎥
⎥⎢ b1 ⎥
⎢ ⎥,
.. ⎥
⎥⎢ . ⎥
. ⎥⎢ .. ⎥
⎦⎣ ⎦
cm
bn
m ≥ n − 1.
⎤⎡ ⎤
0 ⎥⎢ b 1 ⎥
⎥⎢ ⎥
⎢ ⎥
0⎥
⎥⎢ b2 ⎥
⎢ ⎥,
.. ⎥
⎥⎢ . ⎥
. ⎥⎢ .. ⎥
⎦⎣ ⎦
cm bn
m<n−1
b
b0 = 1
⎤
⎡
⎡
⎢ −cm+1 ⎥ ⎢ cm
⎥ ⎢
⎢
⎢ −cm+2 ⎥ ⎢ cm+1
⎥ ⎢
⎢
⎢ . ⎥=⎢ .
⎢ . ⎥ ⎢ .
⎢ . ⎥ ⎢ .
⎦ ⎣
⎣
−cm+n
cm+n−1
−c(m+1)
···
c0
0
···
···
..
.
c1
..
.
c0
..
.
···
..
.
· · · cn−1 cn−2 · · ·
H
b(1)
⎡
⎤
⎡
···
⎢ −cm+1 ⎥ ⎢ cm
⎢
⎥ ⎢
⎢ −cm+2 ⎥ ⎢ cm+1
⎢
⎥ ⎢
⎢ . ⎥=⎢ .
⎢ . ⎥ ⎢ .
⎢ . ⎥ ⎢ .
⎣
⎦ ⎣
−cm+n
cm+n−1
···
..
.
···
−c(m+1)
H
⎤⎡ ⎤
cm−n+1 ⎥⎢ b1 ⎥
⎥⎢ ⎥
⎢ ⎥
cm−n+2 ⎥
⎥⎢ b2 ⎥
⎥
⎥,
.. ⎥⎢
⎢ .. ⎥
. ⎥⎢ . ⎥
⎦⎣ ⎦
cm
bn
m ≥ n − 1.
b(1)
bi
ai
f (x) =
ex = 1 + x + x2 /2! + x3 /3! + ...
Rf,[2,3] (x)
f (x)
bi
⎡
−c3
⎡
− 16
⎤
⎡
⎥ ⎢
⎢
⎥ ⎢
⎢
⎢−c4 ⎥ = ⎢c3 c2
⎦ ⎣
⎣
−c5
c4 c3
⎤
⎤⎡ ⎤
b
⎥ ⎢ 1⎥
⎥⎢ ⎥
c1 ⎥ ⎢ b2 ⎥ .
⎦⎣ ⎦
c2
b3
c2 c1 c0
⎡
1
2
⎥ ⎢
⎢
⎢ 1 ⎥ ⎢1
⎢ − 24 ⎥ = ⎢ 6
⎦ ⎣
⎣
1
1
− 120
24
⎤⎡ ⎤
1 1
b
⎥ ⎢ 1⎥
⎥
⎥
⎢
1
1 ⎥ ⎢ b2 ⎥ .
2
⎦⎣ ⎦
1
1
b3
6
2
(b1 , b2 , b3 )T = (−0.6, 0.15, −0.0167)T
⎡ ⎤ ⎡
⎤⎡
⎤ ⎡
⎤
a
1 0 0
1
1
⎢ 0⎥ ⎢
⎥⎢
⎥ ⎢
⎥
⎢ ⎥ ⎢
⎥⎢
⎥ ⎢
⎥
⎢a1 ⎥ = ⎢ 1 1 0⎥ ⎢−0.6⎥ = ⎢ 0.4 ⎥ .
⎣ ⎦ ⎣
⎦⎣
⎦ ⎣
⎦
1
a2
1 1
0.15
0.5;
2
Rf,[2,3] (x) =
0.5x2 + 0.4x + 1
.
−0.0167x3 + 0.15x2 − 0.6x + 1
△
x[n] n = 1, 2, ..., L m n
i = 0, ..., m bj
ai
m<n−1
−c(m+1) = Hb
a
−c(m+1) = Hb
L>m+n
b = 0, ..., n b0 = 1
H
H
a = Gb′
HBMbQHp2
x[n]
X(z) =
⎤ ⎡
x[m]
−x[m
+
1]
⎢
⎥ ⎢
⎢
⎥ ⎢
⎢ −x[m + 2] ⎥ ⎢ x[m + 1]
⎢
⎥ ⎢
⎢
⎥=⎢
..
.
⎢
⎥ ⎢
.
.
.
⎢
⎥ ⎢
⎣
⎦ ⎣
x[m + n − 1]
−x[m + n]
⎡
−x(m+1)
m<n−1
!∞
j=0
x[j]z −j
X(z)
⎤⎡ ⎤
0 ⎥ ⎢ b1 ⎥
⎥⎢ ⎥
⎢ ⎥
···
x[1]
x[0]
···
0 ⎥
⎥ ⎢ b2 ⎥
⎥,
⎥
..
.. ⎥⎢
..
..
..
⎢.⎥
.
.
.
.
. ⎥⎢ .. ⎥
⎦⎣ ⎦
· · · x[n − 1] x[n − 2] · · · x[m] bn
···
x[0]
H
0
···
b(1)
⎡
⎤
⎡
x[m]
⎢ −x[m + 1] ⎥ ⎢
⎢
⎥ ⎢
⎢ −x[m + 2] ⎥ ⎢ x[m + 1]
⎢
⎥ ⎢
⎢
⎥=⎢
..
..
⎢
⎥ ⎢
.
.
⎢
⎥ ⎢
⎣
⎦ ⎣
−x[m + n]
x[m + n − 1]
−x(m+1)
···
···
..
.
···
H
⎤⎡ ⎤
x[m − n + 1]⎥⎢ b1 ⎥
⎥⎢ ⎥
⎢ ⎥
x[m − n + 2]⎥
⎥ ⎢ b2 ⎥
⎥ ⎢ . ⎥,
..
⎥⎢ . ⎥
.
⎥⎢ . ⎥
⎦⎣ ⎦
bn
x[m]
b(1)
m≥n−1
⎡
⎤
⎡
x[0]
0
0
⎢
a
0
⎢
⎢ ⎥ ⎢ x[1]
x[0]
0
⎢ ⎥ ⎢
⎢ a1 ⎥ ⎢
⎢ ⎥ ⎢
⎢ . ⎥ = ⎢ x[2]
x[1]
x[0]
⎢ . ⎥ ⎢
.
⎢ ⎥ ⎢ ..
..
..
⎣ ⎦ ⎢ .
.
.
⎣
am
x[m] x[m − 1] x[m − 2]
a
G
···
0
···
0
···
..
.
0
..
.
· · · x[0]
⎤
⎡ ⎤
⎥ b
⎥⎢ 0 ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ b1 ⎥
⎥⎢ ⎥,
⎥⎢ .. ⎥
⎥⎢ . ⎥
⎥⎣ ⎦
⎥
⎦ bm
b′
x[n]
x1 [n] = eαn ,
α∈C
X1 (z)
RX1 ,[1/1] (z)
RX1 ,[1/1] (z) =
z
,
z − eα
RX1 ,[0/1] (z −1 ) =
1
.
1 − eα z −1
2
1.5
1
1
Real part
Imaginary part
0.8
0.5
0.6
0
0.4
0.2
−0.5
0
−0.2
−1
−0.4
−1.5
−0.6
−0.8
−1
0
50
100
150
200
250
−2
−2
300
−1.5
−1
−0.5
0
0.5
1
H
X1 (z) ≈
∞
%
(eα z −1 )j =
j=0
1
z
=
1 − eα z −1
z − eα
z
[m/n]
RX1 ,[m/n] (z)
[1/1]
bj
RX1 ,[m/n] (z) =
q1 (z)
H
aj
zq1 (z)
,
(z − eα )q2 (z)
q2 (z)
RX1 ,[1/1] (z)
RX1 ,[7/8] (z)
α = −0.01 + i2π/15
1.5
2
eα
p8
pi = zi
i = 1, 2, ..., 7
RX1 ,[7/8] (z)
q1 (z) = q2 (z) = (1 − z1 z −1 )(1 − z2 z −1 )...(1 − z7 z −1 ).
X1 (z)
p1 = eα
z1 = 0
[m/n]
[1/1]
x1 [n]
m=n
m = n−1
z1 = 0
m < n−1
RX1 ,[m/n] (z) =
ϵ
q(z)
(z −
eα )(z n−m−1
q(z)
− ϵ)
,
m
m>n
eα
x2 [n] = ue−σn
(ωn + φ)
u>0
(ωn + φ) =
eiφ eiωn e−iφ e−iωn
+
.
2
2
x2 [n] = u1 eα1 n + u2 eα2 n ,
u1 = ueiφ ,
u2 = ue−iφ ,
α1 = −σ + iωn,
RX2 ,[2/2] (z) =
p1 = eα1
u1 z
u2 z
+
.
α
1
z−e
z − e α2
p2 = eα2
[2/2]
[2/2]
α2 = −σ − iωn.
H
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
−2
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−1.5
−1
−0.5
0
0.5
1
1.5
2
[m/n]
m<n−1
m>n
m=n
m = n−1
x1 [n]
X(z)
u1
u
u2
eα1
φ
α1
α2
σ
H
H
m
n
ω
e α2
x(t)
x(t) =
∞
%
fi (t).
i=1
fi (t)
{eiωj t | j = 1, 2, ...}
{
(ωj t),
(ωj t) | j = 1, 2, ...
{eσi t eiωi t | i = 1, 2, ...}
{eσi t
(ωi t), eσi t
i = 1, 2, ...
x(t)
L
x(t)
∞ +
a0 %
ak
+
x(t) =
2
k=1
+
2πkt
L
,
+ bk
+
2πkt
L
,,
(ωt ) |
2
ak =
L
2
bk =
L
1
a0 =
L
1
1
1
,
2πkt
dt,
L
,
+
2πkt
dt,
L
+
L
x(t)
0
L
x(t)
0
L
k ∈ N,
k ∈ N,
x(t)dt.
0
!N
k=1
x(t)
t→t̂
A
t̂
∞ +
a0 %
ak
+
2
k=1
+
2πkt
L
,
+
+ bk
x(t)
N
x(t)
x(t)
2πkt
L
,,B
+
2πkt
L
1
=
2
+
t→t̂+
x(t) +
t→t̂−
x(t) .
L
H=
8
+
1,
2πkt
L
,
,
,
|k∈N
9
ω = 2π/L
ak
+
2πkt
L
,
+ bk
+
2πkt
L
,
= ak
=
eikωt + e−ikωt
eikωt − e−ikωt
+ bk
2
2i
ak − ibk ikωt ak + ibk −ikωt
e
e
+
2
2
,
∞
%
f (t) = c0 +
ck =
2
L
1
(ck e
ikωt
+ c−k e
k=1
−ikωt
∞
%
)=
ck eikωt ,
k=−∞
L
x(t)eikωt dt
0
c0 = a0 /2 ck = (ak − ibk )/2 c−k = (ak + ibk )/2
L = 2π/ω → ∞
f (t) =
=
∞
%
-
k=−∞
∞
%
1
π
1
2
L
ω
k=−∞
1
=
2π
P
1
∞
e
1
L/2
f (ξ)e−ikωξ dξ
−L/2
eikωt
L/2
f (ξ)eikω(t−ξ) dξ
−L/2
N
+1
iωt
−∞
∞
OP
f (ξ)e
iωξ
−∞
QN
IF T
x(t)
X(ω) =
.
1
∞
−∞
x(t)e−iωt dt
Q
,
dξ dω
O
1
x(t) =
2π
1
∞
X(ω)eiωt dt
−∞
N /2
N
x(t) =
N /2
%
βj eσj t
(ωj t + φj )
j=1
βj > 0 σj φj
ωj
j
x(t)
T
x(t)
x[k] := x(kT )
(ωj kT + φj ) =
eiφj iωj kT e−iφj −iωj kT
ei(ωj kT +φj ) + e−i(ωj kT +φj )
=
e
e
+
.
2
2
2
x[k] := x(kT ) =
N
%
ρj λ j k ,
j=1
ρj =
⎡
⎤
βj ±iφj
e
,
2
λj = e(σj ±iωj )T .
⎡
1
⎢ x[1] ⎥ ⎢ 1
⎢
⎥ ⎢
⎢ x[2] ⎥ ⎢ λ1
λ2
⎢
⎥ ⎢
⎢ . ⎥=⎢ .
..
⎢ . ⎥ ⎢ .
.
⎢ . ⎥ ⎢ .
⎣
⎦ ⎣
λ1 N −1 λ2 N −1
x[N ]
x0
Λ
···
⎤⎡
⎤
⎥⎢ ρ1 ⎥
⎥⎢ ⎥
⎢ ⎥
···
λN ⎥
⎥⎢ ρ2 ⎥
⎢ ⎥
.. ⎥
..
⎥⎢ . ⎥
.
. ⎥⎢ .. ⎥
⎦⎣ ⎦
N −1
· · · λN
ρN
1
b
βj = 2 |ρj | ,
φj = ±
βj φj σj
(ρj ),
σj =
ρj
λj
ωj
ρj
ℜ{
(λj )}
,
T
ωj = ±
ℑ{
λj
λj
bi
x[k]
N
x[k] = µ1 x[k − 1] + µ2 x[k − 2] + ... + µN x[k − N ] =
x[1] x[2]
(λj )}
,
T
N
%
j=1
µj x[k − j].
x[2N ]
⎡
⎤
⎡
x[N − 1]
⎢x[N + 1]⎥ ⎢ x[N ]
⎢
⎥ ⎢
⎢x[N + 2]⎥ ⎢ x[N + 1]
x[N ]
⎢
⎥ ⎢
=
⎢
⎢
⎥
..
..
..
⎢
⎥ ⎢
.
.
.
⎢
⎥ ⎢
⎣
⎦ ⎣
x[2N − 1] x[2N − 2]
x[2N ]
x(N +1)
Sx,N
···
µ
N ×N
µj
λj
µN
⎤
x[1] ⎥⎢ µ1 ⎥
⎥⎢ ⎥
⎢ ⎥
· · · x[2] ⎥
⎥⎢ µ2 ⎥
⎢ ⎥
.. ⎥
..
⎥⎢ . ⎥
.
. ⎥⎢ .. ⎥
⎦⎣ ⎦
· · · x[N ] µN
µj
µ1
⎤⎡
Sx,N
N
ψ(λ) := λ − µ1 λ
N −1
N −2
− µ2 λ
− · · · − µN −1 λ − µN = −
µ0 = −1
N
%
l=0
µN −l λl
λ1 λ2
λN
λj
x[k − j] =
N
%
j=1
µj x[k − j] =
N
%
j=1
µj x[k − j] =
λj
l=1
ρl
-
N
%
N
%
ρl λl k−j .
l=1
ρl λl k−j
l=1
h = N −j
j=1→N
N
%
µj
j=1
j
N
%
(k − j)
ψ(λ)
λj
.
=
N
%
ρl
l=1
-
N
%
µj λl k−j
j=1
.
j = N −h
h=0→N −1
-N −1
%
h=0
µN −h λl k−N +h
.
=
N
%
l=1
ρl λl k−N
-N −1
%
h=0
µN −h λl h
ψ(λ)
0 = ψ(λl ) = −
N
%
h=0
h
µN −h λl = −
N
−1
%
h=0
µN −h λl h − µ0 λl N .
.
N
−1
%
h=0
N
%
j=0
µN −h λl h = −µ0 λl N = λl N .
µj x[k − j] =
N
%
bl λl
k−N
N
(λl ) =
l=1
!N
x[k]
j=1
µj x[k − j]
ρl λl k = x[k].
l=1
µj
x[k] =
N
%
x[k]
k = N + 1, ..., 2N
k ≥ 2N + 1
µj
k ≥ 2N + 1
x[k]
λj
λj
ψ(λ)
ρj
λj
Λ
µj
Sx,N
iQ2THBix
H2pBMbQM
x[k] = x(kT ) k = 1, 2, ..., L T
βj φj σj ωj
µj
λj
ψ(λ)
ρj
βj φj σj
ωj
ρj
Sx,N
Λ
λj
Θ(n2 )
λj
ψ(λ)
`QQib
O(n2 )
ρj
bj
Λ
Λ
pM/2`
O(n2 )
Θ(n2 )
m=n
N
m=n=N
m
N
m
n
H
Sx,N
⎧
⎨ −x(N +1) = Hb(1)
⎩ x
= S µ = Hµ,
x,N
(N +1)
b(1) = −µ
N
%
j=0
bj x−j =
N
%
(−µj )x−j = x−N
j=0
N
%
k=0
(−µN −k )xk = x−N ψ(x),
psi(x) = 0
!N
j=0 bj x
Q(z)
−j
bj
X(z) ≈ P (z)/Q(z)
X(z)
!N
j=0 bj x
−j
N
%
j=0
bj z −1 = (1 − p1 z −1 )(1 − p2 z −1 )...(1 − pN z −1 )
x[n]
µ
x[n]
ψ(λ)
λj
ψ(λ) = (λ − λ1 )(λ − λ2 )...(λ − λN )
m=n
N =m
m=
n=N
m=n
[m/n]
x[n]
L
t = nT
x[n] =
N /2
%
βj eσj nT
(ωj nT + φj ).
j=1
x[n] =
N
%
ρj yj [n] =
j=1
ρj λ j n ,
j=1
yj [n] := λj n
N
N
%
x[n]
x[n]
x[n]
y
a − bi
a + bi
x[n]
µj
N
N
λj
N =L
L
µj
yj [n]
L
L
L
x[n]
y[n]
x[n]
y[n] = ρλn
Y (z) =
∞
%
y[n]z −n =
n=0
ρ
.
1 − λz −1
p1 = λ
y[n]
x[n]
X(z) =
x[n]
N
%
j=1
ρj
.
1 − λj z −1
x[n] =
(ωn) = eiωn /2 + e−iωn /2
λ1 = eiω
λ2 = e−iω
ρ1 = ρ2 = 1/2
x[n]
X(z) =
1/2
1/2
+
.
iω
−1
1−e z
1 − e−iω z −1
[n − 1/n]
z −1
RX,[n−1/n] (z −1 )
n−1
%
k=0
n
%
ak z −k
bk z −k
≈
n
%
j=1
ρj
1 − λj z −1
k=0
ρj
λj
x[n]
ρj
n=2
n=2
ρ2
ρ1 (1 − λ2 z −1 ) + ρ2 (1 − λ1 z −1 )
ρ1
+
=
1 − λ1 z −1 1 − λ2 z −1
(1 − λ1 z −1 )(1 − λ2 z −1 )
=
(ρ1 + ρ2 ) − (ρ1 λ2 + ρ2 λ1 )z −1
.
(1 − λ1 z −1 )(1 − λ2 z −1 )
n=2
a0 = ρ1 + ρ2 ,
a1 = −ρ1 λ2 − ρ2 λ1 )z −1 .
λj
ρj
⎡ ⎤ ⎡
⎤⎡ ⎤
a
ρ
1
1
⎣ 0⎦ = ⎣
⎦⎣ 1 ⎦.
−λ2 −λ1 ρ2
a1
a(n=2)
ρ(n=2)
L(n=2)
n=3
a0 + a1 z −1 + a2 z −2
= ρ1 (1 − λ2 z −1 )(1 − λ3 z −1 ) + ρ2 (1 − λ1 z −1 )(1 − λ3 z −1 )
+ ρ3 (1 − λ1 z −1 )(1 − λ2 z −1 )
= (ρ1 + ρ2 + ρ3 ) − [ρ1 (λ2 + λ3 ) + ρ2 (λ1 + λ3 ) + ρ3 (λ1 + λ2 )]z −1 + (ρ1 λ2 λ3
+ ρ2 λ1 λ3 + ρ3 λ1 λ2 )z −2 ,
⎤⎡ ⎤
⎡ ⎤ ⎡
ρ
1
1
1
a0
⎥⎢ 1 ⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢ ⎥ ⎢
⎢a1 ⎥ = ⎢−λ2 − λ3 −λ1 − λ3 −λ1 − λ2 ⎥⎢ρ2 ⎥.
⎦⎣ ⎦
⎣ ⎦ ⎣
a2
λ2 λ3
λ1 λ3
λ1 λ2
ρ3
a(n=3)
ρ(n=3)
L(n=3)
n=4
⎡ ⎤ ⎡
1
⎢a0 ⎥ ⎢
⎢ ⎥ ⎢
⎢a1 ⎥ ⎢ −λ2 −λ3 −λ4
⎢ ⎥ ⎢
⎢ ⎥=⎢
⎢a ⎥ ⎢ λ2 λ3 +λ2 λ4
⎢ 2 ⎥ ⎢ +λ3 λ4
⎣ ⎦ ⎣
a3
−λ2 λ3 λ4
a(n=4)
ρj
1
1
−λ1 −λ3 −λ4
−λ1 −λ2 −λ4
λ1 λ3 +λ1 λ4
+λ3 λ4
λ1 λ2 +λ1 λ4
+λ2 λ4
−λ1 λ3 λ4
−λ1 λ2 λ4
L(n=4)
⎤⎡ ⎤
1
⎥⎢ρ1 ⎥
⎥⎢ ⎥
−λ1 −λ2 −λ3 ⎥⎢ρ2 ⎥
⎥⎢ ⎥
⎥⎢ ⎥.
λ1 λ2 +λ1 λ3 ⎥⎢ ⎥
+λ2 λ3 ⎥⎢ρ3 ⎥
⎦⎣ ⎦
−λ1 λ2 λ3
ρ4
ρ(n=4)
L
λj
L
n
n
x1 , · · · , x n
e0 (x1 , · · · , xn ) = 1
e1 (x1 , · · · , xn ) =
e2 (x1 , · · · , xn ) =
e3 (x1 , · · · , xn ) =
%
1≤j1 ≤n
%
x j1 x j 2
1≤j1 <j2 ≤n
%
x j1 x j 2 x j 3
1≤j1 <j2 <j3 ≤n
..
.
en−1 (x1 , · · · , xn ) =
xj 1
%
1≤j1 <j2 <···<jn−1 ≤n
xj1 xj2 · · · xjn−1
en (x1 , · · · , xn ) = x1 x2 · · · xn .
ej (x) := ej (x1 , · · · , xn )
L
e0 (λ)
−e1 (λ)
λj
e2 (λ)
λj
(−1)j+1 ej−1 (λ)
(j, k)
λk
λk
Ljk = (−1)j+1 ej−1 (λ) + λk Lj−1,k ,
j = 2, 3, · · · , n,
k = 1, 2, · · · , n.
L
L
n λj j = 1, · · · , n
L
λ1 , · · · , λ n
//
m=1
//
n
!
em (λ) = j1 <j2 <···<jm λj1 ...λjm
L1k = 1
k
j=2 n
Ljk = (−1)j+1 ej−1 (λ) + λk Lj−1,k ;
L
E
L
n=4
e0 (λ) = 1,
L(n=4)
e1 (λ) = λ1 + λ2 + λ3 + λ4 ,
e2 (λ) = λ1 λ2 + λ1 λ3 + λ1 λ4 + λ2 λ3 + λ2 λ4 + λ3 λ4
e3 (λ) = λ1 λ2 λ3 + λ1 λ2 λ4 + λ1 λ3 λ4 + λ2 λ3 λ4 ,
e4 (λ) = λ1 λ2 λ3 λ4 .
L
R
S
R
S
L1,: = e0 (λ) 1 1 1 1 = 1 1 1 1
R
S
R
S
L2,: = −e1 (λ) 1 1 1 1 + L1,: ◦ λ1 λ2 λ3 λ4
S
R
= −λ2 − λ3 − λ4 −λ1 − λ3 − λ4 −λ1 − λ2 − λ4 −λ1 − λ2 − λ3
R
S
R
S
L3,: = e2 (λ) 1 1 1 1 + L2,: ◦ λ1 λ2 λ3 λ4
S
R
= λ2 λ3 +λ2 λ4 +λ3 λ4 λ1 λ3 +λ1 λ4 +λ3 λ4 λ1 λ2 +λ1 λ4 +λ2 λ4 λ1 λ2 +λ1 λ3 +λ2 λ3
R
S
R
L4,: = −e3 (λ) 1 1 1 1 + L3,: ◦ λ1 λ2 λ3 λ4
S
R
S
= −λ2 λ3 λ4 −λ1 λ3 λ4 −λ1 λ2 λ4 −λ1 λ2 λ3 ,
◦
△
RX,[m/n] (z −1 )
x[n]
n−s
p1 , · · · , pn−s
m−s
[m, n]
z1 , · · · , zm−s
s
RX,[m/n] (z −1 ) =
(1 − z1 z −1 )(1 − z2 z −1 ) · · · (1 − zm−s z −1 )q(z −1 )
,
(1 − p1 z −1 )(1 − p2 z −1 ) · · · (1 − pn−s z −1 )q(z −1 )
q(z −1 )
z −1
x[n] = 0.8e
[0, 2π]
−n/150
q(z −1 )
s
/ πn 0
5
−n/100
+ 0.5e
/ πn 0
4
,
[n − 1/n]
r=1
n
1.5
0.4
Pure signal
Noisy signal
0.3
1
0.2
0.5
0.1
0
0
−0.5
−0.1
−1
−1.5
−0.2
0
10
20
30
40
50
60
70
80
90
100
−0.3
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
[7/8]
[7/8]
2
0
−2
−2
10
−1.5
20
30
−1
40
−0.5
50
0
60
0.5
70
80
1
90
1.5
100
2
PZ diagram of 8 poles and 0 zeros
PZ diagram of 30 poles and 0 zeros
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
[7/8]
[29/30]
n
(r)
[n − 1/n]
ĥn (r)
(r)
ĥn (r)
-
1
ûn (r) =
n
(n)
γj
(n)
(n)
= γn−j 0 ≤ γj
≤1
(n)
γ0
n
%
(n)
γj r2j
j=0
(n)
= γn
.
,
=1
u1 (r)
ĥ(r)
n (r) =
0
1 / ′′
′
rûn (r) + ûn (r)
2
n = 1, · · · , 15
1
n = 1, · · · , 15
n→∞
ĥ(r)
n (r) = δ(r − 1)
[m/n]
x[n]
[mj /nj ]
y[n]
x[n] n = 1, 2, ..., L
[m1 /n1 ], · · · , [ms /ns ]
y[n]
j=1 s
//
P = {p1 , · · · , pnj }
Z = {z1 , · · · , zmj }
[mj /nj ]
[a, b] =
(x[n], m!
j , nj )
mj
−k
={
= 0}
k=0 ak z
!nj Z −k
b
z
=
0}
P ={
k=0 k
//
P
Z
//
y[n]
P = {p1 , · · · , pn }
Z = {z1 , · · · , zm }
p1
ẑp1
p1
ẑp1
p1
ẑp1
p1
ẑp1
p1
p1
ẑp1
ẑp1
p1
p1
p1
p2
ẑp1
P = {p1 , · · · , pn }
dthr
(pπp (1) , zπz (1) ), · · · , (pπp (s) , zπz (s) ) s < m s < n
dmin ← ∞
j=1
|P |
j
Z = {z1 , · · · , zm }
p∗
P
p
∗
Z
d1 , · · · , d|Z|
//
d1 < dmin
dmin ← d1
zp∗ ,1 , · · · , zp∗ ,|Z|
zp∗ ,1
P
//
pzp∗ ,1 ,1 , · · · , pzp∗ ,1 ,|P |
pzp∗ ,1 ,1 = p∗
p∗
zp∗ ,1
P
dmin > dthr
n
n
2n
Z
v1
w(|v1 − v2 |)
v2
|v1 − v2 |
w(x)
w(x)
x
x
w(x) = c
(x),
w(x) = cxγ ,
0 < γ < 1,
c
n
A = {a1 , · · · , an }
aj
(j, k)
bk
k−
aj
n
W
j
wjk
B
wjk
B = {b1 , · · · , bn }
aj
bk
bk
wjk
∞
A
P = {p1 , · · · , pn }
dthr
Z = {z1 , · · · , zn }
(pπp (1) , zπz (1) ), · · · , (pπp (s) , zπz (s) ) s < m
w(x)
W
//
//
//
j=1 n
u←
Wjk ← Wjk − u
j
k = 1, · · · , n
k=1 n
v←
Wjk ← Wjk − v
k
j = 1, · · · , n
nmin
nmin cover < n
//
w←
Wjk
Wjk ← Wjk − w
W
W
W
W
cover
nmin
cover
=n
Wjk
Wjk ← Wjk + w
//
W
n
(πp (1), πz (1)), · · · , (πp (n), πz (n))
(pπp (j) , zπz (j) )
j = 1, · · · , n
6
a1
4
2
a2
b1
6
2
3
b2
3 2
a3
b3
1
1
a4
b4
W
a1
⎡
b1
6
⎢
⎢
a2 ⎢
⎢4
W =
⎢
a3 ⎢
⎢6
⎣
a4
b2
b3
b4
2
3
2
3
2
W
1
1
⎤
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎦
W
a1
W(1)
⎡
b1
b2
4
⎢
⎢
a2 ⎢
⎢2
=
⎢
a3 ⎢
⎢3
⎣
a4
b3
b4
0
1
0
0
1
0
0
⎤
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎦
/
0T
srow = 2 2 3 1 .
W
W(1)
a1
W(2)
⎡
b1
b2
2
⎢
⎢
a2 ⎢
⎢0
=
⎢
a3 ⎢
⎢1
⎣
a4
b3
b4
0
1
0
0
1
0
0
⎤
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎦
W
n
W(2)
n
n
w31 = 1
w11 w14
w23
w43
a1
W(4)
b1
w44
b2
1
⎢
⎢
a2 ⎢
⎢0
=
⎢
a3 ⎢
⎢0
⎣
a4
n
w13 w22 w31
⎡
b3
b4
0
0
0
0
1
1
0
⎤
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎦
6
a1
4
2
a2
b2
a2
b3
a3
1
4
0
6
2
3 2
b3
b4
a4
1
b4
a1
a1
a1
a1
a1
W
j
aj
W
k
bk
W
b2
1
1
a4
b1
1
3 2
a3
4
a1
6
2
3
2
b1
W(1)
W(2)
W
w11 w31
w14
w43
a1
a3
b3
W(4)
(a1 , b3 ) (a2 , b2 ) (a3 , b1 )
(a4 , b4 )
6
a1
a2
4
2
2
2
b2
3
b3
1
2
2
a2
0
1
3
a3
b4
2
0
01
b1
a4
0
a3
3
0
3
0
b2
2
0
a2
4
a3
0
0
0 1
b1
a4
0
2
0
0
b2
b3
1
1
b4
b4
1
a1
0
b3
b3
0
a4
2
b2
0 1
1
0
1
2
0
1
1
a1
a2
b1
1
3 2
a4
4
a1
6
2
3
a3
b1
b4
-1
n
n
n
n×m
(m, n)
(m, n)
O(n4 )
O(n3 )
m
L
x[n]
L = 100
[m/n]
m + n < 100
(x, y)
µ
(x, y)
x[n] = e−0.01n
(2πn/15)
L = 300
x[n]
w(x) =
(x)
[n/n]
n
2
1.5
1
0.5
0
−0.5
−1
−1.5
0
50
100
(−0.01 ± i2π/15)
150
200
250
300
J2Ma?B7i*Hmbi2`
J2Ma?B7i*Hmbi2`
p start
B
P
p start
p start
C
C
B/2
P
P
m ratio
sthr
C1 , · · · , Ccluster count
j = 1, · · · cluster count
cluster count ← 0
|P | > 0
p start ∈ P
M ← p start
C = φ //
P = {p1 , · · · , pN }
B
m ratio
Cj .center
N (p start) ← { p ∈ P
M old ← M
M ← M ean(N (p start))
C ← C ∪ N (p start)
//
|Mold − M | < sthr
//
merge with ← 0
j = 1 cluster count
|Cj .center − M | < B/2
// C
merge with ← j
p start
merge with > 0
//
Cmerge with ← Cmerge with ∪ C
Cmerge with .center ← M ean(Cmerge
with )
cluster count ← cluster count + 1
Ccluster count ← C
Ccluster count .center ← M
//
P ← P \C
P
m ratio
B}
P = {p1 , · · · , pN }
N smooth
K
pthr
r
p 1, · · · , p s
r
D(∆x, ∆y)
p∈P
p
D(∆xp , ∆yp ) = D(∆xp , ∆yp ) + 1
j = 1 N smooth
D = D ∗ K //
pthr
x[n]
(∆xp , ∆yp )
N smooth
D
λj
x[k] =
N
%
ρj λ j k .
j=1
⎡
⎤
⎡
1
⎢ x[1] ⎥ ⎢ 1
⎢
⎥ ⎢
⎢ x[2] ⎥ ⎢ λ1
λ2
⎢
⎥ ⎢
⎢ . ⎥=⎢ .
..
⎢ . ⎥ ⎢ .
.
⎢ . ⎥ ⎢ .
⎣
⎦ ⎣
λ1 N −1 λ2 N −1
x[N ]
x0
Λ
···
1
⎤⎡
⎤
⎥⎢ ρ1 ⎥
⎥⎢ ⎥
⎢ ⎥
···
λN ⎥
⎥ ⎢ ρ2 ⎥
.
⎥
.. ⎥
.. ⎥⎢
..
⎢
⎥
.
. ⎥⎢ . ⎥
⎦⎣ ⎦
N −1
· · · λN
ρN
b
x[n]
x[n]
L>n
⎤ ⎡
1
x[1]
⎢
⎥ ⎢ 1
⎢
⎥ ⎢
⎢ x[2] ⎥ ⎢ λ1
λ2
⎢
⎥ ⎢
⎢ . ⎥=⎢ .
..
⎢ . ⎥ ⎢ .
.
⎢ . ⎥ ⎢ .
⎣
⎦ ⎣
λ1 L−1 λ2 L−1
x[L]
⎡
x0
L
N
Λ
⎤
ρ
⎥⎢ 1 ⎥
⎥⎢ ⎥
⎢ ⎥
···
λN ⎥
⎥ ⎢ ρ2 ⎥
⎥
⎥,
.. ⎥⎢
..
⎢ . ⎥
.
. ⎥⎢ .. ⎥
⎦⎣ ⎦
· · · λN L−1 ρN
···
1
⎤⎡
b
Λ
ρj
b̂ = Λ† x0 ,
Λ† := (ΛT Λ)−1 ΛT
Λ
p1 = λ1 , · · · , pn = λn
!n
j=0
aj z −j
ρj
Tn aj
= j=1 (1 − zj z −1 )
Ln
ρ = Ln −1 a
ρ1 , · · · , ρ n
z1 , · · · , zn
aj
λj
n
Ln
aj
n
%
j=0
aj z
−j
=
n
*
j=1
(1 − zj z −1 ).
Ln
Ln
aj
ρj
[m/n]
m
n
n = m+2
[48/50] [49/51] [50/52]
m=n−4
m=n
m=n
n
[m/n]
n
x
KtUHQ;U/BbiJiV-@`2HKtV
/BbiJi
L
T
{[mj /nj ]}
{[50/50], [51/51],
· · · , [100/100]}
dthr
w(x)
−
(x)
dthr
B
sthr
10−3 B
m ratio
K
N smooth
pthr
r
⎛
⎞
1 2 1
⎟
1 ⎜
⎜
⎟
⎜2 4 2⎟
16 ⎝
⎠
1 2 1
x[n]
e
−n/50
[
(nπ/5) + 0.5
m = n−4
(2n/5)] + v[n]
[m/n]
v[n]
n = 50, · · · , 100
m=n
=
m=n
m = n−4
dthr
n
dthr
dthr
dthr
N smooth
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
−0.5
0
0.5
1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
x[n]
−n/50
e
[
(nπ/5) + 0.5
(2n/5)] + v[n]
v[n]
=
[n/n]
n
s<n
n
s
1.5
1
0.5
0
−0.5
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.5
1
0.5
0
−0.5
−1
15
10
5
0
−5
−10
−15
−20
1.5
1
0.5
0
−0.5
−1
x[n]
e−n/50 [
(nπ/5) + 0.5
(2n/5)] + v[n]
v[n]
=
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.1
0.15
0.2
x[n] = e−n/50 [
0.25
0.3
(nπ/5) + 0.5
0.35
0.4
(2n/5)] + v[n] v[n]
[n/n]
n = 6, 7, · · · , 14
[m/n]
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
x[n] =
e
−n/50
[
(3nπ/10) + 0.5
(n/3)] + v[n]
v[n]
1.4
0.9
1.2
Average mean square error
1
Average correlation
0.8
0.7
0.6
0.5
With 1 component
With 2 components
With 4 components
0.4
0.3
−10
−5
0
[50/50] [51/51]
5
SNR(dB)
10
15
0.8
0.6
0.4
0.2
0
−10
20
0.9
1.2
Average mean square error
1.4
0.7
0.6
0.5
With 1 component
With 2 components
With 4 components
0.4
−5
−5
0
5
SNR(dB)
10
15
20
[100/100]
0.8
Average correlation
1
1
0.3
−10
With 1 component
With 2 components
With 4 components
0
5
SNR(dB)
10
15
With 1 component
With 2 components
With 4 components
1
0.8
0.6
0.4
0.2
20
0
−10
−5
0
5
SNR(dB)
10
15
20
L
aj
ρj
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