࿘Ἴᩘ䝣䜱䝹䝍
᣺ᖜ
䝕䜱䝆䝍䝹ಙྕฎ⌮
䠄㼄㻵㻵䠅
Cutoff Frequency
䠄ゅ䠅࿘Ἴᩘ Z
᣺ᖜ
S/ T
పᇦ㏻㐣䝣䜱䝹䝍䠄 Lowpass filter 䠅
Ꮫ⾡ᅜ㝿᝟ሗ䝉䞁䝍䞊
ᒣཱྀ㞞ᾈ
᣺ᖜ
Cutoff Frequency
䠄ゅ䠅࿘Ἴᩘ Z
䠄ゅ䠅࿘Ἴᩘ Z
ᖏᇦ㏻㐣䝣䜱䝹䝍䠄 Bandpass filter 䠅
S/ T
㧗ᇦ㏻㐣䝣䜱䝹䝍䠄 Highpass filter 䠅
᣺ᖜ
S/ T
Cutoff Frequency
Cutoff Frequency
䠄ゅ䠅࿘Ἴᩘ Z
S/ T
ᖏᇦ㜼Ṇ䝣䜱䝹䝍䠄 Band Elimination filter 䠅
1
3
2
4
䝕䜱䝆䝍䝹䝣䜱䝹䝍
• ┠ⓗ
–
–
–
–
–
–
䝜䜲䝈పῶ
ຎ໬䛧䛯ಙྕ䛾᚟ඖ䠄䝕䝁䞁䝪䝸䝳䞊䝅䝵䞁䠅
≉ᚩᢳฟ
䝇䝨䜽䝖䝹䛾ศᯒ
➢ྕ໬
䝟䝍䞊䞁ㄆ㆑
• ᐇ⿦᪉ἲ
– 䝋䝣䝖䚸䝝䞊䝗
– FFT䛾฼⏝䚸FIR䚸IIR
• ฼⏝ศ㔝䛾౛
– AVᶵჾ䚸ไᚚ䝅䝇䝔䝮䚸䝉䞁䝃䞊䚸
– ㏻ಙ䝅䝇䝔䝮䠄↓⥺䞉᭷⥺䠅䚸⏬ീ䝅䝇䝔䝮䚸་⏝デ᩿ᶵჾ䠊䠊䠊
FIR䝣䜱䝹䝍䛾ᵓᡂ
⥺ᙧ䝅䝣䝖୙ኚ䝅䝇䝔䝮䛻䜘䜛䝕䜱䝆䝍䝹䝣䜱䝹䝍
• 䜲䞁䝟䝹䝇ᛂ⟅ h(k)
• 㞳ᩓ䛯䛯䜏㎸䜏₇⟬
M
f
y ( n)
¦ x ( n k ) h( k )
x ( n) * h ( n )
k 0
• z ኚ᥮䛩䜛䛸䚸
k f
M
H ( z)
• 㞳ᩓ᫬㛫䝣䞊䝸䜶ኚ᥮
¦
x ( n) e
j:n
¦ h( n) z
n
㐜ᘏ⣲Ꮚ
x(n)
z-1
H (:)
X (:) H (:)
z-1
f
¦
h(n) e j:n
h(0)
஌⟬ჾ
h(1)
ຍ⟬ჾ
5
FIR䝣䜱䝹䝍䛸IIR䝣䜱䝹䝍
m
– ఏ㐩㛵ᩘ
H ( z)
¦a
M
H ( z)
m
¦a
n
ධຊ
m
h(M2) 䞉䞉䞉䞉䞉
h(M)
z-1
x(k m) ¦ bn y(k n)
+
z-1
h(1)
h(0)
y(n)
+
z-1
+
ฟຊ
䞉䞉䞉┤᥋ᙧᵓᡂ䛾ධຊ䛸ฟຊ䜢ධ䜜᭰䛘䛯ᙧ
n 1
M
¦a
H ( z)
¦ h( n) z
N
m 0
– ఏ㐩㛵ᩘ
7
ฟຊ
x(n)
z m
• IIR䝣䜱䝹䝍䠄ᕠᅇᙧ䠈෌ᖐᙧ䠅ͤ
– ᕪศ᪉⛬ᘧ y (k )
y(n)
h(0) z 1 [h(1) z 1{h(2) z 1 (h(3) )}]
m 0
M
+
h(M)
n 0
x(k m)
m 0
M
䞉䞉䞉䞉䞉
┤᥋ᙧᵓᡂ䠄㌿⨨ᙧ䠅
• FIR䝣䜱䝹䝍䠄㠀ᕠᅇᙧ䠈㠀෌ᖐᙧ䠅
¦a
z-1
ධຊ
n f
M
┤᥋ᙧᵓᡂ
• ᴟ䛿 z = 0 䛾䜏䠖ᖖ䛻Ᏻᐃ
䝕䜱䝆䝍䝹䝣䜱䝹䝍䛾
࿘Ἴᩘ≉ᛶ
(:=ZT)
– ᕪศ᪉⛬ᘧ y (k )
X ( z)
n 0
n f
Y (:)
n
n 0
M
X (:)
¦ h( n) z
Y ( z) H ( z) X ( z)
h(n) 䠖 䜲䞁䝟䝹䝇ᛂ⟅
f
¦ h( k ) x ( n k )
y ( n ) x ( n) * h ( n )
m
z m
m 0
N
1 ¦ bn z n
n 1
6
ͤ෌ᖐᙧ䛾FIR䝣䜱䝹䝍䜒䛒䜚䛘䜛
8
䜲䞁䝟䝹䝇ᛂ⟅ hA(n) 䛜അ㛵ᩘ
⡆༢䛺FIR䝕䜱䝆䝍䝹䝣䜱䝹䝍䛾タィ౛
䝍䝑䝥ᩘN䛜ወᩘ
• ᡤᮃ䛾࿘Ἴᩘ≉ᛶ HA(:) = A(:)
– 䜎䛪䚸఩┦≉ᛶ䜢䠌䛸䛩䜛
→ ඲䛶䛾࿘Ἴᩘ䛷㐜ᘏ䛜䠌 → Ἴᙧ䛾ṍ䛜↓䛔
→ HA (:) 䛿ᐇᩘ
䝍䝑䝥ᩘN䛜അᩘ
t
t
0
0
→ ᅉᯝⓗ䛷䛺䛔
• 䛭䛣䛷
• ᣺ᖜ䛾䜏䜢ኚㄪ䛩䜛䝣䜱䝹䝍 䠄㞽఩┦≉ᛶ䠅
• ౛䠖⌮᝿ⓗ䛺LPF
hA(n)䜢䚸඲䛶 n t 0 䛾㡿ᇦ䛻཰䜎䜛䜘䛖䛻䝅䝣䝖
– N 䛜ወᩘ䛾䛸䛝 (N-1)/2 䛰䛡ྑ䛻䝅䝣䝖
– ᣺ᖜ≉ᛶ A(:)
h(n) = hA(n - (N-1)/2)
– N 䛜അᩘ䛾䛸䛝 N/2䛰䛡ྑ䛻䝅䝣䝖
ῶ⾶㔞 -20 log A(:) dB
᣺ᖜ
h(n) = hA(n - N/2)
Cutoff frequency
• hA(n) 䛜᭷㝈㛗䛷䛒䜜䜀䚸h(n) 䛿ᅉᯝⓗ䛺䝅䝇䝔䝮䛸䛺䜛
Cutoff frequency
S
䠄ゅ䠅࿘Ἴᩘ :
:
– N䛜ወᩘ䛾䛸䛝䚸
S
9
䜲䞁䝟䝹䝇ᛂ⟅䛿 HA(:) 䛾㏫䝣䞊䝸䜶ኚ᥮䛻䜘䜚ồ䜎䜛䚹
䜲䞁䝟䝹䝇ᛂ⟅h(n)䠖䝊䝻఩┦≉ᛶ䜢ᣢ䛴hA(n)䜢᫬㛫㍈䛷䝅䝣䝖䛧
䛯䜒䛾
• 䜲䞁䝟䝹䝇ᛂ⟅䛿ᐇᩘ → HA(-:) = HA*(:) = HA(:)
H A ( :)
f
¦h
n f
A
f
¦h
(n){cos(:n) j sin( :n)}
n f
ª f
º
« ¦ h A (n){cos :n j sin :n}»
¬n f
¼
A
• 䜲䞁䝟䝹䝇ᛂ⟅h(n)䜢ᣢ䛴䝅䝇䝔䝮䛾࿘Ἴᩘᛂ⟅
(n){cos :n j sin :n}
*
H ( :)
H A (:)
*
1
2S
³
ª 1
« 2S
¬
2S
0
³
j:n
f
1
2S
H A (:){cos :(n) j sin :(n)}d:
2S
0
f
¦ h( n) e
n f
¦h
• HA(:) 䛿ᐇᩘ䠖ᐇᩘ䛾㏫䝣䞊䝸䜶ኚ᥮䛿 hA(n) = hA(-n)
䠄HA(:) 䜒 hA(n) 䜒അ㛵ᩘ䠅
h A ( n)
º
H A (:){cos :n j sin :n}d:»
¼
11
h(n) = h(N-n-1)
³
2S
0
n f
H A (:){cos :n j sin :n}d:
*
*
h A ( n)
h A ( n)
A
( n) e
f
¦h
n f
j:n
(n ( N 1) / 2) e j:n
e j ( N 1) : / 2
H A (:) ˜ e j ( N 1) : / 2
• ᣺ᖜ≉ᛶ䠖 |H(:)| = A(:)
• ఩┦≉ᛶ䠖 (N1) : / 2
䠄ᐇ㝿䛻䛿఩┦≉ᛶ䛿S~S䛾⠊ᅖ䛷ᢡ䜚㏉䛥䜜䜛䠅
఩┦≉ᛶ
• HA(:) 䛿࿘ᮇ㛵ᩘ䛺䛾䛷䚸
A
఩┦≉ᛶ
:
䛣䛾䜘䛖䛺ᙧ䛻䛺䜛
S
᣺ᖜ
2S
:
S
S
S
(N1)S
S
:
S
10
┤⥺఩┦≉ᛶ
12
ලయ౛䠄LPF䛾タィ䠅
• ┤⥺఩┦䜢ᣢ䛴䝅䝇䝔䝮䛷䛿䚸
h(n) = h(Mn)
A(:)
䜢฼⏝䛩䜛䛣䛸䛷䚸஌⟬ჾ䛾ಶᩘ䜢༙ῶ䛷䛝䜛
A(:)
­1 | : | : 0
®
¯0 otherwise
䞉䞉䞉䛭䛾ព࿡䛿䠛
:
S
:0
A(:)
ᩍ⛉᭩p.158
:
ධຊ x(n)
z-1
+
h(0)
z-1
+
z-1
z-1
h(1)
z-1
:0
h(M/2)
䞉䞉䞉䞉䞉
+
1
2S
1
2S
h A ( n)
z-1
y(n)
ฟຊ
13
๓㏙䛾౛䜘䜚䚸
䠙 ᐇᩘ䛷അ㛵ᩘ䛾䜲䞁䝟䝹䝇ᛂ⟅䠇୍ᐃ䛾㐜ᘏ
2S
³
:0
0
S
1 S
A(:) e j:n d:
³
S
2S
sin : 0 n
Sn
A(:) e j:n d:
e j:n d:
: 0
15
sin : 0 n
Sn
h A ( n)
┤⥺఩┦≉ᛶ䜢ᣢ䛴䝅䝇䝔䝮
³
S
:0
䢲䢰䢶
䢲䢰䢵
䢲䢰䢴
• ┤⥺఩┦≉ᛶ䜢ᣢ䛴䝅䝇䝔䝮䛾ఏ㐩㛵ᩘ
H ( z)
䢲䢰䢳
䢲
䢯䢲䢰䢳
䢯䢶䢲
h(0) h(1) z 1 h(2) z 2 n h( N 1) z M
h(0) h(1) z 1 h(2) z 2 n h(2) z ( M 2) h(1) z ( M 1) h(0) z M
䢯䢵䢲
M
䢯䢳䢲
䢯䢴䢲
䢲
M䢳䢲
䢴䢲
䢵䢲
䢶䢲
• n 䜢 ᭷㝈䛾್䠄 M ~ M 䠅䛷ᡴ䛱ษ䜛
䠄䝍䝑䝥ᩘ N = 2M + 1 䠖ወᩘ䛾ሙྜ䠅
h(0)(1 z M ) h(1)( z 1 z ( M 1) ) h(2)( z 2 n z ( M 2) ) • h(n) = hA(nM)
h( M / 2) z M / 2
䢲䢰䢶
䢲䢰䢵
஌⟬䛾ᅇᩘ M+1 → M/2 䛻๐ῶ
䢲䢰䢴
N=17
䢲䢰䢳
䢲
䢯䢲䢰䢳
14
䢲
䢷
䢳䢲
䢳䢷
䢴䢲
16
N
¦ h( n) e
䛣䛾䜘䛖䛻䛧䛶ᚓ䜙䜜䛯䝅䝇䝔䝮䛾࿘Ἴᩘ≉ᛶ H (:)
㐺⏝౛䠄䠎䠅
j:n
n 0
䢴
䢳䢰䢷
䢳
ཎಙྕ
䢲
䢳
᣺ᖜ≉ᛶ
䢯䢳
䢲䢰䢷
䢲
䢯䢴
䢲
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴
S
䢲
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢷䢲䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
䢷䢲䢲
䢷䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
䢷䢲䢲
䢴
Hamming❆䜢
⏝䛔䛯䝣䜱䝹䝍
㐺⏝ᚋ䛾ಙྕ
䢶䢲
䢴䢲
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴
S
䢳
䢲
䢯䢳
䢯䢴
䢲
17
19
❆㛵ᩘ䛾㐺⏝
䝸䝥䝹䜢⏕䛨䜛ཎᅉ䛿䚸䜲䞁䝟䝹䝇ᛂ⟅
䜢ᡴ䛱ษ䛳䛯䛣䛸䚹
→ ᡴ䛱ษ䜚䛾ቃ⏺䜢⁥䜙䛛䛻䛩䜛
→ ❆㛵ᩘ
౛䠖 Hanning, Hamming, Kaiser, etc.
ཎಙྕ
䢳
䢲䢰䢷
䢲
䢳
䢯䢲䢰䢷
䢲䢰䢺
䢳䢲䢲
䢶䢷䢲
䢷䢲
㐺⏝౛
䢲
䢶䢲䢲
S
䢲
䢯䢳
䢵䢷䢲
䢴
䢸䢲
䢯䢴䢲
䢵䢲䢲
䢯䢳
䢺䢲
ῶ⾶≉ᛶ (dB)
20 log10 |H(:)|
䢴䢷䢲
䝸䝥䝹
䢯䢴
䢲
䢲䢰䢴
䢴䢲䢲
䢲
䢯䢴
䢲
䢳䢷䢲
䢳
䝣䜱䝹䝍㐺⏝ᚋ
䛾ಙྕ
䢴
䢯䢶
䢳䢲䢲
䢴
䢶
఩┦≉ᛶ
䢷䢲
䢴䢲䢲
䢵䢲䢲
䢶䢲䢲
Hamming ❆
w(n)
䢲䢰䢸
䢷䢲䢲
䢲䢰䢴
䢲
䢯䢺
䢳
䢲䢰䢷
䢲
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴
఩┦≉ᛶ
䢶
䢲䢰䢶
䝣䜱䝹䝍䞊㐺⏝ᚋ䛾ಙྕ
᣺ᖜ≉ᛶ
䢳䢰䢷
䢴
䢯䢸
䢯䢶
䢯䢴
䢲
䢴
䢶
䢸
䢺
䢲
䢳
䢲䢰䢶
w(n) h(n)
䢲䢰䢷
䢲䢰䢵
䢯䢶
䢲
䢲䢰䢴
䢯䢲䢰䢷
䢯䢳
䢯䢴
䢲
䢳䢲䢲
䢴䢲䢲
䢵䢲䢲
䢶䢲䢲
䢷䢲䢲
䢲䢰䢳
䢺䢲
䢲
䢸䢲
䢯䢲䢰䢳
䢯䢺
18
w(n)
䢯䢸
䢯䢶
䢯䢴
䢲
䢴
䢶
2S n
­
°0.54 0.46 cos
®
N 1
°̄
0
䢸
䢺
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴
ῶ⾶≉ᛶ
䛾ẚ㍑
᪉ᙧ❆
䢶䢲
䢴䢲
for | n | d ( N 1) / 2
䢲
for | n | ! ( N 1) / 2
䢯䢴䢲
Hamming
20
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴
IIR䠄ᕠᅇᙧ䠈෌ᖐᙧ䠅䝕䜱䝆䝍䝹䝣䜱䝹䝍
┤᥋ᙧᵓᡂ䠄᭱䜒⡆༢䛺ሙྜ䠅
• 䝣䜱䞊䝗䝞䝑䜽䝹䞊䝥䜢᭷䛩䜛
M
• ᕪศ᪉⛬ᘧ y (k )
¦ am x(k m) ¦ bn y(k n)
m 0
¦a
H ( z)
m
a
ฟຊ
+
y(k)
x(k)
n 1
M
• ఏ㐩㛵ᩘ
ධຊ
N
z-1
z m
y(n)
ax(n) by(n 1)
Y ( z)
aX ( z ) bY ( z ) z 1
Y ( z)
H ( z) X ( z)
a
H ( z)
1 bz 1
-b
m 0
N
1 ¦ bn z n
n 1
Im
• IIR䝣䜱䝹䝍䛾≉ᚩ
–
–
–
–
•
→
•
•
FIR䜘䜚ᑡ䛺䛔ḟᩘ䛷ᛴᓧ䛺䝣䜱䝹䝍≉ᛶ䛜ᚓ䜙䜜䜛
Ᏻᐃᛶ䛻ὀព䛩䜛ᚲせ䛜䛒䜛
᏶඲䛺┤⥺఩┦≉ᛶ䛿ᐇ⌧䛷䛝䛺䛔
タィ䛿ẚ㍑ⓗ」㞧
ᴟ䛿 z = b
Ᏻᐃ䛺䝅䝇䝔䝮䛸䛩䜛䛻䛿䚸|b| < 1 䛜᮲௳
b < 0 䛾䛸䛝䠄ṇᖐ㑏䠅 LPF
b > 0 䛾䛸䛝䠄㈇ᖐ㑏䠅
HPF
Re
-b
• ⦎⩦ၥ㢟
– ᕪศ᪉⛬ᘧ䛛䜙ୖ䛾ఏ㐩㛵ᩘ䛜ᚓ䜙䜜䜛䛣䛸䜢♧䛫䚹
22
24
┤᥋ᙧᵓᡂ
IIR䝅䝇䝔䝮䛾䜲䞁䝟䝹䝇ᛂ⟅
M
¦a
ax(n) b y(n 1)
• ౛ y(n)
H ( z)
– ึᮇ್ y(-1) = 0 䛸䛩䜛
h( n)
b y (1)
M
¦a
m
z m ˜
m 0
a(b)
n
1
N
1 ¦ bn z n
H 2 ( z)H1 ( z)
n 1
ฟຊ
z-1
z-1
䜉䛝ಙྕ
H(z) 䛾ᴟ䛜඲䛶 z ᖹ㠃ෆ䛾༢఩෇ෆ
䛻Ꮡᅾ䛩䜛䛣䛸
༢఩䜲䞁䝟䝹䝇㛵ᩘ䜢⏝䛔䛶⾲䛩䛸
z-1
a0
+
+
a1 +
+
a2 +
+
䞉
䞉
䞉
y(k)
z-1
b1
z-1
b2
䞉
䞉
䞉
z-1
f
¦ b G (n k )
1 ¦ bn z n
x(k)
| b | < 1 䛺䜙཰᮰
→ Ᏻᐃ
䛭䛖䛷䛺䛔ሙྜⓎᩓ → ୙Ᏻᐃ
h( n)
N
ධຊ
a
ab 2
b y (n 1)
z m
n 1
h(0) aG (0) b y (1)
h(1) b y (0) ab
h(2)
m
m 0
z-1
z-1
k
k 0
23
aM
bN
25
IIR䝕䜱䝆䝍䝹䝣䜱䝹䝍䛾࿘Ἴᩘᛂ⟅䛾౛
┤᥋ᙧᵓᡂ䛾ኚᙧ
M
¦a
m
H ( z)
• 2ḟIIR䝣䜱䝹䝍䛾ఏ㐩㛵ᩘ
z m
m 0
N
1 ¦ bn z n
H 2 ( z)H1 ( z)
H1 ( z)H 2 ( z)
n 1
ධຊ
• ࿘Ἴᩘᛂ⟅
ฟຊ
+
x(k)
z-1
z-1
a0
+
y(k)
x(k)
+
z-1
a0
1
1 2r (cos T ) z 1 r 2 z 2
H ( z)
ᶆ‽ᙧ
䠄㐜ᘏ⣲Ꮚᩘ䛜ᑡ䛺䛔䠅
+
y(k)
H (:)
1
1 2r (cos T )e j: r 2 e j 2:
䢵䢰䢷
+
+
b1
z-1
z-1
b2
z-1
z-1
䞉
䞉
䞉
z-1
a1 +
+
a2 +
+
䞉
䞉
䞉
b1
z-1
z-1
z-1
z-1
᣺ᖜ≉ᛶ
r = 0.75
䢳
r = 0.6
䢲
䢲
䞉
䞉
䞉
T = S / 5;
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
䢴S
䢴
䢳䢰䢷
䢳
aM
䢲䢰䢷
఩┦≉ᛶ
䢲
䢯䢲䢰䢷
䢯䢳
26
bN
28
䢯䢳䢰䢷
䢯䢴
䢲
䢲䢰䢴
䢲䢰䢶
䢲䢰䢸
䢲䢰䢺
䢳
䢳䢰䢴
䢳䢰䢶
䢳䢰䢸
䢳䢰䢺
S
䢴
䜲䞁䝟䝹䝇ᛂ⟅
䠎ḟIIR䝣䜱䝹䝍
ධຊ
䢴
䢳䢰䢷
䢲䢰䢷
aM
bN
r = 0.9
䢴䢰䢷
a2 +
b2
䞉
䞉
䞉
䢵
a1 +
䢳
ฟຊ
䢲䢰䢷
+
x(n)
z-1
+
r = 0.6
y(n)
y(n)
x(n) b1 y(n 1) b2 y(n 2)
H ( z)
1
1
1 b1 z b2 z 2
䢲
䢯䢲䢰䢷
-b1
z-1
-b2
r
T
䢴䢲
䢶䢲
䢸䢲
䢺䢲
䢳䢲䢲
䢳䢴䢲
䢳䢶䢲
䢳䢸䢲
䢳䢺䢲
䢴䢲䢲
䢲
䢴䢲
䢶䢲
䢸䢲
䢺䢲
䢳䢲䢲
䢳䢴䢲
䢳䢶䢲
䢳䢸䢲
䢳䢺䢲
䢴䢲䢲
䢲
䢴䢲
䢶䢲
䢸䢲
䢺䢲
䢳䢲䢲
䢳䢴䢲
䢳䢶䢲
䢳䢸䢲
䢳䢺䢲
䢴䢲䢲
䢳䢰䢷
䢳
1
1 2r (cos T ) z 1 r 2 z 2
1
1
jT
(r z e )(r z 1 e jT )
Im
䢲
r = 0.75
䢲䢰䢷
䢲
䢯䢲䢰䢷
䢳䢰䢷
Re
ᴟ䛿ศẕ=0䛾ゎ䠖 z
䢳
re r jT
䢲䢰䢷
r = 0.9
䢲
䢯䢲䢰䢷
27
29
䢯䢳
䜰䝘䝻䜾䝣䜱䝹䝍
䝕䝆䝍䝹䝣䜱䝹䝍
䢴
䢳
ཎಙྕ
䢲
䢯䢳
䢯䢴
䢲
䢷䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
䠙
sᖹ㠃
zᖹ㠃
࿘Ἴᩘ≉ᛶ
䠙
⹫ᩘ㍈
༢఩෇
䢷䢲䢲
䢶
jZ
䢴
r = 0.6
ఏ㐩㛵ᩘ
Im
λ
1
䢲
䢯䢴
䢯䢶
䢲
䢷䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
V
䢷䢲䢲
Re
0
䢷
r = 0.75
䢲
䢯䢷
1
െλ
䢲
䢷䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
䢷䢲䢲
䢳䢲
䢷
r = 0.9
Z
䢲
S
䢯䢷
䢯䢳䢲
0
S
30
䢲
䢷䢲
䢳䢲䢲
䢳䢷䢲
䢴䢲䢲
䢴䢷䢲
䢵䢲䢲
䢵䢷䢲
䢶䢲䢲
䢶䢷䢲
32
䢷䢲䢲
p.137
䜲䞁䝟䝹䝇୙ኚἲ
IIR䝕䝆䝍䝹䝣䜱䝹䝍䛾タィ᪉ἲ
• 䜰䝘䝻䜾䝣䜱䝹䝍䛾ఏ㐩㛵ᩘ䜢㏆ఝ䛩䜛᪉ἲ
sᖹ㠃 → zᖹ㠃
• 䜰䝘䝻䜾䝣䜱䝹䝍䛾䜲䞁䝟䝹䝇ᛂ⟅䜢ᶆᮏ໬䛧䛶䝕
䝆䝍䝹䝣䜱䝹䝍䛾䜲䞁䝟䝹䝇ᛂ⟅䜢ᚓ䜛䚹
– 䜲䞁䝟䝹䝇୙ኚἲ
– ཮୍ḟኚ᥮ἲ
䜰䝘䝻䜾䝣䜱䝹䝍䠖 䛂䝥䝻䝖䝍䜲䝥䛃
݄ ݊ܶ ൌ ݄ܶ௔ ሺ݊ܶሻ
݄௔ ሺ‫ݐ‬ሻ
• ᡤᮃ䛾䝣䜱䝹䝍≉ᛶ䜢┤᥋㏆ఝ䛩䜛᪉ἲ
࿘Ἴᩘ≉ᛶ
ȁ‫ܪ‬௔ ߱ ȁ
߱
t
– 䝞䝍䝽䞊䝇䝣䜱䝹䝍䚸䝏䜵䝡䝅䜵䝣䝣䜱䝹䝍䛺䛹
• ࿘Ἴᩘኚ᥮
• 䜸䞊䝹䝟䝇ኚ᥮
݄ሺ݊ܶሻ
– ඲䛶䛾࿘Ἴᩘ䛻ᑐ䛧䛶᣺ᖜ≉ᛶ䛜୍ᐃ
࿘Ἴᩘ≉ᛶ
t
31
ȁ ȳ ȁ
ȳ
33
p.137
⦪⥆ᙧᵓᡂ
䜲䞁䝟䝹䝇୙ኚἲ
• ఏ㐩㛵ᩘ H1(z), H2(z)
• 䜲䞁䝟䝹䝇ᛂ⟅ h1(n), h2(n)
䛾䠎䛴䛾䝅䝇䝔䝮䛜⦪⥆䛻᥋⥆䛥䜜䛶䛔䜛ሙྜ
䜰䝘䝻䜾䝣䜱䝹䝍䛾ఏ㐩㛵ᩘ
௠
௠
L
݄௔ ‫ ݐ‬ൌ ෍ ݄௜ ݁ ି௔೔௧
‫ܪ‬௔ ‫ ݏ‬ൌ ෍
௜ୀଵ
௜ୀଵ
௠
௠
݄ ݊ܶ ൌ ܶ ෍ ݄௜
x(n)
䜲䞁䝟䝹䝇୙ኚኚ᥮
䝕䝆䝍䝹䝣䜱䝹䝍䛾ఏ㐩㛵ᩘ
݁ ି௔೔ ௡்
݄௜
‫ ݏ‬൅ ܽ௜
Z
‫ܪ‬௔ ‫ ݏ‬ൌ ෍
௜ୀଵ
௜ୀଵ
h1(n)
y(n)
h2(n)
y(n
y(n
(n) = h2(n) * h1(n) * xx(n
(n)
݄ܶ௜
ͳ െ ݁ ି௔೔் ‫ି ݖ‬ଵ
• z ኚ᥮
ኚ᥮䛩䜛䛸
ኚ
᥮䛩
䛩䜛䛸
䜛䛸
Y(zz) = H2(z)
Y(
z) H1(z)
z) X(z
(z)
ͤ 䜲䞁䝟䝹䝇ᛂ⟅䛜ᖏᇦไ㝈䛥䜜䛶䛔䛺䛔ሙྜ䜶䜲䝸䜰䝆
䞁䜾䜢⏕䛨䜛䚹
34
→ H
H(z
((z
z) 䛜୚䛘䜙䜜䛯䛸䛝䚸
䛜
䚸H
H(z
((zz) = H2(zz)) H1(zz)) 䛾䜘䛖䛻
䛾
ᅉᩘศゎ䛷䛝䜜䜀䚸⦪⥆ᙧᵓᡂ䜢ồ䜑䜛䛣䛸䛜䛷䛝䜛
36
p.133
཮୍ḟኚ᥮
㠀ᕠᅇᙧ䝣䜱䝹䝍䛾⦪⥆ᙧᵓᡂ䛾౛
• 䜰䝘䝻䜾䝣䜱䝹䝍䛾࿘Ἴᩘ≉ᛶ Z f ~ f 䜢
䝕䝆䝍䝹䝣䜱䝹䝍䛾࿘Ἴᩘ≉ᛶ : SaS
Z
䛻෗ീ䛩䜛
ȳ
߱ ൌ –ƒ
ʹ
௝ஐ
௝ஐ
ȳ
•‹
ሺ݁ ଶ െ ݁ ି ଶ ሻȀʹ݆ ͳ ͳ െ ݁ ି௝ஐ
ʹ
ൌ
ൌ
ൌ
௝ஐ
௝ஐ
ȳ
݆ ͳ ൅ ݁ ି௝ஐ
ି
…‘•
ଶ
ଶ
ሻȀʹ
ሺ݁ ൅ ݁
ʹ
→ 䜰䝘䝻䜾䝣䜱䝹䝍䛸䝕䝆䝍䝹䝣䜱䝹䝍䛾
࿘Ἴᩘ䛾㛵ಀ
s 䛛䜙 z 䜈䛾ኚ᥮
ͳ െ ‫ି ݖ‬ଵ
•ൌ
ͳ ൅ ‫ି ݖ‬ଵ
S
ධຊ
x(k)
a(1)
S
z-1
:
a(0)
+
b(1)
b(2)
z-1
z-1
+
b(0)
y(k)
+
ฟຊ
• ⦎⩦ၥ㢟
݆߱ ൌ
ͳ െ ݁ ି௝ஐ
ͳ ൅ ݁ ି௝ஐ
– ୖᅗ䛾ᵓᡂ䛾䝕䜱䝆䝍䝹䝣䜱䝹䝍䛻䛴䛔䛶䚸
ఏ㐩㛵ᩘ H(z)
䜲䞁䝟䝹䝇ᛂ⟅ h(n)
䜢ồ䜑䜘䚹
ゎ
ͤ≉䛻㧗࿘Ἴ䛷࿘Ἴᩘ䛾
ᑐᛂ㛵ಀ䛾ṍ䛜኱䛝䛔35
H ( z ) {a(0) a(1) z 1 }{b(0) b(1) z 1 b(2) z 2 }
a(0)b(0) {a(1)b(0) a(0)b(1)}z 1 {a(0)b(2) a(1)b(1)}z 2 a(1)b(2) z 3
䛣䜜䜘䜚
h(0) a(0)b(0), h(1)
a(1)b(0) a(0)b(1), h(2)
a(0)b(2) a(1)b(1), h(3)
37
a(1)b(2)
䜴䜵䞊䝤䝺䝑䝖ኚ᥮
IIR䝣䜱䝹䝍䛾⦪⥆ᆺᵓᡂ
M
¦a
H ( z)
m
z m
m 0
N
1 ¦ bn z n
H L ( z)
1 a11 z 1 1 a 21 z 1 a 22 z 2
˜
˜ }
a0 {
1 b11 z 1 1 b21 z 1 b22 z 2
H H ( z)
n 1
1
2
1
2
(1 z 1 )
(1 z 1 )
ศᯒ䝣䜱䝹䝍
x(k)
+
+
a0
+
+
y(k)
HL(z)
z-1
z-1
↓2
x(n)
+
HH(z)
+
↓2
HL(z)
↓2
xLL(n)
HH(z)
↓2
xHL(n)
xL(n)
HL(z)
↓2
xLLL(n)
HH(z)
↓2
xLLH(n)
xH(n)
z-1
䜸䜽䝍䞊䝤ศ๭
䜴䜶䞊䝤䝺䝑䝖ኚ᥮ 䞉䞉䞉 JPEG2000䠋MPEG4➼
38
䝣䜱䝹䝍䞉䝞䞁䜽
ᖏᇦศ๭䝣䜱䝹䝍
䠄ศᯒ䝣䜱䝹䝍䠅
ಙྕ
H0(z)
↓N
H1(z)
↓N
䞉
䞉
䞉
x(n)
䜰䝑䝥
䝃䞁䝥䝸䞁䜾
䝎䜴䞁
䝃䞁䝥䝸䞁䜾
HN(z)
x0(n)
x1(n)
䞉
䞉
䞉
↓N
ྜᡂ䝣䜱䝹䝍
↑N
F0(z)
↑N
F1(z)
䞉
䞉
䞉
xN(n)
↑N
䞉
䞉
䞉
ฟຊ
^
x(n)
ྜᡂ
FN(z)
ྜᡂ䝣䜱䝹䝍
H0(z) H1(z) H2(z)
᣺ᖜ
H0(z) F0(z) + H1(z) F1(z) + 䞉䞉䞉 = z-b
䛸䛺䜛䜘䛖䛻Ỵ䜑䜛
࿘Ἴᩘ
40
41

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