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Digital Filter Design
By:
Douglas L. Jones

Digital Filter Design
By:
Douglas L. Jones
Online:
C O N N E X I O N S
Rice University, Houston, Texas

Digital Filter Design
By:
Douglas L. Jones
Online:
C O N N E X I O N S
Rice University, Houston, Texas

Digital Filter Design
By:
Douglas L. Jones
Online:
C O N N E X I O N S
Rice University, Houston, Texas

L ∞
L 2

−π ≤ ω ≤ π
H (ω)
H (ω) = |H (ω) |e −(iθ(ω))
H (ω) = |H (ω) |e −(iωK) e −(iθ0)

H (ω) = M−1
−(iωn)
h=0 h (n) e = h (0) + h (1) e−(iω) + h (2) e−(i2ω) + · · · +
M−1
−(iω M−1
M−1
iω −(iω
h (M − 1) e −(iω(M−1)) = e
2 )
h (0) e
2 + · · · + h (M − 1) e
M−1
−(iω
e 2 ) (h (0) + h (M − 1)) cos M−1
2 ω + (h (1) + h (M − 2)) cos M−3
2 ω + · · · + i (h (0) − h (M − 1))
2 )
e −(iθ0) ω θ0 = 0
h (0) + h (M − 1) =
h (0) − h (M − 1) =
h (1) + h (M − 2) =
h (1) − h (M − 2) =
=

h (k) = h ∗ (M − 1 − k) θ0 = 0
θ0 = π
2 e−(iθ0) = −i
h (0) + h (M − 1) =
h (0) − h (M − 1) =
⇒ h (k) = − (h ∗ (M − 1 − k))
θ0 = 0 h (k) = h (M − 1 − k)
θ0 = π
2
h (k) = − (h (M − 1 − k))

H (ω) = A (ω) e−(iθ0) M−1
−(iω
e 2 ) A (ω)
A (ω)
M−1
−(iω
H (ω) = A (ω) e 2 ) A (ω) A (ω)
A (ω)
1 −(iπ
|H (ω) | = ±A (ω) = A (ω) e 2 (1−signA(ω))) ⎧
⎨ 1 x > 0
signx =
⎩ −1 x < 0
− ` ´ M−1
2
ω
∠ω

A (ω)
ω
∠ω
2π π A (ω)
M−1
2

∀n, 0 ≤ n ≤ M − 1 : (h [n])
π
minh[n] −π
(|Hd (ω) − H (ω) |) 2
dω
π
minh[n] −π (|Hd (ω) − H (ω) |) 2
dω = 2π ∞
n=−∞ (|hd [n] − h [n] |) 2
=
−1
2π n=−∞ (|hd [n] − h [n] |) 2 + M−1
n=0 (|hd [n] − h [n] |) 2 + ∞
n=M (|hd [n] − h [n] |) 2
∀n, n < 0n ≥ M : (= h [n])
π
minh[n] −π (|Hd (ω) − H (ω) |) 2
dω =
M−1
n=0 (|h [n] − hd [n] |) 2 + ∞
n=M (|hd [n] |) 2
h [n]
h [n] = hd [n] w [n]
⎧
⎨ hd [n] 0 ≤ n ≤ M − 1
h [n] =
⎩ 0
⎧
⎨ 1 0 ≤ n (M − 1)
w [n] =
⎩ 0
−1
h=−∞ (|hd [n] |) 2
L2
L∞
H (ω) = Hd (ω) ∗ W (ω)
∀n0 ≤ n ≤ M − 1h [n] = hd [n] w [n]
+

ωk
∀k, k = [o, 1, . . . , N − 1] :
Hd (ωk) =
M−1
n=0
−(iωkn)
h (n) e
Hd (ω) ωc
⎛
⎜
⎝
Hd (ω0)
Hd (ω1)
Hd (ωN−1)
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ = ⎜
⎟ ⎜
⎠ ⎝
Hd (ωk) =
M−1
n=0
−(iωkn)
h (n) e
e −(iω00) e −(iω01) . . . e −(iω0(M−1))
e −(iω10) e −(iω11) . . . e −(iω1(M−1))
e −(iωM−10) e −(iωM−11) . . . e −(iωM−1(M−1))
Hd = W h
h = W −1 Hd
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
h (0)
h (1)
h (M − 1)
W N = M ωi = ωj + 2πl i = j
0 2π ωk = 2πk
M + α
Hd (ωk) =
M−1
n=0
2πkn −(i
h (n) e M ) −(iαn)
e
=
h [n] = eiαn
M
h (n) e −(iαn) = 1
M
M−1
k=0
Hd [ωk] e
M−1
k=0
M−1
n=0
2πnk i M
h (n) e −(iαn)
2πkn −(i
e M )
=
Hd (ωk) e
2πnk +i M
= e iαn IDF T [Hd [ωk]]
⎞
⎟
⎠

h [n] h [n] = h [−1] M
2
M
2
H [ωk] = M−1
−(iωkn)
n=0 h [n] e
⎧
⎨
M
2
=
⎩
−1
−(iωkn) −(iωk(M−n−1))
n=0 h [n] e + e
M− 3
2
n=0 +h [n] e−(iωkn)
−(iωk(M−n−1)) + e h
M−1 −(iωk
2 e M−1
2 )
⎧
M−1
⎨ −(iωk e 2 ) M
2 2
=
⎩
−1
M−1
n=0 h [n] cos ωk 2 − n
M−1 M− −(iωk e 2 ) 2 3
2
M−1
n=0 h [n] cos ωk 2 − n + h
M−1
2
⎧
⎨
2
A (ωk) =
⎩
M
2 −1
M−1
n=0 h [n] cos ωk 2 − n
M− 3
2
M−1
2 n=0 h [n] cos ωk 2 − n + h
M−1
2
M
2 ωk ω ∈ [0, π)
−ωk M
2
h [n]
h [n] ωk
∀k, 0 ≤ k ≤ M − 1 :
ωk = nπk
M
h [n] = IDF T [Hd (ωk)]
= 1
M−1
2πk −(i
M k=0 A (ωk) e M ) M−1
2
2πk
A (k) e
i( M (n− M−1
2 ))
= 1
M
M−1
k=0
A (ω)
h [n] = 1
M
= 1
M
= 1
M
A (0) +
A (0) + 2
A (0) + 2
A (ω) = A (−ω) ⇒ A [k] = A [M − k]
M−1
2
k=1
M−1
2
k=1
M−1
2
k=1
2πnk
ei M
2πk i
A [k] e M (n− M−1
2πk M−1
A [k] cos M n − 2
A [k] (−1) k cos
2πk 1
M n + 2
α = 1
2
2 ) + e −(i2πk(n− M−1
2 ))
h [n]
H (ω)
ω = ωk

H (ωk) 0 ≤ k ≤ M − 1 N > M h [n] 0 ≤ n ≤ M − 1
Hd (ωk) − H (ωk)
l ∞
l 2
l 2 N−1 n=0 (|Hd (ωk) − H (ωk) |)
⎛
⎞
⎛
⎞
⎜
⎝
e −(iω00) . . . e −(iω0(M−1))
e −(iωN−10) . . . e −(iωN−1(M−1))
W h = Hd
⎜
⎟ ⎜
⎟ ⎜
⎟ h = ⎜
⎠ ⎜
⎝
Hd (ω0)
Hd (ω1)
Hd (ωN−1)
h = W W −1 W Hd W W −1 W
Hd (ωk)
1
2
⎧
⎨
H (ω) =
⎩
1 − 1
2 17 ≤ |H (ω) | ≤ 1 + 1
2 17 |ω| ≤ ωp
1
2 17 ≥ |H (ω) | ωs ≤ |ω| ≤ π
⎟
⎠

∞
M
W (ω)
argminargmax|E
(ω) | = argmin
h ω∈F h E (ω) ∞ E (ω) = W (ω) (Hd (ω) − H (ω))
F ω ∈ [0, π] ω
E (ω) ∞ ≤ δ M h
δ M
M M

∞
L ∞
L ∞
∞
F x P (x) L
P (x) =
L
akx k
k=0
D (x) x F W (x)
F E (x) F
E (x) = W (x) (D (x) − P (x))
E (x) ∞ = argmax|E
(x) |
x∈F
P (x) L E (x) ∞
E (x) L + 2 L + 2 x xk ∈ F
k = [0, 1, . . . , L + 1] x0 < x1 < · · · < xL+2 E (xk) = − (E (xk+1)) = ± ( E ∞ )
M
A (ω) =
L
n=0
h (L − n) cos ω n + 1
2
L = M
2 − 1 cos (α + β) = cos (α − β) + 2cos (α) cos (β)
A (ω)
ω
2
A (ω) = cos
ω
L
αkcos
2
k (ω)
k=0

x = cos (ω)
E (ω) = W (ω) (Ad (ω) − A (ω))
= W (ω) Ad (ω) − cos
ω
2 P (ω)
= W (ω) cos
ω
2
Ad(ω)
cos( ω
− P (ω)
2 )
W (x) = W
E (x) = W (x) A d (x) − P (x)
(cos (x)) −1
1
cos (cos (x))−1
2
A d (x) =
Ad (cos (x)) −1
1 cos 2 (cos (x))−1
E (ω)
L + 2 = M
2 + 1 L∞
⎧
⎨ 1 |ω| ≤ ωp
W =
⎩
δs
δp |ωs| ≤ |ω|

∞
L + 3
∂
∂x P (x) = 0 P ′ (x)
(L − 1) L − 1 x = cos (ω)
∂
∂ω A (ω) = 0 ω = 0 ω = π
L − 1 + 2 + 2 = L + 3
ω = 0 ω = π
ωp ωs
ω = 0 ω = π

L + 2
⎛
⎜
⎝
1 cos (ω0) cos (2ω0) ... cos (Lω0) 1
W (ω0)
1
cos (ω1)
cos (2ω1)
...
...
cos (Lω1)
−1
W (ω1)
...
1 cos (ωL+1) cos (2ωL+1) ... cos (LωL+1) ±1
W (ωL+1)
⎛
W ⎝ h
⎞
⎠ = Ad
δ
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
h (L)
h (L − 1)
h (1)
h (0)
δ
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ = ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎜
⎝
Ad (ω0)
Ad (ω1)
Ad (ωL+1)
L + 2 h δ (h, δ) T = W −1 Ad
A (ω) h (n) ωk
A (ω) h (n)
O L 3 Nlog 2N N ≥ 32L
A (ω) L x = cos (ω)
A (ω) L + 1 A (ωk) k = [0, 1, 2, ..., L]
δ A (ω)
A (ωk) = (−1)k(+1)
W (ωk) δ + Ad (ωk)
L + 2 δ
γk =
δ =
L+1
i=0
i=k
L+1
k=0 (γkAd (ωk))
L+1
k=0
(−1) k(+1) γk
W (ωk)
1
cos (ωk) − cos (ωi)
⎞
⎟
⎠

δ O ` L 2´ O (16LL) ≈ O ` 16L 2´ h (n) O ` L 3´

O 16L 2 O L 3 L
L
L + 1
L
P (x) = a0 + a1x + ... + aLx L =
L
akx k
L + 1 P (xk) xk k ∈ {0, 1, ..., L} xi = xj i = j
P (x) =
L
k=0
(x − x1) (x − x2) ... (x − xk−1) (x − xk+1) ... (x − xL)
P (xk)
(xk − x1) (xk − x2) ... (xk − xk−1) (xk − xk+1) ... (xk − xL)
x
ak
⎧
L
x −
⎨
xi
1 x = xk
=
xk − xi ⎩ 0 x = xj ∧ j = k
i=0,i=k
L x
P (xk) xk L L
⎛
1 x0 x0
⎜
2 ... x0 L
1 x1 x1 2 ... x1 L
⎞ ⎛ ⎞ ⎛ ⎞
a0 P (x0)
⎟ ⎜ ⎟ ⎜ ⎟
⎟ ⎜
⎟ ⎜ a1
⎟ ⎜
⎟ ⎜ P (x1) ⎟
⎜
⎝
1 x2 x2 2 ... x2 L
1 xL xL 2 ... xL L
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
a2
aL
k=0
⎟ ⎜
⎟ = ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
P (x2)
P (xL)
⎟
⎠
L L + 1
xj L
L + 1 P (x)

A (ω) A (ω)
⎧
⎨
⎩
M−1
−(iω
e 2 ) − ωc ≤ ω ≤ ωc
0 − π ≤ ω < −ωc ∨ ωc < ω ≤ π
H (ω) = M−1
−(iωn)
n=0 h (n) e
M−1
−(iω
= e 2 )
h M−1
L
M−1
2 + 2 n=1 h 2 − n cos (ωn)
A (ω) = h (L) + 2
L
(h (L − n) cos (ωn))
n=1
L . = M−1
2
cos (nα) = 2cos ((n − 1) α) cos (α) − cos ((n − 2) α)
A (ω)
L
L
A (ω) = h (L) + 2 (h (L − n) cos (ωn)) = αkcos k (ω)
n=1
αk h (n) x = cos (ω)
x ∈ [−1, 1] ω ∈ [0, π] A (ω) L x = cos (ω)
L ∞
k=0

M
L∞ E (ω) = W (ω) (Ad (ω) − A (ω))
L + 2 = M+3
2 h (n) 0 ≤ n ≤ M − 1

y (n) = −
M−1
k=1
(aky (n − k))
M−1
+ (bkx (n − k))
k=0
H (z) = b0 + b1z −1 + b2z −2 + ... + bM z −M
1 + a1z −1 + a2z −2 + ... + aM z −M
{ai} {bi} |Hd (w) | Hd (w)
• L ∞ L ∞
• L 2
• L p 1 < p < ∞
L ∞ L p

H (s) =
∞
ha (t) e
−∞
−(st) dt
H (iλ)
H (s) = b0 + b1s + b2s 2 + ... + bM s M
1 + a1s + a2s 2 + ... + aM s M
L ∞
(|H (λ) |) 2 = B λ 2
H (iλ) H (iλ) = b0+b1iλ+b2(iλ)2 +b3(iλ) 3 +...
1+a1iλ+a2(iλ) 2 H (iλ)
+...
= b0−b2λ2 +b4λ 4 +...+iλ(b1−b3λ 2 +b5λ 4 +...)
1−a2λ2 +a4λ4 +...+iλ(a1−a3λ2 +a5λ4 b0−b2λ
+...)
2 +b4λ4 +...+iλ(b1−b3λ2 +b5λ4 +...)
1−a2λ2 +a4λ4 +...+iλ(a1−a3λ2 +a5λ4 +...)
= (b0−b2λ2 +b4λ 4 +...) 2 +λ 2 (b1−b3λ 2 +b5λ 4 +...) 2
(1−a2λ2 +a4λ4 +...) 2 +λ2 (a1−a3λ2 +a5λ4 +...) 2
= B λ2 s = iλ B − s 2
p1 p1 p1 −p1 − (p1)

s = iλ B λ2 = B − s2 = H (s) H (−s) = H (iλ) H (− (iλ)) = H (iλ) H (iλ)
B − s2 H (s) H (−s) H (s) H (−s)
(|H (s) |) 2 = H (s) H (−s) s = iλ H (s) B λ2
B λ2
B λ2 = (|H (λ) |) 2 1 = 1+F (λ2 ) F
λ2
α = 1+i
√ 2
B − s 2 =
B λ 2 2 + λ2
=
1 + λ4 2 − s2
=
1 + s4 √ 2 − s √ 2 + s
(s + α) (s − α) (s + α) (s − α)
1 + s N N

H (s) = LHP
H (s) =
√ 2 + s
(s + α) (s + α) =
B λ 2 =
1
1 + λ 2M
√ 2 + s
s 2 + √ 2s + 1
λ = 0 λ = ∞
B λ 2 = (|H (λ) |) 2

B λ 2 1
=
1 + ɛ2CM 2 (λ)
CM 2 (λ) M th

B λ 2 1
=
1 + ɛ2CM 2 (λ)
CM 2 (λ) M th

B λ 2 1
=
1 + ɛ2JM 2 (λ)
JM
L ∞ M δp δs λp

L ∞
Ha (s) H (z)
(C → C) s
z s ℜ (s) < 0
z iλ = s e iω z
iλ = α eiω−1 2arctan
λ
α
eiω +1 = α(eiω −1)(e −(iω) +1)
(eiω +1)(e−(iω) +1)
z − 1
s = α
z + 1
z − 1
H (z) = Ha s = α
z + 1
2isin(ω)
= 2+2cos(ω) = iαtan
ω
ω
2 λ ≡ αtan 2 ω ≡

λ ω
ω
H (ω) = Ha αtan 2

L ∞
L ∞ L ∞
L ∞
α λ0

ω0
λ0 = αtan
α =
λ0
tan ω0
2
ω0
2
ω λ = 1
α ω M th
M th
ωs = ωs ωp = ωp δs = δs δp = δp α = α0
δi λi = α0tan
ωi
2
λs = α0tan ` ωs
2
´ ` ωp ´
λp = α0tan
2

∀n : (h (n) = ha (nT ))
|z| > |e skT |
Ha (s) = b0 + b1s + b2s2 + ... + bpsp 1 + a1s + a2s2 A1
= +
+ ... + apsp s − s1
A2
+ ... +
s − s2
Ap
s − sp
ha (t) = A1e s1t + A2e s2t + ... + Ape spt u (t)
h (n) = ha (nT ) = A1e s1nT + A2e s2nT + ... + Ape spnT u (n)
Ake (skT )n u (n) ≡ Akz
z − e skT
H (z) =
p
k=1
Ak
z
z − eskT
skT |z| > maxk |e |
z −1 = g z −1
e −(iω1) = g e −(iω) = |g (ω) |e i∠(g(ω))
|g e −(iω) | = 1
g z −1 =
p
k=1
−1 z − αk
1 − αkz−1

|αK| < 1
z1 −1 = z−1 − a
1 − az −1
ωc ω ′ c
a = sin 1
2 (ωc − ω ′ c)
sin 1
2 (ωc + ω ′ c)
z1 −1 = z−1 + a
1 + az −1
ωc ω ′ c
a = cos 1
2 (ωc − ω ′ c)
cos 1
2 (ωc + ω ′ c)
H (z)
h (n) = 0 n < 0
h = 0 h (0) = b0
H (z) =
b0
1 + M
k=1 (akz −k )
M
h (n) = − (akh (n − k)) + b0δ (n)
k=1
hd (n)
ɛ 2 =
∞
n=0
(|hd (n) − h (n) |) 2
H (z)
(|desired − actual|) 2

n > 0
M
h (n) = − (akh (n − k))
k=1
h (n) M
h (n)
hd (n) ak
⎛
∞
2 M
⎝ |hd (n) + (akhd (n − k)) |
⎞
⎠
ɛp 2 =
n=1
∞ N
⎛
⎜
⎝
k=1
hd (0) 0 ... 0
hd (1) hd (0) ... 0
hd (N − 1) hd (N − 2) ... hd (N − M)
Hda ≈ −hd
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
a1
a2
aM
Hd alp = −
H −1
Hd Hd H
hd
H (z) M th
H (z) =
M
k=0 bkz−k 1 + M k=1 (akz−k )
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ≈ − ⎜
⎟ ⎜
⎠ ⎝
M h (n) = − k=1 (akh
(n − k)) + M k=0 (bkδ (n − k))
⎧ M ⎨ − k=1
=
⎩
(akh
(n − k)) + bn 0 ≤ n ≤ M
M − k=1 (akh
(n − k)) n > M
hd (1)
hd (2)
hd (N)
n > M
⎛
hd (M)
⎜ hd ⎜ (M + 1)
⎜ ⎝
hd (M − 1)
hd (M)
...
...
hd (1)
hd (2)
hd (N − 1) hd (N − 2) ... hd (N − M)
aopt =
ˆHda ≈ ˆ hd
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
a1
a2
aM
H
−1
ˆHd Hd Hd Hˆ hd
hd (N)
⎞
⎟
⎠
⎞ ⎛
⎞
hd (M + 1)
⎟ ⎜
⎟
⎟ ⎜ hd ⎟ ⎜ (M + 2) ⎟
⎟ ≈ ⎜
⎟ ⎜
⎟
⎠ ⎝
⎠

a bn
bn =
M
(akhd (n − k))
k=1
hd (n − k) = 0 n − k < 0
N = 2M ˆ Hd ak bk
M + 1 h (n) N = 2M 2M + 1
h (n) hd (n)
N > 2M hd (n) = h (n) 0 ≤ n ≤ M M + 1 < n ≤ N
n ≥ N + 1
hd (0) hd (M)
hd (n) 1 ≤ n ≤ N
v (n)
v (n) ∗ hz (n) ≈ hd (n)
h (n) ≈ hd (n)
⎧ ⎛
⎨ N
M
minbk
⎝ |hd (n) − (bkv (n − k)) |
⎩
⎛
⎜
⎝
v (0)
v (1)
v (2)
0
v (0)
v (1)
0
0
v (0)
...
...
...
0
0
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
v (N) v (N − 1) v (N − 2) ... v (N − M)
n=0
k=0
⎫
⎬
⎠
⎭
2 ⎞
b0
b1
b2
bM
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ≈ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
hd (0)
hd (1)
hd (2)
hd (N)
⎞
⎟
⎠

opt = V H V −1 V H h
∞
(|hd (n) − h (n) |) 2
mina,b
n=0
⎧ ⎛
⎨ ∞
M
minα,β ⎝ |hd (n) − αie
⎩
βin 2
|
⎞⎫
⎬
⎠
⎭
n=0
αi βi
⎛
⎜
⎝
hd (0) 0 ... 0
hd (1) hd (0) ... 0
i=1
hd (N − 1) hd (N − 2) ... hd (N − M)
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
a1
a2
aM
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ≈ − ⎜
⎟ ⎜
⎠ ⎝
hd (1)
hd (2)
hd (N)
a = Hd H −1
Hd Hd H hd
Hd H Hd = R rij hd
rij =
N−max{ i,j }
k=0
Hd H hd =
rd (i) =
N−i
n=0
(hd (k) hd (k + |i − j|))
⎛
⎜
⎝
rd (1)
rd (2)
rd (3)
rd (M)
⎞
⎟
⎠
(hd (n) hd (n + i))
aopt = − RH
rd Ra = −r R M × M a M × 1 r M × 1
⎞
⎟
⎠

ij ≈ r (i − j) = r (j − i)
rij = r (i − j)
⎛
⎜
⎝
⎞ ⎛
r (0) r (1) r (2) ... r (M − 1)
⎟ ⎜
r (1) r (0) r (1) ...
⎟ ⎜
⎟ ⎜
⎟ ⎜
r (2) r (1) r (0) ...
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
r (M − 1) ... ... ... r (0)
a1
a2
a3
aM
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ = − ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
r (1)
r (2)
r (3)
r (M)
⎞
⎟
⎠
R a O M 2
y (n)
M
y (n) = − (aky (n − k)) + u (n)
minak E
k=1
u (n)
{ak}
y (n) + M k=1 (aky
2
(n − k)) = minak E y2 (n) + 2 M
E [y2 (n)] + 2 M k=1 (akE [y (n) y (n − k)]) +
M M
k=1
minak
y (n)
ɛ2
= r (0) + 2
a1 a2 a3 ... aM
r (1) r (2) r (3) ... r (M)
k=1 (aky (n) y (n − k)) +
l=1 (akalE
[y (n − k) y (n − l)])
⎛
⎜
⎜
⎝
⎛
⎞
⎜
⎜
⎝
r (0)
r (1)
r (2)
r (1)
r (0)
r (1)
r (2)
r (1)
r (0)
...
...
...
r (M − 1)
⎟
⎟
⎟
⎟
⎟
⎟
⎠
r (M − 1) ... ... ... r (0)
a1
a2
a3
aM
⎞
⎟
⎠
+
M
k=1 (ak

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0

∂ 2
ɛ
∂a
= 2r + 2Ra
Ra = −r
y (n) r (n)
r (n) ˆ = 1
N−n
(y (n) y (n + k)) ≈ E [y (k) y (n + k)]
N
k=0