Grundlagen der stochastischen Sprachverarbeitung 9783486595000
Wie kann ein technisches System genauso selbstverständlich wie der Mensch den Inhalt gesprochener Sprache erkennen und v
225 63 13MB
German Pages 285 [311] Year 2009
Recommend Papers
- Author / Uploaded
- Andreas Wendemuth
File loading please wait...
Citation preview
Grundlagen der stochastischen Sprachverarbeitung Von Andreas Wendemuth unter Mitwirkung von Edin Andelic Sebastian Barth Stefan Dobler Marcel Katz Sven Krüger Michael Maiwald Mathias Mamsch Martin Schafföner
Oldenbourg Verlag München Wien
Sapere aude! Immanuel Kant,1784
Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar.
© 2004 Oldenbourg Wissenschaftsverlag GmbH Rosenheimer Straße 145, D-81671 München Telefon: (089) 45051-0 www.oldenbourg-verlag.de Das Werk einschließlich aller Abbildungen ist urheberrechtlich geschützt. Jede Verwertung außerhalb der Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Bearbeitung in elektronischen Systemen. Lektorat: Margit Roth Herstellung: Rainer Hartl Umschlagkonzeption: Kraxenberger Kommunikationshaus, München Gedruckt auf säure- und chlorfreiem Papier Druck: R. Oldenbourg Graphische Betriebe Druckerei GmbH ISBN 3-486-27465-1
β
α
β
α
>
−
−
−
−
•
• •
•
•
• •
•
•
•
Akustisches Modell P(A|W)
Erkannter Text
Wortfolge W*
Intention des Sprechers
Sprachmodell P(W)
Nach der besten Hypothese suchen Ermittlung Maximum von P(A|W)P(W)
Wortfolge W
Akustische Analyse Ermittlung charakteristischer Merkmale Sprachschall Merkmalsfolge A
Linguistische Ausprägung Vorwissen, Stimmung, Umgebung, etc...
Spracherkennungssystem
Sprachproduktion biologische Parameter
Artikulation des Sprechers
Telefon-Netzwerk
Netzwerk Interface und I/O-Kontrolle
Spracherkennung
Aussprache
DialogKontrolle
Verstehen
Sprachausgabe
Datenbank
P (W )P (A|W )
"
!
#
!
-
"
#
/
G
5
1
:
'
"
!
'
!
"
#
!
'
'
: :
!
!
!
!
0.4 der
0.5
0.7 Ball
0.5
0.5
0.3 Baum
0.5
Konfidenzgrenze 0.25 0.2 er
0.0 der
0.2 und
0.2 Meer
0.0 der
0.0 er
0.0 der
0.2 mehr
0.0 er
0.0 Ball
0.0 er
0.0 Ball
0.0 ist
0.0 Baum
0.0 Ball
0.0 Baum
0.0 Meer
0.0 bunt
0.0 Baum
0.0 bunt
0.0 mehr
0.0 Meer
0.0 ist
0.0 ist
0.0 rund
0.0 mehr
0.0 Meer
0.0 rund
0.0 und
0.0 rund
0.0 mehr
0.0 und
"
!
'
!
!
!
!
!
#
!
!
"
!
"
"
#
!
!
!
'
!
!
"
'
!
'
!
"
'
'
!
!
:
!
!
!
#
!
!
!
!
!
'
!
"
"
!
!
'
!
!
'
!
#
"
"
!
!
!
!
#
!
"
-
!
1
"
!
!
!
!
'
"
!
:
!
/
#
"
#
!
!
#
"
!
!
"
!
'
"
"
'
'
!
0.4 rund
0.0 bunt
)
-
0.4 bunt
0.4
0.0 und
0.6
1.0 ist
!
'
Spracheingabe
Akustische Analyse
Sprachzusammenhang
Nach der besten Hypothese suchen
Akustisches Modell
Gitter
Nachbearbeitung
Erkannter Text
17cm
Tg G(z) en
ImpulsGenerator
σ stimmhaft
Vokaltraktparameter
Glottismodell un
VokaltraktModell
Lippenabstrahlung
V(z)
R(z)
σ stimmlos
fn
weißes Rauschen
R(z)
10ms
G(z) V (z)
6 5.5 5 4.5 4 3.5 3 2.5 0
1
2
3
4
5
6
7
t
330
80
300
8
f (kHz)
Nasenraum Rachenraum
Mundraum
Gaumensegel Zunge Stimmbänder und Glottis
Lippen
Lunge
L
L
L
A5
L
A4
L
L
A2
A3
A1
17cm
0
A6
M =7
M
Lippen
A7
Glottis
L
fk = (2k − 1) · 500Hz .
10%
F1 F2 F3
300
c
3000
F5
F1
F4
Schallsignal
ARModellspektrum
Signalspektrum
Anregungsspektrum
F1
Vokaltrakt
Anregungssignal
L
(2k − 1) · c L 4
fk =
30
F1−F3
0, 3%
konnte ihn verstehen.
Niemand, der den Satz, der so lang war, hörte,
statisch
selektiv
dynamisch
adaptiv
LA+DS
LA+DS
LA+DS
LA+DS
ˆ ω
ˆ ω
ˆ ω
ˆ ω
GM
Erkenner
wn
Erkenner
n
GM
GM
wn
Erkenner
w1 w2 . . . wn
wn w1 w2 . . . wn−1 P (wn |w1 . . . wn−1 )
Erkenner
w1 w2 . . . wn−1
P (wn |w1 . . . wn−1 ) ≈ P (wn |wn−m+1 . . . wn−1 ).
m=0
m=1
m−1
P (wn |w1 . . . wn−1 ) ≈ P (wn ); m=2
m=3
P (wn |w1 . . . wn−1 ) ≈ P (wn |wn−2 wn−1 ).
P
w1 w2 . . . wn
n
P (w1 . . . wN ) = P (w1 )P (w2 |w1 ) . . . P (wN |w1 . . . wN −1 ).
P (w1 . . . wN )
P (wn |w1 . . . wn−1 ) ≈ P (wn |wn−1 );
N
P (w1 . . . wN ) ≈ P1G (w1 . . . wN ) :=
P (wn ),
n=1
N
P (w1 . . . wN ) ≈ P2G (w1 . . . wN ) := P (w1 )
P (wn |wn−1 ),
n=2
N
P (w1 . . . wN ) ≈ P3G (w1 . . . wN ) := P (w1 )P (w2 |w1 )
P (wn |wn−2 wn−1 ).
n=3
W
Wm
W
PP
wi
∀i = 1 . . . W
P (wi )
P P := [P (w1 . . . wN )]−1/N .
w1 . . . wN
wn
hn
PP =
−1/N
N
P (wn |hn )
.
n=1
W
PP = W
0.48 0.46 0.44 WER 0.42 [%] 0.4 0.38 0.36 0.34 0.32 0.3
150
200
250
300
350
400
Perplexität
N (h)
m
m−1 N (h, w)
h hw
N (h) =
w1 . . . wN
w
h
N (h, w).
w
P (w|h) =
N (h, w ) N (h)
w
h
P (w|h)
N (h, w)
log P (w|h) = − log P P
N
h,w
log P (wi |hi ) =
i=1
P (w|h)
N
1/N
P (w|h) = 1
w
P (w|h) h f (h, w) := N (h, w)/N F := f (h, w) log P (w|h) − λ(h)[ P (w|h) − 1]
h,w
P (w|h)
w
h
∂F f (h, w) = − λ(h) = 0 ∂P (w|h) P (w|h)
P (w|h) = f (h, w)/λ(h) P (w|h) = 1
w
λ(h) =
f (h, w) = N (h)/N
w
W = 104 W = 108
P (w|h) = N (h, w)/N (h)
P (w|h)
N = 107
2
W 3 = 1012
pk
N
K
hw
W3
Nk∗ K Nk∗
pk =
k
Nk
Nk = N k k = 1...K
k=1
Nk
P (w|h) = N (h, w)/N (h)
K
P (w1 . . . wN ) = 0
Nk = N (h, w)
w1 . . . wN
P (w|h) = N (h, w)/N (h)
k=1
Nk∗ = Nk
Nk∗
pk = 0
k
pk
Nk
ηr = r η1
Nk
Nk∗ = Nk + 1,
η0
k
∞
ηr r = N.
r=0
ηr
N
pk =
r∗ , N
N˜k
Nk
r∗ = (r + 1)
ηr+1 ηr
∀k : Nk = r
∞
r
ηr r/N
ηs s/N = N/N = 1
ηr > 0
k
(m−1)
P (w3 )
r = Nk
w1 w2 w3
P (w3 |w1 w2 )
m Nk = 0 ηr
P (w3 |w2 ) P (w3 |w2 ) = P (w3 |w1 w2 )
Nk = r
∞
Nk /N
r
s=1
ηr+1 (r + 1)/N =
(m + 1)
r=0
r r+1 = ηr+1 . N N
pj = ηr
j|Nj =r
P (w3 |w2 )
1/W
P (w3 |w1 w2 )
w1
P (w|h)
w
P (w|h) β(h)P˜ (w|h )
P˜ (w|h) =
hw
h
[P (w|h)|N (h, w) = 0]
N (h, w ) > 0 . N (h, w) = 0
[P (w|h)|N (h, w) = 0] β(h) = w , [P˜ (w|h )|N (h, w) = 0]
w
P˜ (w|h) = 1
w
N (h)
p(w3 |w1 w2 )
p(w3 ) = 1/W
N (h, w) > 0 N (h, w)
P (w|h)
P (w|h) = N (h, w)/N (h)
N (h, w) > 0
n
P˜ (w|h) = P˜ (wn |w1 . . . wn−1 )
n+ 1
n
λi
n
½
P˜ (w|h) = λ2 P (w|h) + λ1 P (w) + λ0 1/W
λi
P˜ (wn |w1 . . . wn−1 ) = λn P (wn |w1 . . . wn−1 ) + · · · +λ2 P (wn |wn−1 ) + λ1 P (wn ) + λ0 1/W,
n
P (w) P (w1 w2 )
P (w1 w2 ) = P (w2 )
P (w1 w2 ) = P (w1 )
w2
2W
W2
P (w1 w2 )
w1
P(v,w) P(u,*,w)
P(w) P(u,v,w)
V
W U
P (w1 w2 )
H=−
P (w1 w2 ) log P (w1 w2 )
w1 w2
¾
P˜ (w|h) = λ2 P (w|h) + λ1 P (w) + λ0 1/W + λA Pt (w)
Pt (w)
10 ms
0.5
0
-0.5
t/s
0. 8
0.6
0.4
0.2
0
Bandfilter Gleichrichtung Tiefpass A/D Umsetzer höchste Frequenz
...
...
...
...
...
...
A/D ...
... ...
Codiertes Ausgangssignal
Multiplexer
Analoges Sprachsignal
A/D
tiefste Frequenz
A/D
Grundfrequenzanalyse
si (n) 0 si (n) < 0
+1 −1
w(n) =
si (n) = sin(2πF n)
yi (n) = si (n) · w(n)
si(n) 10 a)
1
2
3
4
5
1
2
3
4
n
0 -1
w(n) 1
b)
0 -1
y(n) 1 c)
0.5 0 0
|S(f)| a) f
|W(f)| b) 3F
|(f)|
f
c) 0
2F
f
4F
1F
4
1 ) fa
fa > 2fg
fa = 32
δ(t − n
8
∞
fa
fa = 8
n=−∞
s(n) = s(t)
s(t)
β(k)
N
β(k) =
s(n) exp
2πn N −1
20
0≤n0
N × N)
ai,j = 1
N × K)
)
t=1
K
T
Q
∀i = 1 . . . N
Ot qt
πi = P (q1 = si ) (
bj,k = P (Ot = νk |qt = sj ) (
K = {ν1 , ν2 , . . . , νK }
ai,j = P (qt = sj |qt−1 = si ) (
Q = {s1 , s2 , . . . , sN }
K
K
∀j = 1 . . . N
bj,k = 1
k=1
t=1
N
πj = 1
j=1
m
P (st |st−1 , . . . , s0 ) = P (st |st−1 , . . . , st−m ) P (Ot |st , . . . , s0 ) = P (Ot |st , . . . , st−m+1 )
πs
s0
a∗
P (qt |q1 , . . . , qt−1 ) = P (qt |qt−1 ) P (Ot |s1 , . . . , st ) = P (Ot |st )
P (qi ) = πi
t=1 s
⎛
⎞ 0 0 ... 0 ⎜ π1 a1,1 . . . aN,1 ⎟ ⎜ ⎟ a∗ = ⎜ ⎟. ⎝ ⎠ πN a1,1 . . . aN,N
0
ai,0 = 0
πi
i
t=0
0
t=1
Σ
X1 . . . XN
Y
D(X, S)
Y
s 1 . . . sM
dij = d(xi , sj )
3
1. X(t) → X(t + 1); Y (t) → Y (t + 1) 2. X(t) → X(t); Y (t) → Y (t + 1) 3. X(t) → X(t + 1); Y (t) → Y (t)
D(X(T ), Y (T ))
T
D(X(0), Y (0)) = 0 D(X(T ), Y (T )) = min{D(X(T − 1), Y (T )), D(X(T ), Y (T − 1)), D(X(T − 1), Y (T − 1))} + d(X(T ), Y (T ))
(N ) ×
T
×
T −1
T
(M )
N ·M = 4·3
2
s3
3
5
3
s4
1
2
5
4
6
s2
1
5
1
s1
X3
X2
X1
a)
7
4
2
7
5
2
10
5
3
10
10
3
11
2
5
11
12
5
d) 1
6
7
1
6
7
7
5
7
7
5
7
10
10
8
10
10
8
11
12
13
11
12
13
1
6
1
1
5
1
b)
c)
NM
3·N ·M
Ot
q
T
λ = {π, a, b}
P (O|λ) = P (O|π, a, b)
O
λ
qmax
P (O|λ, q) =
T i=1
O
O
q
bqt (Ot )
bqt (Ot ) q P (O, q|λ) = q P (O|q, λ) · P (q|λ) q
qmax
P (O|λ) =
bqt ,Ot
q
qmax = maxP (q|O, λ)
q
q
q
T
P (q|λ) = πq1 ·
T
1
N
a
NT 5
T N
a∗qt−1 ,qt bqt (Ot )
qT =1 t=1
q
...
N
q1 =1
=
3, 8·1015
a∗qt−1 ,qt bqt (Ot )
aqt−1 ,qt
aq1 ·bq1
q t=1
T
P (O|λ) =
2T · N T T = 20
N
t=1
20
a∗qt−1 ,qt bqt (Ot )
q t=1
T
aqt−1 ,qt =
T
P (O|λ) =
t=2
λ
qi
t=1
P (O|λ) = N
=
N
bqT (OT )(
qT =1
a∗q1 ,q2 bq1 (O1 )a∗0,q1 ) . . .)) $ %& ' q1 =1
qT −1 =1
%&
'
i=1 N i=1 N
=
N
=
α1 (j) = P (O1 |q1 = j) · P (q1 = j) = P (O1 , q1 = j) α2 (j) =
α
λ
'
αT −1 (qT −1 )
αT (qT )
α1 (q1 )
%&
$
N
a∗qT −1 ,qT bqT −1 (OT−1 )(. . . ( $
P (O2 |q2 = j) $ %& '
· P (q2 = j|q1 = i)α1 (i)
=P (O2 |O1 ,q2 =j,q1 =i)
P (O2 |O1 , q2 = j, q1 = i) · P (q2 = j|q1 = i)P (O1 , q1 = i) $ %& ' =P (q2 =j|O1 ,q1 =i)
P (O2 , q2 = j|O1 , q1 = i)P (O1 , q1 = i)
i=1 N
=
P (O1 , O2 , q2 = j, q1 = i)
i=1
= P (O1 , O2 , q2 = j) ... αt (j) = P (O1 , . . . , Ot , qt = j)
t
sj
t = 2...T j = 1...N
j = 1...N α1 (j) = a0,j · bj (O1 ) = πj · bj (O1 )
N αt (j) = ( i=1 αt−1 (i) · ai,j ) · bj (Ot )
N
P (O|λ) =
αT (j)
j=1
· (
N
a∗q1 ,q2 · bq2 (O2 ) · (
q2 =1
...· (
q3 =1
N
qT =1
b
a
%&
%&
'
β2 (q2 )
λ
β
'
'
βT −1 (qT −1 )
%&
β1 (q1 )
$
$
a∗qT −1 ,qT · bqT (OT ))) . . .))
$
a∗0,q1 bq1 (O1 ) ·
q1 =1 N
N
P (O|λ) =
λ
N
βT −1 (i) =
j=1 N
=
P (OT |qT = j) · P (qT = j|qT −1 = i) $ %& '
P (OT |qT =j,qT −1 =i)
P (OT , qT = j|qT −1 = i) = P (OT |qT −1 = i)
j=1 N
βT −2 (i) =
j=1 N
=
P (OT −1 |qT −1 = j) · P (qT −1 = j|qT −2 = i) · βT −1 (j) $ %& '
P (OT −1 |qT −1 =j,qT −2 =i)
P (OT −1 |qT −1 = j, qT −2 = i) · P (qT −1 = j|qT −2 = i) ·
j=1
P (OT |qT −1 = j) $ %& '
·
P (OT |qT −1 =j,qT −2 =i,OT −1 ) N
=
P (OT , OT −1 |qT −1 = j, qT −2 = i) · P (qT −1 = j|qT −2 = i)
j=1 N
=
P (OT , OT −1 , qT −1 = j|qT −2 = i)
j=1
= P (OT , OT −1 |qT −2 = i) [. . .] βt (i) = P (Ot+1 , . . . , OT |qt = i)
βT (j) = 1
t = T − 1...1
qt
(i)
O
j = 1...N
βt (i) Ot+1 · · · OT
t
qt
i = 1...N
N
βt (i) =
ai,j · bj (Ot+1 ) · βt+1 (j)
j=1
N
P (O|λ) =
a0,j · bj (O1 ) · β1 (j)
j=1
λ
αt (j) = P (O1 , . . . , Ot , qt = j|λ) βt (j) = P (Ot+1 , . . . , OT |qt = j, λ)
αt (j)βt (j) = P (O1 , . . . , Ot , qt = j|λ) P (Ot+1 , . . . ,OT |qt = j, λ) $ %& '
P (Ot+1 ,...,OT |qt =j,O1 ,...,Ot ,λ)
= P (O1 , . . . , OT , qt = j|λ)
α
β
N
λ
t
sj
N
αt (j)βt (j) =
P (O1 , . . . , OT , qt = j|λ)
t=T
t
t=0
λ
j=1
= P (O1 , . . . , OT |λ) = P (O|λ)
j=1
t
α
β
2N 2 T T
t+1
sj (t)
1000
Ot
β
O qt∗
t
α
β
O
αt (j)βt (j) P (O, qt = j|λ) = P (O|λ) Σi αt (i)βt (i) q∗t = argmax P (qt = sj |O, λ)
P (qt = j|O, λ) = sj ∈Q
q∗t
λ = (π, A, B)
q∗
t
α
2T · N T
T
3, 8 · 1015
β
α
O1 . . . Ot
β
α
O
qt
t
P (O1 . . . Ot , qt = j|λ) αt (j) = P (O1 . . . Ot |λ) Σi αt (i) q∗t = argmax P (qt = sj |O1 . . . Ot , λ) P (qt = j|O1 . . . Ot , λ) =
sj ∈Q
q
q
P (O, q|λ) . P (O|λ)
P (q|O, λ) =
q∗
P (O, q∗ |λ) = max P (O, q|λ) =: P ∗ (O|λ).
q∈QT
NT T
q
O1 . . . Ot sj [ψt (j)]
qt = j}
½
ϑt (j) = max{P (O1 . . . Ot , q1 . . . qt |λ)|q ∈ QT
j = 1, . . . , N
ϑ1 (j) = πj bj (O1 ),
t>1
j ϑT (j),
j ϑT (j).
K
N
ψt (j) = argmax ϑt−1 (i)aij .
qT =
t = T − 1, . . . , 1.
O = {O1 , . . . , OT } O
λ
∗ qT∗ = ψt+1 (qt+1 ),
j = 1, . . . , N
i
P (O|λ) =
ψ1 (j) = 0.
i
ϑt (j) = max(ϑt−1 (i)aij )bj (Ot ),
λ
P (O|λ)
LHMM = log P (O|λ) = log
P (O, q|λ)
q∈QT ¾
Mλ = {(π, A, B)|πi , aij , bjk ≥ 0
πi =
i
λ
j
aij =
bjk = 1}
k
ˆ λ
ˆ ≥ LHMM (λ) LHMM (λ)
ξt (i, j)
j
t
i
O
λ
t+1
P (qt = i, qt+1 = j, O|λ) . P (O|λ)
ξt (i, j) := P (qt = i, qt+1 = j|O, λ) =
P (qt = i, qt+1 = j, O|λ) = = P (qt = i, O1 , . . . , Ot |λ) P (Ot+1 , . . . , OT , qt+1 = j|λ, O1 , . . . , Ot , qt = i) $ %& '$ %& '
αt (i)
j
t+1
P (Ot+1 ,...,OT ,qt+1 =j|λ,qt =i)
Ot+1
αt (i)
=P (Ot+2 ,...,OT |qt+1 =j,λ)=βt+1 (j)
= βt+1 (j) P (Ot+1 |qt+1 = j, qt = i, λ) P (qt+1 = j|qt = i, λ) $ %& '$ %& ' =aij
= aij bj (Ot+1 )βt+1 (j),
ξt (i, j)
αt (i)aij bj (Ot+1 )βt+1 (j) . N i=1 αt (i)βt (i)
ξt (i, j) =
γt (i) := P (qt = i|O, λ) = αt (i)βt (i)
= N
j=1
αt (j)βt (j)
O
λ
t
i
P (qt = i, O|λ) P (O|λ) =
N j=1
ξt (i, j)
P (Ot+1 , . . . , OT , qt+1 = j|qt = i, λ) = P (Ot+2 , . . . , OT |Ot+1 , qt+1 = j, qt = i, λ) P (Ot+1 , qt+1 = j|qt = i, λ) $ %& '
=P (Ot+1 |qt+1 =j,λ)=bj (Ot+1 )
ξt (i, j)
T −1
a ˆij
t=1
γt (i)
T −1
λ
ˆ λ
O
T −1
P (qt+1 = j, qt = i|O, λ) =
ξt (i, j)
t=1
t=1 T −1
P (qt = j|O, λ) P (Ot = k|qt = j, λ) $ %& '
T −1
P (Ot = k, qt = j|O, λ) =
ˆ λ→λ
t=1|Ot =k γt (j) . T t=1 γt (j)
T
ˆbjk =
t=1 ξt (i, j) , T −1 t=1 γt (i)
γt (i)
T −1 π ˆi = γ1 (i), a ˆij =
γt (j)
t=1|Ot =k
=P (Ot =k|qt =j,O,λ)
t=1
P (qt+1 =j|qt =i,O,λ)
T −1
γt (j) =
=
β
P (qt = i|O, λ) P (qt+1 = j|qt = i, λ) $ %& '
T −1
t=1
ˆbjk
t=1
t=1
=
T −1
γt (i) =
si
γt (i) λ
ξt (i, j)
si → s j
ξt (i, j)
α
Q(ˆ λ, λ) =
[log P (O, q|ˆ λ)] · P (q|O, λ) = E[log P (O, q|ˆ λ)|O, λ].
q∈QT
O
u
x
q
T
log P (O, q|ˆ λ) = log[ˆ πq1
a ˆqt−1 ,qt ˆbqt (Ot )]
t=1 T
= log π ˆq1 +
T
log a ˆqt−1 ,qt +
t=1
t=1
N
P (qt = i|O, λ) = γt (i) =
log ˆbqt (Ot ).
ξt (i, j),
j=1
Q(ˆ λ, λ) = N
=
N N
γ1 (i) log π ˆi +
[
ξt (i, j)] log a ˆi,j +
T N
i=1 j=1 t=1
i=1
T
j=1 t=1
γt (j) log ˆbj (Ot ).
b
1 1 exp[− (x−μk )T Σ−1 k (x−μk )] 2 (2π)D/2 |Σk |1/2
bk (x) = P (x|Ωk , λ) =
λ
bj (x)dx = 1 ∀j
j
RD
i
πi , aij , bj (x) ≥ 0 πi = aij = 1
N T
3
γt (k) log ˆbk (Ot ) =
k=1 t=1 N T
=
ˆ k |1/2 − 1 (x − μ ˆ −1 (x − μ ˆ k )T Σ ˆ k )] γt (k)[− log(2π)D/2 |Σ k 2 t=1
k=1
1 γt (k)xt t γt (k) t ˆk = 1 ˆ k )(xt − μ ˆ k )T . γt (k)(xt − μ Σ t γt (k) t ˆk = μ
k
γt (k)
t
k
γt (k)
t
ˆk μ ˆk Σ
ˆ (i+1) Σ k
ˆk μ
(i + 1)
(i)
ˆk μ
ˆk Σ
(i+1)
t
ˆk μ
λ
λ = {{μk }, Σ}
1 exp[− (x − μk )T Σ−1 (x − μk )] 2
γt (k)
k
k
1
1
(2π)D/2 |Σ|1/2
x
(i)
P (x|Ωk , λ) =
t 1 t ) + t γs (k)xs . T T s=1 γs (k) s=1
t=0 t=T (i + 1)
t
(i)
ˆ k (1 − =μ
(i) ˆk μ
i
ˆk μ
ˆ μ
2
1 γt (k)xt t γt (k) t 1 ˆ = μ xt T t ˆ = 1 ˆ t − μ) ˆ T (xt − μ)(x Σ T t
λ
λ = {{μk }, {σik }}
1 1 exp[− (xi − μki )2 /σik ] 0 D k D/2 1/2 2 (2π) i=1 i=1 (σi )
1 γt (k)xt t γt (k) t 1 ˆ ki )2 σ ˆik = γt (k)(xti − μ t γt (k) t
D
D = 30 . . . 90 D2 /2
D
P (x|Ωk , λ =
ˆk = μ
ˆk = μ
λ
λ = {{μk }, {σi }}
1 1 exp[− (xi − μki )2 /σi ] 0 D 2 i=1 (2π)D/2 i=1 (σi )1/2 D
P (x|Ωk , λ) =
ˆ μ
k
1 γt (k)xt t γt (k) t 1 ˆ = μ xt T t 1 t ˆ i )2 σ ˆi = (xi − μ T t
ˆk = μ
;
k : γt (k) = max{γt (j)}
j
γt (k) = 1
γt (k) = 0
1 − log |Σk |1/2 − (xt − μk )T Σ−1 k (xt − μk ) 2
N
N
x
1 − 5%
j
V
N
K
N =6·V
bj (x) =
10 . . . 100
N = 300 N = 6000
cjk gjk (x)
k=1
K
c
g
K
cjk
kt
K
t
jk 2 jk i=1 (xi − μi ) /σi ] 0D (2π)D/2 i=1 (σijk )1/2
qt
D
exp[− 12
k=1
P (x|q = j, λ) =
cjk
j
xt
q
P (O|λ) =
P (O|q, k, λ) · P (q, k|λ)
k
q
P (O, q, k|λ) =
k
i
t
P (qt = i, O|λ) P (O|λ)
γt (i) := P (qt = i|O, λ) =
αt (i)βt (i) = N ξt (i, j). = j=1 αt (j)βt (j) j=1 N
j
t
k
P (qt = i, kt = k, O|λ) ζt (i, k) := P (qt = i, kt = k|O, λ) = P (O|λ) α (j)a c g (O )β (i) ji ik ik t t j t−1 = . N α (j)β (j) t j=1 t
π
i
aij
k
1 ζt (j, k) t γt (j) t 1 = ζt (j, k)xt t ζt (j, k) t 1 2 ˆ jk = ζt (j, k)(xti − μ i ) . ζ (j, k) t t t
σ ˆijk
t
ˆ jk μ
cˆjk =
k
i
w
p(s|s , w)
p(xt |s, w)
w1N = (w1 , w2 , . . . , wN )
P r(w1N ) = p(w1 ) · p(w2 ) · . . . · p(wN )
s = 1, 2, ...S(w)
γ
[sT 1 ]
[w1N ]opt = argmax{P r(w1N ) · maxP r(xT1 , sT1 |w1N )}. [w1N ]
st
[sT1 , w1N ]
(u, v) (S(u), u)
t
T
wt
sT1
xT1
(1, v)
(2, v)
[sT1 , w1N ]
[sT1 , w1T ]
max{P r(w1N ) · max P r(xT1 , sT1 |w1N )} [w1N ]
[sT 1 ]
= max
N,[w1N ] tN 1 :t1 =1,tN =T T
= max { T [sT 1 ,w1 ]
p˜()
p˜(xt , st ; st−1 , wt ) = p(st |st−1 , wt )·p(xt |st , wt )
10
7
st−1 = 0
330 ·7·107 ≈ 1022
p˜(xt , st ; st−1 , wt )}.
p(xt , st |st−1 , wn )]}
t=tn−1 +1
n−1 +1
p˜(xt , st ; st−1 = 0, wt ) = p(wt )·p(xt , st |st−1 = 0, wt )
stn
n=1
t=1
tn
[p(wn ) · tmax
107
30 330
N
{
max
Q
t
[s1 ,w1 ]
Q(t, s; w) = max { t t
t
s
w
p(xr , sr |sr−1 , wr ) : (st , wt ) = (s, w)}.
r=1
Q(t, s; w) = maxs {Q(t − 1, s ; w) · p(xt , s|s , w)}
D(t, s; w)
t → [st , wt ]
10
(30 · 7) · 10 · 3 =
(s, w)
w
s
30
30 · 7
t = 1, ..., T
t=T
D(t, s; w) = − log Q(t, s; w)
s=0 Q(t, s = 0; w) = p(w) · maxv {Q(t, S(v); v)}
w
v
t=T
7 6300
Q(t, s; w)
w
20 − 60
t2
t1
/o/
3
/t/
o2
o3
3
M
E
B
oder
E
3
M
B
/o/
t3
/t/
o1
50
70
/m/ /a/ /k/ /d/ /@/ /b/ /U/ /r/ /k/
/#/ /m/ /a/
/r/ /k/ /#/
2%
40
503 = 125000
2500
50
/U/ /r/ /k/
/b/ /U/ /r/
/@/ /b/ /U/
/d/ /@/ /b/
/k/ /d/ /@/
/a/ /k/ /d/
/m/ /a/ /k/
50
5
5
1250
5 · 5 = 25
25
100
/m/ /a/ /k/ /d/ /@/ /b/ /U/ /r/ /k/
/mak/
/d@/
Triphone
/#ma/ /mak/ /akd/
/kd@/ /d@b/
Biphone
/ma/
/ak/
/kd/
/d@/
/@b/
/bU/
/Ur/
/rk/
/k#/
Monophone
/m/
/a/
/k/
/d/
/@/
/b/
/U/
/r/
/k/
Silben
(max(γ) → 1
γ
0
γ
/@bU/ /bUr/ /Urk/ /rk#/
/bUrk/
6
1
2
4−6
Sil
Sil A
1
2 B E
C
D 3
Sil
Sil
Sil 1
B
B
Sil
Sil 2
A
A
E
E
Sil
Sil 3
C
C
D
D
t
t
Zeit
akustische Übergänge leere Übergänge Language-Modell-Übergänge
w
w
v → v
v
(v , w )
v
v
v
δ(v , w ) = v
|W |
p(w|v)
v
w
p(w|v) = 1,
w=1
|W |
½
p(w , v|v ) = p(w |v )·p(v|v , w ) p(v|v , w )
w=1 w’1
v’1
w’2
v’2
v
w’3 v’3 w=W
•
wi
vi
v wi
v
w
v
Qv (t, s; w) t
s
σvopt (t, s; w) := argmax{Qv (t − 1, s ; w) · p(xt , s|s , w)}
s
Bv (t, s; w) = Bv (t − 1, σvopt (t, s; w); w)
Qv (t, s; w) = max {Qv (t − 1, s ; w) · p(xt , s|s , w)}
(n − 2)
A
n
s Bv (t, s; w)
=
(n − 2)
AA
AA
AB
AB
AC
AC
BA
BA
BB
BB
BC
BC
CA
CA
CB
CB
CC
CC
t
•
t
Qv (t − 1, 0; w) =
max
Bv (t − 1, 0; w) = t − 1
A
{v ,w :δ(v ,w )=v}
= p(w|v) ·
Zeit
{Qv (t − 1, S(w ); w ) · p(w|v)}
max
{v ,w :δ(v ,w )=v}
{Qv (t − 1, S(w ); w )}
v
v
w
t
(V, W )(v, t) = argmax{v ,w :δ(v ,w )=v} {Qv (t, S(w ); w )}
B(v, t) = B(t, S(W (v, t)); W (v, t))
P (wn |wn−2 , wn−1 ) u v v
H(v, t) = max{v ,w :δ(v ,w )=v} {Qv (t, S(w ); w )}
w
P r(wn |w1n−1 ) = (u, v, w) = (wn−2 , wn−1 , wn ) w Qv (t, s; w)
Qv (t, s; w) = maxs {Qv (t − 1, s ; w) · p(xt , s|s , w)}
Qv (t − 1, 0; w) = maxu {Qu (t − 1, S(v); v) · p(w|u, v)}
15
ist
grün
15
ißt
10
grau
20
9
9
10
20
10
20
Bau
15
15
9
Baum
20
9
20
Er
20
15
10
15
Der
B(v, w, t) = BU (v,w,t) (t, S(v); v)
U (v, w, t) = argmaxu {Qu (t − 1, S(v); v) · p(w|u, v)}
H(v, w, t) = maxu {Qu (t − 1, S(v); v) · p(w|u, v)}
)
#
H
#
"
-
P
H
#
/
'
)
!
&
H
#
#
&
34
-
P
/
#
grün grau
30
ißt
Bau 29
;
#
ißt 30
ißt 29
.
-
/
/
#
10 + 15 + 9 = 34
P
grau
+
ist 24
26
Bau 25
35
39
30
Bau
grün
31
30
/
#
ist 25
30
ist
35
38
*
+
27
Baum
Er
27
30
Er
25
Baumißt 29
30
ist
25
35
Baum 25
30
Bau
Der
25
Baum
Der
#
+ +
H
-
P
/
"
#
#
'
"
-
#
/
'
"
6 '
$
#
#
"
'
$
&
$
P
#
$
+
#
#
#
#
#
H
#
#
#
$
+
#
H
#
#
#
H
#
"
&
.
#
#
H
#
'
.
*
/
;
(
#
H
;
H
#
#
#
#
+
+
H
#
+
#
+
$
+
#
#
&
#
+
#
#
+
+
$
t
-
/
H
#
Q0 (t)
#
&
Q(t, s; w) t
s
w
Q0 (t) := max Q(t, s; w).
s,w
{(t, s; w) : Q(t, s; w) > fAC · Q0 (t)}, 0 < fAC < 1
n
2 · 10 = 20
- komplexeres Sprachmodell (Trigramm ...)
210 = 1024
Testphase
- akustische Erkennung - mit einfachem Sprachmodell (Bigramm)
2.Phase:
n beste Wortketten
Hypothesenbildung
1.Phase:
w
xtτ +1 = xτ +1 . . . xt
h(w; τ, t) = P r(xtτ +1 |w) = max {P r(xtτ +1 , stτ +1 |w) : st = Sw } t
[sτ +1 ]
w1n
t
G(w1n ; t) := P r(w1n ) · P r(xt1 |w1n ) = P r(wn |w1n−1 ) · max{G(w1n−1 ; τ ) · h(wn ; τ, t)} τ
x1 . . . xt
W1 ... Wn-1
Wn
...
t
T Zeit
x1 . . . xτ $ %& '
xτ +1 . . . xt $ %& '
G(w1n−1 ;τ )
m
x1 . . . xt
m−1
h(wn ;τ,t)
xt+1 . . . xT $ %& '
u2 . . . um
...
um 2
t
n H(um max [P r(w1n ) · P r(xt1 |w1n ) : wn−m+2 = um 2 ; t) := n−m+1 2 ].
w1
u1
m−1 H(um ) · max{H(um−1 ; τ ) · h(um ; τ, t)}]. 2 ; t) := max[p(um |u1 1 u1
τ
τ
m−1 τ (t; um ; τ ) · h(um ; τ, t)} 1 ) := argmax{H(u1
τ
m−1 m H(um )·H(um−1 ; τ (t; um 2 ; t) := max[p(um |u1 1 ))·h(um ; τ (t; u1 ), t)]. 1
u1
um−1
um
t
(t; v, w)
um m−1 = (u, v)
v
τ (t; v, w)
w h(w; τ (t; v, w), t)
t
½
Modellinterface
Modellsteuerung
Aktionsschemata
FT Modell
MIC
Einzelwort Spracherkenner
D
K
|·|
D 1 j − |ai − bi | σj i=1
exp
σj D
1
wj
j
ai
bji
j=1
K
d=
L1
i
j
wj
4
Satzfehlerrate
Fehlerrate
3.4 3 2.5 2
1.8
Wortfehlerrate 1.2 0.8
1
1.4 0.6
0.5
0 1
2
4
8
Anzahl der Dichten
|a − b|
Satzfehlerrate 3.8
4 Fehlerrate 3
2.5 2.2
2
Wortfehlerrate 1.3 0.9
1
1.8 0.8 0.6
0 13
12
18
25
Komponenten des Merkmalsvektors
Satzfehlerrate 1.1
1.2
1.1
1.1
0.4
0.4
Fehlerrate 0.8 Wortfehlerrate 0.4
0.4
0 4 Byte
2 Byte
1 Byte Quantisierung des Merkmalsvektors
400
100
8kHz
4kHz
2
RFInterface
HostProzessor
DSP
Antenne
FlashMemory
RAM ROM
RAM
ω
A
ω
Ω
Ω = {1, 2, 3, ..., 6}
B
A ⊂ Ω
Ω
A
P : P (Ω) → [0, 1]
Ω
P (Ω) = 1
1
Ω
A
0 ≤ P (A) ≤ 1
B
A
A
B
B
A∪B
A
P (A ∪ B) = P (A) + P (B)
A
A
Ω
Ω
1 card (Ω)
P (A)
{ω1 , ..., ωN }
A∩B =∅
card (A) card (Ω)
P (A) =
card (Ω)
P (A) =
A
A
(Ω, P )
A
B
P (A, B ) = P (A ∩ B)
Ω
Ω = {1, 2, 3, 4, 5, 6} .
A B
C
A = {3} , B = {2, 4, 5} , C = {1, 3, 5} ,
B
A∪B = {2, 3, 4, 5} A∩B = ∅
B
A
A1 , A2 , . . . , An
P (A1 , A2 , . . . , An ) = P (A1 ) · P (A2 |A1 ) · P (A3 |A1 , A2 ) · . . . . . . ·P (An |A1 , A2 , A3 , . . . An−1 )
P (A, B ) P (B)
card (A ∩ B) card (Ω) card (A ∩ B) = · card (B) card (Ω) card (B)
P (A1 , A2 , . . . , An ) > 0
P (A|B)
P (A|B) =
P (A|B) =
A C A ∩ C = {3}
A
P (A|B) B
N
P (A, B)
P (Aj |B)
N
A1 , . . . , AN
B
P (Ai )P (B|Ai )
A1 , . . . , AN Ω
B
j
P (Aj )P (B|Aj ) P (Aj ∩ B) = N P (B) i=1 P (Ai )P (B|Ai )
P (Aj )
Ω
i=1
P (Ai ∩ B) =
P (Aj |B) =
P (A|B)
i=1
P (B) =
B
P (A, B) = P (A) · P (B).
P (A|B) = P (A)
)
P (A = 3|
A
P (A)
A2
N 95%
0, 7% 3%
A1
P (A1 ) = 0, 007 P (A2 ) = 0, 993 P (B, A1 ) = 0, 95 · 0, 007 = 0, 00665 P (B, A2 ) = 0, 03 · 0, 993 = 0, 02979
P (B, A1 ) P (B, A1 ) = P (B) P (B, A1 ) + P (B, A2 ) 0, 00665 = = 0, 18. 0, 00665 + 0, 02979
P (A1 |B) =
18%
n≥0
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
N n
0! = 1
(x + y)N
N N xn · y (N −n) = n
x , y ∈ R.
n=0
N
n
N −n
n!
n
N! (N − n)!
N n
n
N N −2
N! = (N − n)!n!
n=3
N · (N − 1) · (N − 2) · . . . · (N − n + 1) =
N = 10
3
n
N −1 N −n+1
N · N · N = Nn
N
N
n=3
N · (N − 1) · . . . · (N − n + 1) N! N N = = . = N −n n n!(N − n)! n · (n − 1) · . . . · 2 · 1
n
n
N +n−1
N
n−1
N +n−1 n
n
Nn
N! (N −n)!
N +n−1 n
N n
n 1 , n 2 , . . . , nr
N
n
N , n1
r
{1, . . . , N }
N
r
ω X(ω)
W
Ω
A = {1, 3, 5}
B = {2, 4, 6}
1 0
A
1
0
ω∈A
ω
X(ω) =
N! N − n1 N − n1 − . . . − nr−1 · ·...· = n2 nr n1 !n2 ! . . . nk ! N = n 1 , n 2 , . . . nr
N n1
N − n1 n2
X(ω)
Z = 2 · X(ω) · W
f
B
A
f (x) ≥ 0 x 1∞ −∞ f (x) dx = 1
f (x)
X
a≤b
b
F (a ≤ X ≤ b) =
f (x) dx
a
X
X
F (t)
F (t) = P (X < t) =
(ti )
ti ≤t
F (t)
X
N 10 = = 45 n 2
7
2
N = 10
X
0 ≤ (t) ≤ 1
3
x1 = 0, x2 = 1
x3 = 2
n=2
x1 = 0
2
7
X
2
N 7 = = 21 n 2
f (0) = P (X = 0) = 21/45 = 7/15 x3 = 2
3
2
N 3 = =3 n 2
f (2) = P (X = 2) = 3/45 = 1/15 x2 = 1
7
3
7 3 · = 21. 1 1
f (1) = P (X = 1) = 21/45 = 7/15
x2 = 1
x1
x3
F (X) = f (0) + f (1) + f (2) = 1,
f (1) = 1 − f (0) − f (2) = 7/15
F(x) 1 14/15
7/15
0 1
2
N n 1 , n 2 , . . . , nν
x
4
10
1 2
· pn1 1 · . . . · pnν ν
4 3
P (X1 = n1 , . . . , Xν = nν ) =
3
5
1
6
V = 6 ; p1 = p2 = . . . = p6 = 1/6 P (X1 = 4, X2 = 0, X3 = 0, X4 = 2, X5 = 3, X6 = 1) = 10 = · p41 · p02 · p03 · p24 · p35 · p16 4, 0, 0, 2, 3, 1 10! 10 = · (1/6) 4! · 2! · 3! · 1! 175 = 839808
N
X
P (xi ) E(X)
x1 = 1, ..., x6 = 6 xi ni
6
ni N
xi
i=1
6
ni N
N
X
xi P (xi )
i=1
|xi | · P (xi )
i
X
Y
X σX
Var (X)
X
Y
X
Cov(X, Y ) = 0
X
Y
Cov(X, Y ) = E ((X − E(X)) · (Y − E(Y )))
E(X)
V ar(X)
σX =
V ar(X) = E (X − E(X))2
Θ
ε>0
ε
Θ
P =1
ˆ − Θ| < ε) = 1 lim P (|Θ
N →∞
ˆ 1 , ..., XN ) Θ(X ˆ Θ
ˆ 1 , . . . , XN )) = Θ lim E(Θ(X
Θ
N →∞
ˆ Θ) = E(Θ) ˆ −Θ b(Θ,
ˆ =Θ E(Θ)
ˆ 1 , ..., XN ) Θ(X ˆ Θ
X
P (X = xi ; Θ)
Θi x1 , x2 , . . . , xn n X
i = 1, 2, . . . , m
Ω
Θ
n
L(x1 , . . . , xn ; Θ) =
P (X = xi ; Θ)
i=1
L(Θ)
ln L(Θ)
ϑ · tϑ−1 0
x = (x1 , ..., xn )
f (t) =
2 d ln L(Θ) 22 =0 dΘ 2Θ=Θ ˆ
ˆ Θ
t ∈ [0, 1]
ϑ xi ∈ [0, 1], (i = 1, ..., n)
f (t) = ϑ · t(ϑ−1) log (t) = ln(ϑ) + (ϑ − 1) · ln(t)
xi
n
ln L(x1 , . . . , xn , ϑ) =
f (t)
ln(ϑ) + (ϑ − 1) · ln(xi )
i=1
∂ ln L(x1 , . . . , xn , ϑ) n = + ln(xi ) ∂ϑ ϑ i=1 n
∂ ln L(x1 , . . . , xn , ϑ) ! =0 ∂ϑ
n ˆ ϑ(x) = − n i=1 ln(xi )
x1 , x2 , . . . , xn
k
Θi
i = 1, 2, . . . , k)
Xi
Θ
μk = E(Xik ) = gk (Θ1 , . . . , Θm )
r≥m
Ω
E(|Xi |r ) < ∞
k = 1, . . . , r
k
1 k m ˆk = x n i=1 i n
Θ1 , . . . , Θm
m ˆ k = gk (Θ1 , . . . , Θm )
(k = 1, . . . , r)
(Θ1 , Θ2 ) = (μ, σ )
Xi
2
g1 (μ, σ 2 ) = μ = E(xi ) g2 (μ, σ 2 ) = μ2 + σ 2 = E(x2i )
μ ˆ =
1 xi n i=1
σ ˆ2 + μ ˆ2 =
1 2 x n i=1 i
n
n
1 Xi n i=1 n
μ ˆ(X1 , . . . , Xn ) =
1 (Xi − μ ˆ )2 n i=1 n
σ ˆ 2 (X1 , . . . , Xn ) =
Θ
Θ
(1−α) ˆ1 Θ
ˆ2 Θ
1−α
Θ
ˆ1 < Θ < Θ ˆ 2) = 1 − α P (Θ
θˆ1 < θˆ2
α
α
σ2
μ
δ
ˆ −μ Θ
P (|
√σ N
Θ
√σ N
δ
ˆ − Zα/2 √σ < μ < Θ ˆ + Zα/2 √σ ) = 1 − α P (Θ N N
Zα/2
√ δ N = σ
Zα/2
√ N
δ1 =
P (|Z| < Zα/2 ) = 1 − α ˆ −μ Θ Z = σ
√ δ N )=1−α |< σ
X 1 , . . . , XN
ˆ −δ 5
.
1 T
fA =
∞ n=−∞
δ(t − nT )
xn (t) = x(t)
S "s 5" "s 4" "s 3" "s 2" "s 1" "s 0" x "s -1" "s -2" "s -3" "s -4" "s -5"
q
-1
1 -q
0101 0100 0011 0010 0001 0000 1111 1110 1101 1100 1011
f 0.5q x -0.5q
xn (t)
|ω| > ωmax
Zp =
i=−n
zi r i ,
ωmax
x
q
xn (t)
x(t)
x(t)
v
2πfA = 2π/T
n
X(jω)
zi = {0, 1, . . . , r − 1}
(r = 10)
n
q = 2−n
−1 ≤ z < 1
−1 ≤ z < 1
r=2
zi = {0, 1, . . . , r − 1} ,
(r = 16)
zi
⎫ ⎪ ⎬
(r − 1 − zi ) ri + 1, ⎪ ⎪ %& ' ⎩i=−n $ ⎭
⎧ ⎪ v ⎨
Zn =
r
q
m
1 ≤ |M | < 1. R
Re
1
[k] =
wRe [n]
k = 0...L − 1 0
w
Zf = M R E
2−m
q/2
−q/2
x[n]
x[0] . . . x[L−1]
α + β cos(2π Nk−1 ) + γ cos(4π Nk−1 ) w[k] = 0
α
w [n] w [n] w [n] w [n]
β
0k N −1
γ
15.6 ≈ N/2
amin
ΩS π/16 2π/16 2π/16 3π/16
32
|H(z = ejωt )| =
(ω)
0Q Q(z) q=1 (z − z0q ) = C 0N H(z) = N (z) n=1 (z − z∞n )
H(z)
C|Q(ejωt )| = |N (ejωt )| ∀ω
jwt
Q
0Q
q=1 (e
C
z0
0q )
q=1 (e
− 1/z
jωt
0q ) 0Q
q=1 (e
0q )
− z0q )
jωt
q=1
2 2 0Q jωt 2 2 2 q=1 (e − 1/z0q ) 2 |z0q | 2 0Q 2 2 (ejωt − z0q ) 2
q=1
− z0q ) 0Q
jωt
(−e−jωt z
Q
|H(z)| = |C|
q=1
−z
−jωt
q=1 (e
Q
H(z) = C 0Q
N
= C
)
CQ(z) = N (z)
CQ(ejwt ) = N ∗(e
H(z) = C)
z = ejωt
z = ejωt
1/z0q = z∞q ∀ q = 1 . . . Q
Q
Q
aq = b∗Q−q ∀ q = 1 . . . Q
k
(X[k])
x[k]
+∞
g(X[k])fX (X, k)dx
g(X[k]) = X[k]
X[k] fX () X[k]
μ[k] = E{X(k)}
g()
−∞
E{g(X[k])} =
g(X[k]) = (X[k] − μ[k])2
σ 2 [k] = V ar {X[k]} = E {g(X[k])}
E {g (X[k])} =
< ··· >
(k)
x[k]
X[k]
E {g (X[k])} ≤ g (x[k]) ≥
(k)
+N 1 x[k] N →∞ 2N + 1
μ[k] = lim
k=−N
+N 1 |x[k] − μ|2 N →∞ 2N + 1
σ 2 [k] = lim
k=−N
k2
k1
sXX (k1 , k2 ) = E{X (k1 )X(k2 )} = E{X1 X2 } = E{(X1 R − jX1 I)(X2 R + jX2 I)} = E{X1 RX2 R − X1 IX2 I} + jE{X1 RX2 I − X1 IX2 R}
sXX (k1 , k2 ) = E{X(k1 )X(k2 )} = E{X1 X2 }
k2
k1 = k, k2 = k + κ
k1
k2
k1
s
k1 − k2
sXX (κ) = E{X (k)X(k + κ)}
x[k]
+N 1 x[k]x[k + κ] N →∞ 2N + 1
sXX (κ) = lim
k=−N
k1 = k2 κ = 0
sXX (0) = E{X (k)X(k + 0)} = E{|x(k)|2 } = σ 2 + |μ|2
κ=0
cXX [κ] = E{(x [k] − μ )(x[k + κ] − μ)}
= E{x [k]x[k + κ]} − μE{x [k]} − μ E{x[k + κ]} + μ μ = sXX [κ] − |μ|2
sXY [κ] = E{X (k)Y (k + κ)}
; < cXY [κ] = E [X (k) − μx ][Y (k + κ) − μy ]
= sXY [κ] − μx μy
sXY (κ) = E{X (k)Y (k + κ)} = E{X (k)}E{Y (k + κ)}
cXY [κ] = E{(X (k) − μx )(y(k + κ) − μy )}
= E{X (k)}E{Y (k + κ)} − μy E{X (k)} − μx E{Y (k + κ)} + μx μy
= μx μy − μx μy − μx μy + μx μy = 0
κ
∞
X(ejΦ ) =
∞
x[k]e−jΦk
Φ = ωT
k=−∞
∞
(ejΦ ) =
S
x[k]e−jωT k =
k=−∞
∀κ
sXY [κ] = E{X (k)Y (k + κ)} = 0
[κ]e−jΦκ
s
κ=−∞
κ=0
S
(jω) s [κ]
κ=0
μ = 0
s
[0] = σ 2 + |μ|2 .
κ = 0
s
[κ] =
σ 0
∞
(ejΦ ) =
S
κ=0
[κ]e−jΦκ = s
s
[0]e−jΦ(0) = σ
κ=−∞
s
(ejΦ ) = s
S
= s
[0]
κ
[0] ∀ κ
[κ] = s
Φ = ωT
∞ 2π δ(ω − κΩ) T κ=−∞
[0] 2π
∞
2π T
Ω=
δ(Φ − 2πκ)
κ=−∞
2πs
[0]
(κ) = E {X(k)X(k + κ)}
s
X(k)
N 1 x[k]x[k + κ] N →∞ 2N + 1
s
(κ) = lim
k=−N
k
|κ|
k
N
N
x[k]x[k + κ]
k=0
N −1−|κ|
N −κ
E {ˆ s
{ˆ s
[|κ|]} = s
lim
N →∞
1 (|κ|) = M
E {ˆ s
M = N − |κ|
1/M M =N
sˆ
[0 . . . N − 1]
[|κ|]
N →∞
[|κ|]} = 0
M = N − |κ|
[|κ|] {ˆ s 1 σ2 = N − |κ|
[|κ|]} = s
[|κ|]}
κ
σ2
κ
κ = N −1
{ˆ s
[|κ|]}
|κ|
N
M =N
N − |κ| s [|κ|] N N − |κ| 2 = σ N2
[|κ|]} =
N
E {ˆ s
κ
|κ|
Var {sxx[|κ|] }
0.4 erwartungstreu 0.2 konsistent 0 0
4
8
16
k
κ
12
L (L > N + M, L = 2n ) x[k] L
X[n]
X[n]
1/N
Modellprozess
q(k)
weiße Rauschquelle
Vergleich zur Identifikation der Modellparameter
H(x)
analysierter Prozess
Modellfilter-Koeffizienten und Leistungen der weißen Rauschquelle
1 1 = A(z) 1 + nν=1 aν z −ν
H(z) =
m
H(z) = B(z) = 1 +
bμ z −μ
μ=1
m 1 + μ=1 bμ z −μ B(z) n = H(z) = A(z) 1 + ν=1 aν z −ν
q[k]
x[k] = q[k] −
n ν=1
aν x[k − ν]
Sxx(ej Ω)
rˆ xx ( κ)
Ω
κ
π
AKFFenster
^
AR (ejΩ ) Sxx
A xK x
ˆr ( κ)
Dämp fu ng Ω
κ
κ>0
κ=0
n
κ=0 κ>0
n 2 − ν=1 aν s [ν] σQ − nν=1 aν s [κ − ν]
n
[κ] =
s
π
.
ai
a = −s
S
a
S −1 (jω) > 0 ∀ ω
S
S
⎤ [1] [2] ⎥ ⎥ ⎥ ⎦ [n]
2 σQ
κ=0
n
2 [0] = σQ −
s
a = −S −1 s
⎤⎡ ⎤ ⎡ −s a1 [−(n − 1)] ⎢ ⎥ ⎢ [−(n − 2)]⎥ ⎥ ⎢ a2 ⎥ ⎢ −s ⎥⎢ ⎥ = ⎢ ⎦⎣ ⎦ ⎣ s [0] an −s
s [0] s [−1] · · · s ⎢ s [1] s [0] · · · s ⎢ ⎢ ⎣ s [n − 1] s [n − 2] · · ·
⎡
2 [ν] = σQ − sT a
aν s
ν=1 2 σQ
[0] − sT S −1 s
[0] + sT a = s
= s
z -1
x(k)
+
-
P(z)
e(k)
+
Pe (z)
P (z)
m
P (z) =
pμ z −μ+1
μ=1
e[k]
m
e[k] = x[k] −
pμ x[k − μ]
μ=1
m
Pe (z) = 1 − z −1 P (z) = 1 −
pμ z −μ
μ=1
e[k] e(k)
E
e[k]
n
pν = −aν
m n 1 −μ =1− Pe (z) = pμ z = 1 + aν z −ν A(z) μ=1 ν=1
pi
x[k]
4 5 E e2 (k) → min
p = −ap = S −1 s
r
r
r
arν z −ν =
arν z −ν
ar0 = 1
ν=0
r
r
arν
r
r+1
⎡
γr+1
r
ν=1
Ar (z) = 1 +
(r − 1)
γr+1
⎤
⎡ ⎤ ⎡ ⎤ 1 0 ⎢ar+1 ⎥ ar1 ⎥ ⎢ arr ⎥ ⎢ 1r+1 ⎥ ⎢ ⎢ r ⎥ r⎥ ⎢a ⎥ ⎢ a2 ⎥ ⎢ar−1 ⎥ ⎢ 2 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ = ⎢ ⎥ − γr+1 ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ r+1 ⎥ ⎣ r ⎦ ⎣ r ⎣ ar ⎦ ar a1 ⎦ 0 1 ar+1 r+1
r ar s [r + 1 − ν] r ν r = ν=0 [−ν] ν=0 aν s
1
r
r
r σE =
arν s
[ν]
ν=0
0 σE =s
[0]
(r = 1)
s [1] γ1 = s [0] 1 0 1 = − γ1 0 1 a11 1 = s σE
[0] + a11 s
0 [1] = (1 − |γ1 |2 )σE
(r = 2 . . . n)
r−1
r−1 s [r − ν] ν=0 aν r−1 r−1 aν s [ν] ⎡ ν=0 ⎡ ⎤ ⎤ ⎤ ⎡ 1 0 1 ⎢ar−1 ⎢ar−1 ⎥ ⎥ ⎢ ar1 ⎥ ⎢ 1 ⎥ ⎢ r−1 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ = ⎢ ⎢ ⎢ ⎥ − γr ⎢ ⎥ ⎥ ⎢ ⎢ r−1 ⎥ ⎢ r−1 ⎥ ⎣ar ⎦ ⎣ ⎣ ⎦ ar−1 a1 ⎦ r−1 arr 0 1 r r−1 r σE = arν s [ν] = (1 − |γr |2 )σE
γr =
ν=0
1 − |γ|2 < 1
(r = 0)
a00 = 1
n
An (z) =
n ν=0
anν z −ν
n
Impulsfolge Frequenz Amplitude
stimmhafte Anregung (Vokale, ...)
Amplitude
weißes Rauschen
Sprache
Vokaltrakt
stimmlose Anregung (Konsonanten, ...)
a)
b) e(k)
LDS in dB 20
0.1 0 0 -20 -0.1 0
10
20 kTA /s
0
4 f in kHz
3
2
1
s(n)
3
κ
s(n)
3
κ
θ
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
θ
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"