Análise Mel-Cepstral

Da mesma forma, (F.17) contempla e_E como saída do filtro W

WԣФ¥_ cuja entrada é e. Assim

eE é representada por eE =_{2{ Â}1  €QWÔ_*+,_€EQ _*+,_ 1 1 + *x+,_{ *}+,LI à xà (F.22)

Substituindo (F.21) e (F.22) em (F.18), verifica-se que (F.18) equivale a (F.12).

F.2 Análise Mel-Cepstral

Para o caso mel-cepstral, fazendo a substituição de variáveis  =  em (D.35), obtém-se =̃X> =_{2{ Â}1  €Q*+,_€EΦ) ∗_*+,_{ ™}I I š I à xà > = 0,1, … ,2k (F.23)

Substituindo (F.6) em (F.23), obtém-se =̃_X> na forma

=̃X> = † ‡ ˆ ‡ ‰ 1 2{   €Q*+,_€E 1 − E_ *+,_{+ *}x+,_{+  I} à xà > = 0 1 − E_ 1 2{  _€Q*+,_€E 1 1 + *+,_{ *}+,)I à xà , > = 1,2, … ,2k o (F.24)

182 eXç =_{2{ Â}1  €Q*+,_€E 1 1 + *+,_{ *}+,0I à xà ç = 0,1, … ,2k (F.25) e =̃_X> é obtido por =̃X> = ‘ eX0 − eX1, > = 0 1 − E_e X>, > = 1,2, … ,2k o (F.26)

expressão cuja consistência com (F.24) (para > = 0) pode ser facilmente verificada. Para o cálculo de e_Xç, utiliza-se

Q*+,_{ = exp á1 + *}+,_{ w } : K LzW

*x+,L_ã (F.27)

expressão na qual o fator do somatório pode ser calculado via FFT.

O critério espectral ; é dado por =̃_X0 (o que é verificado em (F.24) para > = 0, que equilave a (F.1)). Como já considerado no Apêndice D, os elementos associados ao vetor gradiente 9₂Ü; são dados por

=̃X>, > = 1,2, … , k e aqueles associados à matriz Hessiana <, por

?X> =  1 − E_=̃ X0 + 2=̃X1, > = 0 =̃X> + =̃X> + 1, > = 1,2, … , k − 1 o expressão (D.46) e @X> = =̃X> + =̃X> − 1, > = 2,3, … ,2k expressão (D.47)

183

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