http://markus-hauenstein.de/sigpro/codec/codec.shtml
formants
http://research.cs.tamu.edu/prism/lectures/sp/l5.pdf
https://ccrma.stanford.edu/~jos/filters/Formant_Filtering_Example.html
http://www.sfu.ca/sonic-studio/handbook/Formant.html
http://home.cc.umanitoba.ca/~robh/howto.html

625 0
1200 -5 dB
2300 -18 dB
3200 -18dB

f0=10000
w1=2 Pi 625./f0
z1=.92 Exp[I w1]

w2=2 Pi 1200./f0
z2=.92 Exp[I w2]


w3=2 Pi 2300./f0
z3=.95 Exp[I w3]

w4=2 Pi 3200./f0
z4=.95 Exp[I w4]

A[z_]=(1/(1-z1/z)+1/(1-Conjugate[z1]/z))+10^(-6./15) (1/(1-z2/z)+1/(1-Conjugate[z2]/z))+
10^(-18./15) (1/(1-z3/z)+1/(1-Conjugate[z3]/z))+10^(-18./15) (1/(1-z4/z)+1/(1-Conjugate[z4]/z))

Plot[20 Log[10, Abs[A[Exp[I 2 Pi f/f0]]]],{f,1,4000}]



((1-z1 zi)(1-Conjugate[z1] zi))((1-z2 zi)(1-Conjugate[z2] zi))*
((1-z3 zi)(1-Conjugate[z3] zi))((1-z4 zi)(1-Conjugate[z4] zi))

A[1/zi]*%

Simplify[Re[%30], Element[zi,Reals]] 


A[1/zi]*((1-z1 zi)(1-Conjugate[z1] zi))((1-z2 zi)(1-Conjugate[z2] zi))*
((1-z3 zi)(1-Conjugate[z3] zi))((1-z4 zi)(1-Conjugate[z4] zi))

Simplify[Re[%], Element[zi,Reals]] 

Plot[20 Log[10, Abs[AA[Exp[I 2 Pi f/f0]]]],{f,1,4000}]

restart math

f0=10000
w1=2 Pi  625./f0; r1=.92
w2=2 Pi 1200./f0; r2=.92
w3=2 Pi 2300./f0; r3=.95
w4=2 Pi 3200./f0; r4=.95

choice={z4->r4 Exp[I w4],z1->r1 Exp[I w1],z2->r2 Exp[I w2],z3->r3 Exp[I w3]}

A[z_]=(1/(1-z1/z)+1/(1-Conjugate[z1]/z))+10^(-6./15) (1/(1-z2/z)+1/(1-Conjugate[z2]/z))+
10^(-18./15) (1/(1-z3/z)+1/(1-Conjugate[z3]/z))+10^(-18./15) (1/(1-z4/z)+1/(1-Conjugate[z4]/z))

A[1/zi]*((1-z1 zi)(1-Conjugate[z1] zi))((1-z2 zi)(1-Conjugate[z2] zi))*
((1-z3 zi)(1-Conjugate[z3] zi))((1-z4 zi)(1-Conjugate[z4] zi))
Together[%]
% /. choice
Expand[%]
Simplify[Re[%], Element[zi,Reals]] 
Expand[%]

Out[13]= 3.0486 - 5.33333 zi + 7.36138 zi  - 8.19557 zi  + 7.53604 zi  - 
 
               5             6             7
>    5.61113 zi  + 3.61221 zi  - 1.51677 zi

((1-z1 zi)(1-Conjugate[z1] zi))((1-z2 zi)(1-Conjugate[z2] zi))*
((1-z3 zi)(1-Conjugate[z3] zi))((1-z4 zi)(1-Conjugate[z4] zi))
% /. choice
Expand[%]
Simplify[Re[%], Element[zi,Reals]] 
Expand[%]
                                   2             3             4
Out[18]= 1 - 2.47039 zi + 3.8492 zi  - 4.69454 zi  + 4.90044 zi  - 
 
               5            6             7              8
>    4.17172 zi  + 3.0649 zi  - 1.72755 zi  + 0.583507 zi


%13/%18 /. zi->1/z
AA[z_]=%
Plot[20 Log[10, Abs[AA[Exp[I 2 Pi f/f0]]]],{f,1,4000}]