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# ca2tf

Convert coupled allpass filter to transfer function form

## Syntax

[b,a]=ca2tf(d1,d2)
[b,a]=ca2tf(d1,d2,beta)
[b,a,bp]=ca2tf(d1,d2)
[b,a,bp]=ca2tf(d1,d2,beta)

## Description

[b,a]=ca2tf(d1,d2) returns the vector of coefficients b and the vector of coefficients a corresponding to the numerator and the denominator of the transfer function

$H\left(z\right)=B\left(z\right)/A\left(z\right)=\frac{1}{2}\left[H1\left(z\right)+H2\left(z\right)\right]$

d1 and d2 are real vectors corresponding to the denominators of the allpass filters H1(z) and H2(z).

[b,a]=ca2tf(d1,d2,beta) where d1, d2 and beta are complex, returns the vector of coefficients b and the vector of coefficients a corresponding to the numerator and the denominator of the transfer function

$H\left(z\right)=B\left(z\right)/A\left(z\right)=\frac{1}{2}\left[-\left(\overline{\beta }\right)\cdot H1\left(z\right)+\beta \cdot H2\left(z\right)\right]$

[b,a,bp]=ca2tf(d1,d2), where d1 and d2 are real, returns the vector bp of real coefficients corresponding to the numerator of the power complementary filter G(z)

$G\left(z\right)=Bp\left(z\right)/A\left(z\right)=\frac{1}{2}\left[H1\left(z\right)-H2\left(z\right)\right]$

[b,a,bp]=ca2tf(d1,d2,beta), where d1, d2 and beta are complex, returns the vector of coefficients bp of real or complex coefficients that correspond to the numerator of the power complementary filter G(z)

$G\left(z\right)=Bp\left(z\right)/A\left(z\right)=\frac{1}{{2}_{j}}\left[-\left(\overline{\beta }\right)\cdot H1\left(z\right)+\beta \cdot H2\left(z\right)\right]$

## Examples

Create a filter, convert the filter to coupled allpass form, and convert the result back to the original structure (create the power complementary filter as well).

 [b,a]=cheby1(10,.5,.4); [d1,d2,beta]=tf2ca(b,a); % tf2ca returns the % denominators of the % allpasses. [num,den,numpc]=ca2tf(d1, d2,beta); % Reconstruct the original % filter plus the power % complementary one. [h,w,s]=freqz(num,den); hpc = freqz(numpc,den); s.plot = 'mag'; s.yunits = 'sq'; freqzplot([h hpc],w,s); % Plot the mag response of the % original filter and the % power complementary one.