Filter Design Toolbox

tf2ca

Transfer function to coupled allpass conversion

Syntax

• [d1,d2] = tf2ca(b,a)
  [d1,d2] = tf2ca(b,a) 
  [d1,d2,beta] = tf2ca(b,a)
  

Description

[d1,d2] = tf2ca(b,a) where b is a real, symmetric vector of numerator coefficients and a is a real vector of denominator coefficients, corresponding to a stable digital filter, returns real vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z) such that

representing a coupled allpass decomposition.

[d1,d2] = tf2ca(b,a) where b is a real, antisymmetric vector of numerator coefficients and a is a real vector of denominator coefficients, corresponding to a stable digital filter, returns real vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z) such that

In some cases, the decomposition is not possible with real H1(z) and H2(z). In those cases a generalized coupled allpass decomposition may be possible, whose syntax is

• [d1,d2,beta] = tf2ca(b,a)
  

to return complex vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z), and a complex scalar beta, satisfying |beta| = 1, such that

representing the generalized allpass decomposition.

In the above equations, H1(z) and H2(z) are real or complex allpass IIR filters given by

where D1(z) and D2(z) are polynomials whose coefficients are given by d1 and d2.

Note    A coupled allpass decomposition is not always possible. Nevertheless, Butterworth, Chebyshev, and Elliptic IIR filters, among others, can be factored in this manner. For details, refer to Signal Processing Toolbox User's Guide.

Examples

• [b,a]=cheby1(9,.5,.4);
  [d1,d2]=tf2ca(b,a); % TF2CA returns denominators of the allpass.
  num = 0.5*conv(fliplr(d1),d2)+0.5*conv(fliplr(d2),d1);
  den = conv(d1,d2); % Reconstruct numerator and denonimator.
  max([max(b-num),max(a-den)]) % Compare original and reconstructed
                               % numerator and denominators.
  

See Also
ca2tf, cl2tf, iirpowcomp, latc2tf, tf2latc

sos

tf2cl