Create an All-Pass Rational Matrix

Create a square, all-pass rational matrix in statespace form with given \(A,B\).

Usage

make_allpass(A, B)

Arguments

A

square \((s,s)\) matrix

B

\((s,m)\) matrix

Value

A stsp object which represents the all-pass rational matrix.

Notes

The function is intended as an internal helper function and thus does not check the inputs. See also reflect_poles and reflect_zeroes.

If \(A,C\) is given we can use t(make_allpass(t(A), t(C))).

Examples

m = 2
s = 6
A = matrix(rnorm(s*s), nrow = s, ncol = s)
B = matrix(rnorm(s*m), nrow = s, ncol = m)

K = make_allpass(A, B)
all.equal(cbind(A,B), cbind(K$A, K$B))
#> [1] TRUE
# check that K(z) is all-pass
Kf = zvalues(K)
all.equal(zvalues(polm(diag(m))) + complex(imaginary = 0), 
          Kf %r% Ht(Kf) + complex(imaginary = 0))
#> [1] TRUE

# if (A,C) is given, we proceed as follows:
C = matrix(stats::rnorm(m*s), nrow = m, ncol = s)
K = t(make_allpass(t(A), t(C)))
all.equal(rbind(A,C), rbind(K$A, K$C))
#> [1] TRUE
# check that K(z) is all-pass
Kf = zvalues(K)
all.equal(zvalues(polm(diag(m))) + complex(imaginary = 0), 
          Kf %r% Ht(Kf) + complex(imaginary = 0))
#> [1] TRUE