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tf2zpk() - Signal Processing

tf2zpk finds the zeros, poles, and gains
of a discrete-time transfer function. Note

You should use tf2zpk when working with transfer
functions expressed in inverse powers (1 + z-1 + z-2), which
is how transfer functions are usually expressed in DSP. A similar
function, tf2zp, is more useful
for working with positive powers (s2 + s + 1),
such as in continuous-time transfer functions.[z,p,k] = tf2zpk(b,a) finds
the matrix of zeros z, the vector of poles p,
and the associated vector of gains k from the transfer
function parameters b and a:The numerator polynomials are represented as columns
of the matrix b. The denominator polynomial is represented in the vector a.Given a single-input, multiple output (SIMO) discrete-time system
in polynomial transfer function formH(z)=B(z)A(z)=b1+b2z−1⋯+bn−1z−n+bnz−n−1a1+a2z−1⋯+am−1z−m+amz−m−1you can use the output of tf2zpk to produce
the single-input, multioutput (SIMO) factored transfer function formH(z)=Z(z)P(z)=k(z−z1)(z−z2)⋯(z−zm)(z−p1)(z−p2)⋯(z−pn)The following describes the input and output arguments for tf2zpk:The vector a specifies the coefficients
of the denominator polynomial A(z)
in descending powers of z. The ith row of the matrix b represents
the coefficients of the ith numerator polynomial
(the ith row of B(s) or B(z)).
Specify as many rows of b as there are outputs. The zero locations are returned in the columns of
the matrix z, with as many columns as there are
rows in b. The pole locations are returned in the column vector p and
the gains for each numerator transfer function in the vector k.

Syntax

[z,p,k] = tf2zpk(b,a)

Example

Poles, Zeros, and Gain of an IIR FilterOpen This Example
Design a 3rd-order Butterworth filter with normalized cutoff frequency
rad/sample. Find the poles, zeros, and gain of the filter. Plot them to verify that they are where expected.
[b,a] = butter(3,.4);
fvtool(b,a,'polezero')
[z,p,k] = tf2zpk(b,a)
text(real(z)-0.1,imag(z)-0.1,'\bfZeros','color',[0 0.4 0])
text(real(p)-0.1,imag(p)-0.1,'\bfPoles','color',[0.6 0 0])

z =

-1.0000 + 0.0000i
-1.0000 + 0.0000i
-1.0000 - 0.0000i

p =

0.2094 + 0.5582i
0.2094 - 0.5582i
0.1584 + 0.0000i

k =

0.0985