tf2zpk

Convert transfer function filter parameters to zero-pole-gain form

Syntax

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

Description

[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 function converts a polynomial transfer-function representation

$$H(z)=\frac{B(z)}{A(z)}=\frac{b_1+b_2{z}^{-1}\cdots +b_n{z}^{-\left(n-1\right)}+b_{n+1}{z}^{-n}}{a_1+a_2{z}^{-1}\cdots +a_m{z}^{-\left(m-1\right)}+a_{m+1}{z}^{-m}}$$

of a single-input/multi-output (SIMO) discrete-time system to a factored transfer function form

$$H(z)=\frac{Z(z)}{P(z)}=k\frac{(z-z_1)(z-z_2)\cdots (z-z_m)}{(z-p_1)(z-p_2)\cdots (z-p_n)}.$$

Note

Use tf2zpk when working with transfer functions expressed in inverse powers (1 + z^–1 + z^–2). A similar function, tf2zp, is more useful for working with positive powers (s² + s + 1), such as in continuous-time transfer functions.

Examples

Poles, Zeros, and Gain of IIR Filter

Design a 3rd-order Butterworth filter with normalized cutoff frequency $$0.4\pi$$ rad/sample. Find the poles, zeros, and gain of the filter.

[b,a] = butter(3,0.4);
[z,p,k] = tf2zpk(b,a)

z = 3×1 complex

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

p = 3×1 complex

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

k = 0.0985

Plot the poles and zeros to verify that they are where expected.

zplane(b,a)
text(real(z)-0.1,imag(z)-0.1,"Zeros")
text(real(p)-0.1,imag(p)-0.1,"Poles")

Input Arguments

b — Transfer function numerator coefficients

vector | matrix

Transfer function numerator coefficients, specified as a vector or matrix. If b is a matrix, then each row of b corresponds to an output of the system. b contains the coefficients in descending powers of z. The number of columns of b must be equal to the length of a. If the numbers differ, make them equal by padding zeros. You can use the function eqtflength to accomplish this.

Data Types: single | double

a — Transfer function denominator coefficients

vector

Transfer function denominator coefficients, specified as a vector. a contains the coefficients in descending powers of z.

Data Types: single | double

Output Arguments

z — Zeros

matrix

Zeros of the system, returned as a matrix. z contains the numerator zeros in its columns. z has as many columns as there are outputs.

p — Poles

column vector

Poles of the system, returned as a column vector. p contains the pole locations of the denominator coefficients of the transfer function

k — Gains

column vector

Gains of the system, returned as a column vector. k contains the gains for each numerator transfer function.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The complexity of outputs, z and k, might be different in MATLAB^® and the generated code.

• The order of outputs, z and p, might be different in MATLAB and the generated code.

Version History

Introduced before R2006a