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


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


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


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 form


you can use the output of tf2zpk to produce the single-input, multioutput (SIMO) factored transfer function form


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.


collapse all

Design a 3rd-order Butterworth filter with normalized cutoff frequency 0.4π 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);
[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
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])

See Also

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Introduced before R2006a