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tf2zpk

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

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Syntax

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

Description

[z,p,k] = tf2zpk(b,a) finds the vector of zeros z, the vector of poles p, and the associated gain k from the transfer function parameters b and a. The function converts a polynomial transfer-function representation

H(z)=B(z)A(z)=b1+b2z−1⋯+bnz−(n−1)+bn+1z−na1+a2z−1⋯+amz−(m−1)+am+1z−m

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

H(z)=Z(z)P(z)=k(z−z1)(z−z2)⋯(z−zm)(z−p1)(z−p2)⋯(z−pn).

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.

example

Examples

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Open Live Script

Design a 3rd-order Butterworth filter with normalized cutoff frequency 0.4π 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")

Figure contains an axes object. The axes object with title Pole-Zero Plot, xlabel Real Part, ylabel Imaginary Part contains 10 objects of type line, text. One or more of the lines displays its values using only markers

Input Arguments

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Transfer function numerator coefficients, specified as a vector. b contains the coefficients in ascending powers of z–1.

Data Types: single | double

Transfer function denominator coefficients, specified as a vector. a contains the coefficients in ascending powers of z–1.

Data Types: single | double

Output Arguments

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System zeros, returned as a column vector.

System poles, returned as a column vector.

System gain, returned as a scalar.

Extended Capabilities

Version History

Introduced before R2006a

See Also

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