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



[z,p,k] = tf2zp(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


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



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


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Generate a system with the following transfer function.


Find the zeros, poles, and gain of the system. Use eqtflength to ensure the numerator and denominator have the same length.

b = [2 3];
a = [1 1/sqrt(2) 1/4];

[b,a] = eqtflength(b,a);
[z,p,k] = tf2zp(b,a)
z = 2×1


p = 2×1 complex

  -0.3536 + 0.3536i
  -0.3536 - 0.3536i

k = 2

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


Figure Filter Visualization Tool - Pole-Zero Plot contains an axes and other objects of type uitoolbar, uimenu. The axes with title Pole-Zero Plot contains 7 objects of type line, text.

Input Arguments

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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 s. The number of columns of b must be less than or equal to the length of a.

Data Types: single | double

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

Data Types: single | double

Output Arguments

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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.

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

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

Extended Capabilities

See Also

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Introduced before R2006a