Convert transfer function filter parameters to zero-pole-gain form
finds the matrix of zeros
k] = tf2zpk(
z, the vector of poles
p, and the associated vector of gains
k from the
transfer function parameters
a. The function
converts a polynomial transfer-function representation
of a single-input/multi-output (SIMO) discrete-time system to a factored transfer function form
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.
Design a 3rd-order Butterworth filter with normalized cutoff frequency rad/sample. Find the poles, zeros, and gain of the filter.
[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
Plot the poles and zeros to verify that they are where expected.
fvtool(b,a,'polezero') 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])
b— Transfer function numerator coefficients
Transfer function numerator coefficients, specified as a vector or matrix. If
b is a matrix, then each row of
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.
a— Transfer function denominator coefficients
Transfer function denominator coefficients, specified as a vector.
a contains the coefficients in descending powers of
Zeros of the system, returned as a matrix.
z contains the
numerator zeros in its columns.
z has as many columns as there are
Poles of the system, returned as a column vector.
the pole locations of the denominator coefficients of the transfer function
Gains of the system, returned as a column vector.
the gains for each numerator transfer function.