ctf2zp

R2026b

Convert cascaded transfer functions to zero-pole-gain form

Since R2024b

Syntax

[z,p] = ctf2zp(B,A)

[z,p] = ctf2zp(B,A,g)

[___,k] = ctf2zp(___)

Description

[z,p] = ctf2zp(B,A) computes the zeros z and poles p of a system represented as Cascaded Transfer Functions (CTF) with numerator coefficients B and denominator coefficients A.

[z,p] = ctf2zp(B,A,g) specifies the scale values, g, across filter sections.

[___,k] = ctf2zp(___) returns the scalar gain k of the system. Use this syntax with any of the input or output arguments in previous syntaxes.

Examples

Zeros and Poles from Two-Section Digital Filter

Specify a two-section digital filter with a transfer function $H\left(z\right)$ in the CTF format. Plot the 1024-point frequency response of the filter.

$$H\left(z\right)=\left(0.1607+0.2414z^{-1}+0.4689z^{-2}\right)\times \frac{0.1607+0.0828z^{-1}+0.0551z^{-2}}{1-1.1940z^{-1}+0.4360z^{-2}}$$

B = [0.1607    0.2414    0.4689
     0.1607    0.0828    0.0551];
A = [     1         0         0;
          1   -1.1940    0.4360];
freqz(B,A,1024,"ctf")

Compute the zeros and poles of the transfer function.

[z,p] = ctf2zp(B,A)

z = 4×1 complex

  -0.7511 + 1.5342i
  -0.7511 - 1.5342i
  -0.2576 + 0.5258i
  -0.2576 - 0.5258i

p = 4×1 complex

   0.0000 + 0.0000i
   0.0000 + 0.0000i
   0.5970 + 0.2821i
   0.5970 - 0.2821i

Zero-Pole-Gain from Cascaded Transfer Function

Create a 10th-order bandpass digital elliptic filter in the CTF form.

[B,A,gS] = ellip(10,0.1,60,[0.35 0.65],"ctf")

B = 5×5

    1.0000    0.0000   -1.3205    0.0000    1.0000
    1.0000    0.0000    0.3308    0.0000    1.0000
    1.0000    0.0000    0.8670         0    1.0000
    1.0000   -0.0000    1.0340   -0.0000    1.0000
    1.0000   -0.0000    1.0853   -0.0000    1.0000

A = 5×5

    1.0000   -0.0000    1.2677   -0.0000    0.4407
    1.0000    0.0000    1.2212    0.0000    0.6374
    1.0000    0.0000    1.1779   -0.0000    0.8251
    1.0000   -0.0000    1.1571    0.0000    0.9300
    1.0000   -0.0000    1.1561   -0.0000    0.9818

gS = 
0.0053

Specify scale values. Because the filter has five fourth-order sections, the vector of scale values must have six elements.

g = [1:size(B,1) gS]

g = 1×6

    1.0000    2.0000    3.0000    4.0000    5.0000    0.0053

Get the zero-pole decomposition of the filter. Plot the zeros and poles in the z-plane.

[z,p,k] = ctf2zp(B,A,g);
zplane(z,p)
title("Pole-Zero Plot, k = "+k)

Input Arguments

B,A — Cascaded transfer function (CTF) coefficients

scalars | vectors | matrices

Cascaded transfer function (CTF) coefficients, specified as scalars, vectors, or matrices. B and A list the numerator and denominator coefficients of the cascaded transfer function, respectively.

B must be of size L-by-(m + 1) and A must be of size L-by-(n + 1), where:

• L represents the number of filter sections.

• m represents the order of the filter numerators.

• n represents the order of the filter denominators.

For more information about the cascaded transfer function format and coefficient matrices, see Specify Digital Filters in CTF Format.

Note

If any element of A(:,1) is not equal to 1, then ctf2zp normalizes the filter coefficients by A(:,1). In this case, A(:,1) must be nonzero.

Example: B = [2 4 2;3 3 0] and A = [6 0 2;6 0 0] specify a third-order Butterworth filter with normalized 3 dB frequency 0.5π rad/sample.

Data Types: double | single
Complex Number Support: Yes

g — Scale values

1 (default) | scalar | vector

Since R2024b

Scale values, specified as a real-valued scalar or as a real-valued vector with L + 1 elements, where L is the number of CTF sections. The scale values represent the distribution of the filter gain across the sections of the cascaded filter representation.

The ctf2zp function applies a gain to the filter sections using the scaleFilterSections function depending on how you specify g:

• Scalar — The function distributes the gain uniformly across all filter sections.

• Vector — The function applies the first L gain values to the corresponding filter sections and distributes the last gain value uniformly across all filter sections.

Data Types: double | single

Output Arguments

z,p — System zeros and poles

column vectors

System zeros and poles, returned as column vectors.

The vectors z and p contain the n zeros and m poles of the transfer function H(z), respectively.

$$H(z)=k\frac{(z-z_1)(z-z_2)\cdots (z-z_n)}{(z-p_1)(z-p_2)\cdots (z-p_m)}.$$

k — System gain

scalar

System gain, returned as a scalar.

The scalar k represents the gain of the transfer function H(z).

$$H(z)=k\frac{(z-z_1)(z-z_2)\cdots (z-z_n)}{(z-p_1)(z-p_2)\cdots (z-p_m)}.$$

Tips

You can obtain filters in CTF format, including the scaling gain. Use the outputs of digital IIR filter design functions, such as butter, cheby1, cheby2, and ellip. Specify the "ctf" filter-type argument in these functions and specify to return B, A, and g to get the scale values. (since R2024b)

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The ctf2zp function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced in R2024b

R2026b: Use thread-based environments

You can use ctf2zp in thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.