filtord

Filter order

Syntax

n = filtord(b,a)

n = filtord(B,A,"ctf")

n = filtord({B,A,g},"ctf")

n = filtord(d)

n = filtord(sos)

Description

n = filtord(b,a) returns the filter order, n, for the specified digital filter. Specify a digital filter as a causal rational system function with numerator coefficients, b, and denominator coefficients, a.

n = filtord(B,A,"ctf") returns the filter order for the digital filter represented as Cascaded Transfer Functions (CTF) with numerator coefficients B and denominator coefficients A. (since R2024b)

n = filtord({B,A,g},"ctf") returns the filter order for the digital filter in CTF format. Specify the filter with numerator coefficients B, denominator coefficients A, and scaling values g across filter sections. (since R2024b)

n = filtord(d) returns the filter order, n, for the digital filter, d. Use the function designfilt to generate d.

n = filtord(sos) returns the filter order for the filter specified by the second-order sections matrix, sos. sos is a K-by-6 matrix. The number of sections, K, must be greater than or equal to 2. Each row of sos corresponds to the coefficients of a second-order filter. The ith row of the second-order section matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].

Examples

Verify Order of FIR Filter

Design a 20th-order FIR filter with normalized cutoff frequency $$0.5\pi$$ rad/sample using the window method. Verify the filter order.

b = fir1(20,0.5);
n = filtord(b)

n = 
19

Design the same filter using designfilt and verify its order.

di = designfilt('lowpassfir','FilterOrder',20,'CutoffFrequency',0.5);
ni = filtord(di)

ni = 
19

Filter Order from Cascaded Transfer Functions

Since R2024b

Design a 40th-order lowpass Chebyshev type II digital filter with a stopband edge frequency of 0.4 and stopband attenuation of 50 dB. Compute the filter order using its coefficients in the CTF format.

[B,A] = cheby2(40,50,0.4,"ctf");

n1 = filtord(B,A,"ctf")

n1 = 
40

Design a 30th-order bandpass elliptic digital filter with passband edge frequencies of 0.3 and 0.7, passband ripple of 0.1 dB, and stopband attenuation of 50 dB. Compute the filter order using its coefficients and gain in the CTF format.

[B,A,g] = ellip(30,0.1,50,[0.3 0.7],"ctf");
n2 = filtord({B,A,g},"ctf")

n2 = 
60

Determine the Order Difference Between FIR and IIR Designs

Design FIR equiripple and IIR Butterworth filters from the same set of specifications. Determine the difference in filter order between the two designs.

fir = designfilt("lowpassfir",DesignMethod="equiripple", ...
    SampleRate=1e3,PassbandFrequency=100,StopbandFrequency=120, ...
    PassbandRipple=0.5,StopbandAttenuation=60);
iir = designfilt("lowpassiir",DesignMethod="butter", ...
    SampleRate=1e3,PassbandFrequency=100,StopbandFrequency=120, ...
    PassbandRipple=0.5,StopbandAttenuation=60);
FIR = filtord(fir)

FIR = 
114

IIR = filtord(iir)

IIR = 
41

Input Arguments

b — Numerator coefficients

vector | scalar

Numerator coefficients, specified as a scalar or a vector. If the filter is an allpole filter, b is a scalar. Otherwise, b is a row or column vector.

Example: b = fir1(20,0.25)

Data Types: single | double
Complex Number Support: Yes

a — Denominator coefficients

vector | scalar

Denominator coefficients, specified as a scalar or a vector. If the filter is an FIR filter, a is a scalar. Otherwise, a is a row or column vector.

Example: [b,a] = butter(20,0.25)

Data Types: single | double
Complex Number Support: Yes

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 filtord 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

scalar | vector

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 sections of the cascaded filter representation.

The filtord 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

d — Digital filter

digitalFilter object

Digital filter, specified as a digitalFilter object. Use designfilt to generate a digital filter based on frequency-response specifications.

Example: d = designfilt("lowpassiir",FilterOrder=3,HalfPowerFrequency=0.5) specifies a third-order Butterworth filter with normalized 3 dB frequency 0.5π rad/sample.

sos — Matrix of second-order sections

matrix

Matrix of second order-sections, specified as a K-by-6 matrix. The system function of the Kth biquad filter has the rational Z-transform

$$H_k(z)=\frac{B_k(1)+B_k(2)z^{-1}+B_k(3)z^{-2}}{A_k(1)+A_k(2)z^{-1}+A_k(3)z^{-2}}.$$

The coefficients in the Kth row of the matrix, sos, are ordered as follows.

$$\left[\begin{array}{cccccc}{B}_k(1)& B_k(2)& B_k(3)& A_k(1)& A_k(2)& A_k(3)\end{array}\right].$$

The frequency response of the filter is the system function evaluated on the unit circle with

$$z=e^{j2\pi f}.$$

Data Types: single | double
Complex Number Support: Yes

Output Arguments

n — Filter order

integer

Filter order, specified as an integer.

More About

Cascaded Transfer Functions

Partitioning an IIR digital filter into cascaded sections improves its numerical stability and reduces its susceptibility to coefficient quantization errors. The cascaded form of a transfer function H(z) in terms of the L transfer functions H₁(z), H₂(z), …, H_L(z) is

$$H(z)={\displaystyle \prod _{l=1}^L{H}_l(z)}=H_1(z)\times H_2(z)\times \cdots \times H_L(z).$$

Specify Digital Filters in CTF Format

You can specify digital filters in the CTF format for analysis, visualization, and signal filtering. Specify a filter by listing its coefficients B and A. You can also include the filter scaling gain across sections by specifying a scalar or vector g.

Filter Coefficients

When you specify the coefficients as L-row matrices,

$$B=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1,m+1}\\ b_{21}& b_{22}& \cdots & b_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ b_{L1}& b_{L2}& \cdots & b_{L,m+1}\end{array}\right],A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1,n+1}\\ a_{21}& a_{22}& \cdots & a_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ a_{L1}& a_{L2}& \cdots & a_{L,n+1}\end{array}\right],$$

it is assumed that you have specified the filter as a sequence of L cascaded transfer functions, such that the full transfer function of the filter is

$$H\left(z\right)=\frac{b_{11}+b_{12}{z}^{-1}+\cdots +b_{1,m+1}{z}^{-m}}{a_{11}+a_{12}{z}^{-1}+\cdots +a_{1,n+1}{z}^{-n}}\times \frac{b_{21}+b_{22}{z}^{-1}+\cdots +b_{2,m+1}{z}^{-m}}{a_{21}+a_{22}{z}^{-1}+\cdots +a_{2,n+1}{z}^{-n}}\times \cdots \times \frac{b_{L1}+b_{L2}{z}^{-1}+\cdots +b_{L,m+1}{z}^{-m}}{a_{L1}+a_{L2}{z}^{-1}+\cdots +a_{L,n+1}{z}^{-n}},$$

where m ≥ 0 is the numerator order of the filter and n ≥ 0 is the denominator order.

• If you specify both B and A as vectors, it is assumed that the underlying system is a one-section IIR filter (L = 1), with B representing the numerator of the transfer function and A representing its denominator.

• If B is scalar, it is assumed that the filter is a cascade of all-pole IIR filters with each section having an overall system gain equal to B.

• If A is scalar, it is assumed that the filter is a cascade of FIR filters with each section having an overall system gain equal to 1/A.

Note

• To convert second-order section matrices to cascaded transfer functions, use the sos2ctf function.

• To convert a zero-pole-gain filter representation to cascaded transfer functions, use the zp2ctf function.

Coefficients and Gain

If you have an overall scaling gain or multiple scaling gains factored out from the coefficient values, you can specify the coefficients and gain as a cell array of the form {B,A,g}. Scaling filter sections is especially important when you work with fixed-point arithmetic to ensure that the output of each filter section has similar amplitude levels, which helps avoid inaccuracies in the filter response due to limited numeric precision.

The gain can be a scalar overall gain or a vector of section gains.

• If the gain is scalar, the value applies uniformly to all the cascade filter sections.

• If the gain is a vector, it must have one more element than the number of filter sections L in the cascade. Each of the first L scale values applies to the corresponding filter section, and the last value applies uniformly to all the cascade filter sections.

If you specify the coefficient matrices and gain vector as

$$B=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1,m+1}\\ b_{21}& b_{22}& \cdots & b_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ b_{L1}& b_{L2}& \cdots & b_{L,m+1}\end{array}\right],A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1,n+1}\\ a_{21}& a_{22}& \cdots & a_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ a_{L1}& a_{L2}& \cdots & a_{L,n+1}\end{array}\right],g=\left[\begin{array}{ccccc}{g}_1& g_2& \cdots & g_L& g_{\text{S}}\end{array}\right],$$

it is assumed that the transfer function of the filter system is

$$H\left(z\right)=g_{\text{S}}\left(g_1\frac{b_{11}+b_{12}{z}^{-1}+\cdots +b_{1,m+1}{z}^{-m}}{a_{11}+a_{12}{z}^{-1}+\cdots +a_{1,n+1}{z}^{-n}}\times g_2\frac{b_{21}+b_{22}{z}^{-1}+\cdots +b_{2,m+1}{z}^{-m}}{a_{21}+a_{22}{z}^{-1}+\cdots +a_{2,n+1}{z}^{-n}}\times \cdots \times g_L\frac{b_{L1}+b_{L2}{z}^{-1}+\cdots +b_{L,m+1}{z}^{-m}}{a_{L1}+a_{L2}{z}^{-1}+\cdots +a_{L,n+1}{z}^{-n}}\right).$$

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)

References

[1] Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

If you input CTF coefficients to the filtord function, you must include the "ctf" input argument in the filtord syntax to generate code.

GPU Code Generation

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

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Version History

Introduced in R2013a

R2024b: Analyze filters using cascaded transfer functions

The filtord function supports inputs in the cascaded transfer function (CTF) format.