isminphase

Determine whether filter is minimum phase

Syntax

flag = isminphase(b,a)

flag = isminphase(B,A,"ctf")

flag = isminphase({B,A,g},"ctf")

flag = isminphase(d)

flag = isminphase(sos)

flag = isminphase(___,tol)

Description

flag = isminphase(b,a) returns a logical output equal to 1 if the specified filter is minimum phase. Specify a filter with numerator coefficients b and denominator coefficients a.

flag = isminphase(B,A,"ctf") returns 1 if the filter specified as Cascaded Transfer Functions (CTF) with numerator coefficients B and denominator coefficients A is minimum phase. (since R2024b)

flag = isminphase({B,A,g},"ctf") returns 1 if the filter specified in CTF format is minimum phase. Specify the filter with numerator coefficients B, denominator coefficients A, and scaling values g across filter sections. (since R2024b)

flag = isminphase(d) returns 1 if the digital filter d is minimum phase. Use designfilt to generate d based on frequency-response specifications.

flag = isminphase(sos) returns 1 if the filter specified by second order sections matrix sos is minimum phase.

flag = isminphase(___,tol) uses the tolerance tol to determine when two numbers are close enough to be considered equal.

Examples

Minimum Phase Filters

Design a sixth-order lowpass Butterworth IIR filter using second order sections. Specify a normalized 3-dB frequency of 0.15. Check if the filter is minimum phase.

[z,p,k] = butter(6,0.15);
SOS = zp2sos(z,p,k);            
min_flag = isminphase(SOS)

min_flag = logical
   1

Redesign the filter using designfilt. Check that the zeros and poles of the transfer function are on or within the unit circle.

d = designfilt("lowpassiir",DesignMethod="butter",FilterOrder=6, ...
               HalfPowerFrequency=0.25);
d_flag = isminphase(d)

d_flag = logical
   1

zplane(d)

Given a filter defined with a set of single-precision numerator and denominator coefficients, check if it is minimum phase for different tolerance values.

b = single([1 1.00001]);  
a = single([1 0.45]);               
min_flag1 = isminphase(b,a)        

min_flag1 = logical
   0

min_flag2 = isminphase(b,a,1e-3)

min_flag2 = logical
   1

Verify Minimum Phase 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. Verify that the filter is minimum phase using the filter coefficients in the CTF format.

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

flag = isminphase(B,A,"ctf")

flag = logical
   1

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. Verify that the filter is minimum phase using the filter coefficients and gain in the CTF format.

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

flag = logical
   1

Input Arguments

b, a — Transfer function coefficients

vector

Transfer function coefficients, specified as vectors.

Data Types: single | double

B,A — Cascaded transfer function (CTF) coefficients

scalars | vectors | matrices

Since R2024b

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 isminphase normalizes the filter coefficients by A(:,1). In this case, A(:,1) must be nonzero.

Data Types: double | single
Complex Number Support: Yes

g — Scale values

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 sections of the cascaded filter representation.

The isminphase 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.

sos — Second order sections

k-by-6 matrix

Second order sections, specified as a k-by-6 matrix where 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 (biquad) filter. The ith row of the sos matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].

Data Types: double

tol — Tolerance

eps^(2/3) (default) | positive scalar

Tolerance, specified as a positive scalar. The tolerance value determines when two numbers are close enough to be considered equal.

Data Types: double

Output Arguments

flag — Logical output

1 (true) | 0 (false)

Logical output, returned as 1 or 0. The function returns 1 when the input is a minimum phase filter.

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.

Version History

Introduced in R2013a

R2024b: Analyze filters using cascaded transfer functions

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