impzlength

Impulse response length

Syntax

len = impzlength(b,a)

len = impzlength(B,A,"ctf")

len = impzlength({B,A,g},"ctf")

len = impzlength(d)

len = impzlength(sos)

len = impzlength(___,tol)

Description

len = impzlength(b,a) returns the impulse response length of the specified filter. Specify a causal discrete-time filter with the rational system function specified by the numerator, b, and denominator, a, polynomials in z^–1. For stable IIR filters, len is the effective impulse response sequence length. Terms in the IIR filter’s impulse response after the len-th term are essentially zero.

len = impzlength(B,A,"ctf")returns the effective impulse response length for the digital filter represented as Cascaded Transfer Functions (CTF) with numerator coefficients B and denominator coefficients A. (since R2024b)

len = impzlength({B,A,g},"ctf") returns the effective impulse response length 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)

len = impzlength(d) returns the impulse response length for the digital filter, d. Use designfilt to generate d based on frequency-response specifications.

len = impzlength(sos) returns the effective impulse response length for the IIR filter specified by the second order sections matrix, sos. sos is a K-by-6 matrix, where the number of sections, K, must be greater than or equal to 2. If the number of sections is less than 2, impzlength considers the input to be the numerator vector, b. 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)].

len = impzlength(___,tol) specifies a tolerance for estimating the effective length of an IIR filter’s impulse response. By default, tol is 5e-5. Increasing the value of tol estimates a shorter effective length for an IIR filter’s impulse response. Decreasing the value of tol produces a longer effective length for an IIR filter’s impulse response.

Examples

IIR Filter Effective Impulse Response Length — Coefficients

Create a lowpass allpole IIR filter with a pole at 0.9. Calculate the effective impulse response length. Obtain the impulse response. Plot the result.

b = 1;
a = [1 -0.9];
len = impzlength(b,a)

len = 
93

[h,t] = impz(b,a);
stem(t,h)

h(len)

ans = 
6.1704e-05

IIR Filter Effective Impulse Response Length — Second-Order Sections

Design a 4th-order lowpass elliptic filter with a cutoff frequency of 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Design the filter in pole-zero-gain form and obtain the second-order section matrix using zp2sos. Determine the effective impulse response sequence length from the second-order section matrix.

[z,p,k] = ellip(4,1,60,.4);
[sos,g] = zp2sos(z,p,k);
len = impzlength(sos)

len = 
80

IIR Filter Effective Impulse Response Length — Digital Filter

Use designfilt to design a 4th-order lowpass elliptic filter with normalized passband frequency 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Determine the effective impulse response sequence length.

d = designfilt("lowpassiir",FilterOrder=4,PassbandFrequency=0.4, ...
               PassbandRipple=1,StopbandAttenuation=60, ...
               DesignMethod="ellip");
len = impzlength(d)

len = 
80

Impulse Response Length 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 impulse response length of the filter using its coefficients in the CTF format. Specify a tolerance of 1e-4.

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

L = impzlength(B,A,"ctf",1e-4)

L = 
1547

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 impulse response length of the filter using its coefficients and gain in the CTF format. Specify a tolerance of 1e-4.

[B,A,g] = ellip(30,0.1,50,[0.3 0.7],"ctf");
L = impzlength({B,A,g},"ctf",1e-4)

L = 
403641

Input Arguments

b — Numerator coefficients

vector | scalar

Numerator coefficients, specified as a scalar (allpole filter) or a 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 (FIR filter) or vector.

Data Types: single | double
Complex Number Support: Yes

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 impzlength 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 impzlength 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 K-th 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

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

tol — Tolerance for IIR filter effective impulse response length

5e-5 (default) | positive scalar

Tolerance for IIR filter effective impulse response length, specified as a positive number. The tolerance determines the term in the absolutely summable sequence after which subsequent terms are considered to be 0. The default tolerance is 5e-5. Increasing the tolerance returns a shorter effective impulse response sequence length. Decreasing the tolerance returns a longer effective impulse response sequence length.

Output Arguments

len — Length of impulse response

positive integer

Length of the impulse response, specified as a positive integer. For stable IIR filters with absolutely summable impulse responses, impzlength returns an effective length for the impulse response beyond which the coefficients are essentially zero. You can control this cutoff point by specifying the optional tol input argument.

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)

Algorithms

To compute the impulse response for an FIR filter, impzlength uses the length of b. For IIR filters, the function first finds the poles of the transfer function using roots.

If the filter is unstable, the length extends to the point at which the term from the largest pole reaches 10⁶ times its original value.

If the filter is stable, the length extends to the point at which the term from the largest-amplitude pole is tol times its original amplitude.

If the filter is oscillatory, with poles on the unit circle only, then impzlength computes five periods of the slowest oscillation.

If the filter has both oscillatory and damped terms, the length extends to the greater of these values:

• Five periods of the slowest oscillation.

• The point at which the term due to the largest pole is tol times its original amplitude.

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 impzlength function, you must include the "ctf" input argument in the impzlength syntax to generate code.

• If the first input to impzlength is a variable-size matrix at compile time, then it must not become a vector at runtime.

Version History

Introduced in R2013a

R2024b: Analyze filters using cascaded transfer functions

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