Octave Filter

Octave-band and fractional octave-band filter

• 

Libraries:
Audio Toolbox / Filters

Description

The Octave Filter block performs octave-band or fractional octave-band filtering independently across each input channel. An octave-band is a frequency band where the highest frequency is twice the lowest frequency. Octave-band and fractional octave-band filters are commonly used to mimic how humans perceive loudness. Octave filters are best understood when viewed on a logarithmic scale, which models how the human ear weights the spectrum.

Examples

Perform Octave Filtering

Examine the Octave Filter block in a Simulink® model and tune parameters.

• Open Model

Tune Center Frequency Using Input Port

Tune the center frequency of an Octave Filter block in Simulink® using the optional input port.

• Open Model

Ports

Input

x — Input signal

matrix | 1-D vector

• Matrix input –– Each column of the input is treated as an independent channel.

• 1-D vector input –– The input is treated as a single channel.

This port is unnamed unless you specify additional input ports.

Data Types: single | double

CF — Center frequency (Hz)

scalar in the range 3 to 22,000 inclusive

Dependencies

To enable this port, select Specify from input port for the Center frequency (Hz) parameter.

Data Types: single | double

Output

Port_1 — Output signal

matrix

The Octave Filter block outputs a signal with the same data type as the input signal. The size of the output depends on the size of the input:

• Matrix input –– The block outputs a matrix the same size and data type as the input signal.

• 1-D vector input –– The block outputs an N-by-1 matrix (column vector), where N is the number of elements in the 1-D vector.

Data Types: single | double

Parameters

If a parameter is listed as tunable, then you can change its value during simulation.

Filter order — Order of octave filter

6 (default) | even integer

Tunable: No

Center frequency (Hz) — Center frequency of octave filter

1000 (default) | scalar in the range 3 to 22,000 inclusive

• The maximum center frequency is the value that causes the upper band edge to be equal to the Nyquist frequency, Fs/2. Frequencies above this value are saturated.

• The minimum center frequency is the value that causes the lower band edge to be equal to 1 Hz. Frequencies below this value are quantized to 1 Hz.

To specify Center frequency (Hz) from an input port, select Specify from input port for the parameter.

Tunable: Yes

Bandwidth — Filter bandwidth in octaves

1 octave (default) | 2/3 octave | 1/2 octave | 1/3 octave | 1/6 octave | 1/12 octave | 1/24 octave | 1/48 octave

Tunable: Yes

Oversample the input by 2 for this filter — Oversample toggle

off (default) | on

• off –– The Octave Filter block runs at the input sample rate.

• on –– The Octave Filter block runs at two times the input sample rate. Oversampling minimizes the frequency warping effects introduced by the bilinear transformation. An FIR halfband interpolator implements oversampling before octave filtering. A halfband decimator reduces the sample rate back the input sampling rate after octave filtering.

Tunable: No

Inherit sample rate from input — Specify source of input sample rate

off (default) | on

When you select this parameter, the block inherits its sample rate from the input signal. When you clear this parameter, you specify the sample rate in Input sample rate (Hz).

Tunable: No

Input sample rate (Hz) — Sample rate of input

44100 (default) | scalar

Tunable: Yes

Dependencies

To enable this parameter, clear the Inherit sample rate from input parameter.

Simulate using — Specify type of simulation to run

Code generation (default) | Interpreted execution

• Code generation –– Simulate the model using generated C code. The first time you run a simulation, Simulink^® generates C code for the block. The C code is reused for subsequent simulations as long as the model does not change. This option requires additional startup time, but the speed of the subsequent simulations is comparable to Interpreted execution.

• Interpreted execution –– Simulate the model using the MATLAB^® interpreter. This option shortens startup time but has a slower simulation speed than Code generation. In this mode, you can debug the source code of the block.

Tunable: No

Mask for attenuation limits — Create a mask for filter response visualization

No mask (default) | Class 0 | Class 1 | Class 2

The mask attenuation limits are defined in the ANSI S1.11-2004 standard.

• If the mask is green, the design is compliant.

• If the mask is red, the design breaks compliance.

Tunable: Yes

Visualize filter response — Open plot to visualize magnitude response and compliance mask

button

A 2048-point FFT is used to calculate the magnitude response.

Tunable: Yes

Variable name — Variable name of exported filter

myFilt (default) | valid variable name

Name of the variable in the base workspace to contain the filter when it is exported. The name must be a valid MATLAB variable name.

Overwrite variable if it already exists — Overwrite variable if it already exists

on (default) | off

When you select this parameter, exporting the filter overwrites the variable specified by the Variable name parameter if it already exists in the base workspace. If you do not select this parameter and the specified variable already exists in the workspace, exporting the filter creates a new variable with an underscore and a number appended to the variable name. For example, if the variable name is var and it already exists, the exported variable will be named var_1.

Export filter to workspace — Export filter to workspace

button

Export the filter to the base workspace in the variable specified by the Variable name parameter.

Tips

• You cannot export the filter if you have enabled the Inherit sample rate from input parameter and the model is not running.

• You cannot export the filter if you are specifying filter characteristics from input ports.

Block Characteristics

Data Types	double | single
Direct Feedthrough	no
Multidimensional Signals	no
Variable-Size Signals	yes
Zero-Crossing Detection	no

More About

Band Edge

A band edge frequency refers to the lower or upper edge of the passband of a bandpass filter.

Center Frequency of Octave Filter

The center frequency of an octave filter is the geometric mean of the lower- and upper-band edge frequencies.

Algorithms

Octave Bandwidth to Band Edge Conversion

The Octave Filter block uses the specified center frequency and filter bandwidth in octaves to determine the band edges [2].

First the block computes the upper and lower band edge frequencies:

$$f_{\text{pa}}=f_{\text{c}}\times G^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2b$}\right.}$$

$$f_{\text{pb}}=f_{\text{c}}\times G^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2b$}\right.}$$

• f_c is the center frequency specified by the Center frequency (Hz) parameter.

• b is the octave bandwidth specified by the Bandwidth parameter. For example, if Bandwidth is specified as 1/3 octave, the value of b is 3.

• G is a conversion constant:

  $$G={10}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}.$$

Digital Filter Design

The Octave Filter block implements a higher-order digital bandpass filter design method as specified in [1].

In this design method, a desired digital bandpass filter maps to a Butterworth lowpass analog prototype, which is then mapped back to a digital bandpass filter:

1. The analog Butterworth filter is expressed as a cascade of second-order sections:

   $$H(s)=H_1(s)H_2(s)\cdots H_{2N}(s),\text{where:}$$

   $$•H_i(s)=\frac{1}{1-2\frac{s}{{\Omega }_0}\cos {\theta }_i+\frac{s^2}{{\Omega }_0^2}},i=1,2,\mathrm{...},2N$$

   $$•{\theta }_i=\frac{\pi }{2N}\left(N-1+2i\right),i=1,2,\mathrm{...},N,\mathrm{...},2N$$

   N is the filter order specified by the Filter order parameter.

2. The analog Butterworth filter is mapped to a digital filter using a bandpass version of the bilinear transformation:

   $$s=\frac{1-c{z}^{-1}+z^{-2}}{1-z^{-2}},$$

   where

   $$c=\frac{\sin \left({\omega }_{\text{pa}}+{\omega }_{\text{pb}}\right)}{\sin {\omega }_{\text{pa}}+\sin {\omega }_{\text{pb}}}.$$

   This mapping results in the following substitution:

   $${\Omega }_0=\frac{c-\cos {\omega }_{\text{pb}}}{\sin {\omega }_{\text{pb}}}$$

3. The analog prototype is evaluated:

   $$H_{\text{i}}(z)={\frac{1}{1-2\frac{s}{{\Omega }_0}\cos {\theta }_{\text{i}}+\frac{s^2}{{\Omega }_0^2}}|}_{s=\frac{1-2c{z}^{-1}+z^{-2}}{1-z^{-2}}}$$

   Because s is second-order in z, the bandpass version of the bilinear transformation is fourth-order in z.

References

[1] Orfanidis, Sophocles J. Introduction to Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 2010.

[2] Acoustical Society of America. American National Standard Specification for Octave-Band and Fractional-Octave-Band Analog and Digital Filters: ANSI S1.11-2004. Melville, NY: Acoustical Society of America, 2009.

Extended Capabilities

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

Version History

Introduced in R2016b