dsp.NotchPeakFilter

Second-order tunable notching and peaking IIR filter

Description

The NotchPeakFilter object filters each channel of the input using IIR filter implementation.

To filter each channel of the input:

1. Create the dsp.NotchPeakFilter object and set its properties.

2. Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?.

Creation

Syntax

npFilter = dsp.NotchPeakFilter

npFilter = dsp.NotchPeakFilter('Specification','Quality factor and center frequency')

npFilter = dsp.NotchPeakFilter('Specification','Coefficients')

npFilter = dsp.NotchPeakFilter(Name,Value)

Description

npFilter = dsp.NotchPeakFilter returns a second-order notching and peaking IIR filter that independently filters each channel of the input over time, using a specified center frequency and 3 dB bandwidth.

npFilter = dsp.NotchPeakFilter('Specification','Quality factor and center frequency') specifies the quality factor (Q factor) of the notch or peak filter instead of the 3 dB bandwidth.

npFilter = dsp.NotchPeakFilter('Specification','Coefficients') specifies the coefficient values that affect bandwidth and center frequency directly, rather than specifying the design parameters in Hz. This removes the trigonometry calculations involved when the properties are tuned.

npFilter = dsp.NotchPeakFilter(Name,Value) returns a notch filter with each specified property name set to the specified value. Unspecified properties have default values.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

Specification — Filter specification

'Bandwidth and center frequency' (default) | 'Quality factor and center frequency' | 'Coefficients'

Set the specification as 'Bandwidth and center frequency', 'Quality factor and center frequency', or 'Coefficients'.

Bandwidth — 3 dB bandwidth

2205 (default) | positive scalar

Specify the filter’s 3 dB bandwidth as a finite positive numeric scalar in Hz. The value must be a scalar between 0 and half the sample rate.

Tunable: Yes

Dependencies

This property is applicable only if Specification is 'Bandwidth and center frequency'.

Data Types: single | double

CenterFrequency — Notch or peak center frequency

11025 (default) | positive scalar

Specify the filter’s center frequency (for both the notch and the peak) as a finite positive numeric scalar in Hz. The value must be a scalar between 0 and half the sample rate.

Tunable: Yes

Dependencies

This property is applicable only if Specification is set to 'Bandwidth and center frequency' or 'Quality factor and center frequency'.

Data Types: single | double

QualityFactor — Quality factor for notch or peak filter

5 (default) | positive scalar

Specify the quality factor (Q factor) for both the notch and the peak filters. The Q factor is defined as the center frequency divided by the bandwidth. A higher Q factor corresponds to a narrower notch or peak. The Q factor should be a scalar value greater than 0.

Tunable: Yes

Dependencies

This property is applicable only if Specification is set to 'Quality factor and center frequency'.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

SampleRate — Sample rate of input

44100 (default) | positive scalar

Specify the sample rate of the input in Hz as a finite numeric scalar.

Data Types: single | double

BandwidthCoefficient — Bandwidth coefficient

0.72654 (default) | real scalar in the range [-1 1]

Specify the value that determines the filter’s 3 dB bandwidth as a finite numeric scalar in the range [-1 1]. The value -1 corresponds to the maximum 3 dB bandwidth (SampleRate/4), and 1 corresponds to the minimum 3 dB bandwidth (0 Hz, an allpass filter).

Tunable: Yes

Dependencies

This property is only applicable if Specification is set to 'Coefficients'.

Data Types: single | double

CenterFrequencyCoefficient — Center frequency coefficient

0 (default) | real scalar in the range [-1 1]

Specify the coefficient that determines the filter’s center frequency as a finite numeric scalar in the range [-1 1]. The value -1 corresponds to the minimum center frequency (0 Hz), and 1 corresponds to the maximum center frequency (SampleRate/2 Hz). The default is 0, which corresponds to SampleRate/4 Hz.

Tunable: Yes

Dependencies

This property is only applicable if Specification is set to 'Coefficients'.

Data Types: single | double

Usage

Syntax

Y = npFilter(x)

[Yn,Yp] = npFilter(x)

Description

Y = npFilter(x) filters each channel (column) of the input signal, x, to produce the notch filter output, Y.

[Yn,Yp] = npFilter(x) filters each channel of the input signal, x, to produce the notch filter output, Yn, and peak filter output, Yp.

Input Arguments

x — Input signal

vector | matrix

Input signal, specified as a vector or a matrix.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

Yn — Notch filter output

vector | matrix

Notch filter output, returned as a vector or a matrix.

Data Types: single | double
Complex Number Support: Yes

Yp — Peak filter output

vector | matrix

Peak filter output, returned as a vector or a matrix.

Data Types: single | double
Complex Number Support: Yes

Object Functions

To use an object function, specify the System object™ as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

Specific to dsp.NotchPeakFilter

getBandwidth	Get 3 dB bandwidth
getCenterFrequency	Get center frequency
getOctaveBandwidth	Bandwidth in number of octaves
getQualityFactor	Get quality factor
tf	Transfer function

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

Examples

Notch Filter

This example shows how to use dsp.NotchPeakFilter as a notch filter with center frequency of 5000 Hz and a 3 dB bandwidth of 500 Hz.

Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent step syntax. For example, myObject(x) becomes step(myObject,x).

npFilter = dsp.NotchPeakFilter('CenterFrequency',5000,'Bandwidth',500);
sa =  dsp.SpectrumAnalyzer('SampleRate',44100,...
    'PlotAsTwoSidedSpectrum',false,'SpectralAverages',50);
for i=1:5000
    y = npFilter(randn(1024,1));
    sa(y);
    if (i==2500)
        % Tune center frequency to 10000
        npFilter.CenterFrequency = 10000;
    end
end
release(npFilter)
release(sa)

Get 3 dB Bandwidth of Notch Peak Filter

Create a dsp.NotchPeakFilter object with the Specification property set to 'Quality factor and center frequency'. The default quality factor Q is 5, and the center frequency Fc is 11,025 Hz.

np = dsp.NotchPeakFilter('Specification','Quality factor and center frequency')

np = 
  dsp.NotchPeakFilter with properties:

      Specification: 'Quality factor and center frequency'
      QualityFactor: 5
    CenterFrequency: 11025
         SampleRate: 44100

Compute the 3 dB bandwidth of the notch peak filter using the getBandwidth function. The bandwidth is computed as the ratio of the center frequency and the quality factor, $\frac{\mathrm{Fc}}{\mathit{Q}}$.

getBandwidth(np)

ans = 2205

Visualize the filter response using fvtool.

fvtool(np)

Get Center Frequency of Notch Peak Filter

Create a dsp.NotchPeakFilter object with the Specification property set to 'Coefficients'.

np = dsp.NotchPeakFilter('Specification','Coefficients')

np = 
  dsp.NotchPeakFilter with properties:

                 Specification: 'Coefficients'
          BandwidthCoefficient: 0.7265
    CenterFrequencyCoefficient: 0
                    SampleRate: 44100

Determine the center frequency of the notch peak filter using the getCenterFrequency function. When the Specification is set to 'Coefficients', the center frequency is determined from the CenterFrequencyCoefficient value and the sample rate.

getCenterFrequency(np)

ans = 11025

Visualize the filter response using fvtool.

fvtool(np)

Get Octave Bandwidth of Notch Peak Filter

Create a dsp.NotchPeakFilter object in the default configuration.

np = dsp.NotchPeakFilter

np = 
  dsp.NotchPeakFilter with properties:

      Specification: 'Bandwidth and center frequency'
          Bandwidth: 2205
    CenterFrequency: 11025
         SampleRate: 44100

Determine the octave bandwidth of the filter using the getOctaveBandwidth function.

getOctaveBandwidth(np)

ans = 0.2881

Visualize the filter response using fvtool.

fvtool(np)

Compute Quality Factor of Notch Peak Filter

Create a dsp.NotchPeakFilter object in the default configuration, where the Specification property is set to 'Bandwidth and center frequency'.

np = dsp.NotchPeakFilter

np = 
  dsp.NotchPeakFilter with properties:

      Specification: 'Bandwidth and center frequency'
          Bandwidth: 2205
    CenterFrequency: 11025
         SampleRate: 44100

Determine the quality factor of the filter using the getQualityFactor function. The quality factor is given by the ratio of the center frequency to the bandwidth.

getQualityFactor(np)

ans = 5

Visualize the filter response using fvtool.

fvtool(np)

Determine Transfer Function of Notch Peak Filter

Create a dsp.NotchPeakFilter System object™. Obtain the coefficients of the transfer function corresponding to the notch and peak filters.

notchpeak = dsp.NotchPeakFilter;
[Bnotch,Anotch,Bpeak,Apeak] = tf(notchpeak)

Bnotch = 1×3

    0.8633   -0.0000    0.8633

Anotch = 1×3

    1.0000   -0.0000    0.7265

Bpeak = 1×3

    0.1367         0   -0.1367

Apeak = 1×3

    1.0000   -0.0000    0.7265

Bnotch and Anotch are the vectors of numerator and denominator coefficients for the equivalent transfer function corresponding to the notch filter. Bpeak and Apeak are the vectors of numerator and denominator coefficients for the equivalent transfer function corresponding to the peak filter.

Algorithms

The design equation for the peak filter is:

$$H(z)=(1-b)\frac{1-z^{-2}}{1-2b\cos w_0{z}^{-1}+(2b-1)z^{-2}}$$

The design equation for the notch filter is:

$$H(z)=b\frac{1-2\cos w_0{z}^{-1}+z^{-2}}{1-2b\cos w_0{z}^{-1}+(2b-1)z^{-2}}$$

with

$$b=\frac{1}{1+\tan (\Delta w/2)}$$

where ω₀ = 2πf₀/f_s is the center frequency in radians/sample (f₀ is the center frequency in Hz and f_s is the sampling frequency in Hz). Δω = 2πΔf/f_s is the 3 dB bandwidth in radians/sample (Δf is the 3 dB bandwidth in Hz). Note that the two filters are complementary:

$$\begin{array}{l}{H}_{\text{notch}}(z)+H_{\text{peak}}(z)=1\\ \text{they can be written as:}\\ H_{\text{peak}}(z)=\frac{1}{2}\left[1-A(z)\right]\\ H_{\text{notch}}(z)=\frac{1}{2}\left[1+A(z)\right]\\ \text{where }A\text{(z) is a }{2}^{\text{nd}}\text{ order allpass filter}\text{.}\\ A(z)=\frac{a_2+a_1{z}^{-1}+z^{-2}}{1+a_1{z}^{-1}+a_2{z}^{-2}}\\ \text{and}\\ a_1=-2b\cos {\omega }_0\\ a_2=2b-1\end{array}$$

The filter is implemented as follows:

where

$$\begin{array}{l}{G}_{\text{3dB}}=a_2=2b-1\\ G_{\text{cf}}=\frac{a_1-a_1{a}_2}{1-a_2{}^2}=-\cos w_0\end{array}$$

Notice that G_cf depends only on the center frequency, and G_3dB depends only on the 3 dB bandwidth.

References

[1] Orfanidis, Sophocles J. Introduction to Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1996.