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BiquadraticFilterModel

BiquadraticFilterModel[{ω,q}]

creates a lowpass biquadratic filter using the characteristic frequency ω and the quality factor q.

BiquadraticFilterModel[{"type",spec}]

creates a filter of a given {"type",spec}.

BiquadraticFilterModel[{"type",spec},var]

expresses the model in terms of the variable var.

Details

Biquadratic filters are second-order filters defined by a ratio of two quadratic polynomials. They are among the most commonly used circuits in analog and digital signal processing.
BiquadraticFilterModel returns the filter as a TransferFunctionModel.
Filter specifications {"type",spec} can be any of the following:

		{"Lowpass",{{ω,q}}}	uses cutoff frequency ω and quality factor q
		{"Highpass",{{ω,q}}}	uses cutoff frequency ω and quality factor q
		{"Allpass",{{ω,q}}}	uses frequency ω and quality factor q
		{"Bandpass",{ω₁,ω₂}}	uses corner frequencies ω₁ and ω₂
		{"Bandpass",{{ω,q}}}	uses center frequency ω and quality factor q
		{"Bandstop",{ω₁,ω₂}}	uses corner frequencies ω₁ and ω₂
		{"Bandstop",{{ω,q}}}	uses center frequency ω and quality factor q

The following filter specifications can be given to create equalizers:

		{"Peaking",{{ω,q}},g}	peaking equalizer using gain value g
		{"LowShelf",{{ω,q}},g}	lowpass shelving equalizer using gain value g
		{"HighShelf",{{ω,q}},g}	highpass shelving equalizer using gain value g

Given the gain value , the attenuation is .

Examples

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Basic Examples (3)

A lowpass biquadratic filter:

Bode plot of the filter:

A bandpass filter using the full specification:

Bode plot of the filter:

Create a lowpass filter and apply it to a dual-tone signal:

Scope (8)

A symbolic lowpass filter with cutoff frequency ω and quality factor :

Use the full specification:

Specify cutoff frequency :

Bode plot of the filter:

A symbolic highpass filter with cutoff frequency and quality factor :

Use cutoff frequency :

A symbolic bandpass filter with center frequency and quality factor :

Use quality factor :

A symbolic bandstop filter with center frequency and quality factor :

Use quality factor :

A symbolic allpass filter with center frequency and quality factor :

Use quality :

A symbolic "Peaking" allpass filter with center frequency , quality factor , and gain value :

Use peak gain value of decibels:

A symbolic "LowShelf" filter with center frequency , quality factor , and gain value :

Use low-shelf gain value of decibels:

A symbolic "HighShelf" filter with center frequency , quality factor , and gain value :

Use low-shelf gain value of decibels:

Generalizations & Extensions (1)

Improve stopband attenuation by connecting two or more filters in series:

Applications (1)

Filter out the high-frequency tone in a pair of sinusoidal tones:

Use a biquadratic lowpass filter:

Create a higher-order filter by combining three filters to improve the filtering quality:

Properties & Relations (7)

Phase responses of the four basic filter types:

Extract the order of a BiquadraticFilterModel:

Stopband attenuation increases by a factor of 40 decibels per decade:

Gain at cutoff frequency increases with increasing values of quality factor :

Width of bandpass filter decreases with increasing quality factor :

Gain values "boost" magnitude response of peaking equalizer:

Gain values "cut" magnitude response of peaking equalizer:

Gain values "boost" magnitude response of the low-shelf filter:

Gain values "cut" magnitude response of the low-shelf filter:

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