iirlp2mb

Transform IIR lowpass filter to IIR multiband filter

Syntax

[num,den,allpassNum,allpassDen] =
iirlp2mb(b,a,wo,wt)

[num,den,allpassNum,allpassDen] =
iirlp2mb(b,a,wo,wt,pass)

Description

[num,den,allpassNum,allpassDen] = iirlp2mb(b,a,wo,wt) transforms an IIR lowpass filter to an IIR multiband filter.

The iirlp2mb function returns the numerator and denominator coefficients of the transformed IIR multiband filter. The function also returns the numerator, allpassNum, and the denominator, allpassDen, of the M^th order allpass mapping filter. The prototype lowpass filter is specified with the numerator b and the denominator a.

The function transforms a real lowpass prototype filter to a multiband filter by applying an M^th-order real lowpass to real multiple bandpass frequency mapping. Parameter M is the number of times an original feature is replicated in the transformed filter. By default the DC feature is kept at its original location. For more details, see IIR Lowpass Filter to IIR Multiband Filter Transformation.

example

[num,den,allpassNum,allpassDen] = iirlp2mb(b,a,wo,wt,pass) allows you to specify an additional parameter, pass as 'pass' or 'stop', which chooses between using the “Nyquist Mobility” and the “DC Mobility”, respectively. In the case of “Nyquist Mobility”, the DC feature is kept at an original frequency and the Nyquist feature is free to move. In the case of “DC Mobility”, the Nyquist feature stays at its original location and the DC feature is free to move.

Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Examples

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Transform Lowpass Filter to Multiband Filter

Open Live Script

Transform a lowpass IIR filter to a multiband IIR filter using the iirlp2mb function.

Input Lowpass IIR Filter

Design a prototype real IIR lowpass elliptic filter with a gain of about –3 dB at 0.5π rad/sample.

[b,a] = ellip(3,0.1,30,0.409);
filterAnalyzer(b,a)

Transform Filter Using iirlp2mb

Transform the real prototype lowpass filter into a real multiband filter with two passbands.

Specify the prototype filter as a vector of numerator and denominator coefficients, b and a respectively.

[num1,den1] = iirlp2mb(b,a,0.5,[2 4 6 8]/10);

Transform the prototype filter into a real multiband filter with two stopbands.

Specify the prototype filter as a vector of numerator and denominator coefficients, b and a respectively.

[num2,den2] = iirlp2mb(b,a,0.5,[2 4 6 8]/10,"stop");

Compare the magnitude response of the filters.

filterAnalyzer(b,a,num1,den1,num2,den2,...
    FilterNames=["PrototypeFilter_TFForm",...
    "TwoPassbands","TwoStopbands"]);

You can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the second-order section matrices as inputs. Change the desired frequency locations to be [0.5 0.6] in normalized frequency units. The transformed 2-band filter is a fourth-order section filter given by the numerator and the denominator coefficients num3 and den3.

ss = tf2sos(b,a);
[num3, den3] = iirlp2mb(ss(:,1:3), ss(:,4:6),...
    0.5,[5 6]/10);

To visualize the magnitude response of the fourth-order section filter, first pass the filter coefficients to the dsp.FourthOrderSectionFilter object. Then, use this object as an input to the filter analyzer.

fos = dsp.FourthOrderSectionFilter(Numerator=num3,...
    Denominator=den3);

Use the sos2ctf function to convert the second-order section matrices to the cascaded transfer function form.

[b_ctf,a_ctf] = sos2ctf(ss);

Compare the magnitude response of the filters.

filterAnalyzer(b_ctf,a_ctf,fos,...
    FilterNames=["PrototypeFilterFromSOSMatrix",...
    "TransformedMultibandFilterFromSOSMatrix"])

Input Arguments

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`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

Numerator coefficients of the prototype lowpass IIR filter, specified as either:

Row vector –– Specifies the values of [b₀, b₁, …, b_n], given this transfer function form:

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
where n is the order of the filter.
Matrix –– Specifies the numerator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

$b = [\begin{matrix} b_{01} & b_{11} & b_{21} & ... & b_{Q 1} \\ b_{02} & b_{12} & b_{22} & ... & b_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ b_{0 P} & b_{1 P} & b_{2 P} & \dots & b_{Q P} \end{matrix}]$
In the transfer function form, the numerator coefficient matrix b_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
where,
- a –– Denominator coefficients matrix. For more information on how to specify this matrix, see a.
- k –– Row index.
- i –– Column index.
When specified in the matrix form, b and a matrices must have the same number of rows (filter sections) Q.

Data Types: single | double
Complex Number Support: Yes

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

Denominator coefficients for a prototype lowpass IIR filter, specified as one of these options:

Row vector –– Specifies the values of [a₀, a₁, …, a_n], given this transfer function form:

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
where n is the order of the filter.
Matrix –– Specifies the denominator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

$a = [\begin{matrix} a_{01} & a_{11} & a_{21} & \dots & a_{Q 1} \\ a_{02} & a_{12} & a_{22} & \dots & a_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ a_{0 P} & a_{1 P} & a_{2 P} & \dots & a_{Q P} \end{matrix}]$
In the transfer function form, the denominator coefficient matrix a_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
where,
- b –– Numerator coefficients matrix. For more information on how to specify this matrix, see b.
- k –– Row index.
- i –– Column index.
When specified in the matrix form, a and b matrices must have the same number of rows (filter sections) P.

Data Types: single | double
Complex Number Support: Yes

`wo` — Frequency value to transform from prototype filter
positive scalar

Frequency value to transform from the prototype filter, specified as a real scalar. Frequency wo must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Data Types: single | double

`wt` — Desired frequency locations in transformed target filter
row vector

Desired frequency locations in the transformed target filter, specified as a row vector. Frequencies in wt must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate. The value of M which is equal to the number of times an original feature is replicated in the transformed filter equals the length of the wt vector.

Data Types: single | double

`pass` — Choice of passband or stopband at DC
`"pass"` (default) | `"stop"`

Choice of passband or stopband at DC, specified as either:

"pass" –– “Nyquist Mobility”. The DC feature is kept at an original frequency and the Nyquist feature is free to move.
"stop" –– “DC Mobility”. The Nyquist feature stays at its original location and the DC feature is free to move.

Output Arguments

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`num` — Numerator coefficients of transformed multiband filter
row vector | matrix

Numerator coefficients of the transformed multiband filter, returned as one of the following:

Row vector of length Mn+1, where M is the number of times an original feature is replicated in the transformed filter and n is the order of the input filter. The value of M equals the length of the wt vector.
The num output is a row vector when the input coefficients b and a are row vectors.
P-by-(QM+1) matrix, where P is the number of filter sections, Q is the order of each section of the transformed filter, and M is the number of times an original feature is replicated in the transformed filter.
The num output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double
Complex Number Support: Yes

`den` — Denominator coefficients of transformed multiband filter
row vector | matrix

Denominator coefficients of the transformed multiband filter, returned as one of the following:

Row vector of length Mn+1, where M is the number of times an original feature is replicated in the transformed filter and n is the order of the input filter. The value of M equals the length of the wt vector.
The den output is a row vector when the input coefficients b and a are row vectors.
P-by-(QM+1) matrix, where P is the number of filter sections, Q is the order of each section of the transformed filter, and M is the number of times an original feature is replicated in the transformed filter.
The den output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double
Complex Number Support: Yes

`allpassNum` — Numerator coefficients of mapping filter
row vector

Numerator coefficients of the mapping filter, returned as a row vector.

Data Types: single | double

`allpassDen` — Denominator coefficients of mapping filter
row vector

Denominator coefficients of the mapping filter, returned as a row vector.

Data Types: single | double

More About

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IIR Lowpass Filter to IIR Multiband Filter Transformation

IIR lowpass filter to IIR multiband filter transformation effectively places one feature of the original filter, located at frequency w_o, at the required target frequency locations, w_t1, … ,w_tM.

Relative positions of other features of the original filter do not change in the target filter. It is possible to select two features of an original filter, F₁ and F₂, with F₁ preceding F₂. Feature F₁ will still precede F₂ after the transformation. However, the distance between F₁ and F₂ will not be the same before and after the transformation.

Choice of the feature subject to this transformation is not restricted to the cutoff frequency of an original lowpass filter. You can choose to transform any feature of the original filter like stopband edge, DC, deep minimum in the stopband, or others.

The IIR lowpass filter to IIR multiband filter transformation can also be used to transform other types of filters, for example, notch filters or resonators can be easily replicated at a number of required frequency locations without redesigning them. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.

References

[1] Franchitti, Jean-Claude. “All-pass filter interpolation and frequency transformation problems.” MSc Thesis, Dept. of Electrical and Computer Engineering, University of Colorado, 1985.

[2] Feyh, G., J.C. Franchitti and C.T. Mullis.“All-pass filter interpolation and frequency transformation problem.” Proceedings 20th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California, pp. 164-168, November 1986.

[3] Mullis, C.T. and R. A. Roberts. Digital Signal Processing, section 6.7, Reading, Mass., Addison-Wesley, 1987.

[4] Feyh, G., W.B. Jones and C.T. Mullis. “An extension of the Schur Algorithm for frequency transformations.” Linear Circuits, Systems and Signal Processing: Theory and Application. C. J. Byrnes et al Eds, Amsterdam: Elsevier, 1988.

Extended Capabilities

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

Version History

Introduced in R2011a

iirlp2mb

Syntax

Description

Examples

Transform Lowpass Filter to Multiband Filter

Input Arguments

`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

`wo` — Frequency value to transform from prototype filter
positive scalar

`wt` — Desired frequency locations in transformed target filter
row vector

`pass` — Choice of passband or stopband at DC
`"pass"` (default) | `"stop"`

Output Arguments

`num` — Numerator coefficients of transformed multiband filter
row vector | matrix

`den` — Denominator coefficients of transformed multiband filter
row vector | matrix

`allpassNum` — Numerator coefficients of mapping filter
row vector

`allpassDen` — Denominator coefficients of mapping filter
row vector

More About

IIR Lowpass Filter to IIR Multiband Filter Transformation

References

Extended Capabilities

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

Version History

See Also

Functions

iirlp2mb

Syntax

Description

Examples

Transform Lowpass Filter to Multiband Filter

Input Arguments

b — Numerator coefficients of prototype lowpass IIR filter row vector | matrix

a — Denominator coefficients of prototype lowpass IIR filter row vector | matrix

wo — Frequency value to transform from prototype filter positive scalar

wt — Desired frequency locations in transformed target filter row vector

pass — Choice of passband or stopband at DC "pass" (default) | "stop"

Output Arguments

num — Numerator coefficients of transformed multiband filter row vector | matrix

den — Denominator coefficients of transformed multiband filter row vector | matrix

allpassNum — Numerator coefficients of mapping filter row vector

allpassDen — Denominator coefficients of mapping filter row vector

More About

IIR Lowpass Filter to IIR Multiband Filter Transformation

References

Extended Capabilities

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

Version History

See Also

Functions

`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

`wo` — Frequency value to transform from prototype filter
positive scalar

`wt` — Desired frequency locations in transformed target filter
row vector

`pass` — Choice of passband or stopband at DC
`"pass"` (default) | `"stop"`

`num` — Numerator coefficients of transformed multiband filter
row vector | matrix

`den` — Denominator coefficients of transformed multiband filter
row vector | matrix

`allpassNum` — Numerator coefficients of mapping filter
row vector

`allpassDen` — Denominator coefficients of mapping filter
row vector

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