iirlp2xc
Transform IIR lowpass filter to IIR complex N-point filter
Description
[
transforms an IIR lowpass filter to an IIR complex N-point
filter.num
,den
,allpassNum
,allpassDen
] =
iirlp2xc(b
,a
,wo
,wt
)
The prototype lowpass filter is specified with the numerator
b
and the denominator a
. The function
returns the numerator and the denominator coefficients of the transformed IIR
complex N-point filter. The function also returns the numerator,
allpassNum
, and the denominator,
allpassDen
, of the allpass mapping filter.
The function transforms a real lowpass prototype filter to an
N-point complex filter by applying an
Nth-order real lowpass to complex
multipoint frequency transformation, where N is the number of
features being mapped and is given by the length of the vector
wo
. For more details, see IIR Lowpass Filter to IIR Complex N-Point Filter Transformation.
Examples
Transform Lowpass Filter to IIR Complex N-Point Filter
Transform a lowpass IIR filter to an N-point IIR complex filter using the iirlp2xc
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 iirlp2xc
Transform the real prototype lowpass filter to an IIR complex N-point filter.
Specify the prototype filter as a vector of numerator and denominator coefficients, b
and a
respectively.
[num,den] = iirlp2xc(b,a,[-0.5 0.5],[-0.25 0.25]);
Compare the magnitude response of the filters. The transformed filter has complex coefficients and is indeed a bandpass filter.
filterAnalyzer(b,a,num,den,FrequencyRange="centered",... FilterNames=["PrototypeFilter_TFForm",... "TransformedComplexMultipointFilter"])
Alternatively, you can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the second-order section matrices as inputs. The numerator and the denominator coefficients of the transformed filter are given by num2
and den2
, respectively.
ss = tf2sos(b,a);
[num2,den2] = iirlp2xc(ss(:,1:3),ss(:,4:6),...
[-0.5 0.5],[-0.25 0.25]);
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,num2,den2,... FrequencyRange="centered",... FilterNames=["PrototypeFilterFromSOSMatrix",... "TransformedComplexMultipointFilterFromSOSMatrix"])
Copyright 2012–2024 The MathWorks, Inc.
Input Arguments
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, …, bn], given this transfer function form:
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.
In the transfer function form, the numerator coefficient matrix bik of the IIR filter can be represented using the following equation:
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, …, an], given this transfer function form:
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.
In the transfer function form, the denominator coefficient matrix aik of the IIR filter can be represented using the following equation:
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 values to transform from prototype filter
row vector
Frequency values to transform from the prototype filter, specified as a
row vector with even number of elements. Frequencies in
wo
should be normalized to be between
0
and 1
, with 1
corresponding to half the sample rate. The value of N
equals the length of the wo
vector. Length of vectors
wo
and wt
must be the
same.
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 with even number of elements. Frequencies in
wt
should be normalized to be between
-1
and 1
, with
1
corresponding to half the sample rate.
Note
Length of wo
and wt
vectors must be the same.
Data Types: single
| double
Output Arguments
num
— Numerator coefficients of transformed multipoint filter
row vector | matrix
Numerator coefficients of the transformed multipoint filter, returned as one of the following:
Row vector of length Nn/2+1, where n is the order of the input filter and N is the number of features being mapped. The value of N equals the length of the
wo
vector.The
num
output is a row vector when the input coefficientsb
anda
are row vectors.P-by-(QN/2+1) matrix, where P is the number of filter sections, Q is the order of each section of the transformed filter, and N is the number of features being mapped and is given by the length of the vector
wo
.The
num
output is a matrix when the input coefficientsb
anda
are matrices.
Data Types: single
| double
Complex Number Support: Yes
den
— Denominator coefficients of transformed multipoint filter
row vector | matrix
Denominator coefficients of the transformed multipoint filter, returned as one of the following:
Row vector of length Nn/2+1, where n is the order of the input filter and N is the number of features being mapped. The value of N equals the length of the
wo
vector.The
den
output is a row vector when the input coefficientsb
anda
are row vectors.P-by-(QN/2+1) matrix, where P is the number of filter sections, Q is the order of each section of the transformed filter, and N is the number of features being mapped and is given by the length of the vector
wo
.The
den
output is a matrix when the input coefficientsb
anda
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
Complex Number Support: Yes
allpassDen
— Denominator coefficients of mapping filter
row vector
Denominator coefficients of the mapping filter, returned as a row vector.
Data Types: single
| double
Complex Number Support: Yes
More About
IIR Lowpass Filter to IIR Complex N-Point Filter Transformation
IIR lowpass filter to IIR complex N-point
filter transformation effectively places N features of the
original filter, located at frequencies
wo1
, …
,woN
, at the required target
frequency locations, wt1
, …
,wtM
. The function
iirlp2xc
requires that N and
M are equal.
Relative positions of other features of the original filter are the same in the target filter for the Nyquist mobility and are reversed for the DC mobility. For the Nyquist mobility this means that 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. For DC mobility feature F2 will precede F1 after the transformation.
Choice of the feature subject to this transformation is not restricted to the cutoff frequency of an original lowpass filter. In general it is possible to select any feature; e.g., a stopband edge, DC, the deep minimum in the stopband, or other ones. Select features such that there is no band overlap when creating N bands around the unit circle.
IIR lowpass filter to IIR complex N-point 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. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.
References
[1] Krukowski, A., and I. Kale, “High-order complex frequency transformations,” Internal report No. 27/2001, Applied DSP and VLSI Research Group, University of Westminster.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced in R2011a
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