CN110086452B - Design method of low-complexity sparse FIR notch filter - Google Patents

Design method of low-complexity sparse FIR notch filter Download PDF

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CN110086452B
CN110086452B CN201811508545.8A CN201811508545A CN110086452B CN 110086452 B CN110086452 B CN 110086452B CN 201811508545 A CN201811508545 A CN 201811508545A CN 110086452 B CN110086452 B CN 110086452B
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徐微
赵耀
李安宇
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Tianjin Polytechnic University
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    • H—ELECTRICITY
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Abstract

The invention discloses a design method of a linear phase FIR notch filter for realizing small ripple, low tap number and low complexity. Based on the SPT coding method, the method maximizes the common sub-formula in the shared MCM module based on the common sub-formula elimination technology, re-synthesizes the filter coefficient set according to the sensitivity of the sub-formula, and further reduces the number of adders. Simulation results show that the number of adders of the sparse FIR notch filter designed by the invention is more than 51 percent less than that of the adders of the existing similar filters at home and abroad under the requirement of the same design index.

Description

Design method of low-complexity sparse FIR (finite impulse response) notch filter
Technical Field
The invention belongs to the technical field of digital signal processing, and provides a design method of a sparse, efficient and low-complexity linear phase Finite Impulse Response (FIR) notch filter.
Background
The notch filter can effectively filter out frequency component interference of a specific extremely narrow frequency band, and can efficiently transmit signals outside the frequency band, so that the notch filter is widely applied to the fields of communication technology, bioengineering, radar sonar, measuring instruments and the like. In digital signal processing, digital filtering techniques are an important component thereof, and digital signal filters can be generally classified into Finite Impulse Response (FIR) filters and (infinite impulse response) IIR filters from the standpoint of impulse response. From the perspective of frequency response, the following can be classified: low pass filter, high pass filter, band reject filter, notch filter. Notch filters are used to remove certain frequencies from the input signal and have relatively no effect on the remaining frequencies. The method is widely applied to the fields of signal processing, communication, biomedical engineering and the like. A sparse linear phase FIR (finite impulse response) notch filter is a notch filter whose coefficients have a sparse characteristic (the number of non-zero tap coefficients is less than the filter order). The number of addition and multiplier used for realizing the sparse filter is far less than that of the similar filter with the equivalent filtering effect, so that the sparse filter has the advantages of high operation speed, small operation error, low energy consumption and the like.
The methods proposed at present to reduce the hardware implementation complexity of FIR filters mainly fall into two categories. One method is to design an FIR filter by using a sparse algorithm, and on the premise of meeting the design requirement of the frequency domain of the filter, the time domain impulse response of the filter has zero-value tap coefficients as much as possible, so that the number of adder multipliers used for realizing the FIR filter is greatly reduced. The second method for effectively reducing the complexity of hardware implementation is FIR filter optimization design based on multi-constant multiplication (MCM) technology. In the proposed design method of the notch filter, the design methods of chen.jiajiaia and chang.c are classical, and based on the SPT coding method, the common subforms in the shared MCM module are maximized based on the common subforms elimination technique, so that the number of adders is further reduced.
Disclosure of Invention
The invention aims to design a notch filter for realizing small ripple and low tap number, and provides a brand new design method, namely a method for designing a linear phase FIR notch filter with sparseness, high efficiency and low complexity.
The design method of the low-complexity sparse FIR notch filter provided by the invention comprises the following specific steps:
1, obtaining an FIR trapped wave original filter coefficient meeting the design requirement of frequency domain performance by adopting a sparse FIR trapped wave filter design algorithm;
2, converting the design problem of the sparse FIR notch filter into a mathematical optimization problem, and solving by adopting an IROMP algorithm to obtain an FIR notch original filter coefficient;
3, CSD coding is carried out on the coefficients of the FIR trapped wave original filter;
and 4, introducing a sensitivity function of 2-term sub-type sensitivity and solitons, and sequentially selecting a reasonable 2-term sub-type or solitons according to the sensitivity to re-synthesize a filter coefficient set.
(taking type I linear phase FIR filter as an example, the invention is also applicable to other type II, III, IV linear phase FIR filters):
constructing initial parameters according to design requirements:
the invention comprises a notch frequency set according to the design parameter requirements
Figure BSA0000175606180000021
The notch depth d (dB), notch bandwidth delta omega, pass band attenuation alpha (dB) (or pass band ripple delta), the pass band attenuation alpha and the pass band ripple delta can be mutually converted, and the two relations are as follows:
Figure BSA0000175606180000022
given the above design criteria, the frequency response H (e) of the FIR notch filter jω ) It should satisfy:
Figure BSA0000175606180000023
wherein
Figure BSA0000175606180000024
(II) obtaining an FIR trapped wave original filter coefficient H (N) meeting the frequency domain performance design requirement by utilizing a sparse filter design algorithm, wherein N is more than or equal to 0 and less than or equal to N, and the frequency response H (e) of the FIR trapped wave filter is determined by assuming that the trapped wave filter is an I-type linear phase FIR filter, namely the order N is an even number, and H (N) is a real number and has even symmetry jω ) Satisfies the condition of H (e) jω )＝e -jMω H 0 (ω) wherein M = N/2,H 0 (ω) is a zero phase response, which can be expressed as:
Figure BSA0000175606180000031
the design problem of the sparse FIR notch filter is transformed into the following mathematical optimization problem:
min||h|| 0 (4a)
s.t.|c(ω)h-1|＜δ，ω∈[0，π]-Ω 0 (4b)
Figure BSA0000175606180000032
formula (4) is p 0 Solving an objective function under norm, wherein the optimization problem is a complex NP-hard problem, solving the formula (4) by adopting an IROMP algorithm to obtain a sparse filter coefficient vector h = [ h (0) h (1) … h (N) ]] T ；
And (III) assuming that the coding bit number is B, carrying out CSD coding on each coefficient in the original coefficient vector h of the FIR filter. If the CSD coding of the nth coefficient h (n) is expressed as
Figure BSA0000175606180000033
Then h (n) and h q (n) has the following transformation relationship:
Figure BSA0000175606180000034
wherein [. ]]Represents a rounding operation; determining quantization word length B, performing CSD coding on a sparse FIR notch filter coefficient vector h, and if CSD coding of the nth coefficient h (n) is expressed as
Figure BSA0000175606180000035
Then h (n) and h q (n) the transformation relation is as follows
Figure BSA0000175606180000036
Wherein [. Cndot ] represents a rounding operation, thereby obtaining an nxb binary coding coefficient matrix:
Figure BSA0000175606180000037
in the coefficient
Figure BSA0000175606180000038
In, if
Figure BSA0000175606180000039
And
Figure BSA00001756061800000310
is a non-zero bit, then
Figure BSA00001756061800000311
The polynomial corresponding to (j 0 s in the middle) is called j order 2 item subformula, if the non-zero digit of the head and tail bits of the j order 2 item subformula
Figure BSA0000175606180000041
And
Figure BSA0000175606180000042
the same sign, called j order even 2-term subformulae, such as 101,
Figure BSA0000175606180000043
if of different sign, it is called the odd-2 sub-formula of j order, e.g.
Figure BSA0000175606180000044
If a quantized coefficient only contains an independent non-zero number and cannot form any type of 2-term subformula, such as {00100000}, the quantized coefficient is called a soliton, and if 2-term subformulae with the same order exist inside the coefficient or among the coefficients, the quantized coefficient is called a common subformula. After the input signal is multiplied by a sub-formula with j order 2, the result can be directly used by sub-formulas with other positions and the same order j as the sub-formula, which is the basic principle of the CSE technology, and the number of adders can be reduced by the CSE technology. For example, assume a 12-bit quantized word length coefficient
Figure BSA0000175606180000045
If the coefficient h is directly realized q (1) As shown in FIG. 2 (a), 5 adders are required for h q (1) All the 1 st order 2 sub-units 101 perform CSE, and as shown in FIG. 2 (b), the number of adders is reduced to 3; CSE techniques can also be applied between different coefficients, if
Figure BSA0000175606180000046
As shown in fig. 3, all the common 2-term sub-formulas 101 are extracted and the coefficients h are directly realized q (1) And h q (2) In contrast, the number of adders is reduced from 11 to 6.
To count h q Introducing B-2 NxB position matrixes CSP into all j-order 2-item sub-type appearing positions (j) ，j＝1，2，…B-2，CSP (j) Is marked with h q All of them have the form b i 0 … 0b i-j-1 (j 0 s in the middle) of the position where the j-order 2-item sub-formula appears, and the value of the element in the matrix is 0,1 or
Figure BSA0000175606180000047
If CSP (j) (n, i) =1 or
Figure BSA0000175606180000048
Then represents h q (n) of
Figure BSA0000175606180000049
And
Figure BSA00001756061800000410
form j even (odd) 2-term sub-type if CSP (j) (n, i) =0, then it means in h q (n) there is no j order 2 sub-formula at the corresponding position, we
Figure BSA00001756061800000411
And
Figure BSA00001756061800000412
for example, when N =2,b =12,h q The corresponding 1 st, 2 nd, 9 th and 10 th order 2-term position matrices are as follows:
Figure BSA00001756061800000413
Figure BSA00001756061800000414
Figure BSA00001756061800000415
Figure BSA00001756061800000416
to count h q Introducing an NxB position matrix CSP at the positions where all solitons appear (0) If CSP (0) (n, i) =1 or
Figure BSA0000175606180000051
Represents the nth quantization factor h q (n) there is an independent non-zero digit 1 or
Figure BSA0000175606180000052
(IV) under the design parameters of a given notch filter, a desired coefficient set can be obtained more quickly by reasonably selecting 2-term sub-formula or soliton for reconstruction, and for this purpose, the invention introduces the sensitivity function of the soliton and the j-order 2-term sub-formula so as to ensure that
Figure BSA0000175606180000053
Expressed as a set of coefficients h q The nth coefficient, the soliton on the ith bit,
Figure BSA0000175606180000054
j =1,2, … B-2 for h q (n) of
Figure BSA0000175606180000055
And
Figure BSA0000175606180000056
formed j order 2-term sub-formula, the sensitivity function
Figure BSA0000175606180000057
Is shown as
Figure BSA0000175606180000058
Where L is the number of sample points, H q (ω) is the set of quantization coefficients h q The frequency response of the corresponding filter is such that,
Figure BSA0000175606180000059
is to use j order 2 item sub-formula or soliton as slave coefficient set h q Wherein the sensitivity function represents a sub-formula of j order 2 term
Figure BSA00001756061800000510
Or solitons
Figure BSA00001756061800000511
The resulting frequency response error is removed.
Calculating a set of coefficients h according to equation (8) q The sensitivity of each j-order 2-term sub-formula and soliton is put into B-1 NxB sensitivity matrixes SEN (j) J =0,1, … B-2, let h r A reconstruction coefficient matrix representing dimension NxB is initialized to be a full 0 matrix, and in each iteration, reasonable j-order 2-item subforms or solitons are sequentially selected according to the sensitivity and copied to a reconstruction coefficient matrix h r In (1), up to h r Corresponding filter passband ripple delta r And depth d of trap r Meets the requirements.
The invention has the following beneficial effects:
1. the invention provides a design method of a sparse, efficient and low-complexity linear phase Finite Impulse Response (FIR) notch filter for the first time.
2. The invention can design the linear phase FIR trapped wave filter with low nonzero tap number, and the sparsity of the filter can greatly reduce the number of the adder multipliers used for realizing the filter, thereby improving the operation speed, reducing the operation error and the hardware realization complexity and further saving the hardware resources.
3. Simulation results show that under the requirement of the same design index, the number of adders of the linear phase FIR notch filter realized by the invention is more than 51 percent less than that of the adders of the existing similar filters at home and abroad.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention;
FIG. 2 is a graph of the frequency response of a FIR notch filter implementation design;
fig. 3 is a graph of the frequency response of a notch filter obtained in accordance with the present invention.
Detailed Description
Example 1:
the design method of the low-complexity sparse FIR notch filter provided by the invention comprises the following specific steps:
1, obtaining an FIR trapped wave original filter coefficient meeting the design requirement of frequency domain performance by adopting a sparse FIR trapped wave filter design algorithm;
2, converting the design problem of the sparse FIR notch filter into a mathematical optimization problem, and solving by adopting an IROMP algorithm to obtain an FIR notch original filter coefficient;
3, CSD coding is carried out on the coefficients of the FIR trapped wave original filter;
and 4, introducing a sensitivity function of 2-term sub-type sensitivity and solitons, and sequentially selecting reasonable 2-term sub-type or solitons according to the sensitivity to re-synthesize a filter coefficient set.
In order to verify the effectiveness of the filter design method, computer simulation was performed on the method.
The design requirement is as follows: the use of the literature: (CHEN Jianjiajia, TAN Jinghong, chang C, et α l.A new cost-aware sensitivity-drive algorithm for the design of FIR filters [ J]IEEE Transactions on Circuits and Systems-I: regular Papers,2017, 64 (6): 1588-1598.), a notch frequency point set {0.1 pi, 0.25 pi, 0.76 pi }, a notch bandwidth delta omega =0.061 pi, a pass band ripple delta = -0.95dB and a notch depth d = -60dB, the invention designs a sparse FIR notch filter by using an IROMP algorithm, and in the process of obtaining a sparse filter coefficient vector h, a filter reconstruction coefficient matrix h is obtained by adopting CSD coding and common elimination thought iterative computation according to the calculation of weight values r 。
The method comprises the following steps: according to the design parameter requirement of the FIR notch filter, the passband attenuation alpha and the passband ripple delta can be mutually converted, and the relationship between the passband attenuation alpha and the passband ripple delta is as follows:
Figure BSA0000175606180000071
frequency response H of an ideal FIR notch filter d (e jω ) The following were used:
Figure BSA0000175606180000072
wherein
Figure BSA0000175606180000073
Step two: obtaining an FIR notch original filter coefficient h (N) meeting the frequency domain performance design requirement by utilizing a sparse filter design algorithm, wherein N is more than or equal to 0 and less than or equal to N, and converting the design problem of the sparse FIR notch filter into the following mathematical optimization problem:
min ||h|| 0 (3a)
s.t.|c(ω)h-1|＜δ，ω∈[0，π]-Ω 0 (3b)
Figure BSA0000175606180000074
the invention adopts IROMP algorithm to solve the formula, and obtains the sparse filter coefficient vector h = [ h (0) h (1) … h (N) by calculation] T ；
Step three: under the condition of determining the quantization word length B =14, 16 and 18, CSD coding is carried out on each coefficient in the sparse filter coefficient vector h obtained by the solution in the step two, if CSD coding of the nth coefficient h (n) is expressed as
Figure BSA0000175606180000075
Then h (n) and h q The transformation relation of (n) is as follows
Figure BSA0000175606180000076
Wherein [. Cndot. ] represents a rounding operation, thereby obtaining an nxb binary coded coefficient matrix:
Figure BSA0000175606180000081
step four: by reasonably selecting the 2-item sub-type or soliton for reconstruction, a desired coefficient set can be obtained more quickly, and therefore the invention introduces the soliton and the sensitivity function of the j-order 2-item sub-type, so that
Figure BSA0000175606180000082
Expressed as a set of coefficients h q The nth coefficient, the soliton at the ith bit,
Figure BSA0000175606180000083
j =1,2, … B-2 for h q (n) of
Figure BSA0000175606180000084
And
Figure BSA0000175606180000085
formed j order 2-term sub-formula, the sensitivity function
Figure BSA0000175606180000086
Is shown as
Figure BSA0000175606180000087
Where L is the number of sample points, H q (ω) is the set of quantization coefficients h q The frequency response of the corresponding filter is such that,
Figure BSA0000175606180000088
is to select the j order 2 item sub-formula or soliton from the coefficient set h q Wherein the sensitivity function represents a sub-formula of j order 2
Figure BSA0000175606180000089
Or solitons
Figure BSA00001756061800000810
The resulting frequency response error is removed.
Calculating a coefficient set h according to equation (6) q In each j-order 2-item equation and solitonSensitivity, corresponding to the result, into B-1 NxB sensitivity matrices SEN (j) J =0,1, … B-2, let h r A reconstruction coefficient matrix representing dimension NxB is initialized to be a full 0 matrix, and in each iteration, reasonable j-order 2-item subforms or solitons are sequentially selected according to the sensitivity and copied to a reconstruction coefficient matrix h r In (1), up to h r Corresponding filter passband ripple delta r And depth d of trap r The requirement is met, and the coefficient reconstruction process is as follows:
(1) Using sensitivity matrix SEN (j) J =0,1, … B-2, finding the 2-term sub-formula or soliton with the maximum sensitivity and taking the non-zero number of the corresponding bit from h q Copy to h r In (h) q The corresponding position is zero;
(2) Judgment h r Corresponding filter passband ripple delta r And depth d of trap r Whether the requirements are met. And if so, jumping to the step 5. If not, continuing the next step;
(3) If the copy is to h in the step 1 r If the bit in the sequence is a soliton, continuing to step 4; if the copy is to h in the step 1 r If the bit in the sequence is k order 2 sub-formula, the corresponding k order 2 sub-formula is sequentially changed from h to low according to the sensitivity q Copy to h r ，h q Corresponding position zero until h r And if the requirement is met, jumping to the step 5. If h is q All k order 2 sub-formulas are copied to h r After neutralization, h r If the requirements are not met, continuing to step 4;
(4) Update coefficient set h q Corresponding position matrix CSP (j) And sensitivity matrix SEN (j) J =0,1, … B-2, jump back to step 1;
(5) Since the k order 2 item is sub-copied to h in step 3 r Several non-zero digits of very low sensitivity are introduced, so from h r The elimination of these redundant non-zero numbers can further reduce the number of adders. H is calculated according to equation (7) r Sensitivity of all non-zero numbers, at h r On the premise of meeting design requirements, from h r In turn, the least sensitive non-zero numbers are removed.
The order of the filter, the number of non-zero coefficients, the number of non-zero numbers in the reconstructed set of coefficients (# NZ) and the number of adders required for the implementation, which are designed by the two algorithms in the present invention and the literature, respectively, for quantization word lengths B =14, 16 and 18, are given in table 1.
TABLE 1
Figure BSA0000175606180000091

Claims (1)

1. A design method of low-complexity sparse FIR notch filter is characterized in that the method can design the notch filter with low nonzero tap number, and the number of adder multipliers used for realizing the method is reduced, thereby improving the operation speed, reducing operation error and reducing energy consumption, and the method comprises the following specific steps:
1, according to the design parameter requirement, comprising a notch frequency set
Figure FSA0000175606170000011
Notch depth d (dB), notch bandwidth delta omega, pass band attenuation alpha (dB) (or pass band ripple delta), determining the correlation between pass band attenuation alpha and pass band ripple delta, and frequency response H (e) of FIR notch filter jω ) It should satisfy:
Figure FSA0000175606170000012
wherein
Figure FSA0000175606170000013
Obtaining an FIR trapped wave original filter coefficient h (N) meeting the design requirement of frequency domain performance by utilizing a sparse filter design algorithm, wherein N is more than or equal to 0 and less than or equal to N, and converting the design problem of the sparse FIR trapped wave filter into the following mathematical optimization problem:
min ||h|| 0 (4a)
s.t. |c(ω)h-1|＜δ，ω∈[0，π]-Ω 0 (4b)
Figure FSA0000175606170000014
the invention adopts IROMP algorithm to solve the formula to obtain a sparse filter coefficient vector h = [ h (0) h (1) … h (N)] T ；
Determining a quantization word length B, and carrying out CSD coding on the coefficient vector h of the sparse FIR notch filter to obtain an NxB binary coding coefficient matrix:
Figure FSA0000175606170000015
to count h q Introducing B-2 NxB position matrixes CSP (j) ，j＝1，2，…B-2，CSP (j) Is marked with h q All of which have the form b i 0…0b i-j-1 (j 0 s in the middle) of the position where the j-order 2-item sub-formula appears, and the value of the element in the matrix is 0,1 or
Figure FSA0000175606170000021
If CSP (j) (n, i) =1 or
Figure FSA0000175606170000022
Then represents h q (n) of
Figure FSA0000175606170000023
And
Figure FSA0000175606170000024
form j even (odd) 2-term sub-type if CSP (j) (n, i) =0, then it means in h q (n) there is no j order 2 item sub-formula at the corresponding position, which is statistic h q Introducing an NxB position matrix for the positions where all solitons appearCSP (0) If CSP (0) (n, i) =1 or
Figure FSA0000175606170000025
Represents the nth quantization factor h q (n) there is an independent non-zero digit 1 or
Figure FSA0000175606170000026
4, under the design parameters of a given notch filter, a desired coefficient set can be obtained more quickly by reasonably selecting 2-term sub-formula or soliton for reconstruction, and for this purpose, the invention introduces the sensitivity function of the soliton and the j-order 2-term sub-formula so as to ensure that
Figure FSA0000175606170000027
Expressed as a set of coefficients h q The nth coefficient, the soliton on the ith bit,
Figure FSA0000175606170000028
j =1,2, … B-2 for h q (n) of
Figure FSA0000175606170000029
And
Figure FSA00001756061700000210
a constructed j-order 2-term sub-formula, then a sensitivity function
Figure FSA00001756061700000211
Is shown as
Figure FSA00001756061700000212
Where L is the number of sample points, H q (ω) is the set of quantization coefficients h q The frequency response of the corresponding filter is such that,
Figure FSA00001756061700000213
is to select the j order 2 item sub-formula or soliton from the coefficient set h q Wherein the sensitivity function represents a sub-formula of j order 2 term
Figure FSA00001756061700000214
Or solitons
Figure FSA00001756061700000215
Removing the frequency response error caused by the frequency response error, and calculating a coefficient set h according to the formula (8) q The sensitivity of each j-order 2-term sub-formula and soliton is put into B-1 NxB sensitivity matrixes SEN (j) J =0,1, … B-2, let h r A reconstruction coefficient matrix representing dimension NxB is initialized to be a full 0 matrix, and in each iteration, reasonable j-order 2-item subforms or solitons are sequentially selected according to the sensitivity and copied to a reconstruction coefficient matrix h r In (c) up to h r Corresponding filter passband ripple delta r And depth d of trap r Meets the requirements.
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