CN107241081B - Design method of sparse FIR prototype filter of cosine modulation filter bank - Google Patents

Design method of sparse FIR prototype filter of cosine modulation filter bank Download PDF

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CN107241081B
CN107241081B CN201710429939.3A CN201710429939A CN107241081B CN 107241081 B CN107241081 B CN 107241081B CN 201710429939 A CN201710429939 A CN 201710429939A CN 107241081 B CN107241081 B CN 107241081B
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徐微
李怡
缪竟鸿
李安宇
张瑞华
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Tianjin Polytechnic University
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Abstract

The invention discloses a design method of a sparse linear phase FIR prototype filter of a cosine modulation filter bank. The method comprises the following specific steps: 1, initializing design parameters of a sparse linear phase FIR prototype filter of a cosine modulation filter bank; and 2, iteratively calculating the sparse linear phase FIR prototype filter of the cosine modulation filter bank meeting the complete reconstruction condition, wherein the sparse linear phase FIR prototype filter comprises the determination of the number, the position and the value of the coefficient of the nonzero tap of the unit impulse response. The invention can design a prototype filter with low nonzero tap number, and the sparsity of the filter can reduce the number of the adder multipliers used for realizing the filter, thereby improving the operation speed, reducing the operation error and the energy consumption and further reducing the production cost.

Description

Design method of sparse FIR prototype filter of cosine modulation filter bank
Technical Field
The invention belongs to the technical field of digital signal processing, and provides a design method of a linear phase FIR (finite impulse response) prototype filter of a sparse and efficient cosine modulation filter bank.
Background
The theory and design of multi-rate filter banks is of great interest because of their wide application in the fields of communications, speech and image coding/compression, system recognition, fast computing, etc. And the filter bank can be generally divided into a DFT filter bank and a cosine modulation filter bank. The cosine modulation filter bank is obtained by optimally designing a low-pass prototype filter and by fast Discrete Cosine Transform (DCT), and has wide application in the fields of signal processing, communication, biomedical engineering and the like because of the advantages of low computational complexity, simple design process and the like. A cosine modulated filter bank with a sparse linear phase FIR (finite impulse response) prototype filter is a filter bank in which the filter coefficients of each channel 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 design method of the cosine modulation filter bank is mainly divided into the steps of respectively designing an analysis filter bank and a comprehensive filter bank in the filter bank and independently designing a low-pass prototype filter and then carrying out cosine modulation to obtain the filter bank, and in the proposed design method of the cosine filter bank, the design methods of P.P.Vaidyanathan and R.D.Koilpilai are classic, and the filter bank which meets the minimum standard of amplitude distortion and aliasing distortion is estimated through an analysis method.
Disclosure of Invention
The invention aims to design a linear phase FIR prototype filter of a cosine modulation filter bank for realizing small ripples, low tap number, low amplitude distortion and aliasing distortion, and provides a brand new design method, namely a method for designing the linear phase FIR prototype filter of the cosine modulation filter bank with sparseness and high efficiency.
The design method of the sparse linear phase FIR prototype filter of the cosine modulation filter bank provided by the invention comprises the following specific steps:
1, initializing design parameters of a sparse linear phase FIR prototype filter of a cosine modulation filter bank;
and 2, iteratively calculating the sparse linear phase FIR prototype filter of the cosine modulation filter bank meeting the complete reconstruction condition, wherein the sparse linear phase FIR prototype filter comprises the determination of the number, the position and the value of the coefficient of the nonzero tap of the unit impulse response.
(type II linear phase FIR filter is exemplified below):
constructing initial parameters according to design requirements:
according to the invention, the sampling numbers L respectively corresponding to the pass band, the transition band and the stop band are selected according to the channel number M of the cosine modulation filter bankp，Lt，LsSum ripple valuep，t，sDetermining an initial order N of the linear phase FIR prototype filter, wherein a tap coefficient of the linear phase FIR prototype filter is expressed by a vector h as:
h＝2[h1,h2,…,hm…,hN/2]T(1)
wherein h ism(1 m is less than or equal to N/2) represents the m-th tap coefficient of the FIR prototype filter; the design problem of a sparse linear phase FIR prototype filter of a cosine modulation filter bank is converted into the following mathematical optimization problem:
Figure BDA0001317133980000021
s.t.|Bh-d|≤e (2b)
Figure BDA0001317133980000022
wherein h does not calculation0Represents 0-norm operation, i.e. represents the number of nonzero taps in the tap coefficient vector; the combined equations (2a) - (2c) of "min" and "s.t." represent solving for | | h | | computational complexity that satisfies the requirements of (2b) and (2c)0Minimum value of (d); the sampling matrix B is expressed as B ═ Bp；Bt；Bs]In which B isp、BtAnd BsThe sampling matrices, representing pass band, transition band and stop band, respectively, are represented as
Figure BDA0001317133980000023
Figure BDA0001317133980000024
Figure BDA0001317133980000025
Wherein
Figure BDA0001317133980000031
To represent
Figure BDA0001317133980000032
Dimension of row vector; (L)p+Lt+Ls) The x 1-dimensional vector d is the discretized ideal frequency response and is expressed as:
Figure BDA0001317133980000033
where f (ω) is the frequency response function to which the transition band is to be approximated, expressed as:
Figure BDA0001317133980000034
wherein ω is0α pi/2M (0 ≦ α ≦ 1), equation (7) satisfies the complete reconstruction condition,
Figure BDA00013171339800000310
frequency sampling points representing a transition band; the error vector e is (L)p+Lt+Ls) A x 1-dimensional column vector, expressed as:
e＝[p…p t…t s…s]T, (8)
(II) setting (L)p+Lt+Ls) The initial value of the x 1-dimensional weight vector is w(1)＝[1,1,…,1]TIn the invention, in the kth iteration (k is more than or equal to 1 and less than or equal to N/2), the column vector of the matrix B is normalized:
Figure BDA0001317133980000035
wherein
Figure BDA0001317133980000036
The following problem is solved using the OMP algorithm:
Figure BDA0001317133980000037
s.t.||h(k)||0≤k (10b)
and calculating (L)p+Lt+Ls) X 1-dimensional residual vector r(k)Expressed as:
r(k)＝Φ(k)s(k)-d (11)
where k x 1 dimensions of s(k)As a result of the operation of equation (10),
Figure BDA0001317133980000038
representing OMP algorithm from B(k)Set of column vectors, set Λ selected from(k)＝{n1,n2,…,nkDenotes an index set of non-zero tap coefficients.
(III) the invention utilizes the index set Lambda of the obtained nonzero tap coefficient(k)Solving the following linear programming problem:
Figure BDA0001317133980000039
s.t.|B(k)h(k)-d|≤e+μ·1L×1(11b)
Figure BDA0001317133980000041
Figure BDA0001317133980000042
judging whether mu is smaller than zero, if so, updating the weight vector w, wherein the updating formula is as follows:
Figure BDA0001317133980000043
wherein w(k+1)(l) Representing a new weight vector w(k+1)Value of (1), rl (k)Representing a residual vector r(k)A value of (1); new weight vector w(k+1)Carrying out cyclic calculation in the step 2; stopping iterative operation if mu is less than or equal to zero, and calculating
Figure BDA0001317133980000044
The final sparse linear phase FIR prototype filter is obtained.
The invention has the following beneficial effects:
1. the invention provides a design method of a linear phase FIR prototype filter of a sparse and efficient cosine modulation filter bank for the first time.
2. The invention can design a prototype filter with low nonzero tap number, and the sparsity of the filter can reduce the number of the adder multipliers used for realizing the filter, thereby improving the operation speed, reducing the operation error and the energy consumption and further reducing the production cost.
3. Simulation results show that under the requirement of the same design index, the number of the nonzero tap coefficients of the filter is less than that of the optimal similar filter at home and abroad by more than 35%.
Drawings
FIG. 1 is a flow chart of a sparse linear phase FIR prototype filter design method implementing a cosine modulated filter bank of the present invention;
FIG. 2 according to function eam(omega) calculating to obtain an amplitude distortion figure of an approximately completely reconstructed cosine modulation filter bank;
FIG. 3 is a graph according to function ea(omega) calculating an aliasing distortion map of the approximately completely reconstructed cosine modulation filter bank;
fig. 4 is a frequency domain response plot of the sparse linear phase FIR prototype filter of the cosine modulated filter bank of table-2.
Detailed Description
Example 1:
the method for designing the sparse linear phase FIR prototype filter of the cosine modulation filter bank comprises the following steps:
1, initializing design parameters of a sparse linear phase FIR prototype filter of a cosine modulation filter bank;
and 2, iteratively calculating the sparse linear phase FIR prototype filter of the cosine modulation filter bank meeting the complete reconstruction condition, wherein the sparse linear phase FIR prototype filter comprises the determination of the number, the position and the value of the coefficient of the nonzero tap of the unit impulse response.
In order to verify the effectiveness of the filter bank design method, computer simulation was performed on the method.
The design requirement is as follows: the use of the literature: (F.Tan, et al.: optical design of cosine modulated filter banks using quality-corrected discrete optimization algorithm, "4 th International construction on Image and Signal Processing, vol.5, pp.2280-2284,2011.) (F.Tan, et al.: Quantum particle swarm optimization algorithm-based cosine modulated Filter Bank optimization design," fourth International Image and Signal Processing conference, prototype.5, pp.2280-2284,2011.), wherein the number of filter Bank channels M is 16, the number of Filter initial coefficients N is 256, and the sampling numbers L corresponding to the pass band, transition band and stop band are designedp＝4，Lt＝7，Ls94, ripple valuep＝t＝s＝1·10-8Carry-over into the computation. The invention designs a prototype filter of a cosine filter bank by using an IROMP algorithm, and selects a column vector B corresponding to a nonzero coefficient position in an obtained variable set B according to the calculation of a weight valuejAnd carrying the filter coefficients obtained by iterative computation into an IROMP algorithm.
The method comprises the following steps: according to the design parameter requirement of a sparse linear phase FIR prototype filter of a cosine modulation filter bank, substituting each design parameter into an initialization condition to obtain a problem to be solved:
Figure BDA0001317133980000051
s.t.|Bh-d|≤e (2b)
Figure BDA0001317133980000052
the designed pass band, transition band and stop band are respectively corresponding to the sampling number Lp＝4，Lt＝7，LsSubstituting equation (6) with 94 yields the discretized ideal frequency response d in dimension (4+7+94) × 1, expressed as:
Figure BDA0001317133980000053
wherein the values of f (ω) are shown in Table-1:
TABLE-1
Figure BDA0001317133980000061
Will design the ripple valuep＝t＝s＝1·10-8Substituting into the formula (7), a (4+7+94) × 1-dimensional error column vector e is obtained, each value in the vector being 1 · 10-8Setting the initial value of the (4+7+94) × 1-dimensional weight vector as w(1)＝[1,1,…,1]T；
Step two: in the k (k is more than or equal to 1 and less than or equal to N/2) iteration, the column vector of the matrix B is normalized:
Figure BDA0001317133980000062
the following problem is then solved using the OMP algorithm in appendix 1:
Figure BDA0001317133980000063
s.t.||h(k)||0≤k (10b)
index set Lambda of non-zero tap coefficient is obtained through calculation(k)；
Step three: index set Lambda of nonzero tap coefficient obtained by using step two for solving(k)The method is carried into the solution of the following linear programming problem:
Figure BDA0001317133980000064
s.t.|B(k)h(k)-d|≤e+μ·1L×1(11b)
Figure BDA0001317133980000065
Figure BDA0001317133980000066
further obtaining the tap coefficient of the sparse linear phase FIR prototype filter of the cosine modulation filter bank
Figure BDA0001317133980000071
The values are given in Table-2.
TABLE-2
Figure BDA0001317133980000072
Since the impulse response of a type II FIR filter is symmetrical, the filter tap coefficients found by the present invention
Figure BDA0001317133980000073
Half of the tap coefficients of the prototype filter to be solved are symmetrical and the other half is equal, namely the total tap coefficients of the sparse FIR prototype filter to be solved are expressed as:
Figure BDA0001317133980000081
wherein
Figure BDA0001317133980000082
Represents the vector
Figure BDA0001317133980000083
And turning over the upper part and the lower part.
And selecting the variable matrix B by using the algorithm. For a 256-order filter, the algorithm selects all cases that result in half of the tap coefficients (the other half of the coefficients are symmetric with respect to the tap coefficients), i.e., the algorithm selects the tap coefficients for half of the tap coefficients (the other half of the coefficients are symmetric with respect to the tap coefficients), i.e., the filter selects the tap coefficients
Figure BDA0001317133980000084
Will be provided with
Figure BDA0001317133980000085
The total tap coefficient of the prototype filter is obtained by substituting the total tap coefficient expression of the filter
Figure BDA0001317133980000086
The finally obtained sparse linear phase FIR prototype filter of the cosine modulation filter bank is a filter with the nonzero coefficient of 166 orders, and compared with the particle swarm optimization algorithm, the sparse linear phase FIR prototype filter saves 35.6 percent.
Calculating and analyzing filter bank h by using the tap coefficient of the prototype filter obtained in the step threem(n) and a synthesis filter bank gm(n)：
Figure BDA0001317133980000087
Figure BDA0001317133980000088
Wherein M is more than or equal to 1 and less than or equal to M, and calculating the amplitude distortion value e of the cosine modulation filter bankamAnd an aliasing distortion value eaThe calculation formula is expressed as:
eam(ω)＝1-|A0(ejω)
Figure BDA0001317133980000089
Figure BDA00013171339800000810
Figure BDA00013171339800000811
wherein A is0(ejω) And Al(ejω) Expressed as:
Figure BDA00013171339800000812
Figure BDA00013171339800000813
Hk(ejω) For analyzing the frequency domain response of the filter bank, Gk(ejω) The frequency domain response of the synthesis filter bank.
In table-3, the order of the FIR notch filter, the number of non-zero tap weights, the amplitude distortion and aliasing distortion of the filter bank, which are obtained by the algorithm of the present invention and the particle swarm optimization algorithm, are respectively compared, and as shown in table-3, the amplitude distortion value e is obtainedamAnd an aliasing distortion value eaSimilarly, the prototype filter order of the invention is significantly less than the particle swarm optimization algorithm.
TABLE-3
Figure BDA0001317133980000091
In FIG. 2, according to function eam(omega) calculating to obtain an amplitude distortion figure of an approximately completely reconstructed cosine modulation filter bank, wherein the maximum value of the amplitude distortion figure is equal to the amplitude distortion value e obtained by the algorithm of the invention in the table-3amThe correspondence is equal; in FIG. 3, according to function ea(omega) calculating to obtain an aliasing distortion figure of the approximately completely reconstructed cosine modulation filter bank, wherein the maximum value of the aliasing distortion figure is equal to an aliasing distortion value e obtained by the algorithm of the invention in the table-3aThe correspondence is equal; fig. 4 is a plot of the frequency domain response of the sparse linear phase FIR prototype filter of the cosine modulated filter bank of table-2.
Appendix 1
Formula (13) OMP Algorithm calculation Process
Using OMP algorithm calculation formula (10), M × N matrix B is the sensing matrix of OMP algorithm, N × 1 d is the observed value, riRepresenting the residual, t represents the number of iterations,
Figure BDA0001317133980000092
represents the empty set, ΛtSet of indices, λ, representing t iterationstDenotes the index found in the t-th iteration, ajJ-column, B, representing matrix BETAtIndicating by indexΛtSelected set of columns, θ, of matrix BtIs a column vector of t × 1, the symbol @ represents a union operation,<rt-1,aj>the method is shown in the step of calculating the j-th column vector inner product of the residual error and the matrix BETA before the t-th iteration updating. The method comprises the following concrete steps:
1. initializing the residual to be equal to
r0＝d； (1)
2. By the formula
Figure BDA0001317133980000101
Calculating to obtain an index lambdat；
3. To atAnd BtUnion operation, order
Λt＝Λt-1∪{λt}，
Figure BDA0001317133980000102
4. Finding new observed value d ═ BtθtLeast squares solution of (c):
Figure BDA0001317133980000103
5. least squares solution obtained by (4)
Figure BDA0001317133980000104
Updating residual riThe calculation is as follows:
Figure BDA0001317133980000105
6. if t is smaller than a preset value, returning to the step (2), otherwise, stopping iteration and entering the step 7;
7. reconstructing the resultant
Figure BDA0001317133980000106
At ΛtWith non-zero terms having values obtained in the last iteration
Figure BDA0001317133980000107

Claims (3)

1. A method for designing a sparse linear phase FIR prototype filter of a cosine modulated filter bank, characterized in that the method comprises the following steps:
1, selecting sampling numbers L respectively corresponding to a pass band, a transition band and a stop band according to the number M of channels of a cosine modulation filter bankp，Lt，LsSum ripple valuep，t，sDetermining an initial order N of the linear phase FIR prototype filter, wherein a tap coefficient of the linear phase FIR prototype filter is expressed by a vector h as:
h＝2[h1,h2,…,hm…,hN/2]T(1)
wherein h ism(1 m is less than or equal to N/2) represents the m-th tap coefficient of the FIR prototype filter; the design problem of a sparse linear phase FIR prototype filter of a cosine modulation filter bank is converted into the following mathematical optimization problem:
Figure FDA0002618293810000011
s.t.|Bh-d|≤e (2b)
Figure FDA0002618293810000012
wherein h does not calculation0Represents 0-norm operation, i.e. represents the number of nonzero taps in the tap coefficient vector; the combined equations (2a) - (2c) of "min" and "s.t." represent solving for | | h | | computational complexity that satisfies the requirements of (2b) and (2c)0Minimum value of (d); the sampling matrix B is expressed as B ═ Bp；Bt；Bs]In which B isp、BtAnd BsThe sampling matrices, representing pass band, transition band and stop band, respectively, are represented as
Figure FDA0002618293810000013
Figure FDA0002618293810000014
Figure FDA0002618293810000015
Wherein
Figure FDA0002618293810000016
To represent
Figure FDA0002618293810000017
Dimension of row vector; wherein ω is0＝απ/2M(0≤α≤1)，(Lp+Lt+Ls) The x 1-dimensional vector d is the discretized ideal frequency response and is expressed as:
Figure FDA0002618293810000021
where f (ω) is the frequency response function to which the transition band is to be approximated,
Figure FDA0002618293810000022
frequency sampling points representing a transition band; the error vector e is (L)p+Lt+Ls) A x 1-dimensional column vector, expressed as:
e＝[p…p t…t s…s]T, (7)
2 nd, setting (L)p+Lt+Ls) The initial value of the x 1-dimensional weight vector is w(1)＝[1,1,…,1]TIn the k (k is more than or equal to 1 and less than or equal to N/2) iteration, the column vector of the matrix B is normalized:
Figure FDA0002618293810000023
wherein
Figure FDA0002618293810000024
The following problem is solved using the OMP algorithm:
Figure FDA0002618293810000025
s.t.||h(k)||0≤k (9b)
and calculating (L)p+Lt+Ls) X 1-dimensional residual vector r(k)Expressed as:
r(k)＝Φ(k)s(k)-d (10)
where k x 1 dimensions of s(k)As a result of the joint operation of equations (9a) and (9b),
Figure FDA0002618293810000026
representing OMP algorithm from B(k)Set of column vectors, set Λ selected from(k)＝{n1,n2,…,nkDenotes an index set of non-zero tap coefficients;
3, index set Lambda of obtained non-zero tap coefficient(k)Solving the following linear programming problem:
Figure FDA0002618293810000027
s.t.|B(k)h(k)-d|≤e+μ·1L×1(11b)
Figure FDA0002618293810000028
Figure FDA0002618293810000029
judging whether mu is smaller than zero, if so, updating the weight vector w,new weight vector w(k+1)Carrying out cyclic calculation in the step 2; if mu is less than or equal to zero, stopping iterative operation and calculating
Figure FDA0002618293810000031
The final sparse linear phase FIR prototype filter is obtained.
2. The method of claim 1, wherein the frequency response function f (ω) to be approximated by the transition band in equation (6) is expressed as:
Figure FDA0002618293810000032
equation (12) satisfies the complete reconstruction condition.
3. The method according to claim 1, wherein the formula for updating the weight vector w in step 3 is expressed as:
Figure FDA0002618293810000033
wherein w(k+1)(l) Representing a new weight vector w(k+1)Value of (1), rl (k)Representing a residual vector r(k)A value of (1).
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