CN107241081A

CN107241081A - The design method of the sparse FIR ptototype filters of cosine modulated filters group

Info

Publication number: CN107241081A
Application number: CN201710429939.3A
Authority: CN
Inventors: 徐微; 李怡; 缪竟鸿; 李安宇; 张瑞华
Original assignee: Tianjin Polytechnic University
Current assignee: Tianjin Polytechnic University
Priority date: 2017-06-09
Filing date: 2017-06-09
Publication date: 2017-10-10
Anticipated expiration: 2037-06-09
Also published as: CN107241081B

Abstract

The invention discloses a kind of design method of the sparse linear Phase FIR ptototype filter of cosine modulated filters group.Comprise the following steps that：1st, the initialization of the sparse linear Phase FIR ptototype filter design parameter of cosine modulated filters group；2nd, iterative calculation meets the sparse linear Phase FIR ptototype filter for the cosine modulated filters group for rebuilding condition completely, includes non-zero tap number of coefficients, position and the determination for having factor v of unit impulse response.The present invention can design the ptototype filter of low non-zero tap number, and the openness of wave filter achieves adder number of multipliers reduction used, so as to improve its arithmetic speed, reduce arithmetic eror and reduction energy consumption, and then reduces production cost.

Description

The design method of the sparse FIR ptototype filters of cosine modulated filters group

Technical field

The invention belongs to digital signal processing technique field, there is provided sparse, the efficient cosine modulated filters group of one kind Linear phase fir (finite impulse response (FIR)) ptototype filter design method.

Background technology

Multirate filter group theory and design because its communication, voice and Image Coding/compression, System Discrimination, The fields such as quick calculating are widely used and received much concern.And the form of the composition of wave filter group generally can be divided into DFT wave filters Group and cosine modulated filters group.Cosine modulated filters group is passed through by optimizing design to lowpass prototype filter Fast discrete cosine transform (DCT) obtains analysis and synthesis wave filter group, because it has computation complexity low and design process letter Single the advantages of, it is widely used in fields such as signal transacting, communication and biomedical engineerings.With sparse linear phase fir The cosine modulated filters group of (finite impulse response (FIR)) ptototype filter is that the filter coefficient of each passage has sparse characteristic The wave filter group of (number of non-zero tap coefficient is less than filter order).Sparse wave filter its realize used addition and multiply Musical instruments used in a Buddhist or Taoist mass number is far fewer than the similar wave filter suitable with its filter effect, and therefore, sparse wave filter has arithmetic speed height, fortune Calculate that error is small and low power consumption and other advantages.

The design method for cosine modulated filters group is broadly divided into the analysis separately designed in wave filter group and filtered at present After ripple device group and synthesis filter group and individually designed lowpass prototype filter wave filter group is obtained by cosine-modulation, and for In the design method of the cosine filtering group proposed, P.P.Vaidyanathan and R.D.Koilpillai design method are finally It is classical, by analysis method, estimates and meet amplitude distortion and the minimum wave filter group of aliased distortion standard.

The content of the invention

The present invention seeks to design to realize small ripple, low tap number, the cosine-modulation filter of low amplitude distortion and aliased distortion The linear phase fir ptototype filter of ripple device group, and a kind of brand-new design method is provided --- it can design sparse, it is efficient remaining The method of the linear phase fir ptototype filter of string modulated filter group.

The design method of the sparse linear Phase FIR ptototype filter for the cosine modulated filters group that the present invention is provided is specific Step is as follows：

1st, the initialization of the sparse linear Phase FIR ptototype filter design parameter of cosine modulated filters group；

2nd, iterative calculation meets the sparse linear Phase FIR prototype filter for the cosine modulated filters group for rebuilding condition completely Ripple device, includes non-zero tap number of coefficients, position and the determination for having factor v of unit impulse response.

(below by taking II Linear phase FIR filters as an example)：

(1) initial parameter is constructed according to design requirement：

The present invention is according to the port number M of cosine modulated filters group, selection passband, intermediate zone and stopband is corresponding respectively adopts Sample number L_p, L_t, L_sWith ripple value δ_p, δ_t, δ_s, the initial exponent number N of linear phase fir ptototype filter is determined, linear phase fir is former The tap coefficient of mode filter is expressed as with vector h：

H=2 [h₁,h₂,…,h_m…,h_N/2]^T (1)

Wherein h_m(1≤m≤N/2) represents m-th of tap coefficient of FIR ptototype filters；By cosine modulated filters group Sparse linear Phase FIR ptototype filter design problem be converted into following mathematical optimization problem：

s.t.|Bh-d|≤e (2b)

Wherein | | h | |₀0- norm computings are represented, that is, represent the number of non-zero tap in tap coefficient vector；" min " with Formula (2a)-(2c) of " s.t. " combination, which represents to solve, meets (2b) and (2c) requirement | | h | |₀Minimum value；Sampling matrix B It is expressed as B=[B_p；B_t；B_s], wherein B_p、B_tAnd B_sThe sampling matrix of passband, intermediate zone and stopband is represented respectively, is expressed as

WhereinRepresentDimension row Vector；(L_p+L_t+L_sThe vectorial d that) × 1 is tieed up is the ideal frequency response of discretization, is expressed as：

Wherein f (ω) is the intermediate zone frequency response function to be approached, and is expressed as：

Wherein ω₀=α pi/2s M (0≤α≤1), formula (7) is met rebuilds condition completely,Represent transition The stepped-frequency signal of band；Error vector e is (L_p+L_t+L_sThe column vector that) × 1 is tieed up, is expressed as：

E=[δ_p … δ_p δ_t … δ_t δ_s … δ_s]^T, (8)

(2) (L is set_p+L_t+L_sThe vectorial initial value of the dimensional weight of) × 1 is w⁽¹⁾=[1,1 ..., 1]^T, the present invention is in kth (1 ≤ k≤N/2) in secondary iteration, normalized is done to B matrix column vectors：

WhereinUtilize The following problem of OMP Algorithm for Solving：

s.t.||h^(k)||₀≤k (10b)

And calculate (L_p+L_t+L_sThe residual vector r that) × 1 is tieed up^(k), it is expressed as：

r^(k)=Φ^(k)s^(k)-d (11)

The s that wherein k × 1 is tieed up^(k)For the operation result of formula (10),Represent OMP algorithms From B^(k)In the column vector set selected, set Λ^(k)={ n₁,n₂,…,n_kRepresent non-zero tap coefficient index set.

(3) present invention utilizes the index set Λ of obtained non-zero tap coefficient^(k), solve following linear programming problem In：

s.t.|B^(k)h^(k)-d|≤e+μ·1_L×1 (11b)

Judge whether μ is less than zero, if it is greater than zero, then updating weight vectors w, its more new formula is expressed as：

Wherein w^(k+1)(l) new weight vectors w is represented^(k+1)In value, r_l ^(k)Represent residual vector r^(k)In value；To newly it weigh Weight vector w^(k+1)It is brought into cycle calculations in the 2nd step；If μ is less than or equal to zero, stop interative computation, calculate what is obtainedAs final sparse linear phase fir ptototype filter.

The present invention has the advantages that：

1st, the present invention provides a kind of sparse, efficient cosine modulated filters group linear phase fir prototype filter first Ripple device design method.

2nd, the present invention can design the ptototype filter of low non-zero tap number, wave filter it is openness achieve it is used Adder number of multipliers is reduced, and so as to improve its arithmetic speed, reduce arithmetic eror and reduction energy consumption, and then reduces production Cost.

3rd, simulation result shows, under the requirement of same design index, and the number of non-zero tap coefficient of the invention compares state The number few more than 35% of inside and outside optimal similar wave filter.

Brief description of the drawings

Fig. 1 is the sparse linear Phase FIR ptototype filter design method for realizing the cosine modulated filters group of the present invention Flow chart；

Fig. 2 is according to function e_am(ω) calculates the amplitude distortion figure of the approximate cosine modulated filters group rebuild completely；

Fig. 3 is according to function e_a(ω) calculates the aliased distortion of the approximate cosine modulated filters group rebuild completely Figure；

Fig. 4 is the frequency domain response for the sparse linear Phase FIR ptototype filter for drawing cosine modulated filters group in table -2 Figure.

Embodiment

Embodiment 1：

The sparse linear Phase FIR ptototype filter design method for the cosine modulated filters group that the present invention is provided specifically is walked It is rapid as follows：

In order to verify the validity of the filter set designing method, computer simulation emulation has been carried out to this method.

Design requirement：Utilize document：(F.Tan,et al.:“Optimal design of cosine modulated filter banks using quantum-behaved particle swarm optimization algorithm,”4th International Congress on Image and Signal Processing,vol.5,pp.2280-2284, 2011.)(F.Tan,et al.:" the cosine modulated filters group optimization design based on quantum particle swarm optimization, " the 4th boundary International image and signal transacting meeting, vol.5, pp.2280-2284,2011.) given in the design objective that goes out, wave filter group leads to Road number M=16, ptototype filter initial coefficients number N=256, and design passband, intermediate zone and the corresponding sampling of stopband difference Number L_p=4, L_t=7, L_s=94, ripple value δ_p=δ_t=δ_s=110^-8Bring calculating into.Present invention IROMP algorithms design cosine The ptototype filter of wave filter group, according to the calculating of weighted value, in obtained variables set B, selects the coefficient positions pair of non-zero The column vector B answered_j, it is brought into the filter coefficient for iterating to calculate and obtaining in IROMP algorithms.

Step one：Required according to the design parameter of the sparse linear Phase FIR ptototype filter of cosine modulated filters group Bring each design parameter into initialization condition, obtain problem to be solved：

s.t.|Bh-d|≤e (2b)

The passband of design, intermediate zone and stopband are distinguished into corresponding hits L_p=4, L_t=7, L_s=94 bring formula (6) into The ideal frequency response d of the discretization of the dimension of (4+7+94) × 1 is obtained, is expressed as：

Wherein f (ω) numerical value is as shown in Table-1：

Table -1

Will design ripple value δ_p=δ_t=δ_s=110^-8Bring formula (7) into, obtain the dimension error column vector of (4+7+94) × 1 Each value is 110 in e, vector^-8, the vectorial initial value of the dimensional weight of (4+7+94) × 1 is set as w⁽¹⁾=[1,1 ..., 1]^T；

Step 2：In kth (1≤k≤N/2) secondary iteration, normalized is done to B matrix column vectors：

Then the following problem of OMP Algorithm for Solving in annex 1 is used：

s.t.||h^(k)||₀≤k (10b)

Calculate the index set Λ for obtaining non-zero tap coefficient^(k)；

Step 3：The index set Λ of obtained non-zero tap coefficient is solved using step 2^(k), bring into and solve following line In property planning problem：

s.t.|B^(k)h^(k)-d|≤e+μ·1_L×1 (11b)

And then obtain the sparse linear Phase FIR ptototype filter tap coefficient of cosine modulated filters groupIts Numerical value is listed in table -2.

Table -2

Because the impulse response of II type FIR filters has symmetry, therefore by the filter tap system of the invention tried to achieve NumberTo need the half of the ptototype filter tap coefficient solved, second half is symmetry equivalent, i.e., required sparse FIR prototypes filtering The total tap coefficient of device is expressed as：

WhereinRepresent vectorSpin upside down.

Variables choice is carried out to matrix of variables B using inventive algorithm.For 256 rank wave filters, this algorithm is selected The all situations (second half coefficient is symmetry equivalent with it) of half tap coefficient are can obtain afterwards, i.e., WillBring into the total tap coefficient expression formula of wave filter, obtain the total tap coefficient of ptototype filterThe cosine-modulation finally given The sparse linear Phase FIR ptototype filter of wave filter group is the wave filter that nonzero coefficient is 166 ranks, with particle swarm optimization algorithm Compared to saving 35.6%.

The ptototype filter tap coefficient obtained using step 3 calculates analysis filter group h_m(n) with synthesis filter group g_m(n)：

Wherein 1≤m≤M, and calculate the amplitude distortion value e of cosine modulated filters group_amWith aliased distortion value e_a, it is calculated Formula is expressed as：

e_am(ω)=1- | A₀(e^jω)

Wherein A₀(e^jω) and A_l(e^jω) be expressed as：

H_k(e^jω) be analysis filter group frequency domain response, G_k(e^jω) be synthesis filter group frequency domain response.

In table -3, the rank for the FIR notch filter that inventive algorithm is obtained with particle swarm optimization algorithm has been respectively compared Several key indexs such as number, the quantity of non-zero tap weight, the amplitude distortion of wave filter group and aliased distortion, such as institute in table -3 Show, in amplitude distortion value e_amWith aliased distortion value e_aUnder similar situation, ptototype filter exponent number of the invention is considerably less than particle Colony optimization algorithm.

Table -3

In fig 2, according to function e_am(ω) calculates the amplitude of the approximate cosine modulated filters group rebuild completely Distortion map, the amplitude distortion value e that its maximum is obtained with inventive algorithm in table -3_amCorrespondent equal；In fig. 3, according to letter Number e_a(ω) calculates the aliased distortion figure of the approximate cosine modulated filters group rebuild completely, its maximum and this in table -3 The aliased distortion value e that invention algorithm is obtained_aCorrespondent equal；Fig. 4 is the sparse linear for drawing cosine modulated filters group in table -2 The frequency domain response of Phase FIR ptototype filter.

Annex 1

Formula (13) OMP algorithm calculating process

Using OMP algorithms calculating formula (10), M × N matrix B is the sensing matrix of OMP algorithms, and the d of N × 1 is observation Value, r_iResidual error is represented, t represents iterations,Represent empty set, Λ_tRepresent the index set of t iteration, λ_tRepresent the t times repeatedly The index that generation is found, a_jRepresenting matrix Β jth row, B_tRepresent by index Λ_tThe row set for the matrix B selected, θ_tFor t × 1 Column vector, symbol ∪ represents collection union operation,<r_t-1,a_j>Represent to seek the residual error and matrix Β jth before the t times iteration renewal Column vector inner product.Implement step as follows：

1st, initialization residual error makes it equal to

r₀=d； (1)

2nd, formula is used

Calculating obtains indexing λ_t；

3rd, to Λ_tAnd B_tUnion operation, order

Λ_t=Λ_t-1∪{λ_t,

4th, observation of looking for novelty d=B_tθ_tLeast square solution：

5th, the least square solution tried to achieve with (4)Update residual error r_i, it is calculated as：

6th, t=t+1 is made, (2) step is returned if t is less than preset value, otherwise stops iteration and enters the 7th step；

7th, reconstruct gainedIn Λ_tThere is nonzero term at place, and its value is respectively obtained by last time iteration

Claims

1. the design method of the sparse linear Phase FIR ptototype filter of a kind of cosine modulated filters group, it is characterised in that should Method is carried out as steps described below：

1st, according to the port number M of cosine modulated filters group, selection passband, intermediate zone and stopband distinguish corresponding hits L_p, L_t, L_sWith ripple value δ_p, δ_t, δ_s, determine the initial exponent number N of linear phase fir ptototype filter, the filtering of linear phase fir prototype The tap coefficient of device is expressed as with vector h：

H=2 [h₁,h₂,…,h_m…,h_N/2]^T (1)

Wherein h_m(1≤m≤N/2) represents m-th of tap coefficient of FIR ptototype filters；By the sparse of cosine modulated filters group Linear phase fir ptototype filter design problem is converted into following mathematical optimization problem：

s.t.|Bh-d|≤e (2b)

Wherein | | h | |₀0- norm computings are represented, that is, represent the number of non-zero tap in tap coefficient vector；" min " and " s.t. " Formula (2a)-(2c) of combination, which represents to solve, meets (2b) and (2c) requirement | | h | |₀Minimum value；Sampling matrix B is expressed as B=[B_p；B_t；B_s], wherein B_p、B_tAnd B_sThe sampling matrix of passband, intermediate zone and stopband is represented respectively, is expressed as

<mrow> <msub> <mi>B</mi> <mi>p</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo>&lsqb;</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>&omega;</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mo>&rsqb;</mo> <mo>&Element;</mo> <mo>&lsqb;</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>B</mi> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo>&lsqb;</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>&omega;</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>&rsqb;</mo> <mo>&Element;</mo> <mo>&lsqb;</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>,</mo> <mfrac> <mi>&pi;</mi> <mi>M</mi> </mfrac> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>&rsqb;</mo> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>B</mi> <mi>s</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo>&lsqb;</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>&omega;</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msubsup> <mo>&rsqb;</mo> <mo>&Element;</mo> <mo>(</mo> <mfrac> <mi>&pi;</mi> <mi>M</mi> </mfrac> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&pi;</mi> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

WhereinRepresentTie up row vector；

(L_p+L_t+L_sThe vectorial d that) × 1 is tieed up is the ideal frequency response of discretization, is expressed as：

<mrow> <mi>d</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>L</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

Wherein f (ω) is the intermediate zone frequency response function to be approached,Represent the frequency sampling of intermediate zone Point；Error vector e is (L_p+L_t+L_sThe column vector that) × 1 is tieed up, is expressed as：

E=[δ_p … δ_p δ_t … δ_t δ_s … δ_s]^T, (7)

2nd, (L is set_p+L_t+L_sThe vectorial initial value of the dimensional weight of) × 1 is w⁽¹⁾=[1,1 ..., 1]^T, it is secondary in kth (1≤k≤N/2) In iteration, normalized is done to B matrix column vectors：

WhereinL=L_p+L_t+L_s；Utilize OMP Algorithm for Solving Following problem：

s.t.||h^(k)||₀≤k (9b)

r^(k)=Φ^(k)s^(k)-d (10)

The s that wherein k × 1 is tieed up^(k)For the operation result of formula (9),Represent OMP algorithms from B^(k) In the column vector set selected, set Λ^(k)={ n₁,n₂,…,n_kRepresent non-zero tap coefficient index set；

3rd, the index set Λ of obtained non-zero tap coefficient is utilized^(k), solve in following linear programming problem：

s.t.|B^(k)h^(k)-d|≤e+μ·1_L×1 (11b)

<mrow> <msubsup> <mi>h</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>&Element;</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>N</mi> <mo>/</mo> <mn>2</mn> <mo>}</mo> <mo>-</mo> <msup> <mi>&Lambda;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mi>d</mi> <mo>)</mo> </mrow> </mrow>

Judge whether μ is less than zero, if it is greater than zero, weight vectors w is then updated, by new weight vectors w^(k+1)It is brought into the 2nd step Cycle calculations；If μ is less than or equal to zero, stop interative computation, will calculate what is obtainedAs final sparse line Property Phase FIR ptototype filter.

2. the sparse linear Phase FIR ptototype filter of cosine modulated filters group according to claim 1 a kind of is set Meter method, it is characterised in that the intermediate zone frequency response function f (ω) to be approached is expressed as in formula (6)：

<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mfrac> <mi>M</mi> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>(</mo> <mrow> <mi>&omega;</mi> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

Wherein ω₀=α pi/2s M (0≤α≤1), formula (12) is met rebuilds condition completely.

3. the sparse linear Phase FIR ptototype filter of cosine modulated filters group according to claim 1 a kind of is set Meter method, it is characterised in that the formula of the renewal weight vectors w described in the 3rd step is expressed as：

Wherein w^(k+1)(l) new weight vectors w is represented^(k+1)In value, r_l ^(k)Represent residual vector r^(k)In value.