CN105677957B - Design method and device for approximate accurate reconstruction cosine modulation filter bank - Google Patents

Abstract

The invention discloses a design method and a device of an approximate accurate reconstruction cosine modulation filter bank, wherein the device comprises the following steps: calculating to obtain approximate accurate reconstruction conditions of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output-input relation of the cosine modulation filter bank; establishing a mathematical model designed by an approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition; calculating to obtain a group of prototype filter coefficients according to the mathematical model; and calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the group of prototype filter coefficients. The stopband attenuation performance of the quasi-precise reconstruction cosine modulation filter bank obtained through the scale transformation is obviously improved, the convergence speed is high, the numerical stability is high, and the method has very important theoretical guidance significance and engineering practical value for the design of satellite-borne signal processing in a future satellite communication system.

Description

Design method and device for approximate accurate reconstruction cosine modulation filter bank

Technical Field

The invention relates to the technical field of communication satellites, in particular to a design method and a device of an approximate accurate reconstruction cosine modulation filter bank.

Background

The cosine modulation filter bank is widely applied to the field of satellite communication, flexibly realizes the multiplexing and demultiplexing functions of signals, and reduces the complexity of satellite-borne signal forwarding. Therefore, the cosine modulation filter bank has important engineering application value, and the researchers have great interest in the design method.

Depending on the application, the cosine modulated filter bank is divided into an accurately reconstructed cosine modulated filter bank and an approximately accurately reconstructed cosine modulated filter bank. For the design of the approximate accurate reconstruction cosine modulation filter bank, the accurate reconstruction of the filter bank is relaxed within the range allowed by the application accuracy, so that the design complexity is relatively low. In general, the design of an approximate exact reconstruction cosine modulated filter bank is represented as a non-convex quadratic constraint quadratic programming problem, so the design process is to solve the non-convex quadratic constraint quadratic programming.

So far, such planning problems are very difficult to solve, and most of the planning problems turn to some heuristic algorithms. The algorithm is not strong in timeliness, even global optimal solutions of problems cannot be found under some conditions, therefore, in recent years, researchers relax the problem to be solved into a semi-definite programming problem, and then the semi-definite programming is efficiently solved by utilizing the existing algorithm, so that the approximate accurate reconstruction cosine modulation filter bank is obtained. However, the filter bank designed by the method has very large reconstruction error, may not meet application requirements, and is fatal because the convergence and numerical stability of the algorithm are not considered in the algorithm, so that a method with stable and convergent numerical values is found and proposed to design an approximate accurate reconstruction cosine modulation filter bank, and the method has very important theoretical guidance significance and engineering practical value for the design of satellite-borne signal processing in a future satellite communication system.

Disclosure of Invention

The invention provides a design method and a device of an approximate accurate reconstruction cosine modulation filter bank, which solve the problem that the convergence and the numerical stability of an algorithm are not considered in the approximate accurate reconstruction cosine modulation filter bank designed by the existing method.

In a first aspect, the present invention provides a method for designing an approximate exact reconstruction cosine modulated filter bank, including:

calculating to obtain approximate accurate reconstruction conditions of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output-input relation of the cosine modulation filter bank;

establishing a mathematical model designed by an approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

calculating to obtain a group of prototype filter coefficients according to the mathematical model;

and calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the group of prototype filter coefficients.

Preferably, the calculating to obtain the approximate accurate reconstruction condition of the cosine modulated filter bank according to the definition of the cosine modulated filter bank and the output-input relation of the cosine modulated filter bank includes:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

means "for all belonging to the interval [0, π]Upper frequency ω means "a frequency.

Preferably, the establishing a mathematical model of the approximate exact reconstruction cosine modulated filter bank design according to the approximate exact reconstruction condition includes:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

Delta (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

Preferably, said calculating a set of prototype filter coefficients according to said mathematical model comprises:

calculating W according to equations (3), (4), (5) and (6)^(k)、g^(k)、

And

and determining the scale factor d according to equation (7)_j ^(k)(j ═ 1,2, …, m), and a was calculated by performing scaling according to equations (8) and (9)^(k)And c^(k)；

According to formula (10) to A^(k)QR decomposition to give Y^(k)And Z^(k)；

Obtaining y according to formulae (12), (13) and (14)^(k)、s^(k)And λ^(k+1)；

If k ≧ k_maxOr | | | s^(k)||_∞≤ε₁Or c^(k)||_∞≤ε₂Then get the prototype filter coefficient b ═ b^(k+1)(ii) a Otherwise, let k be k +1, recalculate equations (3) - (14);

g^(k)＝2Ub^(k)(4)

d^(k)＝diag(d₁,d₂,…,d_m) (8)

Y^(k)＝Q₁R^-T，Z^(k)＝Q₂(11)

Z^(k)TW^(k)Z^(k)y^(k)＝-Z^(k)T(g^(k)-W^(k)Y^(k)c^(k)) (12)

s^(k)＝-Y^(k)c^(k)+Z^(k)y^(k)(13)

λ^(k+1)＝-Y^(k)T(W^(k)s^(k)+g^(k)(14)

where "diag" is the diagonal matrix identifier, λ^(k)Lagrange multipliers are iterated for the kth time; initial value b of vector b⁽⁰⁾Initial estimated value λ of Lagrange multiplier of equation (2)⁽⁰⁾Maximum number of iterations k_maxLower limit epsilon of change amount of decision variable₁And tolerance of constraint violation ε₂All are preset values, k is 0, b^(k+1)＝b^(k)+s^(k)，||·||_∞Infinite norm, k, representing a vector_maxDenotes the maximum number of iterations, ∈₁Representing the lower bound, epsilon, of the amount of change of the decision variable₂Indicating the tolerance of the constraint violation.

Preferably, the calculating to obtain the approximate exact reconstruction cosine modulated filter bank according to the definition of the cosine modulated filter bank and the set of prototype filter coefficients includes:

according to the dual characteristics h (N) h (N-1-N) of the filters in equations (1) and (2) and the set of prototype filter coefficients b[2h(L-1),…,2h(1),2h(0)]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

In a second aspect, the present invention further provides a device for designing an approximate exact reconstruction cosine modulated filter bank, including:

the approximate accurate reconstruction condition calculation module is used for calculating and obtaining the approximate accurate reconstruction condition of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output and input relation of the cosine modulation filter bank;

the mathematical model establishing module is used for establishing a mathematical model designed by the approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

the filter coefficient calculation module is used for calculating a group of prototype filter coefficients according to the mathematical model;

and the modulation filter bank calculation module is used for calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the group of prototype filter coefficients.

Preferably, the approximate exact reconstruction condition calculation module includes:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

means "for all belonging to the interval [0, π]Upper frequency ω means "a frequency.

Preferably, the mathematical model building module comprises:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

Delta (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

Preferably, the filter coefficient calculation module is configured to:

calculating W according to equations (3), (4), (5) and (6)^(k)、g^(k)、

And

and determining the scale factor d according to equation (7)_j ^(k)(j ═ 1,2, …, m), and a was calculated by performing scaling according to equations (8) and (9)^(k)And c^(k)；

According to formula (10) to A^(k)QR decomposition to give Y^(k)And Z^(k)；

Obtaining y according to formulae (12), (13) and (14)^(k)、s^(k)And λ^(k+1)；

If k ≧ k_maxOr | | | s^(k)||_∞≤ε₁Or c^(k)||_∞≤ε₂Then get the prototype filter coefficient b ═ b^(k+1)(ii) a Otherwise, let k be k +1, recalculate equations (3) - (14);

g^(k)＝2Ub^(k)(4)

d^(k)＝diag(d₁,d₂,…,d_m) (8)

Y^(k)＝Q₁R^-T，Z^(k)＝Q₂(11)

Z^(k)TW^(k)Z^(k)y^(k)＝-Z^(k)T(g^(k)-W^(k)Y^(k)c^(k)) (12)

s^(k)＝-Y^(k)c^(k)+Z^(k)y^(k)(13)

λ^(k+1)＝-Y^(k)T(W^(k)s^(k)+g^(k)) (14)

where "diag" is the diagonal matrix identifier, λ^(k)Lagrange multipliers are iterated for the kth time; initial value b of vector b⁽⁰⁾Initial estimated value λ of Lagrange multiplier of equation (2)⁽⁰⁾Maximum number of iterations k_maxLower limit epsilon of change amount of decision variable₁And tolerance of constraint violation ε₂All are preset values, k is 0, b^(k+1)＝b^(k)+s^(k)，||·||_∞Infinite norm, k, representing a vector_maxDenotes the maximum number of iterations, ∈₁Representing the lower bound, epsilon, of the amount of change of the decision variable₂Indicating the tolerance of the constraint violation.

Preferably, the modulation filter bank calculation module includes:

according to the dual characteristics h (N) h (N-1-N) of the filter in the equations (1) and (2) and the set of prototype filter coefficients b [2h (L-1), …,2h (1),2h (0) ]]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

According to the technical scheme, the stopband attenuation performance of the quasi-precise reconstruction cosine modulation filter bank obtained through the scale transformation is obviously improved, the convergence rate is high, the numerical stability is high, and the method has very important theoretical guidance significance and engineering practical value for the design of satellite-borne signal processing in a future satellite communication system.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.

Fig. 1 is a schematic flowchart of a design method of an approximate exact reconstruction cosine modulated filter bank according to an embodiment of the present invention;

fig. 2 is a general structural diagram of an M-channel cosine modulated filter bank according to an embodiment of the present invention;

FIG. 3 is a flow chart of a method for designing an approximate exact reconstruction cosine modulated filter bank according to an embodiment of the present invention;

fig. 4 is a flowchart illustrating a solving process of a non-convex quadratic constraint quadratic programming model according to an embodiment of the present invention;

fig. 5 is an amplitude-frequency response curve of a prototype filter, an amplitude-frequency response curve of an analysis filter, an amplitude-frequency response curve of a distortion transfer function, and an amplitude-frequency response curve of an aliasing error function according to an embodiment of the present invention;

fig. 6 is a schematic structural diagram of a device for designing an approximate exact reconstruction cosine modulated filter bank according to an embodiment of the present invention.

Detailed Description

The following further describes embodiments of the invention with reference to the drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.

Fig. 1 shows a schematic flowchart of a design method of an approximate exact reconstruction cosine modulated filter bank provided in this embodiment, including:

s1, calculating to obtain approximate accurate reconstruction conditions of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output and input relation of the cosine modulation filter bank;

s2, establishing a mathematical model designed by an approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

s3, calculating to obtain a group of prototype filter coefficients according to the mathematical model;

and S4, calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the prototype filter coefficient.

According to the embodiment, firstly, approximate accurate reconstruction conditions are deduced according to cosine modulation definitions and the relation between cosine modulation filter bank input and cosine modulation filter bank output; secondly, establishing a non-convex quadratic constraint quadratic programming model which takes the coefficient of the prototype filter of the cosine modulation filter bank as a decision variable, namely a mathematical model of the design problem of the approximate accurate reconstruction cosine modulation filter bank by taking the deduced approximate accurate reconstruction condition as a constraint condition and the stop band energy of the prototype filter of the cosine modulation filter bank as a target; then, designing an algorithm to stably solve the established non-convex quadratic constraint quadratic programming mathematical model to obtain a group of optimal prototype filter coefficients; and finally, obtaining a cosine modulation filter bank according to the cosine modulation definition, namely, an approximate accurate reconstruction cosine modulation filter bank.

The stopband attenuation performance of the quasi-precise reconstruction cosine modulation filter bank obtained through the scale transformation is obviously improved, the convergence rate is high, the numerical stability is high, and the method has very important theoretical guidance significance and engineering practical value for the design of satellite-borne signal processing in a future satellite communication system.

As a preferable aspect of the present embodiment, S1 includes:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

means "for all belonging to the interval [0, π]Upper frequency ω means "a frequency.

Specifically, S2 includes:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

δ (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

Further, S3 includes:

calculating W according to equations (3), (4), (5) and (6)^(k)、g^(k)、

And

and determining the scale factor d according to equation (7)_j ^(k)(j ═ 1,2, …, m), and a was calculated by performing scaling according to equations (8) and (9)^(k)And c^(k)；

According to formula (10) to A^(k)QR decomposition to give Y^(k)And Z^(k)；

Obtaining y according to formulae (12), (13) and (14)^(k)、s^(k)And λ^(k+1)；

If k ≧ k_maxOr | | | s^(k)||_∞≤ε₁Or c^(k)||_∞≤ε₂Then get the prototype filter coefficient b ═ b^(k+1)(ii) a Otherwise, let k be k +1, recalculate equations (3) - (14);

g^(k)＝2Ub^(k)(4)

d^(k)＝diag(d₁,d₂,…,d_m) (8)

Y^(k)＝Q₁R^-T，Z^(k)＝Q₂(11)

Z^(k)TW^(k)Z^(k)y^(k)＝-Z^(k)T(g^(k)-W^(k)Y^(k)c^(k)) (12)

s^(k)＝-Y^(k)c^(k)+Z^(k)y^(k)(13)

λ^(k+1)＝-Y^(k)T(W^(k)s^(k)+g^(k)) (14)

where "diag" is the diagonal matrix identifier, λ^(k)Lagrange multipliers are iterated for the kth time; initial value b of vector b⁽⁰⁾Initial estimated value λ of Lagrange multiplier of equation (2)⁽⁰⁾Maximum number of iterations k_maxLower limit epsilon of change amount of decision variable₁And tolerance of constraint violation ε₂All are preset values, k is 0, b^(k+1)＝b^(k)+s^(k)，||·||_∞Infinite norm, k, representing a vector_maxDenotes the maximum number of iterations, ∈₁Representing the lower bound, epsilon, of the amount of change of the decision variable₂Indicating the tolerance of the constraint violation.

Further, S4 includes:

according to the dual characteristics h (N) h (N-1-N) of the filter in the equations (1) and (2) and the set of prototype filter coefficients b [2h (L-1), …,2h (1),2h (0) ]]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

The following takes an M-channel cosine modulated filter bank as an example, and details an implementation process of the present embodiment are described with reference to the drawings.

First, approximately accurate reconstruction conditions are derived.

FIG. 2 shows the structure of an M-channel cosine modulated filter bank, where M is the number of channels of the cosine modulated filter bank, H_k(z) (k-0, 1, …, M-1) is the z-transform of the analysis filter, F^k(z) (k ═ 0,1, …, M-1) is the z transform of the synthesis filter, ↓ M denotes the M times decimation operation, ↓ M denotes the M times interpolation operation, x (z) denotes the z transform of the input signal of the M-channel cosine modulated filter bank,

representing the z-transform of the output signal of an M-channel cosine modulated filter bank.

Let h (N) (0, 1, …, N-1), length N, and z transform (also called transfer function) h (z) for the prototype filter of the cosine modulated filter bank; analysis filter is h_k(N) (k is 0,1, …, M-1, N is 0,1, …, N-1), and the synthesis filter is f_k(N) (k is 0,1, …, M-1, N is 0,1, …, N-1), and equations (15) and (16) are defined according to a cosine modulated filter bank.

At the same time, the cosine modulated filterbank inputs X (z) and outputs

The relationship between them is:

wherein:

respectively, the distortion transfer function and the aliasing distortion function of the cosine modulated filterbank. From the theory of signal processing: when in use

T_l(z)≈0,l＝1,2,…,M-1 (21)

The cosine modulated filter bank has an approximate exact reconstruction property, i.e. the cosine modulated filter bank is an approximate exact reconstruction cosine modulated filter bank. Here, | · | represents a modulo operation. Due to the fact that each analysis filter h in the cosine modulation filter bank_k(N) (k is 0,1, …, M-1, N is 0,1, …, N-1) and each synthesis filter f_k(N) (k-0, 1, …, M-1, N-0, 1, …, N-1) has narrow transition and high impedance attenuation, according to signal processing theory, T_l(z) ≈ 0, l ═ 1,2, …, M-1. From the corresponding relationship among the time domain, the frequency domain and the z domain, the equations (15), (16) and (18) can be obtained

Wherein:

denotes the unit of imaginary number, ω denotes the angular frequency, e denotes the base of the exponential function, T₀(e^jω) Is T₀Amplitude-frequency response of (z), H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency is shifted by k pi/M. Therefore, the approximate exact reconstruction condition of the cosine modulated filter bank is equation (1).

And secondly, establishing a mathematical model of the design problem of the approximate accurate reconstruction cosine modulation filter bank.

Let the length N of the prototype filter h (N) be an even number (this is more applicable in engineering), i.e. N is 2L. Let vector b be [2h (L-1), …,2h (1),2h (0)]^TMatrix of

Here [ ·]^TRepresenting a conjugate transpose operation, matrix D_nThe (i, j) th element of (a) is

δ (-) is a dirac function, the square of the mode of the prototype filter's amplitude-frequency response can be expressed as | H (e)^jω)|²＝b^TT (ω) b, is substituted for formula (15) to give

Wherein:

indicating a rounding down operation.

The engineering usually requires the prototype filter to have the minimum stop-band energy, and the stop-band energy of the prototype filter is

Order matrix

Then E ═ b^TUb, therefore, engineering application targets minimizing E ═ b^TUb, the approximate exact reconstruction cosine filter bank design problem can be abstracted as the following mathematical model, as shown in equation (2).

And thirdly, solving the mathematical model.

Equation (2) is the key to obtain the prototype filter h (N) (0, 1, …, N-1), but it is a nonlinear optimization problem. The following algorithm is designed to solve the problem.

Let b⁽⁰⁾，λ⁽⁰⁾The solution of equation (2) and the initial estimate of the Lagrange multiplier, respectively. Let k equal to 0, solve the quadratic programming subproblem in k +1 step

Wherein: s is a decision variable which is a function of the decision variable,

in solving the subproblem (24), the superscript (k) is omitted for simplicity of description. If the sub-problem (24) is solved directly, as in equations (3) - (6), the coefficient matrix is due to constraints

And constraint violation degree

The elements are unevenly distributed, and the numerical value is unstable in the solving process, even the divergence phenomenon occurs. To prevent this, equation (24) is scaled, i.e., in a constrained equation

Two-sided simultaneous left-multiplication diagonal matrix d ═ diag (d)₁,d₂,…,d_m) Here "diag" is the diagonal matrix identifier, d_j(j ═ 1,2, …, m) is the scale factor, which takes the value of equation (7), resulting in a new constraint equation of a^Ts ═ c, where

The quadratic programming sub-problem after the scaling is thus

Next, the matrix A is QR decomposed, i.e.

Wherein: q is an orthogonal matrix of order L, R is an invertible upper triangular matrix of order (m +1) × (m +1), Q₁And Q₂Respectively, the front m +1 column and the rear L-m-1 column of Q.

Let Y be Q₁R^-T，Z＝Q₂So that A is^TY＝I，A^TZ is 0, where I is a unit matrix, then A^TThe solution of s-c is s-Yc + Zy, where y ∈ R^L-m-1Is an approximate variable. Substituting it into the objective function of equation (24) for 0.5s^TWs+s^TIn g, equation (24a) is transformed into the following unconstrained optimization problem:

order to

To obtain

Z^TWZy＝-Z^T(g-WYc) (27)

Solving the system of linear equations (14) yields the minimum point y of ψ (y)^(k)Thus, the solution of the quadratic programming subproblem (24a), which is the solution of equation (24) according to the theory of mathematical optimization, is

s^(k)＝-Yc+Zy^(k)(28)

Corresponding Lagrange multipliers are

λ^(k+1)＝-Y^T(Ws^(k)+g) (29)

The solution to the problem to be solved is then obtained by operating according to steps 4 and 5 given below. The detailed steps of the algorithm are given below and the flow is shown in fig. 4.

Step 1: given an initial value b of a vector b⁽⁰⁾Initial estimated value λ of Lagrange multiplier of equation (2)⁽⁰⁾Maximum number of iterations k_maxDecision variable changeLower limit of amount ε₁And tolerance of constraint violation ε₂Let k be 0;

step 2: w is calculated according to equations (3), (4), (5) and (6), respectively^(k)、g^(k)、

And

and determining the scale factor d according to equation (7)_j ^(k)(j ═ 1,2, …, m), and then scaled according to equation (25), i.e., a is calculated^(k)And c^(k)；

And step 3: to A^(k)QR decomposition to give Y^(k)And Z^(k)；

And 4, step 4: solving the system of linear equations (27) to obtain y^(k)Thereby obtaining s according to the formulae (28) and (29)^(k)And λ^(k+1)And let b^(k+1)＝b^(k)+s^(k)；

And 5: if k ≧ k_max、||s^(k)||_∞≤ε₁、||c^(k)||_∞≤ε₂If any of these three conditions is satisfied, the algorithm terminates and the solution b-b of equation (2) is obtained^(k+1)(ii) a Otherwise, let k be k +1, jump to step 2, and re-execute the procedure from step 2 to step 5. Here, | · | luminance_∞Representing an infinite norm of the vector.

And fourthly, carrying out cosine modulation to obtain an approximate accurate reconstruction cosine modulation filter bank.

After vector b is obtained, b is [2h (L-1), …,2h (1),2h (0)]^TAccording to the even-length filter, i.e. h (N) ═ h (N-1-N), where N ═ 2L is the length of the prototype filter, the prototype filter coefficients h (N) ═ 1,2, …, N-1 constitute the vector [ h (0), h (1), …, h (L-1), …, h (1), h (0)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Wherein b is_i(i-0, 1, …, L-1) is an element of the vector b.

According to the formulas (15) and (16), each analysis filter h of the approximate accurate reconstruction cosine modulation filter bank can be obtained_k(n) and a synthesis filter f_k(N), where k is 0,1, …, M-1, N is 0,1, …, N-1, the parameter M is the number of channels in the filter bank, and N is the length of the prototype filter. So far, the design process of the approximate accurate reconstruction cosine modulation filter bank is completed. The detailed design flow is shown in fig. 3.

The following describes an embodiment of the design method of an approximate exact reconstruction cosine modulated filter bank according to the present invention with specific data.

For a filter bank, three performance indexes of the filter bank, namely stopband attenuation, reconstruction error peak-to-peak value and maximum aliasing error, are generally considered in engineering, and are defined as follows:

stopband attenuation A_s＝20log₁₀δ_sWherein

log₁₀Denotes a logarithmic function with base 10, ω_sFor stop band edge frequency, "max" represents max operation, | · | represents modulo operation, H (e)^jω) For the magnitude-frequency response of the filter bank prototype filter,

is an imaginary unit;

reconstruction of error peak

Wherein "max" represents maximum operation, "min" represents minimum operation, | |, represents modulo operation, |, M is the number of channels of the filter bank, T₀(e^jω) The magnitude-frequency response of the distortion transfer function of the filter bank represented by equation (3),

is an imaginary unit;

maximum aliasing error

Wherein

For aliasing error amplitude-frequency response, "max" represents maximum operation, "min" represents minimum operation, | · | represents modulus operation, M is the number of channels of the filter bank, T_l(e^jω) The magnitude-frequency response of the aliasing distortion function for the filter bank represented by equation (3a),

in units of imaginary numbers.

The parameters to be applied in the following examples are first given:

TABLE 1 basic parameter Table

Maximum number of iterations kmax	200
		Lower bound epsilon of the amount of change of the decision variable1	1e-10
Tolerance of constraint violation ε2	1e-16

An embodiment of a 16-channel approximate precise reconstruction cosine modulated filter bank is designed by adopting a design method of the approximate precise reconstruction cosine modulated filter bank.

The specification parameters of the filter bank to be designed are as follows: number of channels M equals 16, prototype filter length N equals 384, stop band edge frequency

A16-channel approximate exact reconstruction cosine modulation filter bank is designed and obtained by applying the design method of the approximate exact reconstruction cosine modulation filter bank of the embodiment, and the stopband attenuation A of the filter bank is_sReconstruction error peak-to-peak value E of-106 dB_pp3.26E-14, maximum aliasing error E_a8.50e-8, its prototype filter amplitude-frequency response | H (e)^jω) I curve, analysis filter amplitude-frequency response I H_k(e^jω) Curve (k is 0,1, …, M-1), distortion transfer function amplitude-frequency response | T₀(e^jω) I curve and aliasing error amplitude frequency response E (E)^jω) The curves are shown in fig. 5. In addition, the effect of the scaling in the present method is shown in table 2.

TABLE 2 Effect of Scale Change

Without scale conversion, i.e. d in equation (7)_j＝1,j＝1,2,…,m，

As can be seen from Table 2, after the scale transformation is added in the method, the designed filter bank has obviously improved stop band attenuation performance, namely, the stop band attenuation performance is reduced from-73 dB to-106 dB, because the scale transformation improves the numerical stability of the solving algorithm of the formula (2).

Fig. 6 shows a schematic structural diagram of a design apparatus for an approximate exact reconstruction cosine modulated filter bank provided in this embodiment, including:

the approximate accurate reconstruction condition calculation module 11 is used for calculating and obtaining an approximate accurate reconstruction condition of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output-input relation of the cosine modulation filter bank;

a mathematical model establishing module 12, configured to establish a mathematical model designed by the approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

a filter coefficient calculation module 13, configured to calculate a set of prototype filter coefficients according to the mathematical model;

and the modulation filter bank calculation module 14 is configured to calculate to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the set of prototype filter coefficients.

The stop band attenuation performance of the quasi-exact reconstruction cosine modulation filter bank obtained through the scale transformation is obviously improved, the convergence rate is high, the numerical stability is high, and the method has very important theoretical guidance significance and engineering practical value for the design of satellite-borne signal processing in a future satellite communication system.

As a preferable solution of this embodiment, the approximate accurate reconstruction condition calculation module 11 includes:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

means "for all belonging to the interval [0, π]Upper frequency ω means "a frequency.

Specifically, the mathematical model building module 12 includes:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

Delta (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

Further, the filter coefficient calculation module 13 is configured to:

calculating W according to equations (3), (4), (5) and (6)^(k)、g^(k)、

And

and determining the scale factor d according to equation (7)_j ^(k)(j ═ 1,2, …, m), and a was calculated by performing scaling according to equations (8) and (9)^(k)And c^(k)；

According to formula (10) to A^(k)QR decomposition to give Y^(k)And Z^(k)；

Obtaining y according to formulae (12), (13) and (14)^(k)、s^(k)And λ^(k+1)；

If k ≧ k_maxOr | | | s^(k)||_∞≤ε₁Or c^(k)||_∞≤ε₂Then get the prototype filter coefficient b ═ b^(k+1)(ii) a Otherwise, let k be k +1, recalculate equations (3) - (14);

g^(k)＝2Ub^(k)(4)

d^(k)＝diag(d₁,d₂,…,d_m) (8)

Y^(k)＝Q₁R^-T，Z^(k)＝Q₂(11)

Z^(k)TW^(k)Z^(k)y^(k)＝-Z^(k)T(g^(k)-W^(k)Y^(k)c^(k)) (12)

s^(k)＝-Y^(k)c^(k)+Z^(k)y^(k)(13)

λ^(k+1)＝-Y^(k)T(W^(k)s^(k)+g^(k)) (14)

where "diag" is the diagonal matrix identifier, λ^(k)Lagrange multipliers are iterated for the kth time; initial value b of vector b⁽⁰⁾Initial estimated value λ of Lagrange multiplier of equation (2)⁽⁰⁾Maximum number of iterations k_maxLower limit epsilon of change amount of decision variable₁And tolerance of constraint violation ε₂All are preset values, k is 0, b^(k+1)＝b^(k)+s^(k)，||·||_∞Infinite norm, k, representing a vector_maxDenotes the maximum number of iterations, ∈₁Representing the lower bound, epsilon, of the amount of change of the decision variable₂Indicating the tolerance of the constraint violation.

Still further, the modulation filter bank calculation module 14 includes:

according to the dual characteristics h (N) h (N-1-N) of the filter in the equations (1) and (2) and the set of prototype filter coefficients b [2h (L-1), …,2h (1),2h (0) ]]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

In the description of the present invention, numerous specific details are set forth. It is understood, however, that embodiments of the invention may be practiced without these specific details. In some instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.

Claims (6)

1. A design method of approximate accurate reconstruction cosine modulated filter bank is characterized by comprising the following steps:

calculating to obtain approximate accurate reconstruction conditions of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output-input relation of the cosine modulation filter bank;

establishing a mathematical model designed by an approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

calculating to obtain a group of prototype filter coefficients according to the mathematical model;

calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the group of prototype filter coefficients;

wherein, the approximate accurate reconstruction condition of the cosine modulation filter bank is obtained by calculation according to the definition of the cosine modulation filter bank and the output and input relation of the cosine modulation filter bank, and the approximate accurate reconstruction condition comprises the following steps:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

denotes for all the belonging intervals [0, π]Upper frequency omega.

2. The method of claim 1, wherein the building a mathematical model of an approximate exact reconstruction cosine modulated filter bank design based on the approximate exact reconstruction condition comprises:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

i is 0,1, …, L-1, j is 0,1, …, L-1, δ (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

3. The method of claim 2, wherein computing the approximate exact reconstruction cosine modulated filter bank based on the definition of the cosine modulated filter bank and the set of prototype filter coefficients comprises:

according to the dual characteristics h (N) h (N-1-N) of the filter in the equations (1) and (2) and the set of prototype filter coefficients b [2h (L-1), …,2h (1),2h (0) ]]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

4. An apparatus for designing an approximate exact reconstruction cosine modulated filter bank, comprising:

the approximate accurate reconstruction condition calculation module is used for calculating and obtaining the approximate accurate reconstruction condition of the cosine modulation filter bank according to the definition of the cosine modulation filter bank and the output and input relation of the cosine modulation filter bank;

the mathematical model establishing module is used for establishing a mathematical model designed by the approximate accurate reconstruction cosine modulation filter bank according to the approximate accurate reconstruction condition;

the filter coefficient calculation module is used for calculating a group of prototype filter coefficients according to the mathematical model;

the modulation filter bank calculation module is used for calculating to obtain an approximate accurate reconstruction cosine modulation filter bank according to the definition of the cosine modulation filter bank and the prototype filter coefficient;

wherein the approximate exact reconstruction condition calculation module comprises:

the approximate exact reconstruction conditions are:

wherein M is the channel number of the cosine modulated filter bank, k is an integer greater than or equal to 0,

denotes the unit of imaginary number, ω denotes angular frequency, e denotes the base of the exponential function, H (e)^j(ω-kπ/M)) Amplitude-frequency response H (e) of H (z)^jω) The frequency shift is k pi/M and h (z) is the transfer function of the prototype filter of the cosine modulated filter bank, | · | denotes the modulo operation,

denotes for all the belonging intervals [0, π]Upper frequency omega.

5. The apparatus of claim 4, wherein the mathematical model building module comprises:

the mathematical model is as follows:

wherein: "minize" and "subject to" are mathematical programming terms, meaning "minimize" and "constraint", respectively,

as an objective function, representing the design goal of minimizing the stopband energy of the prototype filter,

to be constrained, it is shown that approximately exact reconstruction conditions must be maintained in the process of approaching the design goal, b ═ 2h (L-1), …,2h (1),2h (0)]^T，[·]^TH (N) is a prototype filter of length N, L is N/2, N is an integer, and the matrix is

Matrix array

Matrix D_nThe (i, j) th element of (a) is

i is 0,1, …, L-1, j is 0,1, …, L-1, δ (·) is a dirac function,

denotes a rounding-down operation, R^LRepresenting a real number set of length L.

6. The apparatus of claim 5, wherein the modulation filter bank computation module comprises:

according to the dual characteristics h (N) h (N-1-N) of the filter in the equations (1) and (2) and the set of prototype filter coefficients b [2h (L-1), …,2h (1),2h (0) ]]^TCalculating to obtain a vector [ h (0), h (1), …, h (L-1), h (L-1), …, h (1), h (0) formed by prototype filter coefficients h (N) (1, 2, …, N-1)]＝0.5[b_L-1,b_L-2,…,b₁,b₀,b^T]Thereby obtaining an approximately accurately reconstructed cosine modulated filter bank of equations (15) and (16):

wherein k is 0,1, …, M-1, N is 0,1, …, N-1.

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