CN108363042B - Waveform realization method for self-adaptive spectrum template constraint - Google Patents

Waveform realization method for self-adaptive spectrum template constraint Download PDF

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CN108363042B
CN108363042B CN201711340018.6A CN201711340018A CN108363042B CN 108363042 B CN108363042 B CN 108363042B CN 201711340018 A CN201711340018 A CN 201711340018A CN 108363042 B CN108363042 B CN 108363042B
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CN108363042A (en
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景阳
梁军利
范旭慧
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
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Abstract

The invention discloses a wave constrained by a self-adaptive frequency spectrum templateThe method comprises the following steps: step 1: constructing a mathematical model of the waveform; step 2: adding inequality constraints in the functional expression obtained in the step 1 into the target function in the form of a penalty term, wherein the penalty term is a non-differentiable function; approximating the non-differentiable penalty term by using a continuous differentiable function to obtain a mathematical model capable of being solved by adopting a gradient descent method; and step 3: solving the function formula constructed in the step 2 by adopting a gradient descent method; and 4, step 4: repeating the step 3 until x converges, and when the convergence condition is met, stopping iteration and outputting the finally obtained waveform x ═ x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)And (6) obtaining the finished product. The waveform obtained by the method has diversity, so the method also has anti-reconnaissance characteristics.

Description

Waveform realization method for self-adaptive spectrum template constraint
Technical Field
The invention belongs to the technical field of waveform control, and relates to a waveform implementation method for self-adaptive spectrum template constraint.
Background
The electromagnetic spectrum is a precious, limited resource that is widely used in electromagnetic devices such as communications, radio, television broadcast, radio navigation, and radar, which typically have bandwidth requirements. As the demand of national defense and civil facilities is continuously increased and the demand of bandwidth is continuously enlarged, the originally limited radio frequency spectrum is more and more crowded, and the main influence is as follows: many electromagnetic services are forced to coexist in a limited frequency band, significantly increasing the likelihood of mutual interference. Therefore, the radar transmit waveform should be compatible with other electromagnetic applications: 1) electromagnetic waves of different services should use different frequency bands; 2) even if some services use the same frequency band, they should reduce mutual interference to a tolerable level.
In order to adapt to crowded radio frequency spectrum, the international famous radar signal processing expert at the university of florida and professor IEEE Fellow Jian Li in the united states propose a SHAPE design algorithm of SHAPE meeting the spectrum constraint (namely, the energy of a radar transmission SHAPE in an operating frequency band is larger, and the energy of a radar transmission SHAPE in a non-operating frequency band or other service occupied frequency bands is small), the SHAPE design algorithm of SHAPE introduces an auxiliary variable, and then the SHAPE sequence and the value of the auxiliary variable are alternately updated. In order to reduce the signal energy of the radar in a non-working frequency band or a frequency band occupied by other services and reduce the interference of the radar to other services, the professor of the Limb et military of northwest university of industry proposes an LPNN method for gradually updating an iterative waveform sequence by using the idea of a neural network. In addition, in consideration of the actual requirement that the radar energy is uniformly distributed on the working frequency band and the signal energy of the radar energy in the non-working frequency band or the occupied frequency band of other services, the professor in the Limb military further provides a waveform implementation method which is designed based on the ADMM framework and better meets the requirement, and the frequency spectrum of the waveform designed by the method is shown in figure 1.
The waveform implementation method of the spectrum constraint performs waveform design on the premise of giving a radar signal spectrum template (namely, the minimum energy of an operating frequency band and the maximum energy of a non-operating frequency band). However, a given spectrum template is not the most suitable template, and the following are the disadvantages: 1) there are no waveforms that satisfy the spectrum template; 2) when the frequency spectrum requirement of the working frequency band is met, the maximum energy of the non-working frequency band can be designed to be lower; 3) there is a higher minimum energy that can be designed for the operating band when meeting the spectrum requirements of the non-operating band. Therefore, there is a limitation in designing waveforms using the above method.
Disclosure of Invention
The invention aims to provide a method for realizing a waveform constrained by a self-adaptive frequency spectrum template, which solves the problems that in the prior art, the minimum energy of a working frequency band and the maximum energy of a non-working frequency band cannot be determined in a self-adaptive mode, and the frequency spectrum template is difficult to be automatically updated under the condition that the frequency spectrum template is unknown and a waveform meeting the requirement of the frequency spectrum template is generated.
The technical scheme adopted by the invention is that a waveform implementation method for self-adaptive spectrum template constraint is implemented according to the following steps:
step 1, constructing a mathematical model of a waveform,
establishing a function formula (1) of a mathematical model:
Figure BDA0001508200150000021
considering that the objective function is the ratio of two unknown variables alpha and beta, which is not easy to solve, a substitute variable is introduced
Figure BDA0001508200150000022
The above function (1) is changed to function (2):
Figure BDA0001508200150000031
step 2, adding inequality constraints in the function formula (2) obtained in the step 1 into the target function in the form of a penalty term, wherein the penalty term is a non-differentiable function; approximating the non-differentiable penalty term by using a continuous differentiable function to obtain a mathematical model capable of being solved by adopting a gradient descent method;
step 3, solving the function formula constructed in the step 2 by adopting a gradient descent method;
step 4, repeating the step 3 until x converges,
when the convergence condition is met, stopping iteration and outputting the finally obtained waveform x ═ x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)And (6) obtaining the finished product.
The invention has the advantages that under the condition that a frequency spectrum template is not given, a proper template is obtained in a self-adaptive mode, and the ratio of the energy upper limit of the obtained waveform in a non-working frequency band to the energy lower limit of the working frequency band is very small, so that the energy of the radar is concentrated in the working frequency band, and the requirements of not wasting radar signal energy information and reducing interference to other electromagnetic services are met; meanwhile, the waveform obtained by the method has diversity, so the method also has anti-reconnaissance characteristics.
Drawings
FIG. 1 is a spectral plot of a waveform obtained by the ADMM method of the prior art;
FIG. 2 is a schematic diagram of the approximation factor and approximation function to the approximation degree of the penalty function in the method of the present invention;
FIG. 3 is a graph of the results of an iteration of a waveform in the method of the present invention with a ratio of the maximum energy in the non-operating band to the minimum energy in the operating band;
fig. 4 is a spectrum diagram of a waveform obtained by the method of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a waveform realization method of self-adaptive spectrum template constraint, which comprises the steps of firstly constructing a mathematical model of a waveform; then, replacing inequality constraints of the model with proper continuous differentiable functions, and transferring the inequality constraints to a target function to obtain a new mathematical model; finally, solving the new model by adopting a gradient descent method to obtain a detection waveform x meeting the constraint condition, namely finally obtaining a signal sequence x with the length of N and the mode of 1, wherein the signal sequence x is { x ═ x }1,...,xNAnd the ratio of the highest energy beta of the signal sequence in the non-working frequency band to the lowest energy alpha of the working frequency band is minimized.
The invention relates to a waveform implementation method for self-adaptive spectrum template constraint, which is implemented according to the following steps:
step 1, constructing a mathematical model of a waveform,
in the actual transmission process, the waveform needs to satisfy the following conditions:
1) constant modulus: in order to maximize the transmission efficiency and the detection sensitivity, the power amplifier is required to work in a saturation stage, namely, a transmission signal constant modulus; therefore, it is necessary to design a single-mode sequence, i.e. x satisfies | x (N) | 1, N ═ 1, …, N, where x ═ x (1), …, x (N)]TThe vector expression form of the sequence x to be designed is shown, and T is a transposition symbol;
2) the frequency bands of interest are normalized to:
Figure BDA0001508200150000041
according to the frequency domain theory of the signal system, the frequency domain signal of x is: y ═ FHx＝[y1,…,yN]T，
Wherein H is a conjugate transpose symbol, and F ═ F1,… fN]，fn＝[1,ej2πn,…,ej2πn(N-1)]TN is the normalized frequency and takes the value of
Figure BDA0001508200150000042
3) The coexistence of multiple applications is resistant to interference: due to the increasing demand of national defense and civil facilities, many electromagnetic services are forced to coexist in a limited frequency band, and the radar emission waveform is compatible with other electromagnetic applications, namely:
the energy of the radar signal is greater than a certain level alpha, i.e. over the designated operating band P
Figure BDA0001508200150000043
Figure BDA0001508200150000044
Representing any variable in the set; in the frequency band S used by other services, the energy of the radar signal is reduced below a certain level β, i.e.
Figure BDA00015082001500000510
The frequency bands P and S satisfy
Figure BDA00015082001500000511
4) The spectrum template is unknown: that is, α and β are unknown to be calculated, in order to increase the energy of the radar signal in the working frequency band as much as possible and reduce the interference of the radar signal to other services, it is required to satisfy that the ratio of β to α is as small as possible, that is, α and β are unknown to be calculated
Figure BDA0001508200150000054
The smaller the size, the better the quality,
and (3) establishing a functional formula of a mathematical model shown in the following formula (1) by combining the four conditions:
Figure BDA0001508200150000055
considering that the objective function is the ratio of two unknown variables alpha and beta, which is not easy to solve, a substitute variable is introduced
Figure BDA0001508200150000056
The above function (1) is changed to function (2):
Figure BDA0001508200150000057
step 2, adding inequality constraints in the function formula (2) obtained in the step 1 into the target function in the form of a penalty term, wherein the penalty term is a non-differentiable function; and then approximating the non-differentiable penalty term by using a continuous differentiable function to obtain a mathematical model capable of being solved by adopting a gradient descent method, wherein the specific process is as follows:
2.1) adding an inequality constraint to the objective function in the form of a penalty term,
defining a penalty function gsAnd gpRespectively as follows:
Figure BDA0001508200150000058
Figure BDA0001508200150000059
then, the function (2) is equivalent to the following objective function (4):
Figure BDA0001508200150000061
wherein the content of the first and second substances,
Figure BDA0001508200150000062
and
Figure BDA0001508200150000063
is a penalty term corresponding to the constraint of two inequalities in the formula (2),
obviously, when the energy of the radar signal on the non-working frequency band is not more than k alpha, namely x, alpha and k, the constraint is satisfied
Figure BDA0001508200150000064
When g iss0; otherwise, gs>0 and radar energy
Figure BDA0001508200150000065
Exceeding a given energy value k alphaThe greater the degree of (g), the penalty function gsThe larger the child g in the penalty terms 2The greater the value of (A);
when the energy of the radar signal on the working frequency band is not less than alpha, namely x and beta meet the constraint
Figure BDA0001508200150000066
When g isp0; otherwise, gp>0 and radar energy
Figure BDA0001508200150000067
The greater the degree of falling below a given energy value α, the greater the penalty function gpThe larger the child g in the penalty termp 2The larger the value of (c).
Therefore, when the energy of the radar signal satisfies the inequality constraint in the function (2), the penalty term
Figure BDA0001508200150000068
Otherwise, the penalty term is positive and the greater the deviation degree of the unsatisfied term is, the greater the sub-term value of the corresponding penalty term is.
Since functional expressions (3-1) and (3-2) are non-differentiable functions, the objective function in functional expression (4) is also a non-differentiable function; the method of the invention approximates the corresponding punishment items in the functional formulas (3-1, 3-2) and (4) by introducing a new continuous differentiable function, so that an optimization problem which can be solved by using a gradient descent method is constructed, and the problem is conveniently solved.
2.2) a continuous micro-approximable function of the penalty function,
for continuous differentiable functions
Figure BDA0001508200150000069
And continuous differentiable function
Figure BDA00015082001500000610
S ∈ S, P ∈ P, and then:
Figure BDA00015082001500000611
Figure BDA0001508200150000071
where μ >0, then the following two corollaries are made:
inference 1:
Figure BDA0001508200150000072
and gsIn a relationship of
Figure BDA0001508200150000073
s∈S；
Figure BDA0001508200150000074
And gp gsIn a relationship of
Figure BDA0001508200150000075
p∈P；
And (3) proving that: order to
Figure BDA0001508200150000076
Figure BDA0001508200150000077
Figure BDA0001508200150000078
F is then1And f2Can be viewed as relating to variable ysThe function (5) is obtained from the functional expressions (3-1) and (5-1):
1) when y issWhen the content is less than or equal to 0,
Figure BDA0001508200150000079
2) when 0 is present<ysWhen the concentration is less than or equal to mu,
Figure BDA00015082001500000710
then it is determined that,
Figure BDA00015082001500000711
Figure BDA00015082001500000712
if and only if ysWhen is equal to μ1'＝f2' when equal to 0, then there are
Figure BDA00015082001500000713
If and only if ysWhen equal to 0 f 10, if and only if ysWhen is equal to μ2＝0；
3) When mu is<ysWhen the temperature of the water is higher than the set temperature,
Figure BDA00015082001500000714
in summary,
Figure BDA00015082001500000715
s belongs to S; the same theory can prove
Figure BDA00015082001500000716
Inference 2: when parameter mu>The smaller the size of 0 is, the smaller,
Figure BDA00015082001500000717
the closer to gs(s∈S)、
Figure BDA00015082001500000718
The closer to gp(p∈P)；
And (3) proving that: order to
Figure BDA0001508200150000081
Figure BDA0001508200150000082
Obtained according to inference 1
Figure BDA0001508200150000083
Can obtain
Figure BDA0001508200150000084
Thus, when μ → 0, f1→ 0, i.e.
Figure BDA0001508200150000085
Similarly, when μ → 0,
Figure BDA0001508200150000086
based on the result of inference 2, the method of the invention calls the parameter mu as an approximation factor, and in order to have more intuitive understanding of inference 2, the value and function of the approximation factor
Figure BDA0001508200150000087
For gsA schematic of the approximation is shown in fig. 2.
According to the results of inference 1 and inference 2, when the approximation factor mu is small, the irreducible penalty function g is addedsAnd gpReplacement by continuous differentiable function
Figure BDA0001508200150000088
And
Figure BDA0001508200150000089
in the above aspect, the functional formula (4) is changed approximately to the following functional formula (7):
Figure BDA00015082001500000810
wherein the objective function
Figure BDA00015082001500000811
Is a continuous differentiable function;
step 3, solving a function formula (7) constructed in the step 2 by adopting a gradient descent method,
the gradient of the function h (x, k, α) is:
Figure BDA00015082001500000812
the values of the relevant variables are respectively as follows,
Figure BDA00015082001500000813
Figure BDA0001508200150000091
Figure BDA0001508200150000092
Figure BDA0001508200150000093
Figure BDA0001508200150000094
Figure BDA0001508200150000095
using a gradient descent method, updating a solving variable alpha, k:
Figure BDA0001508200150000096
Figure BDA0001508200150000097
considering the constant modulus constraint on x: 1, …, N, variable x (N) | 1, N(t+1)The update of (1) is:
Figure BDA0001508200150000098
step 4, repeating the step 3 until x is converged to obtain a detection waveform x and adaptive frequency spectrum templates alpha and beta,
considering when the value of the objective function h (x)(t+1),k(t+1),α(t+1)) And h (x)(t),k(t),α(t)) When the difference is small, h (x) can be reduced(t+1),k(t+1),α(t+1)) Approximately, the local minimum of the objective function h (x, k, α) is considered, i.e., approximately, the waveform x satisfying the model function formula (1) is obtained as x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)Therefore, the present invention determines the convergence condition as:
Figure BDA0001508200150000101
when the convergence condition is met, stopping iteration and outputting the finally obtained waveform x ═ x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)And (6) obtaining the finished product.
Finally, the invention obtains a detection waveform x with the length of N and the power amplifier working in a saturation stage, namely the mode of 1, and adaptive spectrum templates alpha and beta, so that the highest energy beta of the waveform x in a non-working frequency band is small enough to reduce the interference of the waveform x to coexisting services, and the ratio of the beta to the lowest energy alpha of the working frequency band is minimum, thereby realizing the purpose of concentrating radar energy in the working frequency band and not wasting the energy detected by the radar.
Examples
The method of the invention simulates and realizes a waveform with the length of N162.
The corresponding parameters are: approximation factor mu is 10-3The radar sampling frequency is 810kHz, the pulse interval is 200 mu s, the occupied frequency band is set as shown in the table 1, and the pass band is all the frequency bands outside the occupied frequency band in the table 1.
TABLE 1 occupied band settings
Figure BDA0001508200150000102
Fig. 3 is a graph of an iteration result of a ratio of the maximum energy of a waveform in a non-operating frequency band to the minimum energy of an operating frequency band according to an embodiment of the present invention. The spectrogram of the resulting waveform is shown in FIG. 4. The ratio of the maximum energy of the waveform in the non-working frequency band to the minimum energy of the working frequency band can be as low as-18.80 dB.

Claims (5)

1. A waveform implementation method for self-adaptive spectrum template constraint is characterized by comprising the following steps:
step 1, constructing a mathematical model of a waveform, and establishing a functional formula (1) of the mathematical model:
Figure FDA0002948881220000011
considering that the objective function is the ratio of two unknown variables alpha and beta, which is not easy to solve, a substitute variable is introduced
Figure FDA0002948881220000012
The above function (1) is changed to function (2):
Figure FDA0002948881220000013
step 2, adding inequality constraints in the function formula (2) obtained in the step 1 into the target function in the form of a penalty term, wherein the penalty term is a non-differentiable function; approximating the non-differentiable penalty term by using a continuous differentiable function to obtain a mathematical model capable of being solved by adopting a gradient descent method;
step 3, solving the function formula constructed in the step 2 by adopting a gradient descent method;
step 4, repeating the step 3 until x converges,
when the convergence condition is met, stopping iteration and outputting the finally obtained waveform x ═ x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)And (6) obtaining the finished product.
2. The method according to claim 1, wherein in step 1, the waveform is required to satisfy the following condition:
1) constant modulus: it is necessary to design a single-mode sequence, i.e., x satisfies | x (N) | 1, N ═ 1, …, N, where x ═ x (1), …, x (N)]TThe vector expression form of the sequence x to be designed is shown, and T is a transposition symbol;
2) the frequency bands of interest are normalized to:
Figure FDA0002948881220000021
according to the frequency domain theory of the signal system, the frequency domain signal of x is: y ═ FHx＝[y1,…,yN]T，
Wherein H is a conjugate transpose symbol, and F ═ F1,…fN]，fn＝[1,ej2πn,…,ej2πn(N-1)]TN is the normalized frequency and takes the value of
Figure FDA0002948881220000022
3) The coexistence of multiple applications is resistant to interference: the energy of the radar signal is greater than a certain level alpha, i.e. over the designated operating band P
Figure FDA0002948881220000023
Figure FDA0002948881220000024
Representing any variable in the set; in the frequency band S used by other services, the energy of the radar signal is reduced below a certain level β, i.e.
Figure FDA0002948881220000025
The frequency bands P and S satisfy
Figure FDA0002948881220000026
4) The spectrum template is unknown: that is, alpha and beta are unknown to be calculated, and the requirement that the ratio of beta to alpha is as small as possible is satisfied, that is
Figure FDA0002948881220000027
The smaller the better.
3. The method for realizing adaptive spectrum template constrained waveform according to claim 2, wherein in the step 2, the specific process is as follows:
2.1) adding an inequality constraint to the objective function in the form of a penalty term,
defining a penalty function gsAnd gpRespectively as follows:
Figure FDA0002948881220000028
Figure FDA0002948881220000029
then, the function (2) is equivalent to the following objective function (4):
Figure FDA00029488812200000210
wherein the content of the first and second substances,
Figure FDA00029488812200000211
and
Figure FDA00029488812200000212
is a penalty term corresponding to the constraint of two inequalities in the formula (2),
when the energy of the radar signal on the non-working frequency band is not more than k alpha, namely x, alpha and k meet the constraint
Figure FDA0002948881220000031
When g issNot, otherwise gs>0 and radar energy
Figure FDA0002948881220000032
The greater the degree to which a given energy value k α is exceeded, the greater the penalty function gsThe larger the child g in the penalty terms 2The greater the value of (A);
when the energy of the radar signal on the working frequency band is not less than alpha, namely x and beta meet the constraint
Figure FDA0002948881220000033
When g ispNot, otherwise gp>0 and radar energy
Figure FDA0002948881220000034
The greater the degree of falling below a given energy value α, the greater the penalty function gpThe larger the child g in the penalty termp 2The greater the value of (a) is,
therefore, when the energy of the radar signal satisfies the inequality constraint in the function (2), the penalty term
Figure FDA0002948881220000035
If not, the punishment item is a positive value and the deviation degree of the unsatisfied item is larger, the sub-item value of the corresponding punishment item is larger;
2.2) a continuous micro-approximable function of the penalty function,
for continuous differentiable functions
Figure FDA0002948881220000036
And continuous differentiable function
Figure FDA0002948881220000037
S ∈ S, P ∈ P, and then:
Figure FDA0002948881220000038
Figure FDA0002948881220000039
where μ >0, the following two corollaries are made:
inference 1:
Figure FDA00029488812200000310
and gsIn a relationship of
Figure FDA00029488812200000311
s∈S；
Figure FDA00029488812200000312
And gp gsIn a relationship of
Figure FDA00029488812200000313
p∈P；
Inference 2: when the smaller the non-negative parameter mu is,
Figure FDA00029488812200000314
the closer to gs(s∈S)、
Figure FDA00029488812200000315
The closer to gp(p∈P)；
Based on the result of inference 2, this step calls the parameter μ as an approximation factor,
according to the results of inference 1 and inference 2, when the approximation factor mu is small, the irreducible penalty function g is addedsAnd gpReplacement by continuous differentiable function
Figure FDA0002948881220000041
And
Figure FDA0002948881220000042
in a manner such thatThe functional expression (4) is changed approximately to the following functional expression (7):
Figure FDA0002948881220000043
wherein the objective function
Figure FDA0002948881220000044
Is a continuously differentiable function.
4. The method for realizing the waveform constrained by the adaptive spectrum template according to claim 3, wherein in the step 3, the function (7) constructed in the step 2 is solved by a gradient descent method, and the specific process is as follows: the gradient of the function h (x, k, α) is:
Figure FDA0002948881220000045
the values of the relevant variables are respectively as follows,
Figure FDA0002948881220000046
Figure FDA0002948881220000047
Figure FDA0002948881220000048
Figure FDA0002948881220000051
Figure FDA0002948881220000052
Figure FDA0002948881220000053
using a gradient descent method, updating a solving variable alpha, k:
Figure FDA0002948881220000054
Figure FDA0002948881220000055
considering the constant modulus constraint on x: 1, …, N, variable x (N) | 1, N(t+1)The update of (1) is:
Figure FDA0002948881220000056
5. the method for realizing adaptive spectrum template constrained waveform according to claim 4, wherein in the step 4, the specific process is as follows:
considering when the value of the objective function h (x)(t+1),k(t+1),α(t+1)) And h (x)(t),k(t),α(t)) When the difference is small, h (x)(t +1),k(t+1),α(t+1)) Approximately, the local minimum of the objective function h (x, k, α) is considered, i.e., approximately, the waveform x satisfying the model function formula (1) is obtained as x(t+1)And spectrum template alpha ═ alpha(t+1)、β＝k(t+1)α(t+1)The convergence condition is determined as:
Figure FDA0002948881220000057
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