CN107085564A - Higher order polynomial phase signal method for parameter estimation based on depression of order kernel function - Google Patents
Higher order polynomial phase signal method for parameter estimation based on depression of order kernel function Download PDFInfo
- Publication number
- CN107085564A CN107085564A CN201710299589.3A CN201710299589A CN107085564A CN 107085564 A CN107085564 A CN 107085564A CN 201710299589 A CN201710299589 A CN 201710299589A CN 107085564 A CN107085564 A CN 107085564A
- Authority
- CN
- China
- Prior art keywords
- mrow
- msub
- order
- depression
- msup
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
- G06F17/141—Discrete Fourier transforms
- G06F17/142—Fast Fourier transforms, e.g. using a Cooley-Tukey type algorithm
-
- 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
- G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
- G01S7/40—Means for monitoring or calibrating
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/15—Correlation function computation including computation of convolution operations
- G06F17/156—Correlation function computation including computation of convolution operations using a domain transform, e.g. Fourier transform, polynomial transform, number theoretic transform
Landscapes
- Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Mathematical Physics (AREA)
- Data Mining & Analysis (AREA)
- Computational Mathematics (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Pure & Applied Mathematics (AREA)
- Theoretical Computer Science (AREA)
- General Engineering & Computer Science (AREA)
- Databases & Information Systems (AREA)
- Software Systems (AREA)
- Algebra (AREA)
- Computing Systems (AREA)
- Computer Networks & Wireless Communication (AREA)
- Radar, Positioning & Navigation (AREA)
- Remote Sensing (AREA)
- Discrete Mathematics (AREA)
- Medicines Containing Antibodies Or Antigens For Use As Internal Diagnostic Agents (AREA)
- Complex Calculations (AREA)
Abstract
The present invention proposes a kind of higher order polynomial phase signal method for parameter estimation based on depression of order kernel function, the technical problem low for solving the estimated accuracy of method for parameter estimation presence of existing higher order polynomial phase signal, and implementation process is：To the higher order polynomial phase signal uniform sampling of mixed Gaussian white noise, phase signal sequence is obtained；Construct the depression of order kernel function of phase signal sequence；Spline interpolation is carried out to phase signal sequence, non-uniform intervening sequence is obtained；Calculate depression of order signal sequence；FFT is carried out to depression of order signal sequence；Calculate the estimator of parameter to be estimated and output in phase signal sequence；Phase signal sequence is demodulated, depression of order demodulated sequence is obtained；Phase signal sequence is updated, looping construct depression of order kernel function calculates estimator and the output of the phase signal sequence parameter to be estimated after updating successively；Depression of order signal sequence is updated, single order phase signal sequence is obtained, and calculate first order parameter estimator and export.
Description
Technical field
The invention belongs to Non-stationary Signal Analysis technical field, it is related to a kind of Parameterization estimate method of non-stationary signal,
Specifically related to a kind of higher order polynomial phase signal method for parameter estimation based on depression of order kernel function, available for maneuvering target radar
The field such as the detection and estimation of echo-signal and ISAR imaging techniques.
Background technology
Polynomial Phase Signals (PPS) oneself be widely used in non-stationary signal such as radar (pulse Doppler radar, SAR and
ISAR), in sonar, communication, biological medicine, earthquake analysis and the modeling of animal acoustic applications.Linearly, gaussian sum is steadily traditional letter
Number processing main aspect.And the characteristics of modern signal processing it is non-linear, non-gaussian and non-stationary signal.Non-stationary signal
In research, Polynomial Phase Signals are relatively common one kind.Under noise background (the particularly complex environment such as coloured noise, clutter)
The time frequency analysis and Parameter Estimation Problem of Polynomial Phase Signals are in radar, communication, sonar, biomedicine, nature, earthquake letter
Number analysis etc. field letter it is to be solved.For example, in moving communicating field, there is relative motion in emitter, can cause hair with receiver
The phase for penetrating signal changes over time.Instantaneous phase can change with the consecutive variations of distance between transmitting-receiving, now receive letter
Number it is exactly the approximate representation of Polynomial Phase Signals.The signal that time-varying fading channels estimation is used in communication is FM signal (two
Rank PPS is linear FM signal, and high-order PPS is NLFM signal).
As the important signal model of a class, Polynomial Phase Signals can be modeled to many actual signals, multinomial
Phase coefficient in phase signal all has great importance in different applications, can provide abundant information, therefore, multinomial
The parameter Estimation of formula phase signal has broad application prospects and important Research Significance.Under normal circumstances, a white noise
In discrete P rank multinomials phase signal can be expressed as：
X (n)=s (n)+w (n) ,-N/2≤n≤N/2,
Wherein w (n) is white Gaussian noise, and Δ is signal sampling interval, and the sampling number of signal is N+1, b0For signal width
Degree, is a constant, { a1,a2,...,aPP rank parameters are arrived for the 1 of signal phase.Need the polynomial-phase with mixed noise
The N+1 sampling point value estimation parameter { a of signal1,a2,...,aP}.The parameter Estimation of Polynomial Phase Signals has a variety of methods,
Maximal possibility estimation can obtain the mean square error (MSE) of minimum, yet with which employs multi-dimensional search, the meter of this method
Calculation amount is huge.In order to reduce amount of calculation, the method for Higher-Order Ambiguity Function (HAF) and cube phase function (CPF) is using MSE as generation
Valency, employs linear search, greatly reduces computation complexity.HAF recycles phase difference (PD) operator, multinomial to P ranks
Signal is changed into simple signal by formula phase signal using P-1 PD operator depression of order, estimates to believe using fast Fourier transform method
Number most high order parameters, then depression of order demodulation phase signal, repeats cycle calculations low order parameter Estimation amount；But HAF methods
Using P-1 PD operator depression of order, nonlinearity is higher so that the thresholding of estimation is very high, and algorithm is less than thresholding in signal to noise ratio (SNR)
When can thoroughly fail, when being directed to higher order polynomial phase signal, due to error multiplication effect, the wealthy family of HAF methods
Limit can have a strong impact on estimation performance, and then cause the precision of estimation low.And cube phase function and mixing Higher-Order Ambiguity Function
(CPF-HAF) method is improved HAF methods, first passes through phase difference by higher order polynomial phase signal depression of order to 3 ranks,
The phase parameter of 3 rank multinomial phase signals is estimated using CPF methods, at the different time point by setting, spectrum peak search is carried out,
Set up equation group, calculating parameter estimator.Although performance of the CPF-HAF methods in low SNR is better than HAF, non-linear journey
Degree is still higher, and noise component(s) is more, influences larger to parameter Estimation, causes the precision of method of estimation relatively low.
The content of the invention
It is an object of the invention to overcome the defect that above-mentioned prior art is present, it is proposed that a kind of based on depression of order kernel function
Higher order polynomial phase signal method for parameter estimation, the method for parameter estimation for solving existing higher order polynomial phase signal is deposited
The low technical problem of estimated accuracy.
To achieve the above object, the technical scheme that the present invention takes comprises the following steps：
(1) uniform sampling is carried out to the higher order polynomial phase signal x (t) of mixed Gaussian white noise, obtains high order polynomial
Formula phase signal sequence x (n)：
Wherein, n is phase signal sequence x (n) discrete-time variable, and w (n) is Gaussian sequence, and Δ is phase
Signal x (t) sampling interval, A0For phase signal x (t) amplitude, { a1,...,ar,...,aPFor needed for exponent number r is corresponding
The phase parameter of estimation, P is phase signal sequence x (n) top step number；
(2) using higher order polynomial phase signal sequence x (n) top step number P, higher order polynomial phase signal sequence is constructed
X (n) depression of order kernel function ker [x (n)] is arranged, realizes that step is：
2a) the parameter needed for constructing definitions higher order polynomial phase signal sequence x (n) depression of order kernel function ker [x (n)],
And these parameters are initialized：Define phase parameter a to be estimatedPCorresponding exponent number variable P ', strange depression of order iterations i,
Even depression of order iterations j and label vector l(i+j), these parameters are initialized, initial exponent number variable P '=P are obtained, initially
Strange depression of order iterations i=0, initial idol depression of order iterations j=0, initial markers vector l(0)=[]；
2b) match exponents variable P ' is judged；If P ' ≠ 1 and for even number, step 2c is performed), if P ' ≠ 1 and be odd number,
Perform step 2d), if P '=1, output token vector l(M+H), and perform step 2e), wherein l(M+H)=[l1,...,lk,...,
lM+H], numbering k=1 ..., M+H, M are label vector l(M+H)In 0 element number, H be label vector l(M+H)In 1 element number；
2c) match exponents variable P ' carry out depression of order, obtains P '=P '/2, the even depression of order iterations j=j+1 of order, mark to
Measure l(i+j-1)End increases element 0j, obtain label vector l(i+j)=[l(i+j-1),0j], and perform step 2b), wherein 0jFor mark
Remember vector l(i+j)0 element that middle even depression of order iterations is obtained when being j；
2d) match exponents variable P ' carry out depression of order, obtains P '=P ' -1, makes strange depression of order iterations i=i+1, strange depression of order rank
Number Pi=P ', in label vector l(i+j-1)End increases element 1i, obtain label vector l(i+j)=[l(i+j-1),1i], and perform step
Rapid 2b), wherein, 1iFor label vector l(i+j)In 1 element that obtains when being i of strange depression of order iterations, PiFor strange depression of order iteration time
Exponent number when number is i；
2e) according to label vector l(M+H)The value order of middle element, construction higher order polynomial phase signal sequence x's (n)
Depression of order kernel function ker [x (n)]；
(3) the need for according to depression of order kernel function ker [x (n)], spline interpolation is carried out to phase signal sequence x (n), obtained
Phase signal sequence x (n) non-uniform intervening sequence
(4) depression of order signal sequence x is calculatedp：By non-uniform intervening sequenceSubstitute into depression of order core
Function ker [x (n)], obtains depression of order signal sequence xp：
(5) to depression of order signal sequence xpFFT is carried out, depression of order signal sequence x is obtainedpFrequency-domain function f
(Ω)；
(6) estimator of parameter to be estimated in phase signal sequence x (n) is calculatedAnd export：Using linear search method,
Calculate frequency-domain function f (Ω) amplitude function | f (Ω) | the corresponding Frequency point of maximumAnd according to frequency
PointCalculate the estimator of parameter to be estimated in phase signal sequence x (n)And export；
(7) phase signal sequence x (n) is demodulated, obtains depression of order demodulated sequence x ' (n)：
(8) phase signal sequence is updated, x (n)=x ' (n) is made, the phase signal sequence x (n) after depression of order is obtained, structure is circulated
Depression of order kernel function is made, the estimator of the phase signal sequence parameter to be estimated after updating is calculated successivelyAnd export：Order
Top step number subtracts 1, step (2)~(6) is performed, until P=1；
(9) depression of order signal sequence x is madep=x (n), performs step (4)~(6), and calculate first order parameter estimatorAnd it is defeated
Go out.
The present invention compared with prior art, has the following advantages that：
1) present invention employs depression of order kernel function, to higher order polynomial phase signal when estimating phase parameter
Odd even depression of order is carried out, the nonlinear degree of kernel function is reduced, prior art estimation thresholding height is eliminated, and mean square error is larger
Defect, be effectively improved the precision of parameter Estimation.
2) present invention is when calculating phase parameter estimator, using FFT, and linear search method, drop
The low complexity of parameter Estimation.
Brief description of the drawings
Fig. 1 is the implementation process figure of the present invention；
Fig. 2 is 8 ranks respectively to 8 rank phase Polynomial signals, 7 using the present invention, HAF methods and CPF-HAF methods
Rank, 6 ranks and 5 rank parameter a8,a7,a6,a5The simulation result figure of the least mean-square error contrast of estimation.
Embodiment
Below in conjunction with the drawings and specific embodiments, the present invention is described in further detail.
Reference picture 1, the higher order polynomial phase signal method for parameter estimation based on depression of order kernel function, comprises the following steps：
Step 1, uniform sampling is carried out to the higher order polynomial phase signal x (t) of mixed Gaussian white noise, obtains high-order many
Item formula phase signal sequence x (n)：
Wherein, n is phase signal sequence x (n) discrete-time variable, and w (n) is Gaussian sequence, and Δ is phase
Signal x (t) sampling interval, A0For phase signal x (t) amplitude, a1,...,ar,...,aPIt is corresponding to be estimated for exponent number r
Phase parameter, P=8 is phase signal sequence x (n) top step number；
Step 2, using higher order polynomial phase signal sequence x (n) top step number P, construction higher order polynomial phase letter
Number sequence x (n) depression of order kernel function ker [x (n)], realizes that step is：
2a) the parameter needed for constructing definitions higher order polynomial phase signal sequence x (n) depression of order kernel function ker [x (n)],
And these parameters are initialized：Define phase parameter a to be estimatedPCorresponding exponent number variable P ', strange depression of order iterations i,
Even depression of order iterations j and label vector l(i+j), these parameters are initialized, initial exponent number variable P '=P are obtained, initially
Strange depression of order iterations i=0, initial idol depression of order iterations j=0, initial markers vector l(0)=[]；
2b) match exponents variable P ' is judged；If P ' ≠ 1 and for even number, step 2c is performed), if P ' ≠ 1 and be odd number,
Perform step 2d), if P '=1, output token vector l(M+H), and perform step 2e), wherein l(M+H)=[l1,...,lk,...,
lM+H], numbering k=1 ..., M+H, M are label vector l(M+H)In 0 element number, H be label vector l(M+H)In 1 element number；
2c) match exponents variable P ' carry out depression of order, obtains P '=P '/2, the even depression of order iterations j=j+1 of order, mark to
Measure l(i+j-1)End increases element 0j, obtain label vector l(i+j)=[l(i+j-1),0j], and perform step 2b), wherein 0jFor mark
Remember vector l(i+j)0 element that middle even depression of order iterations is obtained when being j；
2d) match exponents variable P ' carry out depression of order, obtains P '=P ' -1, makes strange depression of order iterations i=i+1, strange depression of order rank
Number Pi=P ', in label vector l(i+j-1)End increases element 1i, obtain label vector l(i+j)=[l(i+j-1),1i], and perform step
Rapid 2b), wherein, 1iFor label vector l(i+j)In 1 element that obtains when being i of strange depression of order iterations, PiFor strange depression of order iteration time
Exponent number when number is i；
2e) according to label vector l(M+H)The value order of middle element, construction higher order polynomial phase signal sequence x's (n)
Depression of order kernel function ker [x (n)] its expression formula is：
Wherein, a0,a0+1,...b0For the span of discrete-time variable n in phase signal sequence x (n),For
Sequence operator, comprisingWithTwo kinds of computings, lkWhen=0, signal is carried outComputing, and renewal sequenceDiscrete-time variable m span, lkWhen=1, signal is carried outComputing, and renewal sequence
Discrete-time variable n span：
N=ak,ak+,...,bkak=ak-1+τi,bk=bk-1-τi,
J=1,2 ..., M, i=1,2 ..., H, k=1 ..., M+H,
c1,c2,...,cM-1=1,
Wherein, ak,ak+1,...,bkForThe span of the discrete-time variable of sequence after computing,To be upward
Round,To round downwards,The non-uniform intervening sequence introduced for calculating process.
Step 3, the need for according to depression of order kernel function ker [x (n)], spline interpolation is carried out to phase signal sequence x (n), obtained
To phase signal sequence x (n) non-uniform intervening sequenceFinal depression of order kernel function ker [x
(n) it is] 2M+HIndividual non-uniform intervening sequence multiplication form, and these sequences need interpolation to obtain, in interpolation calculation, interpolation point
Obtained using its immediate signal sequence x (n) sampling-point interpolations, interpolation formula is：
Wherein, x0The interpolation point of (n ') for needed for, n ' discrete times for needed for, ceil (n ') be take more than n ' and close to
Discrete time, floor (n ') be take less than n ' and close to discrete time.
Step 4, depression of order signal sequence x is calculatedp：By non-uniform intervening sequenceSubstitute into depression of order
Kernel function ker [x (n)], obtains depression of order signal sequence xp：
Step 5, to depression of order signal sequence xpFFT is carried out, depression of order signal sequence x is obtainedpFrequency-domain function f
(Ω), its expression formula is：
Wherein, NPFor depression of order signal sequence xPLength, m is discrete-time variable, and Ω is frequency variable.
Step 6, the estimator of parameter to be estimated in phase signal sequence x (n) is calculatedAnd export：Using linear search side
Method, calculate frequency-domain function f (Ω) amplitude function | f (Ω) | the corresponding Frequency point of maximumAnd according to
Frequency pointCalculate the estimator of parameter to be estimated in phase signal sequence x (n)And export, it calculates public
Formula is：
Wherein, NPFor depression of order signal sequence xPLength,For amplitude function | f (Ω) | maximum pair
The Frequency point answered, Ω is frequency variable.
Step 7, phase signal sequence x (n) is demodulated, obtains depression of order demodulated sequence x ' (n)：
Step 8, phase signal sequence is updated, x (n)=x (n) ' is made, obtains the phase signal sequence x (n) after depression of order, this
When, phase signal sequence x (n) top step number is changed into P-1, i.e., parameter correspondence exponent number to be estimated subtracts 1；Looping construct depression of order core letter
Number, and phase signal sequence is updated, the estimator of the parameter to be estimated of the phase signal sequence after updating is calculated successively
And export：Make top step number subtract 1, step (2)~(7) are performed, until P=1；
Step 9, depression of order signal sequence x is madep=x (n), performs step (4)~(6), and calculate first order parameter estimatorAnd
Export, its calculation formula is：
Wherein, NPFor depression of order signal sequence xPLength,For amplitude function | f (Ω) | maximum pair
The Frequency point answered, Ω is frequency variable.
Below in conjunction with emulation experiment, the technology of the present invention effect is further described：
1. simulated conditions and content：
Simulated conditions：The CPU 3.0GHz, Window 7 of MATLAB 7.5.0, Intel (R) Pentium (R) 2
Professional。
Emulation content：The Computer Simulation present invention, HAF methods and CPF-HAF methods are respectively to 8 rank phase Polynomial signals
8 ranks, 7 ranks, 6 ranks and 5 rank parameter a8,a7,a6,a5The least mean-square error of estimation changes with signal to noise ratio, its result such as Fig. 2 institutes
Show.
2. analysis of simulation result：
Reference picture 2, figure (a), figure (b), figure (c) and figure (d) are the present invention, HAF methods and CPF-HAF methods respectively to 8
8 ranks, 7 ranks, 6 ranks and the 5 rank parameter a of rank phase Polynomial signal8,a7,a6,a5The emulation of the least mean-square error contrast of estimation
Result figure, abscissa is signal to noise ratio, and ordinate is the least mean-square error of parameter Estimation；Wherein, CRLB is gram of parameter Estimation
The least mean-square error lower limit of Latin America's sieve lower bound, as parameter Estimation.
The present invention is compared with HAF methods and CPF-HAF methods as can be seen from Figure 2, and when compared with low signal-to-noise ratio, the present invention estimates
The least mean-square error of meter is already close to carat Metro lower bound, i.e., with lower estimation thresholding；Meanwhile, the present invention estimates most
Small mean square error is closer to carat Metro lower bound, with higher estimated accuracy.
Claims (6)
1. a kind of higher order polynomial phase signal method for parameter estimation based on depression of order kernel function, comprises the following steps：
(1) uniform sampling is carried out to the higher order polynomial phase signal x (t) of mixed Gaussian white noise, obtains higher order polynomial phase
Position signal sequence x (n)：
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>A</mi>
<mn>0</mn>
</msub>
<msup>
<mi>e</mi>
<mrow>
<mi>j</mi>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>P</mi>
</munderover>
<msub>
<mi>a</mi>
<mi>r</mi>
</msub>
<msup>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mi>&Delta;</mi>
<mo>)</mo>
</mrow>
<mi>r</mi>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mi>w</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
Wherein, n is phase signal sequence x (n) discrete-time variable, and w (n) is Gaussian sequence, and Δ is phase signal x
(t) sampling interval, A0For phase signal x (t) amplitude, { a1,...,ar,...,aPEstimate for exponent number r needed for corresponding
Phase parameter, P is phase signal sequence x (n) top step number, and P >=3；
(2) using higher order polynomial phase signal sequence x (n) top step number P, construction higher order polynomial phase signal sequence x
(n) depression of order kernel function ker [x (n)], realizes that step is：
2a) the parameter needed for constructing definitions higher order polynomial phase signal sequence x (n) depression of order kernel function ker [x (n)], and right
These parameters are initialized：Define phase parameter a to be estimatedPCorresponding exponent number variable P ', strange depression of order iterations i, even drop
Rank iterations j and label vector l(i+j), these parameters are initialized, initial exponent number variable P '=P is obtained, initial strange drop
Rank iterations i=0, initial idol depression of order iterations j=0, initial markers vector l(0)=[]；
2b) match exponents variable P ' is judged；If P ' ≠ 1 and for even number, step 2c is performed), if P ' ≠ 1 and be odd number, perform
Step 2d), if P '=1, output token vector l(M+H), and perform step 2e), wherein l(M+H)=[l1,...,lk,...,lM+H],
Numbering k=1 ..., M+H, M are label vector l(M+H)In 0 element number, H be label vector l(M+H)In 1 element number；
2c) match exponents variable P ' carry out depression of order, obtains P '=P '/2, the even depression of order iterations j=j+1 of order, in label vector l(i +j-1)End increases element 0j, obtain label vector l(i+j)=[l(i+j-1),0j], and perform step 2b), wherein 0jFor label vector
l(i+j)0 element that middle even depression of order iterations is obtained when being j；
2d) match exponents variable P ' carry out depression of order, obtains P '=P ' -1, makes strange depression of order iterations i=i+1, strange depression of order exponent number Pi=
P ', in label vector l(i+j-1)End increases element 1i, obtain label vector l(i+j)=[l(i+j-1),1i], and perform step 2b),
Wherein, 1iFor label vector l(i+j)In 1 element that obtains when being i of strange depression of order iterations, PiWhen for strange depression of order iterations being i
Exponent number；
2e) according to label vector l(M+H)The value order of middle element, construction higher order polynomial phase signal sequence x (n) depression of order
Kernel function ker [x (n)]；
(3) the need for according to depression of order kernel function ker [x (n)], spline interpolation is carried out to phase signal sequence x (n), phase is obtained
Signal sequence x (n) non-uniform intervening sequence
(4) depression of order signal sequence x is calculatedp：By non-uniform intervening sequenceSubstitute into depression of order kernel function
Ker [x (n)], obtains depression of order signal sequence xp：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>x</mi>
<mi>p</mi>
</msub>
<mo>=</mo>
<mi>ker</mi>
<mo>&lsqb;</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
<mo>...</mo>
<msub>
<mi>x</mi>
<msup>
<mn>2</mn>
<mrow>
<mi>M</mi>
<mo>+</mo>
<mi>H</mi>
</mrow>
</msup>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>;</mo>
</mrow>
(5) to depression of order signal sequence xpFFT is carried out, depression of order signal sequence x is obtainedpFrequency-domain function f (Ω)；
(6) estimator of parameter to be estimated in phase signal sequence x (n) is calculatedAnd export：Using linear search method, calculate
Frequency-domain function f (Ω) amplitude function | f (Ω) | the corresponding Frequency point of maximumAnd according to Frequency pointCalculate the estimator of parameter to be estimated in phase signal sequence x (n)And export；
(7) phase signal sequence x (n) is demodulated, obtains depression of order demodulated sequence x ' (n)：
<mrow>
<msup>
<mi>x</mi>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>&times;</mo>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>(</mo>
<mrow>
<mo>-</mo>
<msub>
<mover>
<mi>a</mi>
<mo>^</mo>
</mover>
<mi>P</mi>
</msub>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mi>&Delta;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>P</mi>
</msup>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
(8) phase signal sequence is updated, x (n)=x ' (n) is made, the phase signal sequence x (n) after depression of order, looping construct drop is obtained
Rank kernel function, calculates the estimator of the phase signal sequence parameter to be estimated after updating successivelyAnd export：Make most high-order
Number subtracts 1, step (2)~(7) is performed, until P=1；
(9) depression of order signal sequence x is madep=x (n), performs step (4)~(6), and calculate first order parameter estimatorAnd export.
2. the higher order polynomial phase signal method for parameter estimation according to claim 1 based on depression of order kernel function, step
The ker of depression of order kernel function described in 2e) [x (n)], its expression formula is：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>ker</mi>
<mo>&lsqb;</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<msub>
<mi>O</mi>
<msub>
<mi>l</mi>
<mrow>
<mi>M</mi>
<mo>+</mo>
<mi>H</mi>
</mrow>
</msub>
</msub>
<mo>&lsqb;</mo>
<msub>
<mi>O</mi>
<msub>
<mi>l</mi>
<mrow>
<mi>M</mi>
<mo>+</mo>
<mi>H</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</msub>
<mo>&lsqb;</mo>
<mo>...</mo>
<msub>
<mi>O</mi>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</msub>
<mo>&lsqb;</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>&rsqb;</mo>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
<mo>...</mo>
<msub>
<mi>x</mi>
<msup>
<mn>2</mn>
<mrow>
<mi>M</mi>
<mo>+</mo>
<mi>H</mi>
</mrow>
</msup>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>,</mo>
<mi>n</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<msub>
<mi>b</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
</mrow>
Wherein, a0,a0+1,...b0For the span of discrete-time variable n in phase signal sequence x (n),Calculated for sequence
Son, comprisingWithTwo kinds of computings, lkWhen=0, signal is carried outComputing, and renewal sequence's
Discrete-time variable m span, lkWhen=1, signal is carried outComputing, and renewal sequenceIt is discrete when
Between variable n span：
<mrow>
<msub>
<mi>O</mi>
<msub>
<mi>l</mi>
<mi>k</mi>
</msub>
</msub>
<mo>&lsqb;</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>q</mi>
<mrow>
<mi>x</mi>
<mo>,</mo>
<msub>
<mi>n</mi>
<mn>0</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>m</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>n</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<msqrt>
<mrow>
<msub>
<mi>c</mi>
<mi>j</mi>
</msub>
<mi>m</mi>
</mrow>
</msqrt>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>n</mi>
<mn>0</mn>
</msub>
<mo>-</mo>
<msqrt>
<mrow>
<msub>
<mi>c</mi>
<mi>j</mi>
</msub>
<mi>m</mi>
</mrow>
</msqrt>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>l</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<msub>
<mn>0</mn>
<mi>j</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>x</mi>
<mo>,</mo>
<msub>
<mi>&tau;</mi>
<mi>i</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>&tau;</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<msup>
<mi>x</mi>
<mo>*</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>&tau;</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>l</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<msub>
<mn>1</mn>
<mi>i</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
N=ak,ak+,...,bkak=ak-1+τi,bk=bk-1-τi,
J=1,2 ..., M, i=1,2 ..., H, k=1 ..., M+H,
c1,c2,...,cM-1=1,
Wherein, ak,ak+1,...,bkForThe span of the discrete-time variable of sequence after computing,To round up,To round downwards,The non-uniform intervening sequence introduced for calculating process.
3. the higher order polynomial phase signal method for parameter estimation according to claim 1 based on depression of order kernel function, step
(3) spline interpolation is carried out to phase signal sequence x (n) described in, its interpolation formula is：
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>-</mo>
<mi>f</mi>
<mi>l</mi>
<mi>o</mi>
<mi>o</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>c</mi>
<mi>e</mi>
<mi>i</mi>
<mi>l</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mi>l</mi>
<mi>o</mi>
<mi>o</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>c</mi>
<mi>e</mi>
<mi>i</mi>
<mi>l</mi>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mi>c</mi>
<mi>e</mi>
<mi>i</mi>
<mi>l</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
</mrow>
<mrow>
<mi>c</mi>
<mi>e</mi>
<mi>i</mi>
<mi>l</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mi>l</mi>
<mi>o</mi>
<mi>o</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>f</mi>
<mi>l</mi>
<mi>o</mi>
<mi>o</mi>
<mi>r</mi>
<mo>(</mo>
<msup>
<mi>n</mi>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
Wherein, x0The interpolation point of (n ') for needed for, n ' discrete times for needed for, ceil (n ') be take more than n ' and close to it is discrete
Time, floor (n ') be take less than n ' and close to discrete time.
4. the higher order polynomial phase signal method for parameter estimation according to claim 1 based on depression of order kernel function, step
(5) the depression of order signal sequence x described inpFrequency-domain function f (Ω), its expression formula is：
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<msub>
<mi>N</mi>
<mi>p</mi>
</msub>
</munderover>
<msub>
<mi>x</mi>
<mi>p</mi>
</msub>
<mo>&times;</mo>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>j</mi>
<mi>&Omega;</mi>
<mi>m</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
2
Wherein, NPFor depression of order signal sequence xPLength, m is discrete-time variable, and Ω is frequency variable.
5. the higher order polynomial phase signal method for parameter estimation according to claim 1 based on depression of order kernel function, step
(6) estimator of parameter to be estimated in the calculating phase signal sequence x (n) described in, its calculation formula is：
<mrow>
<msub>
<mover>
<mi>a</mi>
<mo>^</mo>
</mover>
<mi>P</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>M</mi>
<mo>+</mo>
<mi>H</mi>
</mrow>
</msup>
<msub>
<mi>N</mi>
<mi>P</mi>
</msub>
<msup>
<mi>&Delta;</mi>
<mi>P</mi>
</msup>
<msub>
<mi>c</mi>
<mi>M</mi>
</msub>
<munderover>
<mo>&Pi;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>H</mi>
</munderover>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
<msub>
<mi>&tau;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mi>arg</mi>
<munder>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
<mi>&Omega;</mi>
</munder>
<mo>|</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>,</mo>
</mrow>
Wherein, NPFor depression of order signal sequence xPLength,For amplitude function | f (Ω) | maximum it is corresponding
Frequency point, Ω is frequency variable.
6. the higher order polynomial phase signal method for parameter estimation according to claim 1 based on depression of order kernel function, step
(9) the calculating first order parameter estimator described inIts calculation formula is：
<mrow>
<msub>
<mover>
<mi>a</mi>
<mo>^</mo>
</mover>
<mn>1</mn>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
<mrow>
<msub>
<mi>N</mi>
<mi>P</mi>
</msub>
<msup>
<mi>&Delta;</mi>
<mi>P</mi>
</msup>
</mrow>
</mfrac>
<mi>arg</mi>
<munder>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
<mi>&Omega;</mi>
</munder>
<mo>|</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>&Omega;</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>,</mo>
</mrow>
Wherein, NPFor depression of order signal sequence xPLength,For amplitude function | f (Ω) | maximum it is corresponding
Frequency point, Ω is frequency variable.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710299589.3A CN107085564B (en) | 2017-05-02 | 2017-05-02 | High-order polynomial phase signal parameter estimation method based on reduced kernel function |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710299589.3A CN107085564B (en) | 2017-05-02 | 2017-05-02 | High-order polynomial phase signal parameter estimation method based on reduced kernel function |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN107085564A true CN107085564A (en) | 2017-08-22 |
| CN107085564B CN107085564B (en) | 2019-12-24 |
Family
ID=59612343
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201710299589.3A Active CN107085564B (en) | 2017-05-02 | 2017-05-02 | High-order polynomial phase signal parameter estimation method based on reduced kernel function |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN107085564B (en) |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20040125893A1 (en) * | 2002-12-17 | 2004-07-01 | Saeed Gazor | Methods and systems for tracking of amplitudes, phases and frequencies of a multi-component sinusoidal signal |
| CN106597408A (en) * | 2016-12-16 | 2017-04-26 | 重庆邮电大学 | Method for estimating high-order PPS signal parameter based on time-frequency analysis and instantaneous frequency curve-fitting |
-
2017
- 2017-05-02 CN CN201710299589.3A patent/CN107085564B/en active Active
Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20040125893A1 (en) * | 2002-12-17 | 2004-07-01 | Saeed Gazor | Methods and systems for tracking of amplitudes, phases and frequencies of a multi-component sinusoidal signal |
| CN106597408A (en) * | 2016-12-16 | 2017-04-26 | 重庆邮电大学 | Method for estimating high-order PPS signal parameter based on time-frequency analysis and instantaneous frequency curve-fitting |
Non-Patent Citations (3)
| Title |
|---|
| IGOR DJUROVIĆ ET AL.: "A Hybrid CPF-HAF Estimation of Polynomial-Phase Signals: Detailed Statistical Analysis", 《IEEE TRANSACTIONS ON SIGNAL PROCESSING》 * |
| IGOR DJUROVIĆ ET AL.: "Parameter estimation of non-uniform sampled polynomial-phase signals using the HOCPF-WD", 《SIGNAL PROCESSING》 * |
| 宋玉娥 等: "基于LCT域模糊函数的QFM信号参数估计算法性能分析", 《北京理工大学学报》 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN107085564B (en) | 2019-12-24 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN106487358B (en) | A kind of maneuvering target turning tracking | |
| CN102508206B (en) | Linear frequency modulation (FM) signal parameter estimation method based on small-wave-packet denoising and power spectral entropy | |
| CN106646344B (en) | A kind of Wave arrival direction estimating method using relatively prime battle array | |
| Eichstädt et al. | Deconvolution filters for the analysis of dynamic measurement processes: a tutorial | |
| CN104297740B (en) | Method for estimating Doppler spectrum of radar target on basis of phase analysis | |
| CN107783938B (en) | Method for estimating instantaneous rotating speed of rotating equipment | |
| CN107843894B (en) | A kind of ISAR imaging method of compound movement target | |
| CN108896986A (en) | A kind of measurement conversion Sequential filter maneuvering target tracking method based on predicted value | |
| CN105137373B (en) | A kind of denoising method of exponential signal | |
| CN103973263B (en) | Approximation filter method | |
| CN105354860A (en) | Box particle filtering based extension target CBMeMBer tracking method | |
| CN109507704A (en) | A kind of Double-Star Positioning System frequency difference estimation method based on cross ambiguity function | |
| CN107843875A (en) | Bayes's compressed sensing Radar Data Fusion method based on singular value decomposition noise reduction | |
| CN105676204A (en) | Estimation method for rotation micro Doppler frequency based on radar HRRP (High Resolution Range Profile) | |
| Dash et al. | A hybrid unscented filtering and particle swarm optimization technique for harmonic analysis of nonstationary signals | |
| CN106603036A (en) | Adaptive time delay estimation method based on low-order interpolation filter | |
| CN106872773A (en) | A kind of the multiple-pulse Precision Method of Freuqency Measurement and device of single carrier frequency pulse signal | |
| CN105044698B (en) | Method suitable for micro-Doppler analysis of space target in short-time observation | |
| CN106707279A (en) | Random frequency hopping microwave associated imaging waveform design method | |
| CN110531309A (en) | There are coherent signal angle-of- arrival estimation methods when amplitude phase error based on atom norm | |
| CN104731762A (en) | Cubic phase signal parameter estimation method based on cyclic shift | |
| CN107085564A (en) | Higher order polynomial phase signal method for parameter estimation based on depression of order kernel function | |
| CN105960613A (en) | System identification device | |
| Eappen et al. | Semiparametric algorithm for processing MST radar data | |
| CN113030945B (en) | Phased array radar target tracking method based on linear sequential filtering |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |