CN106597408B - High-order PPS signal parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting - Google Patents
High-order PPS signal parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting Download PDFInfo
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Abstract
The invention discloses a high-order PPS signal parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting, and belongs to the technical field of signal processing. According to the method, received high-order PPS signals are sampled, time-frequency cross item interference is suppressed through smooth pseudo Wigner-Ville transformation, then instantaneous frequency of the signals is obtained through obtained time-frequency distribution through a method of extracting a maximum value, instantaneous frequency curve fitting is carried out through a least square method, meanwhile, the judging of curve fitting orders is carried out, and the purpose of accurately estimating high-order PPS signal parameters can be achieved through multiple groups of tests. The method can effectively inhibit the cross term interference of the high-order PPS signals, estimates the phase parameters of the unknown-order PPS signals under the condition of low signal-to-noise ratio, has better parameter estimation performance, overcomes the influence of the time-frequency cross term interference on the traditional method, and has important significance on the subsequent processing and characteristic analysis of the non-stationary signals.
Description
Technical Field
The invention relates to non-stationary signal processing in communication, in particular to a high-order Polynomial Phase (PPS) signal parameter estimation problem based on time-frequency analysis and instantaneous frequency curve fitting.
Background
A Polynomial Phase Signal (PPS) is a typical non-stationary Signal, has properties of frequency-varying, low interception probability, and the like, and is widely used in the fields of communication, radar, biomedicine, seismic Signal processing, and the like. In various communication and radar systems, the motion characteristics of a target often reflect the phase function of an echo signal, a primary phase function represents that the target does uniform motion, a secondary phase function represents that the target does uniform accelerated motion, and a tertiary phase function represents that the target is in a variable accelerated motion state. Because the relative motion of the transmitter and the receiver can cause the phase of the transmitted signal to change along with the time, and the change of the distance between the receiving and the transmitting inevitably causes the continuous change of the instantaneous phase, according to the Weierstrass theory, the continuous function on the closed interval can be approximated by a polynomial function, and the received signal is approximated by a polynomial phase signal, so the PPS signal parameter estimation method has important significance for research.
High-order PPS signal parameter estimation is always an important content in the field of signal processing, and in recent years, a lot of research and exploration are carried out by many domestic and foreign scholars. In an actual information system, people often face a complex and variable signal environment, the instantaneous frequency of a PPS signal changes along with time, traditional time domain analysis and frequency domain analysis methods cannot analyze non-stationary signals with local change characteristics, and time frequency analysis focuses on the signal time-varying spectrum characteristics and describes the energy distribution of the signal along with time and frequency. The time-frequency analysis is carried out on the signals, the characteristic information of the signals can be intuitively reflected, but the frequency distribution of the non-stationary signals generates serious time-frequency interference due to a large number of cross terms, and further signal processing is hindered. Therefore, various non-stationary signal parameter estimation based on the time-frequency analysis method has become a hotspot of research in the field.
At present, the research aiming at the second-order PPS signal parameter estimation is mature, a perfect signal processing method and a perfect signal processing theory are formed, however, the research on the high-order PPS signal parameter estimation is still in a development stage, and the existing proposed method has certain limitation and is difficult to realize the parameter estimation of the PPS signal with unknown order. The literature ' Wanpu ' polynomial phase signal time-frequency analysis and parameter estimation. electronics report, 2005 ' proposes adaptive time-frequency distribution based on multiplicative fuzzy function, and can inhibit cross terms while enhancing time-frequency aggregation by designing adaptive kernel function, but the calculated amount is large and the estimation performance is influenced by the designed kernel function. The document "jinxiang", polynomial phase number parameter estimation based on fractional fourier transform application, 2010 "processes a high-order PPS signal by using a delay correlation demodulation method, and performs parameter estimation on the signal through fractional fourier transform (FRFT). Therefore, the invention provides a parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting, aiming at the problem of parameter estimation of high-order PPS signals.
Disclosure of Invention
the invention aims to solve the technical problems that the traditional time frequency analysis method has the defects of time frequency cross item interference, poor estimation performance under low signal-to-noise ratio, large calculated amount and the like when processing high-order PPS signals, provides an estimation method based on time frequency analysis and instantaneous frequency curve fitting, and solves the difficult problem of high-order PPS signal parameter estimation. The method overcomes the defects of the traditional high-order PPS signal parameter estimation method, can effectively inhibit the cross term interference of signals, and can accurately estimate the phase parameters of the PPS signals under the condition of low signal-to-noise ratio.
The technical scheme for solving the technical problems is as follows: a high-order PPS signal parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting comprises the steps of firstly carrying out sampling processing on a received high-order PPS signal, then carrying out smooth pseudo Wigner-Ville time-frequency transformation processing on the sample signal to obtain time-frequency distribution without cross term interference, obtaining signal instantaneous frequency by adopting a method of extracting a maximum value, finally carrying out instantaneous frequency curve fitting by utilizing a least square method, and meanwhile, judging through fitting orders until a set judgment condition is met, and realizing parameter estimation of signals with unknown orders.
in an actual communication system, the noisy observed signal model may be expressed as x (t) ═ s (t) + n (t). Where n (t) is white gaussian noise with zero mean and variance σ 2, s (t) is a constant-amplitude PPS signal, and its mathematical expression is that where a is amplitude, usually considering a ═ 1, and is the signal phase, p is the phase order, and a0, a1, a2, …, and ap are the phase coefficients of the signal.
The invention utilizes a parameter estimation method of time-frequency analysis and instantaneous frequency curve fitting to carry out phase parameter estimation on high-order PPS signals, analyzes the problem of time-frequency cross term interference of the signals, deduces a Cramer-Miller limit (CRB) of unbiased estimation of signal parameters, inhibits the cross term interference by carrying out smooth pseudo Wigner-Ville transformation on the signals, and utilizes a least square curve fitting method to realize parameter estimation on the PPS signals with unknown orders.
Drawings
FIG. 1 is a flow chart of a PPS parameter estimation method of the present invention
FIG. 2 is a schematic block diagram of the smooth pseudo Wigner-Ville transform of the present invention
FIG. 3 is a Wigner-Ville distribution diagram of fourth order PPS of the present invention
FIG. 4 is a smooth pseudo Wigner-Ville distribution plot for a fourth order PPS with noise according to the present invention
FIG. 5 RMS error of parameter a0 of the present invention at different SNR
FIG. 6 shows the root mean square error of the parameter a1 of the present invention at different SNR
FIG. 7 RMS error of parameter a2 of the present invention at different SNR
FIG. 8 RMS error of parameter a3 of the present invention at different SNR
FIG. 9 shows the root mean square error of the parameter a4 of the present invention at different SNR
Detailed Description
the invention is further described with reference to the following drawings and specific examples.
Fig. 1 is a flow chart of a method for estimating parameters of a high-order PPS signal according to the present invention, which includes the following steps: firstly, sampling a received signal, wherein the sampling frequency is fs, performing smooth pseudo Wigner-Ville time-frequency transformation processing to obtain signal time-frequency distribution without cross item interference, and then obtaining the instantaneous frequency of the signal by extracting an extreme value from the obtained time-frequency distribution; setting a fitting judgment order P and a threshold value delta, fitting the instantaneous frequency of a signal by using a least square method, obtaining a phase coefficient by fitting when a calculation criterion xi f actually fitted by an instantaneous frequency curve is smaller than the threshold value delta to realize parameter estimation of high-order PPS, performing fitting judgment on the order P to increase by 1 when the calculation criterion xi f actually fitted is larger than the threshold value delta, namely P +1, and performing the least square curve fitting process again until the judgment condition is met.
FIG. 2 is a schematic block diagram of a smooth pseudo Wigner-Ville transform. Firstly, an observation signal is subjected to sampling processing, then a sample signal is subjected to two parts of processing, wherein one part of the signal is subjected to-tau/2 delay and conjugation, the other part of the signal is subjected to tau/2 delay processing and then multiplied by an impulse function delta (s-n), the signal subjected to delay processing is subjected to primary integration processing, the obtained signal and a short time window function h (tau) are processed through a multiplier and subjected to secondary integration processing, and finally the output result is a smooth pseudo Wigner-Ville distribution time-frequency diagram of the signal. The smooth pseudo Wigner-Ville transformation is the smooth windowing treatment of the Wigner-Ville transformation, can effectively inhibit the cross term interference of high-order PPS signals, and keeps the high time-frequency aggregation of the signals.
FIG. 3 is a diagram showing a Wigner-Ville distribution of a fourth order PPS signal of the present invention. As can be seen from the simulation diagram, the Wigner-Ville distribution of the signals has good time-frequency aggregation, the signal energy is concentrated around the instantaneous frequency, but the fourth-order PPS signals have cross term interference, and the cross terms influence the estimation error of the signal parameters, so that the signals cannot be directly analyzed.
The following is a detailed analysis of the Wigner-Ville distribution of a signal for the presence of cross-term interference.
According to the time-frequency analysis theory, the Wigner-Ville transform of a signal is defined as
In the formula, f is the instantaneous frequency of the signal, and the symbol "+" represents the conjugate operation.
Considering a single-component higher-order PPS signal, its discrete mathematical model can be expressed as
where A is the amplitude and the signal phase, p is the signal order (p > 2), a0, a1, a2, …, ap are phase coefficients. Since the instantaneous frequency is the first derivative of the phase function with respect to time, the instantaneous frequency of the signal is expressed as
The Wigner-Ville conversion of the signal can be developed by combining the formula (1) and the formula (2) as follows
and respectively developing the sum according to Taylor formula to obtain:
Let equation (4) be converted into
As can be seen from the equation (7), when the derivative of the phase function of the signal above the second order is zero, the Wigner-Ville distribution is an impulse function located at the instantaneous frequency of the signal, and for example, a single-component chirp signal has no cross-term interference and shows the best time-frequency aggregation. When the derivative of the phase function of the signal is not zero above the second order, such as the higher derivative of the phase of the PPS signal is a polynomial function with respect to time, the WVD of the signal generates self-cross terms due to the higher order term effect of the phase.
FIG. 4 is a graph of the smoothed pseudo Wigner-Ville distribution of a noisy fourth order PPS in accordance with the present invention. The energy of the signal is concentrated, and the time-frequency cross item interference is effectively inhibited. The smooth pseudo Wigner-Ville distribution has better cross term interference suppression, because the smooth pseudo Wigner-Ville transformation is subjected to smooth windowing on the basis of the Wigner-Ville transformation. In an actual communication system, the Wigner-Ville distribution of signals is not only interfered by noise, but also seriously interfered by cross terms of the signals, and the instantaneous frequency of the signals cannot be extracted, so that the Wigner-Ville distribution of the PPS signals cannot be directly analyzed and processed. As can be seen from the figure, the smooth pseudo Wigner-Ville transduction effectively inhibits the interference of cross terms, and inhibits the influence of noise to a certain extent, and has great effect on extracting signal characteristic information.
The smooth pseudo Wigner-Ville transform is a smooth windowing process of the Wigner-Ville transform and is defined as
Where f denotes the instantaneous frequency of the signal, and the pulse function is g (t) δ (t), and h (t) is a smoothing window function. The purpose of smooth pseudo Wigner-Ville transformation is to effectively inhibit the interference of time-frequency cross terms and facilitate the extraction of maximum value of instantaneous frequency.
And performing polynomial fitting on the signal instantaneous frequency curve by using a least square method, and estimating the phase parameter of the PPS signal more accurately when the instantaneous frequency curve is closer to the actual instantaneous frequency curve. The least squares curve fitting basic process is described briefly as follows:
Extracting the maximum value of the time-frequency distribution after the smooth pseudo Wigner-Ville conversion processing to obtain the instantaneous frequency of the signal, namely
IF(n)＝maxSPW(n,f) (9)
And then carrying out polynomial fitting approximation on the roughly estimated instantaneous frequency by using a least square method, wherein the fitting polynomial coefficient is the phase parameter estimation value of the high-order PPS signal after the error meets the judgment condition.
Assuming that m data of signal acquisition are (xi, yi), i is 0,1,2, …, m and x0 < x1 < … < xm, the basic idea of least squares curve fitting is to solve a polynomial to satisfy
Obtaining coefficients to obtain an estimate of a higher order PPS signal parameter
Since the received PPS signal has no prior knowledge, the curve fitting and the fitting judgment of polynomial order are needed to realize accurate estimation of unknown PPS signal parameters. When the actual fitting calculation criterion xi f is smaller than the threshold value delta, obtaining a polynomial coefficient through a least square method, and estimating a phase parameter of a signal; and when the actual fitting calculation criterion xi f is larger than the threshold value delta, the fitting order is increased by 1, namely P is P +1, and the new curve fitting process is repeated until the given judgment condition is met. The difference between two consecutive instantaneous frequencies can be used as a calculation criterion and defined as
Where ifi (n) is the instantaneous frequency of the ith fit, and δ is a predetermined threshold.
The cramer-perot lower bound (CRB) of the PPS signal is derived as follows:
Let the received signal be sn-sn + vn,0 < N-1, vn be zero-mean additive white gaussian noise, and the discretized PPS signal be expressed as
s＝Aexp[j(a+a(nΔ)+a(nΔ)+a(nΔ)+at)] (12)
Where Δ is the sampling interval, let η ═ T (a0, a1, …, aM) and x ═ T (x1, x2, … xN), then the likelihood function is
Elements of the information matrix I (eta)
CRB capable of obtaining estimation precision of each parameter of PPS
Wherein, after calculation, the CRB of each phase parameter estimation order can be approximated as
The effectiveness verification is carried out on the estimation method by utilizing a simulation experiment, a fourth-order PPS signal is adopted, and the parameters are set as follows: the phase coefficient a0 is 0.1, a1 is 0.3, a2 is-0.000561, a3 is 0.000000756, a4 is 1, and the amplitude a is 1. The noise-containing PPS observation signal is x (t) s (t) n (t), n (t) is Gaussian white noise with zero mean and variance sigma 2, and the mathematical expression of s (t) is as follows
s(t)＝exp[j2π(0.1+0.3t-0.000561t+0.000000756t+t)] (17)
Firstly, the fourth-order PPS signal is sampled, wherein the sampling frequency fs is 1kHz, and the number of sampling points is 768. And then, carrying out smooth pseudo Wigner-Ville transformation on the sample signal, simultaneously carrying out signal phase parameter estimation by using a least square curve fitting method, and carrying out 800 Monte Carlo experiments in simulation, wherein the threshold delta is 0.01 Hz. The phase parameter estimation values of the fourth-order PPS signals can be obtained through computer simulation, so that the effectiveness of the method is verified through simulation, and the PPS phase parameters can be accurately estimated.
On the basis of the experiment, the method is utilized to carry out parameter estimation on the simulation signal under different signal-to-noise ratios, the range of the signal-to-noise ratio is-5 dB to 20dB, and the variation interval is 1 dB.
Fig. 5, 6, 7, 8 and 9 represent the performance of the phase parameters a0, a1, a2, a3 and a4 of the fourth-order PPS signal, respectively, under different snr. As can be seen from the figure, when the SNR (signal to noise ratio) is less than 2dB, the error of the phase parameter estimation of the fourth-order PPS signal is reduced along with the increase of the SNR, and the estimation performance is greatly influenced by noise; when the SNR is more than 2dB, the phase parameter estimation error of the signal is gradually converged along with the increase of the SNR, and the estimation errors are all less than-20 dB, so that the method has good estimation performance. Simulation results show that when the signal-to-noise ratio reaches a certain degree, even if the signal-to-noise ratio is further increased, the parameter estimation precision is not improved along with the increase of the signal-to-noise ratio, and the method has robustness.
According to the invention, smooth pseudo Wigner-Ville transformation is carried out on the high-order PPS signal, the energy of the signal is concentrated on the instantaneous frequency, the cross term interference of the signal is effectively inhibited, the time-frequency distribution with high time-frequency aggregation is obtained, the extreme value extraction is further carried out on the instantaneous frequency of the signal, and the phase parameter of the PPS signal can be estimated by adopting a least square curve fitting method. The method effectively inhibits cross term interference caused by high-order phases of signals, has better estimation performance under low signal-to-noise ratio, and solves the problems of poor estimation performance and large computation amount under the conditions of PPS signal parameter estimation with unknown orders and low signal-to-noise ratio. The method can carry out parameter estimation on other non-stationary signals, and has important significance on subsequent processing and fine feature analysis of the signals.
Claims (4)
1. A high-order PPS signal parameter estimation method based on time-frequency analysis and instantaneous frequency curve fitting comprises the following steps: firstly, sampling a received PPS signal with unknown order, obtaining time-frequency distribution without cross item interference through smooth pseudo Wigner-Ville time-frequency transformation, then obtaining an instantaneous frequency curve of the signal by adopting a method of extracting a maximum value from the time-frequency distribution, and obtaining a curve which is the closest to the instantaneous frequency of the actual signal through circular curve fitting by utilizing the principle that the sum of squares of errors between the extracted data and the actual data is the minimum; when the error xi f of the fitting instantaneous frequency for two consecutive times is greater than the threshold value delta, the fitting order is increased by 1, namely P +1, and the curve fitting of the next round is carried out again until the fitting error xi f is less than the threshold value delta, and the curve fitting process is finished; and obtaining the fitting order with the maximum phase parameter estimation value as the phase order P by the fitting polynomial coefficient, thereby realizing the parameter estimation of the high-order PPS signal.
2. The parameter estimation method according to claim 1, wherein the high order PPS signal model is established wherein a is the signal amplitude and is the signal phase, p is the signal order, p > 2, a0, a1, a2, …, ap are the signal phase coefficients; thus, the instantaneous frequency of the PPS signal is expressed as
3. The estimation method as claimed in claim 1, wherein the Wigner-Ville distribution is one of the important tools for analyzing the non-stationary time-varying signal, and has good time-frequency aggregation, for the high-order PPS signal, the Wigner-Ville distribution has cross term interference of high-order phase, the Wigner-Ville transform is defined as the formula in which f is the frequency of the signal, τ is the time delay, and the symbol "×" is the conjugate operation; the smooth pseudo Wigner-Ville transform is a smooth windowing process of the Wigner-Ville transform, which is defined as
Where n is a time domain variable of the signal, g (s-n) is a pulse function, g (s-n) ═ δ (s-n), h (τ) is a smooth window function, and the symbol "+" is a conjugate operation.
4. The estimation method according to any one of claims 1 to 3, and performing polynomial fitting to the instantaneous frequency curve of the signal by a least square method to estimate the phase parameter of the high-order PPS signal, which is divided into two key steps:
Firstly, extracting the maximum value of the time-frequency distribution after the smooth pseudo Wigner-Ville conversion processing to obtain the instantaneous frequency of the signal, namely
IF(n)＝max[SPW(n,f)]
In the formula, "max" represents taking the maximum value, and extracting the maximum value from the roughly estimated instantaneous frequency to realize the fitting approximation of the signal instantaneous frequency curve;
Secondly, the least square curve fitting is carried out on the instantaneous frequency, and the basic idea is as follows: assuming that m data acquired by the signal are (xi, yi), i is 0,1,2, …, m, x0 is more than x1 is more than … is more than xm, solving a polynomial to satisfy
In the formula, min represents taking the minimum value, obtaining polynomial coefficient through curve fitting and further obtaining estimated value of PPS signal parameter
The difference between two consecutive instantaneous frequencies is taken as the fitting error and is defined as
Where ifi (n) is the instantaneous frequency of the ith fitting, | ifi (n) | represents the absolute value of ifi (n), and δ is a predetermined threshold.
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| CN109885805B (en) * | 2019-01-29 | 2022-10-14 | 南京工业职业技术学院 | Instantaneous frequency estimation method for multi-component non-stationary signal |
| CN110346772A (en) * | 2019-08-22 | 2019-10-18 | 上海无线电设备研究所 | A kind of higher-frequency radar significantly ionosphere phase diameter Disturbance Rejection method |
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| CN114488208B (en) * | 2022-02-17 | 2024-04-05 | 合肥工业大学 | Beidou signal anti-interference method combining empirical wavelet and SPWVD conversion |
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