CN107085564B - High-order polynomial phase signal parameter estimation method based on reduced kernel function - Google Patents

High-order polynomial phase signal parameter estimation method based on reduced kernel function Download PDF

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CN107085564B
CN107085564B CN201710299589.3A CN201710299589A CN107085564B CN 107085564 B CN107085564 B CN 107085564B CN 201710299589 A CN201710299589 A CN 201710299589A CN 107085564 B CN107085564 B CN 107085564B
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CN107085564A (en
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李明
孙浩
曹润清
左磊
吴艳
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Xian University of Electronic Science and Technology
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Abstract

The invention provides a high-order polynomial phase signal parameter estimation method based on a reduced order kernel function, which is used for solving the technical problem of low estimation precision of the existing parameter estimation method of high-order polynomial phase signals and comprises the following implementation processes: uniformly sampling a high-order polynomial phase signal of mixed Gaussian white noise to obtain a phase signal sequence; constructing a reduced kernel function of the phase signal sequence; carrying out spline interpolation on the phase signal sequence to obtain a non-uniform interval sequence; calculating a reduced order signal sequence; performing fast Fourier transform on the reduced signal sequence; calculating and outputting an estimator of a parameter to be estimated in the phase signal sequence; demodulating the phase signal sequence to obtain a reduced order demodulation sequence; updating the phase signal sequence, circularly constructing a reduced kernel function, sequentially calculating and outputting estimators of parameters to be estimated of the updated phase signal sequence; and updating the reduced-order signal sequence to obtain a first-order phase signal sequence, and calculating and outputting a first-order parameter estimator.

Description

High-order polynomial phase signal parameter estimation method based on reduced kernel function
Technical Field
The invention belongs to the technical field of non-stationary signal analysis, relates to a parametric estimation method of a non-stationary signal, and particularly relates to a high-order polynomial phase signal parameter estimation method based on a reduced order kernel function, which can be used in the fields of detection and estimation of a radar echo signal of a maneuvering target, ISAR imaging technology and the like.
Background
Polynomial Phase Signals (PPS) have been widely used in non-stationary signals such as radar (pulsed doppler radar, SAR and ISAR), sonar, communications, biomedical, seismic analysis and modeling for animal acoustic applications. Linearity, gaussian and stationary are the main aspects of conventional signal processing. Modern signal processing is characterized by nonlinear, non-gaussian and non-stationary signals. In the study of non-stationary signals, a polynomial phase signal is a common one. The time-frequency analysis and parameter estimation problems of polynomial phase signals under the background of noise (especially complex environments such as color noise, clutter and the like) are to be solved in the fields of radar, communication, sonar, biomedicine, nature, seismic signal analysis and the like. For example, in the field of mobile communications, relative motion between a transmitter and a receiver may cause the phase of a transmitted signal to vary over time. The instantaneous phase varies with the continuous variation of the distance between the transmitter and the receiver, and the received signal is an approximate representation of the polynomial phase signal. The signals adopted by the time-varying fading channel estimation in communication are frequency modulation signals (second-order PPS is a linear frequency modulation signal, and high-order PPS is a non-linear frequency modulation signal).
As an important signal model, the polynomial phase signal can be used for modeling a plurality of actual signals, and phase coefficients in the polynomial phase signal have important significance in different applications and can provide rich information, so that the parameter estimation of the polynomial phase signal has wide application prospect and important research significance. In general, a discrete P-order polynomial phase signal in white noise can be expressed as:
x(n)＝s(n)+w(n),-N/2≤n≤N/2，
where w (N) is white Gaussian noise, Δ is the signal sampling interval, the number of signal samples is N +1, b0Is the signal amplitude, is a constant, { a1,a2,...,aP1 to P of the signal phaseAn order parameter. The parameter { a } needs to be estimated using N +1 sample point values of the noise-mixed polynomial phase signal1,a2,...,aP}. There are several methods for estimating the parameters of polynomial phase signals, and maximum likelihood estimation can obtain the minimum Mean Square Error (MSE), however, since it employs multidimensional search, this method is computationally expensive. In order to reduce the calculation amount, the methods of a high-order fuzzy function (HAF) and a Cubic Phase Function (CPF) adopt one-dimensional search at the cost of MSE, and the calculation complexity is greatly reduced. The HAF repeatedly utilizes a Phase Difference (PD) operator, reduces the order of the P-order polynomial phase signal by adopting the PD operator for P-1 times, changes the signal into a single-frequency signal, estimates the highest-order parameter of the signal by adopting a fast Fourier change method, demodulates the phase signal in a reduced order, and repeatedly operates to circularly calculate the low-order parameter estimator; however, the HAF method adopts P-1 PD operator order reduction, the nonlinearity is high, so that the estimated threshold is very high, the algorithm completely fails when the signal-to-noise ratio (SNR) is lower than the threshold, and when aiming at a high-order polynomial phase signal, the estimation performance is seriously affected by the high threshold of the HAF method due to the error multiplication effect, thereby causing low estimation accuracy. A cubic phase function and mixed higher-order fuzzy function (CPF-HAF) method is used for improving the HAF method, the higher-order polynomial phase signal is reduced to 3 orders through phase difference, the phase parameters of the 3-order polynomial phase signal are estimated by using the CPF method, spectral peak search is carried out by setting different time points, an equation set is established, and parameter estimation amount is calculated. Although the performance of the CPF-HAF method is better than that of the HAF method at low SNR, the degree of nonlinearity is still higher, the noise component is more, the influence on parameter estimation is larger, and the accuracy of the estimation method is lower.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a high-order polynomial phase signal parameter estimation method based on a reduced order kernel function, and is used for solving the technical problem of low estimation precision of the existing high-order polynomial phase signal parameter estimation method.
In order to achieve the purpose, the technical scheme adopted by the invention comprises the following steps:
(1) uniformly sampling a high-order polynomial phase signal x (t) of mixed Gaussian white noise to obtain a high-order polynomial phase signal sequence x (n):
wherein n is the discrete time variable of the phase signal sequence x (n), w (n) is the Gaussian white noise sequence, Delta is the sampling interval of the phase signal x (t), A0Amplitude of the phase signal x (t) { a1,...,ar,...,aPThe P is the highest order of the phase signal sequence x (n);
(2) utilizing the highest order P of the high-order polynomial phase signal sequence x (n) to construct a reduced order kernel function ker [ x (n) ] of the high-order polynomial phase signal sequence x (n), the implementation steps are:
2a) defining a reduced order kernel ker [ x (n) ] for constructing a high order polynomial phase signal sequence x (n)]The required parameters and the initialization of these parameters: defining a phase parameter a to be estimatedPCorresponding order variable P', odd reduced iteration times i, even reduced iteration times v and mark vector l(i+v)Initializing the parameters to obtain an initial order variable P', an initial odd-order reduction iteration number i equal to 0, an initial even-order reduction iteration number v equal to 0, and an initial label vector l(0)＝[]；
2b) Judging the order variable P'; if P ' ≠ 1 and is even, execute step 2c), if P ' ≠ 1 and is odd, execute step 2d), if P ' ≠ 1, output token vector l(M+H)And performing step 2e) wherein l(M+H)＝[l1,...,lk,...,lM+H]The number k is 1.. multidot.m + H, M is a marker vector l(M+H)The number of middle 0 elements, H is a mark vector l(M+H)The number of the middle 1 element;
2c) reducing the order of the order variable P 'to obtain P'/2, making the even order-reducing iteration number v ═ v +1, and marking the vector l(i+v-1)End addition of element 0vTo obtain a mark vector l(i+v)＝[l(i+v-1),0v]And performing step 2b) of 0vIs a mark vector l(i+v)0 element obtained when the number of the even-middle reduced order iterations is v;
2d) reducing the order of the variable P 'to obtain P' -1, making the odd order-reducing iteration number i equal to i +1, and making the odd order-reducing order PiP' in the mark vector l(i+v-1)End addition element 1iTo obtain a mark vector l(i+v)＝[l(i+v-1),1i]And performing a step 2b), wherein 1iIs a mark vector l(i+v)1 element, P, obtained when the number of iterations of the odd-middle order reduction is iiThe order is the order when the odd order-reducing iteration times are i;
2e) according to the mark vector l(M+H)The value sequence of the medium elements is used for constructing a reduced kernel function ker [ x (n) of a high-order polynomial phase signal sequence x (n)]；
(3) According to a reduced order kernel function ker [ x (n)]Performing spline interpolation on the phase signal sequence x (n) to obtain a non-uniform interval sequence of the phase signal sequence x (n)
(4) Calculating a reduced order signal sequence xp: non-uniform interval sequenceSubstituting a reduced order kernel ker [ x (n)]To obtain a reduced order signal sequence xp：
(5) For reduced order signal sequence xpPerforming fast Fourier transform to obtain reduced signal sequence xpThe frequency domain function f (Ω);
(6) calculating an estimate of a parameter to be estimated in a sequence of phase signals x (n)And outputs: calculating the frequency domain function f (by one-dimensional search method)Ω) of the amplitude function | f (Ω) | is calculated based on the frequency point corresponding to the maximum value of the amplitude function | f (Ω) |And according to frequency pointCalculating an estimate of a parameter to be estimated in a sequence of phase signals x (n)And outputting;
(7) demodulating the phase signal sequence x (n) to obtain a reduced demodulation sequence x' (n):
(8) updating the phase signal sequence, making x (n) equal to x' (n), obtaining the phase signal sequence x (n) after reducing, circularly constructing a reducing kernel function, and sequentially calculating the estimators of the parameters to be estimated of the updated phase signal sequenceAnd outputs: subtracting 1 from the highest order number, and executing the steps (2) to (6) until P is equal to 1;
(9) order-reduced signal sequence xpExecuting the steps (4) to (6), and calculating a first-order parameter estimatorAnd output.
Compared with the prior art, the invention has the following advantages:
1) when phase parameters are estimated, the invention adopts the order-reducing kernel function to carry out odd-even order reduction on high-order polynomial phase signals, reduces the nonlinearity degree of the kernel function, eliminates the defects of high estimation threshold and larger mean square error in the prior art, and effectively improves the precision of parameter estimation.
2) When the phase parameter estimator is calculated, the invention adopts fast Fourier transform and a one-dimensional search method, thereby reducing the complexity of parameter estimation.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 shows the 8 th, 7 th, 6 th and 5 th order parameters a for an 8 th order phase polynomial signal using the present invention, the HAF method and the CPF-HAF method, respectively8,a7,a6,a5And (4) a simulation result graph of the comparison of the estimated minimum mean square error.
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments.
Referring to fig. 1, the method for estimating the phase signal parameters of the higher order polynomial based on the reduced order kernel function includes the following steps:
step 1, uniformly sampling a high-order polynomial phase signal x (t) mixed with Gaussian white noise to obtain a high-order polynomial phase signal sequence x (n):
wherein n is the discrete time variable of the phase signal sequence x (n), w (n) is the Gaussian white noise sequence, Delta is the sampling interval of the phase signal x (t), A0Is the amplitude, a, of the phase signal x (t)1,...,ar,...,aPThe phase parameter to be estimated corresponding to the order r, and P ═ 8 is the highest order of the phase signal sequence x (n);
step 2, constructing a reduced order kernel function ker [ x (n) ] of the high order polynomial phase signal sequence x (n) by using the highest order P of the high order polynomial phase signal sequence x (n), and the implementation steps are as follows:
2a) defining a reduced order kernel ker [ x (n) ] for constructing a high order polynomial phase signal sequence x (n)]The required parameters and the initialization of these parameters: defining a phase parameter a to be estimatedPCorresponding order variable P', odd reduced iteration times i, even reduced iteration times v and mark vector l(i+v)Initializing the parameters to obtain an initial order variable P' ═ P, and performing initial odd-order reduction iterationThe number i is 0, the initial even-order-reduced iteration number v is 0, and the initial token vector l(0)＝[]；
2b) Judging the order variable P'; if P ' ≠ 1 and is even, execute step 2c), if P ' ≠ 1 and is odd, execute step 2d), if P ' ≠ 1, output token vector l(M+H)And performing step 2e) wherein l(M+H)＝[l1,...,lk,...,lM+H]The number k is 1.. multidot.m + H, M is a marker vector l(M+H)The number of middle 0 elements, H is a mark vector l(M+H)The number of the middle 1 element;
2c) reducing the order of the order variable P 'to obtain P'/2, making the even order-reducing iteration number v ═ v +1, and marking the vector l(i+v-1)End addition of element 0vTo obtain a mark vector l(i+v)＝[l(i+v-1),0v]And performing step 2b) of 0vIs a mark vector l(i+v)0 element obtained when the number of the even-middle reduced order iterations is v;
2d) reducing the order of the variable P 'to obtain P' -1, making the odd order-reducing iteration number i equal to i +1, and making the odd order-reducing order PiP' in the mark vector l(i+v-1)End addition element 1iTo obtain a mark vector l(i+v)＝[l(i+v-1),1i]And performing a step 2b), wherein 1iIs a mark vector l(i+v)1 element, P, obtained when the number of iterations of the odd-middle order reduction is iiThe order is the order when the odd order-reducing iteration times are i;
2e) according to the mark vector l(M+H)The value sequence of the medium elements is used for constructing a reduced kernel function ker [ x (n) of a high-order polynomial phase signal sequence x (n)]The expression is as follows:
wherein, a0,a0+1,...b0The value range of the discrete-time variable n in the phase signal sequence x (n),being a sequence operator, comprisingAndtwo kinds of operation,/kWhen equal to 0, the signal is processedOperate on and update the sequenceThe value range of the discrete-time variable m,/, iskWhen 1, the signal is processedOperate on and update the sequenceThe value range of the discrete time variable n is as follows:
n＝ak,ak+1,...,bk ak＝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,...,bkIs composed ofThe value range of the discrete time variable of the sequence after operation,in order to get the whole upwards,in order to get the whole downwards,non-uniform spacer sequences introduced for the calculation process.
Step 3, according to the reduced kernel function ker [ x (n)]Performing spline interpolation on the phase signal sequence x (n) to obtain a non-uniform interval sequence of the phase signal sequence x (n)Final reduced order kernel function ker [ x (n)]Is 2M+HThe non-uniform interval sequence multiplication forms are obtained by interpolation, when interpolation is calculated, interpolation points are obtained by utilizing sampling points x (n) of signal sequences adjacent to the interpolation points, and the interpolation formula is as follows:
wherein x is0(n ') is the desired interpolation point, n' is the desired discrete time, ceil (n ') is the discrete time taken to be greater than n' and immediately adjacent, floor (n ') is the discrete time taken to be less than n' and immediately adjacent.
Step 4, calculating the reduced order signal sequence xp: non-uniform interval sequenceSubstituting a reduced order kernel ker [ x (n)]To obtain a reduced order signal sequence xp：
Step 5, for the reduced order signal sequence xpPerforming fast Fourier transform to obtain reduced signal sequence xpThe frequency domain function f (Ω) of (a), whose expression is:
wherein N isPFor reducing the order of the signal sequence xPM is a discrete time variable and Ω is a frequency variable.
Step 6, calculating the estimation quantity of the parameter to be estimated in the phase signal sequence x (n)And outputs: calculating a frequency point corresponding to the maximum value of the amplitude function | f (omega) | of the frequency domain function f (omega) by adopting a one-dimensional search methodAnd according to frequency pointCalculating an estimate of a parameter to be estimated in a sequence of phase signals x (n)And outputting, wherein the calculation formula is as follows:
wherein N isPFor reducing the order of the signal sequence xPThe length of (a) of (b),is the frequency point corresponding to the maximum value of the amplitude function | f (Ω) |, where Ω is the frequency variable.
And 7, demodulating the phase signal sequence x (n) to obtain a reduced demodulation sequence x' (n):
step 8, updating the phase signal sequence, and making x (n) ═ x' (n) to obtain a phase signal sequence x (n) after the order reduction, where at this time, the highest order of the phase signal sequence x (n) becomes P-1, that is, the order corresponding to the parameter to be estimated is subtracted by 1; circularly constructing a reduced order kernel function, updating the phase signal sequence, and sequentially calculating the estimators of the parameters to be estimated of the updated phase signal sequenceAnd outputs: subtracting 1 from the highest order number, and executing the steps (2) to (7) until P is equal to 1;
step 9, let the reduced order signal sequence xpExecuting the steps (4) to (6), and calculating a first-order parameter estimatorAnd outputting, wherein the calculation formula is as follows:
wherein N isPFor reducing the order of the signal sequence xPThe length of (a) of (b),is the frequency point corresponding to the maximum value of the amplitude function | f (Ω) |, where Ω is the frequency variable.
The technical effects of the invention are further explained by combining simulation experiments as follows:
1. simulation conditions and contents:
simulation conditions are as follows: MATLAB 7.5.0, Intel (R) Pentium (R)2CPU 3.0GHz, Window 7 Professional.
Simulation content: computer simulation invention, HAF method and CPF-HAF method respectively perform 8-order, 7-order, 6-order and 5-order parameters a on 8-order phase polynomial signals8,a7,a6,a5The estimated minimum mean square error varies with the signal to noise ratio, the result of which is shown in fig. 2.
2. And (3) simulation result analysis:
referring to fig. 2, graphs (a), (b), (c) and (d) are 8-, 7-, 6-and 5-order parameters a of the 8-order phase polynomial signal of the present invention, the HAF method and the CPF-HAF method, respectively8,a7,a6,a5The horizontal coordinate is the signal-to-noise ratio, and the vertical coordinate is the minimum mean square error of parameter estimation; wherein, CRLB is the cramer-mello lower bound of the parameter estimation, i.e. the minimum mean square error lower bound of the parameter estimation.
From fig. 2, it can be seen that compared with the HAF method and the CPF-HAF method, at a lower signal-to-noise ratio, the minimum mean square error estimated by the present invention is already close to the cramer-circle lower bound, i.e. has a lower estimation threshold; meanwhile, the minimum mean square error estimated by the method is closer to the lower boundary of Cramer-Rao, and the method has higher estimation precision.

Claims (6)

1. A high-order polynomial phase signal parameter estimation method based on a reduced kernel function comprises the following steps:
(1) uniformly sampling a high-order polynomial phase signal x (t) of mixed Gaussian white noise to obtain a high-order polynomial phase signal sequence x (n):
wherein n is the discrete time variable of the phase signal sequence x (n), w (n) is the Gaussian white noise sequence, Delta is the sampling interval of the phase signal x (t), A0Amplitude of the phase signal x (t) { a1,...,ar,...,aPP is the phase parameter to be estimated corresponding to the order r, and P is the highest phase parameter of the phase signal sequence x (n)The order is more than or equal to 3;
(2) utilizing the highest order P of the high-order polynomial phase signal sequence x (n) to construct a reduced order kernel function ker [ x (n) ] of the high-order polynomial phase signal sequence x (n), the implementation steps are:
2a) defining a reduced order kernel ker [ x (n) ] for constructing a high order polynomial phase signal sequence x (n)]The required parameters and the initialization of these parameters: defining a phase parameter a to be estimatedPCorresponding order variable P', odd reduced iteration times i, even reduced iteration times v and mark vector l(i+v)Initializing the parameters to obtain an initial order variable P', an initial odd-order reduction iteration number i equal to 0, an initial even-order reduction iteration number v equal to 0, and an initial label vector l(0)＝[]；
2b) Judging the order variable P'; if P ' ≠ 1 and is even, execute step 2c), if P ' ≠ 1 and is odd, execute step 2d), if P ' ≠ 1, output token vector l(M+H)And performing step 2e) wherein l(M+H)＝[l1,...,lk,...,lM+H]The number k is 1.. multidot.m + H, M is a marker vector l(M+H)The number of middle 0 elements, H is a mark vector l(M+H)The number of the middle 1 element;
2c) reducing the order of the order variable P 'to obtain P'/2, making the even order-reducing iteration number v ═ v +1, and marking the vector l(i +v-1)End addition of element 0vTo obtain a mark vector l(i+v)＝[l(i+v-1),0v]And performing step 2b) of 0vIs a mark vector l(i+v)0 element obtained when the number of the even-middle reduced order iterations is v;
2d) reducing the order of the variable P 'to obtain P' -1, making the odd order-reducing iteration number i equal to i +1, and making the odd order-reducing order PiP' in the mark vector l(i+v-1)End addition element 1iTo obtain a mark vector l(i+v)＝[l(i+v-1),1i]And performing a step 2b), wherein 1iIs a mark vector l(i+v)1 element, P, obtained when the number of iterations of the odd-middle order reduction is iiThe order is the order when the odd order-reducing iteration times are i;
2e) according to the mark vector l(M+H)The value sequence of the medium elements is used for constructing a reduced kernel function ker [ x (n) of a high-order polynomial phase signal sequence x (n)]；
(3) According to a reduced order kernel function ker [ x (n)]Performing spline interpolation on the phase signal sequence x (n) to obtain a non-uniform interval sequence of the phase signal sequence x (n)
(4) Calculating a reduced order signal sequence xp: non-uniform interval sequenceSubstituting a reduced order kernel ker [ x (n)]To obtain a reduced order signal sequence xp：
(5) For reduced order signal sequence xpPerforming fast Fourier transform to obtain reduced signal sequence xpThe frequency domain function f (Ω);
(6) calculating an estimate of a parameter to be estimated in a sequence of phase signals x (n)And outputs: calculating a frequency point corresponding to the maximum value of the amplitude function | f (omega) | of the frequency domain function f (omega) by adopting a one-dimensional search methodAnd according to frequency pointCalculating an estimate of a parameter to be estimated in a sequence of phase signals x (n)And outputting;
(7) demodulating the phase signal sequence x (n) to obtain a reduced demodulation sequence x' (n):
(8) updating the phase signal sequence, making x (n) equal to x' (n), obtaining the phase signal sequence x (n) after reducing, circularly constructing a reducing kernel function, and sequentially calculating the estimators of the parameters to be estimated of the updated phase signal sequenceAnd outputs: subtracting 1 from the highest order number, and executing the steps (2) to (7) until P is equal to 1;
(9) order-reduced signal sequence xpExecuting the steps (4) to (6), and calculating a first-order parameter estimatorAnd output.
2. The method for estimating phase signal parameters of higher order polynomials based on reduced order kernels according to claim 1, wherein the reduced order kernels ker [ x (n) ] in step 2e) are expressed as:
wherein, a0,a0+1,...b0The value range of the discrete-time variable n in the phase signal sequence x (n),being a sequence operator, comprisingAndtwo kinds of fortuneCalculating, lkWhen equal to 0, the signal is processedOperate on and update the sequenceThe value range of the discrete-time variable m,/, iskWhen 1, the signal is processedOperate on and update the sequenceThe value range of the discrete time variable n is as follows:
n＝ak,ak+1,...,bk ak＝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,...,bkIs composed ofThe value range of the discrete time variable of the sequence after operation,in order to get the whole upwards,in order to get the whole downwards,non-uniform spacer sequences introduced for the calculation process.
3. The method for estimating parameters of higher-order polynomial phase signal based on reduced-order kernel function as claimed in claim 2, wherein said step (6) of calculating the estimation quantity of the parameter to be estimated in the phase signal sequence x (n)The calculation formula is as follows:
wherein N isPFor reducing the order of the signal sequence xPThe length of (a) of (b),is the frequency point corresponding to the maximum value of the amplitude function | f (Ω) |, where Ω is the frequency variable.
4. The method for estimating phase signal parameters of higher order polynomial based on reduced order kernel function as claimed in claim 1, wherein said step (3) is performed by spline interpolation for phase signal sequence x (n), and the interpolation formula is:
wherein x is0(n ') is the desired interpolation point, n' is the desired discrete time, ceil (n ') is the discrete time taken to be greater than n' and immediately adjacent, floor (n ') is the discrete time taken to be less than n' and immediately adjacent.
5. The method for estimating parameters of higher order polynomial phase signal based on reduced order kernel function as claimed in claim 1, wherein said reduced order signal sequence x in step (5)pThe frequency domain function f (Ω) of (a), whose expression is:
wherein N isPFor reducing the order of the signal sequence xPM is a discrete time variable and Ω is a frequency variable.
6. The method for phase signal parameter estimation of higher order polynomials based on reduced order kernels according to claim 1, said calculating the first order parameter estimate in step (9)The calculation formula is as follows:
wherein N isPFor reducing the order of the signal sequence xPThe length of (a) of (b),is the frequency point corresponding to the maximum value of the amplitude function | f (Ω) |, where Ω is the frequency variable.
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