CN103412188A - SFM signal parameter estimation method based on Bessel function and Toeplitz algorithm - Google Patents

Abstract

The invention provides a method combining a Bessel function with a Toeplitz algorithm, and belongs to the technical field of signal processing. According to the method, an SFM signal is processed according to the character of the Bessel function and Jacobi transformation, and then parameters of the SFM signal are detected and estimated in combination with an array signal processing algorithm. The method solves the problem that a signal covariance matrix is not approximate to a true value after the transformation of the SFM signal. The method not only can accurately and comprehensively estimate all the parameters of the signal but also can estimate multi-component SFM signal parameters.

Description

SFM modulated parameter estimating method based on Bessel's function and Toeplitz algorithm

Technical field

The present invention relates to signal processing technology, relate in particular to multiple sinusoidal frequency modulation (SFM) modulated parameter estimating method.

Background technology

Multiple sinusoidal frequency modulation (SFM) signal is a kind of in non-stationary signal, and its instantaneous frequency is a sine function along with the time nonlinearities change.The non-linear of phase function caused the non-stationary of signal, so the frequency of sinusoidal frequency modulation (SFM) signal has a greater change in time again, and presents stronger time local characteristics.The analytic approach of tradition time domain and frequency domain all can not be suitable for by the picture analysis stationary signal like that, therefore needs effective research method process this type of signal.

Time frequency analysis is the effective means of processing non-stationary signal, the most frequently used method is that Wigner-Ville distribution (WVD) is combined with Array Signal Processing, but in the situation that many signals WVD calculates very complex, to sampling rate, require very high, and easily being subject to the interference of cross term, these have all reduced the practicality of these class methods.For this reason, the people such as Li Liping (Li Liping, Huang Keji etc. the Coherent Wideband FM signal 2-D angle of arrival based on STFT is estimated [J]. electronics and information journal, 2005.27(11): 1760-1764.) adopt Short Time Fourier Transform (STFT) to set up spatial time-frequency and distribute, avoided the interference of WVD cross term, calculated simple.But the method limits length of window when the computer memory time-frequency distributions, to meet the approximate constant hypothesis of signal transient frequency.(the PELEGS such as Peleg, FRIEDLANDERB.The discrete poly-nomial-phase transform[J] .IEEE Trans.on Signal Processing, 1995,43(8): 1901-1904.) utilize discrete polynomial-phase conversion to realize the Waveform Reconstructing of sinusoidal FM signal, but parameter estimation algorithm is not provided.The people such as Lv Yuan (Lv Yuan, Zhu Jun, Tang Bin. the nonlinear frequency modulation signal parameter based on DPT is estimated [J]. electronic surveying and instrument journal, 2009,23(6): 63-67.) by discrete polynomial transformation, realize the parameter estimation of multiple sinusoidal FM signal, but in the depth modulation situation, algorithm performance is poor.

In sum, less to the research of the method for parameter estimation of multiple sinusoidal FM signal, or be subjected to parameter limit or can only the estimating part parameter, therefore, the general Processing Algorithm of applicable multiple sinusoidal FM signal also at present.

Summary of the invention

Technical matters to be solved by this invention is that in to multiple sinusoidal frequency modulation (SFM) signal parameter estimation procedure, multiple sinusoidal frequency modulation (SFM) signal exists the covariance matrix after non-stationary and decomposition to keep off the problem of actual value.The SFM modulated parameter estimating method that Bessel's function combines with Teoplitz (Toeplitz) algorithm is proposed, in signal processing, can effectively to the SFM signal, decompose, namely its resolve into one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency and form.Thereby, in conjunction with Toeplitz algorithm reconstruct covariance matrix, then utilize the parameter of the method estimation SFM of signal subspace.The method not only frequency-measurement accuracy is high, and estimated performance is good.

The technical scheme that the present invention solves the problems of the technologies described above is, a kind of method for parameter estimation of SFM signal, a kind of method for parameter estimation of SFM signal, it is characterized in that, adopt Bessel's function treatment S FM signal, obtain the arrowband sinusoidal signal, according to the arrowband sinusoidal signal, set up the covariance matrix that receives signal

By the Toeplitz algorithm to covariance matrix

Carry out pre-service, obtain the covariance matrix R of reconstruction signal _T, to covariance matrix R _TCarry out Eigenvalues Decomposition, obtain signal subspace and the noise subspace of data, utilize the MUSIC algorithm to try to achieve humorous angular frequency, thereby obtain carrier frequency and modulating frequency is estimated, then according to the recurrence relation J of Bessel function of the first kind _V+1(m _f)=2vJ _v(m _f)/m _f-J _V-1(m _f), estimate the index of modulation and harmonic amplitude.

Described Bessel's function treatment S FM signal is specially: by the SFM signal decomposition become one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency and form, its harmonic amplitude is:

Each harmonic frequency is: f ₁=f _c-Vf _m, L, f _V+1=f _c+ 0f _m, L, f _2V+1=f _c+ Vf _m, wherein, A is the amplitude of signal, f _cFor signal(-) carrier frequency, f _mFor signal modulating frequency, m _fFor coefficient of frequency modulation, J _v(k) be first kind v rank Bessel's functions, v=1 ..., V is the Bessel function of the first kind exponent number.

By the Toeplitz pre-service, be specially: call formula: Make matrix near real data covariance matrix R, the element on the clinodiagonal of data covariance matrix is averaged, wherein, S _TIt is the Toeplitz matrix stack.

Call formula

The data covariance matrix of 0≤n<M pair array

Carry out the evaluation computing, obtain a new vector $\hat{r}=({\hat{r}}_T(-(M-1)),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(-1),{\hat{r}}_T\left(0\right),{\hat{r}}_T\left(1\right),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(M-1)),$ By this vector reconstruct data covariance matrix R _T, wherein, M is array number,

For

In element,

For

On clinodiagonal, element is average.To covariance matrix R _TCarry out Eigenvalues Decomposition and be specially, to R _TCarry out Eigenvalues Decomposition, obtain the diagonal matrix D=diag[λ formed by large eigenwert and little eigenwert ₁λ ₂L λ _M], wherein, eigenwert meets relation: λ ₁>=λ ₂>=L>=λ _Q>λ _Q+1=L=λ _M=σ ², according to formula R _n=σ ²I tries to achieve noise power, calls formula: R _T=BR _sB ^H+ R _nTry to achieve its signal covariance matrix R _s, get its real part evolution and namely obtain its harmonic amplitude.Described acquisition arrowband sinusoidal signal specifically comprises: to given SFM signal s (t), constantly it is carried out to discretize at t=n Δ t, obtain the discrete sampling sequence and be { s (n) }; Call formula: $\exp \left(\mathrm{jk}\sin \mathrm{\&beta;}\right)=\stackrel{\&infin;}{\mathrm{\&Sigma;}}{J}_v\left(k\right)\exp \left(\mathrm{jv\&beta;}\right)$ Signal after discretize is carried out to conversion, obtain infinite multinomial sinusoidal signal sum with harmonic amplitude harmonic frequency, according to formula: J _v(m _f)=(-1) ^vJ _-v(m _f) obtain one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency.Recurrence relation according to Bessel's function: J _V+1(m _f)=2vJ _v(m _f)/m _f-J _V-1(m _f), from the top step number of Bessel function of the first kind successively down, to set up all recurrence Relations and form the overdetermined equation group, the least square solution of group of equations, be the index of modulation

The present invention is on the basis of Bessel's function treatment S FM signal, and the method (Toeplitz algorithm, MUSIC algorithm) of employing Array Signal Processing detects and estimate the parameter of SFM signal.The method is treatment S FM signal well, and the Frequency Estimation effect is very good, and is not subjected to parameter limit, and calculated amount is little, can more comprehensively estimate the parameter of SFM signal.

The accompanying drawing explanation

Fig. 1 SFM signal parameter of the present invention is estimated process flow diagram;

The time domain waveform figure of Fig. 2 SFM signal;

The frequency-domain waveform figure of Fig. 3 SFM signal;

Fig. 4 SFM signal transient frequency and the variation relation of time;

Fig. 5 m _fThe harmonic amplitude of=1 o'clock;

Fig. 6 m _fThe harmonic amplitude of=2 o'clock;

Narrow band signal real part after Fig. 7 conversion of the present invention;

The spectrum of each harmonic component after Fig. 8 conversion;

Each humorous angular frequency that Fig. 9 Bessel's function of the present invention combines with the Toeplitz algorithm is estimated.

Embodiment

The present invention carries out frequency detecting and estimation in the method for on the basis of Bessel's function treatment S FM signal, the arrowband sinusoidal signal after decomposing being estimated according to direction of arrival, has solved the problem that the SFM poor signal is processed and limited by other parameter estimation.

Below directly adopt the method for the derivation of equation, derive the carrier frequency f of multiple sinusoidal FM signal _c, modulating frequency f _m, index of modulation m _fAnd harmonic amplitude parameter.

The parameter estimation of SFM signal is specially, and multiple sinusoidal FM signal is expressed as:

s(t)＝Aexp(j(2πf _ct+m _fsin(2πf _mt))),0≤t≤T

（1）

Wherein, A is the amplitude of signal, f _cFor signal(-) carrier frequency, f _mFor signal modulating frequency, m _fFor coefficient of frequency modulation.At t=n Δ t, constantly given signal s (t) discretize is obtained to the discrete sampling sequence and is { s (n) }:

s(n)＝Aexp(j(2πf _c(nΔt)+m _fsin(2πf _m(nΔt)))),0≤n≤N-1

（2）

Wherein, N is sampling length, and Vt is sampling interval.

Receiving signal can be expressed as:

x(n)＝s(n)+W(n)

＝Aexp(j(2πf _c(nΔt)+m _fsin(2πf _m(nΔt))))+W(nΔt),0≤n≤N-1

（3）

Wherein, W is the white Gaussian noise of zero-mean.

To the signal after discretize according to the Jacobi expansion identical relation:

$\exp \left(\mathrm{jk}\sin \mathrm{\&beta;}\right)=\underset{v=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{J}_v\left(k\right)\exp \left(\mathrm{jv\&beta;}\right)---\left(4\right)$

Wherein, J _v(k) be first kind v rank Bessel's functions.Wushu (4) substitution formula (3) can obtain infinite a plurality of form with sinusoidal signal sum of harmonic amplitude harmonic frequency, desirable Vt=1 generally, that is:

$x\left(n\right)=s\left(n\right)+W\left(n\right)$

$=\mathrm{Aexp}\left(j(2\mathrm{\&pi;}{f}_{c}n+m_f\sin \left(2\mathrm{\&pi;}{f}_{m}n\right))\right)+W\left(n\right)$

$=\underset{v=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{AJ}}_v\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{\mathrm{vf}}_m)n\right)+W\left(n\right),0\&le;n\&le;N-1---\left(5\right)$

By formula (5), can be found out, nonlinear multiple sinusoidal FM signal become infinite multinomial linearity sinusoidal signal and form.Wherein,

Symmetric property by Bessel's function: J _v(m _f)=(-1) ^vJ _-v(m _f), and when | v|>| during V|, J _v(m _f) ≈ 0, s (n) part component is little to be disregarded to being left in the basket.Therefore formula (5) can be write as:

$x\left(n\right)\&ap;\underset{v=-V}{\overset{V}{\mathrm{\&Sigma;}}}{\mathrm{AJ}}_v\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{\mathrm{vf}}_m)n\right)+W\left(n\right),0\&le;n\&le;N-1$

$={\mathrm{AJ}}_{-V}\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c-{\mathrm{Vf}}_m)n\right)+L+{\mathrm{AJ}}_0\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+0f_m)n\right)$

$+L+{\mathrm{AJ}}_V\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{\mathrm{Vf}}_m)n\right)+W\left(n\right)---\left(6\right)$

Wherein, V is the top step number that component of signal cannot uncared-for Bessel function of the first kind.Namely work as m _f>1 o'clock, V ≈ m _f+ 1, however m worked as _fDuring ∈ [0.14,1], V=1 or 2; Work as m _f∈ [0,0.14) time, V=0.

By formula (6), can be found out, after Bessel's function decomposes, the SFM signal be broken down into one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency and form, its harmonic amplitude is:

Each harmonic frequency is: f ₁=f _c-Vf _m, L, f _V+1=f _c+ 0f _m, L, f _2V+1=f _c+ Vf _m.Here it is, and Bessel's function decomposes the process of SFM signal, and the method can be good at the SFM signal is processed into to the narrow band signal form that the enough conventional algorithms of energy are estimated.

By formula (6), known, the SFM signal through Bessel's function character and Jacobi's transformation be transformed into one group of arrowband sinusoidal signal and form.Then according to its narrow band signal model, obtain its real data auto-covariance matrix R.In the ideal case, the covariance matrix of array received data

Have Toeplitz character, and in actual conditions, due to the impact that is subjected to error, the covariance matrix of array received data

It is generally diagonally dominant matrix.Therefore by the Toeplitz pre-service, make matrix near real data covariance matrix R at this.

Namely

$\underset{R_T\&Element;S_T}{\min }|R_T-R|---\left(7\right)$

Here S _TIt is the Toeplitz matrix stack.The Toeplitz algorithm is exactly that the element on the clinodiagonal of data covariance matrix is averaged.Can carry out equivalence by following two formulas, that is:

${\hat{r}}_T(-n)=\frac{1}{M-n}\underset{i=1}{\overset{M-n}{\mathrm{\&Sigma;}}}{\hat{r}}_{i(i+n)},0\&le;n<M---\left(8\right)$

${\hat{r}}_T\left(n\right)={\hat{r}}_T^{*}(-n)---\left(9\right)$

Wherein, M is array number, For

In element,

For

On clinodiagonal, element is average.

Utilize the data covariance matrix of formula (8) pair array Carry out the evaluation computing, obtain a new vector $\hat{r}=({\hat{r}}_T(-(M-1)),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(-1),{\hat{r}}_T\left(0\right),{\hat{r}}_T\left(1\right),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(M-1)),$ By this vector reconstruct data covariance matrix R _TThereby, to R _TEigenvalues Decomposition, obtain signal subspace and noise subspace, then utilizes the MUSIC algorithm to carry out the estimation of humorous angular frequency, thereby estimate its carrier frequency and modulating frequency.

More than the carrier frequency of signal, the estimation of modulating frequency.

In order to solve, be subjected to parameter limit or problem that can only the estimating part parameter, the present invention gives the index of modulation and harmonic amplitude is estimated.

By Bessel function of the first kind, can obtain following recurrence relation:

J _v+1(m _f)＝2vJ _v(m _f)/m _f-J _v-1(m _f) （10）

We do following derivation by the recurrence relation of formula (10), suppose that the top step number that Bessel's function decomposes is V, and each exponent number of Bessel's function is respectively-V-V+1 L V.Have following formula to set up:

And formula (11) can resolve into following form:

$J=\left[\begin{array}{c}\\ J_V\left(m_f\right)\\ J_{V-2}\left(m_f\right)\\ M\\ J_{-V+2}\left(m_f\right)\end{array}\right]=\left[\begin{array}{cc}2(V-1)J_{V-1}\left(m_f\right)& {-J}_{V-2}\left(m_f\right)\\ 2(V-2)J_{V-2}\left(m_f\right)& -J_{V-3}\left(m_f\right)\\ M& M\\ M& M\\ 2(-V+1)J_{-V+1}\left(m_f\right)& {-J}_{-V}\left(m_f\right)\end{array}\right]\&CenterDot;\left[\begin{array}{c}\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="m\; f"\; filenames="HDA0000372690970000011.png,HDA0000372690970000031.png"\; state="\{\{state\}\}">m</figure-callout>}}_f}\\ 1\end{array}\right]=\mathrm{Zb}---\left(12\right)$

As can be known according to the definition of overdetermined equation group, formula (12) is the overdetermined equation group, and so-called overdetermined equation group refers to that the equation number is greater than the system of equations of unknown quantity number.

Work as Z ^TBut the Z inverse time, there is least square solution in overdetermined equation group (12), and is system of equations Z ^TZb=Z ^TThe solution of J, i.e. b=(Z ^TZ) ^-1Z ^TJ.So have: $\left[\begin{array}{c}1/{\mathrm{<figure-callout\; id="1"\; label="m\; f"\; filenames="HDA0000372690970000011.png,HDA0000372690970000031.png"\; state="\{\{state\}\}">m</figure-callout>}}_f\\ 1\end{array}\right]={\left(Z^{T}Z\right)}^{-1}{Z}^{T}J.$ Thereby can calculate the index of modulation, least square solution is the index of modulation.

The data covariance matrix R that reconstruct is obtained _TCarry out Eigenvalues Decomposition, thereby obtain the diagonal matrix D=diag[λ formed by large eigenwert and little eigenwert ₁λ ₂L λ _M], and the eigenwert in following formula meets following relation: λ ₁>=λ ₂>=L>=λ _Q>λ _Q+1=L=λ _M=σ ²Thereby, try to achieve noise power R _n=σ ²I; Utilize formula: R _T=E[XX]=BR _sB ^H+ R _n, try to achieve its signal covariance matrix R _s, get its real part evolution and namely obtain its harmonic amplitude.

Below in conjunction with concrete accompanying drawing and simulation example, enforcement of the present invention is done further to describe in detail, but embodiments of the present invention are not limited in this.

Be illustrated in figure 1 the method for parameter estimation process flow diagram that Bessel's function of the present invention and Toeplitz algorithm combine, the SFM signal parameter estimated to concrete steps are as follows, suppose to have the broadband SFM signal of two identical time-frequency distributions:

s ₁(t)＝A ₁exp{j[2π*f _ct+m _f*sin(2π*f _mt)]}

s ₂(t)＝A ₂exp{j[2π*f _ct+m _f*sin(2π*f _mt)]}

Signal parameter is set to: m _f=1, f _c=0.35, f _m=0.03, signal amplitude is A ₁=1, A ₂=2A ₁, SNR=20dB, fast umber of beats N=256.

Step 1: at t=n Δ t constantly to given signal s ₁(t), s ₂(t) discretize obtains the discrete sampling sequence for { s ₁(n), s ₂(n) }.

Step 2: utilize the Jacobi expansion formula: $\exp \left(\mathrm{jk}\sin \mathrm{\&beta;}\right)=\underset{v=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{J}_v\left(k\right)\exp \left(\mathrm{jv\&beta;}\right),$ And after according to the symmetric property of Bessel's function, can obtaining conversion, narrow band signal x (n) can be expressed as:

$x\left(n\right)\&ap;\underset{v=-2}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{AJ}}_v\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{\mathrm{vf}}_m)n\right)+W\left(n\right),0\&le;n\&le;N-1---\left(13\right)$

Through decomposing, formula (13) can be write as following expression-form:

$x\left(n\right)\&ap;\underset{v=-2}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{AJ}}_v\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{\mathrm{vf}}_m)n\right)+W\left(n\right),0\&le;n\&le;N-1$

$={\mathrm{AJ}}_{-2}\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c-2f_m)n\right)+{\mathrm{AJ}}_{-1}\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c-1f_m)n\right)$

$+{\mathrm{AJ}}_0\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+{0f}_m)n\right)+{\mathrm{AJ}}_1\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+1f_m)n\right)$

$+{\mathrm{AJ}}_2\left(m_f\right)\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(f_c+2f_m)n\right)+W\left(n\right)---\left(14\right)$

Step 3: try to achieve by formula (14) data covariance matrix that Array Model receives, adopt the Toeplitz algorithm to carry out pre-service to it, obtain the data covariance matrix of reconstruct.

Step 4: to the data covariance matrix Eigenvalues Decomposition of reconstruct, thereby obtain signal subspace and noise subspace with and the eigenwert space.

Step 5: utilize the spectrum estimation formulas of MUSIC algorithm to try to achieve each humorous angular frequency.

Step 6: thus according to the symmetry of Bessel's function and each the humorous angular frequency estimated, estimate carrier frequency and modulating frequency.

Step 7: as can be known by formula (10), formula (11), work as m _f=1 o'clock, namely the top step number of Bessel function of the first kind was 2 o'clock, and this recurrence relation can be expressed as:

$\left\{\begin{array}{cc}v=1,& J_2\left(m_f\right)={2J}_1\left(m_f\right)/m_f-J_0\left(m_f\right)\\ v=0,& J_1\left(m_f\right)=2{\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="HDA0000372690970000011.png,HDA0000372690970000021.png"\; state="\{\{state\}\}">g</figure-callout>}0\mathrm{gJ}}_0\left(m_f\right)/m_f-J_{-1}\left(m_f\right)\\ v=-1,& J_0\left(m_f\right)=2g(-1){\mathrm{gJ}}_{-1}\left(m_f\right)/m_f-J_{-2}\left(m_f\right)\end{array}\right.---\left(15\right)$

So can obtain following relation by formula (15):

$J=\left[\begin{array}{cc}{2J}_1\left(m_f\right)& -J_0\left(m_f\right)\\ {2\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="HDA0000372690970000011.png,HDA0000372690970000021.png"\; state="\{\{state\}\}">g</figure-callout>}0\mathrm{gJ}}_0\left(m_f\right)& -J_{-1}\left(m_f\right)\\ 2g(-1){\mathrm{gJ}}_{-1}\left(m_f\right)& -J_{-2}\left(m_f\right)\end{array}\right]g\left[\begin{array}{c}\frac{1}{m_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000372690970000011.png,HDA0000372690970000031.png"\; state="\{\{state\}\}">f</figure-callout>}}}\\ 1\end{array}\right]=J_Z\left[\begin{array}{c}\frac{1}{m_f}\\ 1\end{array}\right]---\left(16\right)$

Therefore by $\left[\begin{array}{c}1/{\mathrm{<figure-callout\; id="1"\; label="m\; f"\; filenames="HDA0000372690970000011.png,HDA0000372690970000031.png"\; state="\{\{state\}\}">m</figure-callout>}}_f\\ 1\end{array}\right]={\left({J_Z}^T{J}_Z\right)}^{-1}{J_Z}^{T}J$ Thereby J estimates the index of modulation

Step 8: according to little eigenwert and the equal characteristics of noise power, be R thereby try to achieve its noise power _n=σ ²The little eigenwert of I=, then utilize formula: R _T-R _n=BR _sB ^H, try to achieve its signal covariance matrix R _s, get its real part evolution and namely obtain harmonic amplitude.

In order better to understand the non-stationary property of SFM signal, below from time domain, frequency domain, time-frequency domain, analyze this signal respectively, its oscillogram is shown in respectively Fig. 2, Fig. 3, shown in Figure 4.

By the time-domain expression of the multiple sinusoidal FM signal shown in formula (1), can obtain its instantaneous frequency expression formula and be:

$f_{\mathrm{SFM}}\left(t\right)=f_c+m_f\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000372690970000031.png,HDA0000372690970000032.png"\; state="\{\{state\}\}">m</figure-callout>}}}{2\mathrm{\&pi;}}\cos \left({\mathrm{\&omega;}}_{m}t\right)---\left(17\right)$

As shown in Figure 2, as can be seen from the figure this signal is the signal of a complexity to the time domain waveform of signal, and can find out its duration and signal amplitude relation over time; From frequency domain, see to be Fig. 3, we only know its frequency content, but can't see its time information; Yet the instantaneous frequency of signal is as shown in Figure 4, the information of signal time and frequency can not only be found out, and the frequency of signal and the variation relation of time can be expressed, provided the directviewing description of the T/F of a two dimension.

Fig. 5, Fig. 6 are the harmonic amplitude figure of different modulating frequency of the present invention, and this picture group has proved symmetric property and the v ≈ m of Bessel function of the first kind _f+ 1, when | v|>| m _f| the time, J _v(m _f) ≈ 0, namely s (n) part component is little disregards to being left in the basket.Fig. 5 gets m _f=1, when V>=3, can obviously find out J ₃(1) ≈ 0; Fig. 6 gets m _f=2, when V>=4, can obviously find out J ₄(2) ≈ 0.

As can be known by formula (6), (14), by Bessel's function and Jacobi's transformation signal, become one group of arrowband sinusoidal signal.Its simulation result as shown in Figure 7, Figure 8.Fig. 7 has provided the narrow band signal after the conversion respectively when there is no noise and SNR=20dB, the real part amplitude characteristic figure of signal.As can be seen from the figure after conversion, when there is no noise, signal has been transformed into the signal of arrowband; When SNR=20dB, the signal of reception has been transformed into narrow band signal and noise.Thereby the signal in broadband has been become to the signal of arrowband, be conducive to carry out parameter estimation by the method for comparative maturity.Fig. 8 has provided the spectrum of each harmonic component after the conversion.Namely corresponding with the harmonic frequency of each non-null part in Fig. 5.

As can be known by above analysis, the SFM signal, after the Bessel's function conversion, in conjunction with Toeplitz algorithm reconstruct data covariance matrix, thereby adopts the MUSIC algorithm to carry out parameter estimation.As shown in Figure 9, in the known situation of signal parameter in front, the theoretical value of each harmonic frequency should be its simulation result:

f ₁＝(f _c-2f _m)＝0.29,f ₂＝(f _c-1f _m)＝0.32,f ₃＝(f _c+0f _m)＝0.35。Compose f for these 5 ₄=(f _c+ 1f _m)=0.38, f ₅=(f _c+ 2f _m)=0.41

The peak humorous angular frequency theoretical value of correspondence respectively is:

ω ₁＝2πf ₁＝(f _c-2f _m)*2π＝0.58π,ω ₂＝2πf ₂＝(f _c-f _m)*2π＝0.64π

ω ₃＝2πf ₃＝(f _c)*2π＝0.7π,ω ₄＝2πf ₄＝(f _c+f _m)*2π＝0.76π

ω ₅＝2πf ₅＝(f _c+2f _m)*2π＝0.82π

By Fig. 9 and above Data Comparison, the frequency estimating methods clearly combined with the Toeplitz algorithm can be good at estimating each humorous angular frequency, thereby utilize formula ω=2 π f to estimate each harmonic frequency, and then according to the symmetry of these harmonic frequencies, namely all harmonic frequency sums integral multiple that is carrier frequencies, obtain

The difference of any two adjacent harmonic frequencies is modulating frequencies, obtains

Perhaps

Wherein the actual value of carrier frequency is f _c=0.35, what simulation result obtained is

And the modulating frequency actual value is f _m=0.03, what simulation result obtained is

Therefore in the error allowed band, can estimate accurately carrier frequency and modulating frequency.

Then utilize the recursive nature of Bessel's function, form the overdetermined equation group, according to the theorem of overdetermined equation group, estimate the coefficient of frequency modulation of SFM signal.For the validity of sufficient proof the method, allow the index of modulation be respectively m _f=1 and m _f=2.

As can be known according to the theorem of overdetermined equation group, what estimate should be the inverse of modulating frequency, and the result therefore estimated is: With As can be known by simulation result, the coefficient of frequency modulation estimated and actual value are identical basically, so coefficient of frequency modulation has obtained good estimation.

Due to the method for parameter estimation of SFM signal unsound imperfection also, the parameter estimation of a lot of SFM signals all is subjected to the restriction of partial parameters, perhaps can not be complete estimate its whole parameter, for the integrality of parameter estimation, be below the estimation of its harmonic amplitude.According to theory, calculating each harmonic amplitude is respectively: (A ₁+ A ₂) J ₂, (A ₁+ A ₂) (J ₁), (A ₁+ A ₂) J ₀, (A ₁+ A ₂) J ₁, (A ₁+ A ₂) J ₂, know A by the hypothesis of front ₁=1, A ₂=2, by actual emulation and the Bessel's function table found, can be obtained the actual value of harmonic amplitude.Each rank Bessel's function is respectively: J ₂=0.1149, J ₁=0.4401, J ₀=0.7652.What each harmonic amplitude was estimated the results are shown in Table shown in 1.

Table 1: each harmonic amplitude estimated result

As can be seen from Table 1, the method can not only estimate each harmonic frequency, but also can easier estimate each harmonic amplitude.And each harmonic amplitude and actual theoretical value differ seldom, therefore higher estimated accuracy is arranged.

What the present invention adopted is the SFM signal with 2 components of identical time-frequency distributions, and the method is easy to be generalized to the situation of the SFM signal of a plurality of components.And the method not only can estimate parameters accurately, and not limited by partial parameters, and can more completely estimate parameters of SFM signal etc.

Claims (7)

1. the method for parameter estimation of a SFM signal, is characterized in that, adopts Bessel's function treatment S FM signal, obtains the arrowband sinusoidal signal, according to the arrowband sinusoidal signal, sets up the covariance matrix that receives signal By the Toeplitz algorithm to covariance matrix

Carry out pre-service, obtain the covariance matrix R of reconstruction signal _T, to covariance matrix R _TCarry out Eigenvalues Decomposition, obtain signal subspace and the noise subspace of data, thereby try to achieve humorous angular frequency, obtain carrier frequency and modulating frequency and estimate, according to the recurrence relation J of Bessel function of the first kind _V+1(m _f)=2vJ _v(m _f)/m _f-J _V-1(m _f), estimate the index of modulation and harmonic amplitude.

2. method according to claim 1, is characterized in that, described Bessel's function treatment S FM signal is specially: by the SFM signal decomposition become one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency and form, its harmonic amplitude is: C=[AJ _-V(m _f) L AJ ₀(m _f) L AJ _V(m _f)], each harmonic frequency is: f ₁=f _c-Vf _m, L, f _V+1=f _c+ 0f _m, L, f _2V+1=f _c+ Vf _m, wherein, A is the amplitude of signal, f _cFor signal(-) carrier frequency, f _mFor signal modulating frequency, m _fFor coefficient of frequency modulation, J _v(k) be first kind v rank Bessel's functions, v=1 ..., V is the Bessel function of the first kind exponent number.

3. method according to claim 1, is characterized in that, by the Toeplitz pre-service, is specially: call formula:

Make matrix near real data covariance matrix R, the element on the clinodiagonal of data covariance matrix is averaged, wherein, S _TIt is the Toeplitz matrix stack.

4. method according to claim 1, is characterized in that, calls formula

The data covariance matrix of 0≤n<M pair array

Carry out the evaluation computing, obtain a new vector $\hat{r}=({\hat{r}}_T(-(M-1)),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(-1),{\hat{r}}_T\left(0\right),{\hat{r}}_T\left(1\right),\&CenterDot;\&CenterDot;\&CenterDot;,{\hat{r}}_T(M-1)),$ By this vector reconstruct data covariance matrix R _T, wherein, M is array number,

For

In element,

For

On clinodiagonal, element is average.

5. method according to claim 1, is characterized in that, to covariance matrix R _TCarry out Eigenvalues Decomposition and be specially, to R _TCarry out Eigenvalues Decomposition, obtain the diagonal matrix D=diag[λ formed by large eigenwert and little eigenwert ₁λ ₂L λ _M], wherein, eigenwert meets relation:

λ ₁>=λ ₂>=L>=λ _Q>λ _Q+1=L=λ _M=σ ², according to formula R _n=σ ²I tries to achieve noise power, calls formula: R _T=BR _sB ^H+ R _nTry to achieve its signal covariance matrix R _s, get its real part evolution and namely obtain its harmonic amplitude.

6. method according to claim 1, is characterized in that, described acquisition arrowband sinusoidal signal specifically comprises: to given SFM signal s (t), constantly it is carried out to discretize at t=n Δ t, obtain the discrete sampling sequence and be { s (n) }; Call formula: $\exp \left(\mathrm{jk}\sin \mathrm{\&beta;}\right)=\underset{v=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{J}_v\left(k\right)\exp \left(\mathrm{jv\&beta;}\right)$ Signal after discretize is carried out to conversion, obtain infinite multinomial sinusoidal signal sum with harmonic amplitude harmonic frequency, according to formula: J _v(m _f)=(-1) ^vJ _-v(m _f) obtain one group of arrowband sinusoidal signal with harmonic amplitude harmonic frequency.

7. method according to claim 1, is characterized in that, the estimation of the described index of modulation specifically comprises, according to the recurrence relation of Bessel's function: J _V+1(m _f)=2vJ _v(m _f)/m _f-J _V-1(m _f), from the top step number of Bessel function of the first kind successively down, to set up all recurrence Relations and form the overdetermined equation group, the least square solution of group of equations, be the index of modulation

Images