CN102832942A - Method for extracting features of triangular linear frequency modulation continuous wave based on fractional order Fourier transform - Google Patents

Abstract

The invention relates to a method for extracting features of a triangular linear frequency modulation continuous wave (TLFMCW) based on fractional order Fourier transform, and belongs to the technical field of information countermeasure. The method uses discrete sampling type fractional order Fourier transform as a basic tool, the TLFMCW is transformed to a multiple component linear frequency modulation (LFM) signal. Using the multiple component LFM signal has an energy shock feature in different fractional order domains, characteristic parameter of each LFM signal component is extracted. According to geometrical relationships of the LFM signal components on a time-frequency domain, the method combines a mathematical statistics method and realizes extraction of characteristic parameters of the TLFMCW signal such as modulation period, adjusted frequency, original frequency, bandwidth, etc. The method provided in the invention improves estimated accuracy of the TLFMCW signal parameters in a low signal to noise ratio condition, reduces restriction on signal sampling time, and the method can be used in a LFM pulse signal and a LFM continuous wave signal. The method provides a new approach for design of a future signal characteristic extraction technology.

Description

Triangle linear frequency modulation continuous wave feature extracting method based on Fractional Fourier Transform

Technical field

The present invention relates to a kind of triangle linear frequency modulation continuous wave characteristic parameter extraction method, belong to the information countermeasure technical field based on Fractional Fourier Transform.

Background technology

In recent years; Deepen continuously along with what the LFM signal was studied, LFM pulse signal (LFMP), LFM CW with frequency modulation signal (LFMCW), symmetric triangular linear frequency modulation continuous wave signal (STLFMCW) and asymmetric triangle linear frequency modulation continuous wave signal (NSTLFMCW) have occurred.In a sense, LFMP signal, LFMCW signal, STLFMCW signal and NSTLFMCW signal all belong to triangle linear frequency modulation continuous wave signal (TLFMCW) (seeing table 1).How research effectively detects and to discern its characteristic parameter to this type signal has important theoretical research meaning and engineering application value.

The notion of Fractional Fourier Transform promptly was suggested as far back as nineteen twenty-nine, applied to optical field in the eighties in 20th century, became one of signal processing hot research fields from the nineties.Fractional Fourier Transform is appreciated that to the Chirp base decomposes, therefore is particularly suitable for handling Chirp class signal.Utilize linear frequency modulation (LFM) signal (Chirp signal just) to demonstrate the characteristic of different energy aggregation in the Fractional Fourier territory of different orders, just can realize detection and parameter Estimation the LFM signal through do the peak value two-dimensional search in the Fractional Fourier territory.And the triangle linear frequency modulation continuous wave can be thought to be made up of many LFM Signal, and Fractional Fourier Transform is linear transformation, does not have cross term to disturb, and has more advantage having under the situation of additive noise.

Present method for parameter estimation to the triangle linear frequency modulation continuous wave; Can only estimate the partial parameters of signal mostly; What have requires signal to noise ratio higher, and the requirement that also has just in time is a modulation period to signal sampling, and these methods all do not have well to solve the characteristic parameter extraction problem of this type of signal.

Four types of table 1 triangle linear frequency modulation continuous wave signal

Summary of the invention

The present invention is a kind of triangle linear frequency modulation continuous wave characteristic parameter extraction method based on Fractional Fourier Transform; With discrete sampling type Fractional Fourier Transform is basic tool; The triangle linear frequency modulation continuous wave is converted into many LFM Signal, utilizes many LFM Signal to have the energy impact characteristic, realize the modulating frequency of triangle linear FM signal in different fractional orders territory; The frequency modulation rate; Characteristic parameter such as original frequency and bandwidth is estimated, has been improved the estimated accuracy under the low signal-to-noise ratio situation, and reduced the signal sampling time constraints.

The present invention is divided into the linear FM signal extraction and characteristic parameter is estimated two parts, and the expression formula of said triangle linear frequency modulation continuous wave (TLFMCW) signal (see figure 2) s (t) in one-period T does

$s_{+}\left(t\right)={\mathrm{<figure-callout\; id="2"\; label="Ae\; j"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">Ae</figure-callout>}}^j,$ 0≤t≤T ₊，μ ₊≥0 (1-a)

$s_{-}\left(t\right)={\mathrm{<figure-callout\; id="2"\; label="Ae\; j"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">Ae</figure-callout>}}^j,$ 0≤t≤T _-，μ _-≤0 (1-b)

Wherein, f _Upper, f _LowerBe original frequency; μ ₊, μ _-Be positive negative frequency modulation rate; T ₊, T _-Be the positive negative frequency modulation duration, A is a signal amplitude, cycle T=T ₊+ T _-, T＞0.

The step that realizes the linear FM signal extraction is following:

Step 1, observation signal x (t)=s (t)+w (t) of any one section TLFMCW is sampled sample frequency f _s, sampling time T _sThen sampled signal x (n) is carried out the Fourier conversion of discrete fraction rank.

$X_{\mathrm{\&alpha;}}\left(\frac{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">m</figure-callout>}}{2\mathrm{\&Delta;u}}\right)=\frac{\sqrt{1-j\cot \mathrm{\&alpha;}}}{2\mathrm{\&Delta;u}}\exp \left\{\frac{\mathrm{j\&pi;}[\cot \mathrm{\&alpha;}-\csc \mathrm{\&alpha;}]m^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right\}$

(2)

$\underset{n=-N}{\overset{N}{\mathrm{\&Sigma;}}}\exp \left[\frac{\mathrm{j\&pi;}\cot \mathrm{\&alpha;}{(m-n)}^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right]\exp \left\{\frac{\mathrm{j\&pi;}[\cot \mathrm{\&alpha;}-\csc \mathrm{\&alpha;}]n^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right\}x\left(\frac{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">n</figure-callout>}}{2\mathrm{\&Delta;u}}\right)$

Wherein, α=π p/2,

N=f _sT _s, w (t) is an observation noise.The calculating energy spectrum | X _α(m) | ²

Step 2, employing are worth LFM signal component time-frequency line frequency modulation rate and the intercept valuation on time frequency plane of method to the TLFMCW signal most, as shown in the formula description,

$\{{\mathrm{\&alpha;}}_i,m_i\}=\arg \underset{\mathrm{\&alpha;},m}{\max }{\left|X_{\mathrm{\&alpha;}}\left(m\right)\right|}^2\&GreaterEqual;D$

$\left\{\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_i={-f}_s{\cot \mathrm{\&alpha;}}_i/T_s\\ {\hat{f}}_i=m_i\csc {\mathrm{\&alpha;}}_i/\sqrt{T_s/f_s}\end{array}\right.---\left(3\right)$

Wherein D is a threshold value, get D=10E [| x (n) | ²].

Adopt the CLEAN technology, an every LFM signal component, renewal s (t) and D value of successfully detecting.

The step that realizes the characteristic parameter estimation is following:

Step 1, intercept valuations of all corresponding positive negative frequency modulation rates press the vector of descending composition

With

Corresponding

The vector formed respectively of frequency modulation rate valuation

With

N _-And N ₊The positive negative frequency modulation rate LFM signal component number that detects of expression respectively.Structure time-frequency linear equation group

$\left\{\begin{array}{c}Y={\hat{f}}_{+}+{\widehat{\mathrm{\&mu;}}}_{+}X\\ Y={\hat{f}}_{-}+{\widehat{\mathrm{\&mu;}}}_{-}X\end{array}\right.---\left(4\right)$

Trying to achieve straight-line intersection is (X ₀, Y ₀), to Y ₀Merge identical point, after the descending, obtain new vector f ₀

Positive negative frequency modulation rate, duration, signal period, original frequency and the bandwidth of step 2, calculating TLFMCW signal.

${\widehat{\mathrm{\&mu;}}}_{+}=E\left[{\widehat{\mathrm{\&mu;}}}_{+}\right]=\frac{1}{N_{+}}\underset{i=1}{\overset{N_{+}}{\mathrm{\&Sigma;}}}{\widehat{\mathrm{\&mu;}}}_{+}\left(i\right),$ ${\widehat{\mathrm{\&mu;}}}_{-}=E\left[{\widehat{\mathrm{\&mu;}}}_{-}\right]=\frac{1}{N_{-}}\underset{i=1}{\overset{N_{-}}{\mathrm{\&Sigma;}}}{\widehat{\mathrm{\&mu;}}}_{-}\left(i\right)---\left(5\right)$

$\hat{T}=\frac{1}{2{\widehat{\mathrm{\&mu;}}}_{+}(N_{+}-1)}\underset{i=1}{\overset{N_{+}-1}{\mathrm{\&Sigma;}}}|{\hat{f}}_{+}(i+1)-{\hat{f}}_{+}\left(i\right)|+\frac{1}{2{\widehat{\mathrm{\&mu;}}}_{-}(N_{-}-1)}\underset{j=1}{\overset{N_{-}-1}{\mathrm{\&Sigma;}}}|{\hat{f}}_{-}(j+1)-{\hat{f}}_{-}\left(j\right)|---\left(6\right)$

${\hat{T}}_{+}=\frac{{-\widehat{\mathrm{\&mu;}}}_{-}\hat{T}}{{\widehat{\mathrm{\&mu;}}}_{+}-{\widehat{\mathrm{\&mu;}}}_{-}},$ ${\hat{T}}_{-}=\frac{{\widehat{\mathrm{\&mu;}}}_{+}\hat{T}}{{\widehat{\mathrm{\&mu;}}}_{+}-{\widehat{\mathrm{\&mu;}}}_{-}}---\left(7\right)$

Confirm signal transient frequency f (t ₀) the place frequency range, if f (t ₀) ∈ [f ₀(i), f ₀(i+1)], then the original frequency of TLFMCW signal and bandwidth are respectively

${\hat{f}}_{\mathrm{lower}}=f_0\left(i\right),$ ${\hat{f}}_{\mathrm{upper}}=f_0(i+1)---(8-a)$

$\mathrm{\&Delta;}\hat{f}={\hat{f}}_{\mathrm{upper}}-{\hat{f}}_{\mathrm{lower}}={\hat{T}}_{+}{\widehat{\mathrm{\&mu;}}}_{+}=\hat{T}{\widehat{\mathrm{\&mu;}}}_{-}---(8-b)$

Beneficial effect

1. the triangle linear frequency modulation continuous wave characteristic parameter extraction method of the present invention's proposition can estimate the cycle of signal, positive negative frequency modulation rate, positive negative frequency modulation duration, original frequency and bandwidth.

2. the present invention does not have specific (special) requirements basically to the signals sampling time, can begin from any time of signal to analyze, and under low signal-to-noise ratio, still has high estimation accuracy.

3. the triangle linear frequency modulation continuous wave characteristic parameter extraction method of the present invention's proposition can apply to symmetry and asymmetric triangle linear frequency modulation continuous wave, can also apply to LFM pulse signal and LFM continuous wave signal.

Description of drawings

Fig. 1 is based on the triangle linear frequency modulation continuous wave characteristic parameter extraction scheme sketch map of Fractional Fourier Transform

Fig. 2 triangle linear frequency modulation continuous wave signal time-frequency figure

The TLFM signal in three cycles of Fig. 3 is at the energy spectrum distribution map in Fractional Fourier territory

Fig. 4 triangle linear frequency modulation continuous wave signal time-frequency straight-line intersection figure

Embodiment

A kind of triangle linear frequency modulation continuous wave characteristic parameter extraction method that the present invention proposes based on Fractional Fourier Transform, concrete performing step is following:

(1) the triangle linear frequency modulation continuous wave is sampled, obtains observation signal x (n), x (n) is carried out the Fourier conversion of discrete fraction rank according to formula (2), ask for energy spectrum | X _α(m) | ²

(2) exist | X _α(m) | ²Constitute (α m) carries out two-dimensional search in the plane, combine the CLEAN technology, utilizes frequency modulation rate and the intercept valuation of formula (3) extraction LFM signal component on time frequency plane.

(3) the intercept valuation that step (two) is obtained all corresponding positive negative frequency modulation rates is formed new vector by descending With

The vector that the valuation of corresponding frequency modulation rate is formed respectively

With

According to formula (4) structure linear equation group, calculate intersecting point coordinate (X ₀, Y ₀), to Y ₀After merging identical point and descending, obtain new vector f ₀

(4) calculate positive negative frequency modulation rate, duration and the signal period of TLFMCW signal according to formula (5)-(7).Utilize the single order Fractional Fourier Transform to confirm signal transient frequency f (t ₀) the place frequency range, utilize formula (8) to calculate the original frequency and the bandwidth of TLFMCW signal.

Basic principle below in conjunction with continuous Fractional Fourier Transform is done detailed theoretical demonstration to the practical implementation procedural mode.

Suppose that any one period TLFMCW signal s (t) duration is T _s, comprise N LFM signal,

$s_i\left(t\right)=\left\{\begin{array}{cc}A{\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">e</figure-callout>}}^j,& t\&Element;[t_{i-1},t_i]\\ 0,& t\&NotElement;[t_{i-1},t_i]\end{array}\right.---\left(9\right)$

In the formula, f _0i, μ _iRepresent values of intercept and the frequency modulation rate of i LFM signal on time frequency plane respectively.The Fractional Fourier Transform of s (t) does

$X_{\mathrm{\&alpha;}}\left(u\right)=F_p\left[s\left(t\right)\right]=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{F}_p\left[s_i\left(t\right)\right]---10$

Formula (10) shows that the TLFMCW signal is one section duration T _dIn FRFT can be expressed as a plurality of LFM signal FRFT with.Can get by formula (10)

${\left|X_{\mathrm{\&alpha;}}\left(u\right)\right|}^2={\left|\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{F}_p\right[s_i\left(t\right)\left]\right|}^2---\left(11\right)$

$=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{F}_p^{*}\left[s_i\left(t\right)\right]F_p\left[s_j\left(t\right)\right]$

In the modus ponens (11) any one, can turn to

$F_p^{*}\left[s_i\left(t\right)\right]F_p\left[s_j\left(t\right)\right]$

$={\left|A_{\mathrm{\&alpha;}}\right|}^2{\&Integral;}_{t_{i-1}}^{t_i}{K}_p^{*}(u,t)s_i^{*}\left(t\right)\mathrm{dt}{\&Integral;}_{t_{j-1}}^{t_j}{K}_p(u,t)s_j\left(t\right)\mathrm{dt}$

$=A^2\left|\csc \mathrm{\&alpha;}\right|{\&Integral;}_{t_{i-1}}^{t_i}\exp \{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}[(f_{0i}-u\csc \mathrm{\&alpha;})t+\frac{{\mathrm{\&mu;}}_i+\cot \mathrm{\&alpha;}}{2}{t}^2\left]\right\}\mathrm{dt}---\left(12\right)$

${\&Integral;}_{t_{j-1}}^{t_j}\exp \left\{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000524546400021.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\right[(f_{0j}-u\csc \mathrm{\&alpha;})t+\frac{{\mathrm{\&mu;}}_j+\cot \mathrm{\&alpha;}}{2}{t}^2\left]\right\}\mathrm{dt}$

Wherein 0＜α＜π, and α ≠ 0, pi/2.

1) when i=j, if anglec of rotation α satisfies: f _0i=u csc α _i, μ _i=-cot α _iThe time, formula (12) will obtain maximum A ²T _iT _j| csc α |, T wherein _i=t _i-t _I-1, T _j=t _j-t _J-1This moment mark exponent number p _i=2 α _i/ π.Promptly at two dimensional surface (α, the point (p in u) _i, u _i) locate energy impact of appearance, (α is u) through generating behind the FRFT that signal is carried out different mark exponent numbers on the plane.

2) when i ≠ j, the condition that formula (12) obtains ultimate maximum is: | f _0i-f _0j| → 0, | μ _i-μ _j| → 0.But, have for the TLFMCW signal | f _0i-f _0j| ≠ 0, (α, can not assemble be an energy impact to this up-to-date style (12) in the arbitrfary point in u) on the plane.

This shows that for the one section TLFMCW that comprises N LFM signal, (α N energy pulse (see figure 3) will occur in p) on the plane.(α u) goes up and the LFM signal to be detected more effective, receives the influence of cross term very little on the plane.At this moment, can think to detect and estimation problem to the TLFMCW problem of signal detection based on many LFM Signal of FRFT.

Adopt Fractional Fourier Transform algorithm and CLEAN technology; Through at two dimensional surface (α; P) pulse scans to the LFM signal energy in; Can utilize formula (13) to carry out parameter Estimation, obtain the positive negative frequency modulation rate of TLFMCW signal and the estimated value (see figure 2) of each linear frequency modulation component intercept in the time-frequency coordinate system.

$\{{\widehat{\mathrm{\&alpha;}}}_i,{\hat{u}}_i\}=\arg \underset{\mathrm{\&alpha;},u}{\max }{\left|X_{\mathrm{\&alpha;}}\left(u\right)\right|}^2\&GreaterEqual;D$

$\left\{\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_i=-\cot {\widehat{\mathrm{\&alpha;}}}_i\\ {\hat{f}}_i={\hat{u}}_i\csc {\widehat{\mathrm{\&alpha;}}}_i\end{array}\right.---\left(13\right)$

Wherein D is a threshold value.

Suppose that actual observation triangle linear frequency modulation continuous wave signal model is:

x(t)＝s(t)+w(t) (14)

Wherein, s (t) is by formula (9) decision, and w (t) is a white complex gaussian noise.

Be prone to know Y=|x (t) | ²Obeys index distribution, distribution function does

$p=F\left(y\right|\mathrm{\&lambda;})=\underset{0}{\overset{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HSA00000524546400021.png,HSA00000524546400022.png"\; state="\{\{state\}\}">y</figure-callout>}}{\&Integral;}}\frac{1}{\mathrm{\&mu;}}{e}^{-\frac{y}{\mathrm{\&lambda;}}}\mathrm{dt}=1-e^{-\frac{y}{\mathrm{\&lambda;}}}---\left(15\right)$

Because Fractional Fourier Transform (FRFT) has energy accumulating property to the LFM signal at FrFD, and white complex gaussian noise, it still has a gaussian characteristics the Fractional Fourier territory.Therefore, can satisfy by given threshold D: F (D| λ)=1, desirable λ ≈ E [| x (t) | ²], D=10 λ.When using the CLEAN technology that the LFM signal is detected, whenever successfully detect a LFM signal component, x (t) is replaced by x ' (t), needs this moment upgrade the λ value.That is to say that threshold value M is variable, can guarantee that like this a weak component LFM input comes out.

Formula (13) shows, when the TLFMCW signal is carried out the FRFT detection and estimates, can estimate the frequency modulation rate of signal and a plurality of values of intercept of same frequency modulation rate.Utilize the geometrical relationship between a plurality of values of intercept of same frequency modulation rate, can further try to achieve other parameter of TLFMCW signal.Suppose in the one-period T of TLFMCW signal,, can get following relational expression based on Fig. 2

$\left\{\begin{array}{c}{T}_{+}+T_{-}=T\\ T_{+}/T_{-}=-\tan {\mathrm{\&theta;}}_{+}/\tan {\mathrm{\&theta;}}_{-}\end{array}\right.---\left(16\right)$

Again by tan θ ₊=μ _-, tan θ _-=μ _-Can get duration T ₊, T _-

$T_{+}=\frac{{-\mathrm{\&mu;}}_{-}T}{{\mathrm{\&mu;}}_{+}-{\mathrm{\&mu;}}_{-}},$ $T_{-}=\frac{{\mathrm{\&mu;}}_{+}T}{{\mathrm{\&mu;}}_{+}-{\mathrm{\&mu;}}_{-}}---\left(17\right)$

The bandwidth of TLFMCW signal does

Δf＝μ ₊T ₊＝μ _-T _- (18)

Known μ ₊And μ _-, according to formula (17) and formula (18), can be by parameter T, T ₊, T _-Obtain other parameter value with among the Δ f any one.To analyze the method for asking cycle T and bandwidth deltaf f respectively below.

The intercept of all corresponding positive negative frequency modulation rates is pressed the vector that descending is formed

With

Corresponding

The vector formed respectively of frequency modulation rate

With

N _-And N ₊Represent positive negative frequency modulation rate LFM signal component number respectively.The cycle T of TLFMCW signal does

$T=\frac{1}{2{\mathrm{\&mu;}}_{+}(N_{+}-1)}\underset{i=1}{\overset{N_{+}-1}{\mathrm{\&Sigma;}}}|f_{+}(i+1)-f_{+}\left(i\right)|+\frac{1}{2{\mathrm{\&mu;}}_{-}(N_{-}-1)}\underset{j=1}{\overset{N_{-}-1}{\mathrm{\&Sigma;}}}|f_{-}(j+1)-f_{-}\left(j\right)|---\left(19\right)$

Use and

structure equations

$\left\{\begin{array}{c}Y=f_{+}+{\mathrm{\&mu;}}_{+}X\\ Y=f_{-}+{\mathrm{\&mu;}}_{-}X\end{array}\right.---\left(20\right)$

If straight-line intersection is (X ₀, Y ₀), to Y ₀Merge identical point, and, obtain new vector f by descending ₀

Intercept form according to linear equation can know that what in fact obtain is the rule straight line in time-frequency domain.The straight line of people having the same aspiration and interest frequency is parallel to each other, and any two straight lines of different frequency modulation rates have the intersection point (see figure 4).Because the positive negative frequency modulation rate of TLFMCW signal has identical bandwidth, positive and negative modulation frequency line segment joins end to end, the ordinate value f of all these intersection points ₀, frequency domain is divided into different zones.Confirm signal transient frequency f (t ₀) the place frequency range, if f (t ₀) ∈ [f ₀(i), f ₀(i+1)], then the original frequency of TLFMCW signal does

f _lower＝f ₀(i)，f _upper＝f ₀(i+1) (21)

Signal bandwidth can be written as again

Δf＝f ₀(i+1)-f ₀(i) (22)

Convolution (13-22) can come out the signal characteristic parameter Estimation.

Below in conjunction with instance the present invention is done the emulation explanation:

Suppose the original frequency f of TLFMCW signal ₊=70MHz, f _-=130MHz; Positive negative frequency modulation rate is respectively μ ₊=30MHz/ μ s, μ _-=-20MHz/ μ s; The positive negative frequency modulation rate duration is respectively T ₊=2 μ s, T _-=3 μ s; Signal bandwidth Δ f=60MHz, sample frequency is f _s=300MHz, sampling number N=4096, the sampling time is about T _s=13.653 μ s.

The absolute error that the correct criterion that detects is defined as initial frequency estimation does not surpass 10%.When satisfying criterion, correctly detect times N and add 1, correct detection probability is defined as P=N/M.Utilize Monte Carlo method to carry out emulation, establishing additive noise is white complex gaussian noise, simulates M=1200 time respectively under the different input signal-to-noise ratios, obtains estimated value, standard deviation and the correct detection probability (see Table 2 with table 3) of parameter.

Above-mentioned simulation result shows that the present invention is higher to the estimated accuracy of each parameter of TLFMCW signal; Under the low signal-to-noise ratio condition, the TLFMCW signal still had higher correct detection probability.

Table 2 triangle linear frequency modulation continuous wave signal parameter is estimated average

The standard variance of table 3 triangle linear frequency modulation continuous wave parameter Estimation

Claims (1)

1. based on the triangle linear frequency modulation continuous wave characteristic parameter extraction of Fractional Fourier Transform, be divided into linear FM signal extraction and characteristic parameter and estimate two parts, amount to four treatment steps.The expression formula of said triangle linear frequency modulation continuous wave (TLFMCW) signal s (t) in one-period T does

$s_{+}\left(t\right)={\mathrm{Ae}}^{j2\mathrm{\&pi;}(f_{\mathrm{lower}}+{\mathrm{\&mu;}}_{+}t/2)t},$ 0≤t≤T ₊，μ ₊≥0 (1-a)

$s_{-}\left(t\right)={\mathrm{Ae}}^{j2\mathrm{\&pi;}(f_{\mathrm{upper}}+{\mathrm{\&mu;}}_{-}t/2)t},$ 0≤t≤T _-，μ _-≤0 (1-b)

Wherein, f _Upper, f _LowerBe original frequency; μ ₊, μ _-Be positive negative frequency modulation rate; T ₊, T _-Be the positive negative frequency modulation duration, A is a signal amplitude, cycle T=T ₊+ T _-, T＞0.

The step that realizes the linear FM signal extraction is following:

Step 1, observation signal x (t)=s (t)+w (t) of any one section TLFMCW is sampled sample frequency f _s, sampling time T _sThen sampled signal x (n) is carried out the Fourier conversion of discrete fraction rank.

$X_{\mathrm{\&alpha;}}\left(\frac{m}{2\mathrm{\&Delta;u}}\right)=\frac{\sqrt{1-j\cot \mathrm{\&alpha;}}}{2\mathrm{\&Delta;u}}\exp \left\{\frac{\mathrm{j\&pi;}[\cot \mathrm{\&alpha;}-\csc \mathrm{\&alpha;}]m^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right\}$

(2)

$\underset{n=-N}{\overset{N}{\mathrm{\&Sigma;}}}\exp \left[\frac{\mathrm{j\&pi;}\cot \mathrm{\&alpha;}{(m-n)}^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right]\exp \left\{\frac{\mathrm{j\&pi;}[\cot \mathrm{\&alpha;}-\csc \mathrm{\&alpha;}]n^2}{{\left(2\mathrm{\&Delta;u}\right)}^2}\right\}x\left(\frac{n}{2\mathrm{\&Delta;u}}\right)$

Wherein, α=π p/2,

N=f _sT _s, w (t) is an observation noise.The calculating energy spectrum | X _α(m) | ²

Step 2, employing are worth LFM signal component time-frequency line frequency modulation rate and the intercept valuation on time frequency plane of method to the TLFMCW signal most, as shown in the formula description,

$\{{\mathrm{\&alpha;}}_i,m_i\}=\arg \underset{\mathrm{\&alpha;},m}{\max }{\left|X_{\mathrm{\&alpha;}}\left(m\right)\right|}^2\&GreaterEqual;D$

$\left\{\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_i={-f}_s{\cot \mathrm{\&alpha;}}_i/T_s\\ {\hat{f}}_i=m_i\csc {\mathrm{\&alpha;}}_i/\sqrt{T_s/f_s}\end{array}\right.---\left(3\right)$

Wherein D is a threshold value, get D=10E [| x (n) | ²].

Adopt the CLEAN technology, an every LFM signal component, renewal s (t) and D value of successfully detecting.

The step that realizes the characteristic parameter estimation is following:

Step 1, intercept valuations of all corresponding positive negative frequency modulation rates press the vector of descending composition

With Corresponding

The vector formed respectively of frequency modulation rate valuation

With N _-And N ₊The positive negative frequency modulation rate LFM signal component number that detects of expression respectively.Structure time-frequency linear equation group

$\left\{\begin{array}{c}Y={\hat{f}}_{+}+{\widehat{\mathrm{\&mu;}}}_{+}X\\ Y={\hat{f}}_{-}+{\widehat{\mathrm{\&mu;}}}_{-}X\end{array}\right.---\left(4\right)$

Trying to achieve straight-line intersection is (X ₀, Y ₀), to Y ₀Merge identical point, after the descending, obtain new vector f ₀

Positive negative frequency modulation rate, duration, signal period, original frequency and the bandwidth of step 2, calculating TLFMCW signal.

${\widehat{\mathrm{\&mu;}}}_{+}=E\left[{\widehat{\mathrm{\&mu;}}}_{+}\right]=\frac{1}{N_{+}}\underset{i=1}{\overset{N_{+}}{\mathrm{\&Sigma;}}}{\widehat{\mathrm{\&mu;}}}_{+}\left(i\right),$ ${\widehat{\mathrm{\&mu;}}}_{-}=E\left[{\widehat{\mathrm{\&mu;}}}_{-}\right]=\frac{1}{N_{-}}\underset{i=1}{\overset{N_{-}}{\mathrm{\&Sigma;}}}{\widehat{\mathrm{\&mu;}}}_{-}\left(i\right)---\left(5\right)$

$\hat{T}=\frac{1}{2{\widehat{\mathrm{\&mu;}}}_{+}(N_{+}-1)}\underset{i=1}{\overset{N_{+}-1}{\mathrm{\&Sigma;}}}|{\hat{f}}_{+}(i+1)-{\hat{f}}_{+}\left(i\right)|+\frac{1}{2{\widehat{\mathrm{\&mu;}}}_{-}(N_{-}-1)}\underset{j=1}{\overset{N_{-}-1}{\mathrm{\&Sigma;}}}|{\hat{f}}_{-}(j+1)-{\hat{f}}_{-}\left(j\right)|---\left(6\right)$

${\hat{T}}_{+}=\frac{{-\widehat{\mathrm{\&mu;}}}_{-}\hat{T}}{{\widehat{\mathrm{\&mu;}}}_{+}-{\widehat{\mathrm{\&mu;}}}_{-}},$ ${\hat{T}}_{-}=\frac{{\widehat{\mathrm{\&mu;}}}_{+}\hat{T}}{{\widehat{\mathrm{\&mu;}}}_{+}-{\widehat{\mathrm{\&mu;}}}_{-}}---\left(7\right)$

Confirm signal transient frequency f (t ₀) the place frequency range, if f (t ₀) ∈ [f ₀(i), f ₀(i+1)], then the original frequency of TLFMCW signal and bandwidth are respectively

${\hat{f}}_{\mathrm{lower}}=f_0\left(i\right),$ ${\hat{f}}_{\mathrm{upper}}=f_0(i+1)---\left(8\right)$

$\mathrm{\&Delta;}\hat{f}={\hat{f}}_{\mathrm{upper}}-{\hat{f}}_{\mathrm{lower}}={\hat{T}}_{+}{\widehat{\mathrm{\&mu;}}}_{+}=\hat{T}{\widehat{\mathrm{\&mu;}}}_{-}---\left(9\right)$

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