KR20110116537A - Signal estimating apparatus and signal estimating method thereof - Google Patents

Abstract

Disclosed is a signal estimation method. The signal estimation method includes measuring a fractional Fourier transform amplitude spectrum for a target signal, starting at the maximum value and the coarse level of the fractional Fourier transform spectrum, and ending at a fine level. Determining an optimal transform order of a Fractional Fourier transform (FRFT) by applying a maximum-amplitude-based coarse-to-fine (MACF) algorithm using a coarse to fine technique, and a fractional Fourier transform Estimating a compact fractional Fourier domain using an optimal transform order of FRFT: fractional Fourier transform. This makes it possible to estimate the compact fractional Fourier domain quickly and effectively.

Description

Signal estimating apparatus and signal estimating method

The present invention relates to a signal estimating apparatus and a signal estimating method thereof, and more particularly, to a signal estimating apparatus and a signal estimating method for estimating a compact fractional Fourier domain using a maximum amplification spectrum of a signal.

The fractional Fourier transform is a concept proposed by David Mendlovic and Haldun M.Ozaktas, which assumes that the ordinary Fourier transform is of order 1 and is a common Fourier transform The transform to be transformed is defined as a fractional Fourier transform with an order of 1 / p.

The general Fourier transform is used as a very useful tool for interpreting temporal or spatial information in the frequency domain in various changes. The concept of further expanding the domain of the fractional Fourier transform. The concept of the Fourier transform can be used to handle frequency and temporally or spatially mixed information using the fractional Fourier transform. Accordingly, the fractional Fourier transform is expected to provide a more versatile method in the signal processing domain. Attempts have been made to apply the fractional Fourier transform to various transforms based on the Fourier transform.

On the other hand, more compact signals are useful in some applications. For example, when compressing a signal in the Fourier domain, the simplest method is to store frequencies with very large magnitudes in the spectrum. In this case, the signal energy can only be concentrated on some components. In addition, a recent investigation by Stankovie et al. Demonstrated that compact representation of the signal helps to improve the performance of suppressing cross-terms of Winger variance (V. Namias, "The fractional order Fourier transform"). and its applications to quantum mechanics, "J. Inst. Mathemat. and its Applic., vol. 25, pp. 241-65, 1980).

Fractional Fourier transform (FRFT), on the other hand, is a common form of typical Fourier transform. Furthermore, many of the characteristics that solidify the foundation for various applications are derived from the following papers.

1.H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier

Transform: With Applications in Optics and Signal Processing. New York: Wiley, 2001.

2. A. C. McBride and F. H. Kerr, On Namias's fractional Fourier transforms, IMA J. Appl. Mathemat., Vol. 39, no. 2, pp. 159-175, 1987.

3. H. M. Ozaktas et al., "Digital computation of the fractional Fourier transform," IEEE Trans. Signal Process., Vol. 44, pp. 2141-2150, 1996.

Importantly, if the frequency of the signal varies over time, there is at least one optimal fractional fourier (FRF) domain with the smallest signal bandwidth. The optimal domain is called the compact domain when the signal is only within a predetermined (small) range centered at its origin.

In practice, however, effectively estimating the compact FRFT domain is not easy. Criteria for measuring compactness and searching strategies are two important techniques for doing this. Vijaya and Bhat employed percent root-mean-square difference (PRD) criteria and exhaustive search to calculate compact domains (C. Vijaya and J. S. Bhat, "Signal compression using discrete fractional

Fourier transform and set partitioning in hierarchical tree, "Signal

Process., Vol. 86, pp. 1976-1983, 2006), Stankovie et al. Employs FRF moments to accomplish this task (L. Stankovie, T. Alieva, and MJ Bastiaans, "Time-freuqency signal analysis based on the windowed fractional Fourier transform," Signal Process., Vol. 83, pp. 2459-2468, 2003).

Serbes and Durak proposed two criteria: maximum fractional time-bandwidth ratio and minimum required-bandwidth (A. Serbes and L. Durak, "Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains," Commun.Nonlin.Sci. And Numer.Simul., Vol. 15, no.3, pp. 675-689, Mar. 2009). Instead of a thorough search to find the optimal conversion order, a GA (genetic algorithm) was employed by Serbes and Durak. In general, the best of these methods is time consuming. Theoretically, although the moment-based method is fast, there is a problem in that power is weak for signals composed of multichirp elements.

SUMMARY OF THE INVENTION The present invention has been made to solve the above problems, and an object of the present invention is to provide a signal estimating apparatus and a signal estimating method for estimating a compact FRFT domain effectively.

In accordance with another aspect of the present invention, a signal estimation method includes measuring a fractional Fourier transform amplitude spectrum of a target signal, and a maximum of the fractional Fourier amplitude spectrum. Fractional Fourier Transform (FRFT) by applying a maximum-amplitude-based coarse-to-fine (MACF) algorithm that uses a coarse to fine technique starting at the value and coarse levels and ending at the fine level. Determining an optimal transform order of the Fourier transform and estimating a compact fractional Fourier domain using the optimal transform order of the Fractional Fourier transform (FRFT).

Here, the determining of the optimal transform order, the maximum value of the fractional Fourier amplitude spectrum is determined by Equation 4 below,

 The optimal conversion order may be determined by the following equation.

Where J (α) is the maximum value of the fractional Fourier amplitude spectrum, F is the fractional Fourier transform, α is the order of the fractional Fourier transform, α _opt is the optimal order of the fractional Fourier transform, f (u) is the target signal Indicates.

In addition, the coarse to fine technique, (a) α _begin = 0, α _end = 1, Δα = Δα ₀ , (b) α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end }, (C) performing FRFT of the input signal for each α, (d) obtaining α _opt using Equations 4 and 5, and (e) α _begin = max {0, α _opt -0.5Δα}, α _end = min {1, α _opt It may include repeating the step after step (b) until the conditions of + 0.5Δα}, Δα = Δα × λ, Δα ≦ ε.

On the other hand, the signal estimation device according to an embodiment of the present invention, the spectrum measurement unit for measuring the fractional Fourier transform amplitude spectrum (target fraction) for the target signal, the maximum value and the wide range of the fractional Fourier amplitude spectrum ( Optimal fractional Fourier transform (FRFT) by applying a maximum-amplitude-based coarse-to-fine (MACF) algorithm that uses a coarse to fine technique starting at the coarse level and ending at the fine level. And an operation estimator for determining a transform order, and a signal estimator for estimating a compact fractional Fourier domain using the determined optimal transform order.

Here, the calculation unit, the maximum value of the fractional Fourier amplitude spectrum is determined by the following equation,

The optimal conversion order may be determined by the following equation.

Where J (α) is the maximum value of the fractional Fourier amplitude spectrum, F is the fractional Fourier transform, α is the order of the fractional Fourier transform, α _opt is the optimal order of the fractional Fourier transform, f (u) is the target signal Indicates.

In addition, the coarse to fine technique, (a) α _begin = 0, α _end = 1, Δα = Δα ₀ , (b) α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end }, (C) performing FRFT of the input signal for each α, (d) obtaining α _opt using Equations 4 and 5, and (e) α _begin = max {0, α _opt -0.5Δα}, α _end = min {1, α _opt It may include repeating step (b) after the condition of + 0.5Δα}, Δα = Δα × λ, Δα ≦ ε.

This makes it possible to estimate fractional Fourier domains quickly and effectively.

1 is a block diagram illustrating a configuration of a signal estimating apparatus according to an exemplary embodiment.
2 is a diagram for explaining the physical meaning of the fractional Fourier transform.
3A-3C schematically illustrate the main idea of the coare-to-fine procedure.
4 is a diagram for describing a signal estimation method according to an exemplary embodiment.
5A to 5C are diagrams illustrating amplification spectra in the compact domain according to an embodiment of the present invention.
6 shows amplification of the collective signal in the compact domain.

Hereinafter, with reference to the accompanying drawings will be described in detail with respect to the present invention.

1 is a block diagram illustrating a configuration of a signal estimating apparatus according to an exemplary embodiment.

According to FIG. 1, the apparatus 100 for estimating a signal includes a spectrum measuring unit 110, a calculating unit 120, and a signal estimating unit 130.

Meanwhile, since the signal estimating apparatus 100 according to an exemplary embodiment uses a Fractional Fourier Transform, it will first be briefly described.

The Fractional Fourier Transform is a concept proposed by David Menlovic and Haldun Ozaktas, which considers a typical Fourier transform as the case of order 1, and converts the normal Fourier transform to order 1 / It is defined as the fractional Fourier transform of α.

The general Fourier transform is a very useful tool for interpreting temporal or spatial information in the frequency domain, and the fractional Fourier transform is an extension of the general Fourier transform domain, which allows frequency and temporal or spatial information to be appropriately ordered. Can handle mixed information

2 is a diagram for explaining the physical meaning of the fractional Fourier transform.

In general, the physical meaning of a Fourier transform is a transform from a time domain signal to a frequency domain signal, and the physical meaning of an inverse Fourier transform is a transform from a frequency domain signal to a time domain signal. Similarly, a fractional Fourier transform can transform a signal in the time domain or frequency domain into a domain between time and frequency.

 According to FIG. 2, when the signal in the time domain is a rectangular signal, it may be a sinc function in the frequency domain. However, when applying a fractional Fourier transform with a square signal, the transform output is in the domain between time and frequency.

The spectrum measuring unit 110 measures a fractional Fourier transform amplitude spectrum for the target signal.

The calculating unit 120 determines the optimal transform order of the Fractional Fourier transform (FRFT) by applying a MACF (maximum-amplitude-based coarse-to-fine) algorithm. Here, MACF means an algorithm using a coarse to fine technique starting at the maximum value and the coarse level of the fractional Fourier amplitude spectrum and ending at the fine level. Specifically, MACF employs a maximum value of the fractional Fourier amplitude spectrum to measure the compactness of a signal, starting at a relatively coarse level and ending at a relatively fine level to increase the speed of the search process. Has characteristics.

Meanwhile, the maximum value of the fractional Fourier amplitude spectrum used in the MACF algorithm may be measured by the spectrum measuring unit 110.

The signal estimator 130 estimates a compact fractional Fourier domain by using an optimal transform order of a fractional Fourier transform (FRFT) determined by the calculator 120. Here, the FRFT is a single transform and may have continuous characteristics for the transform order.

Hereinafter, the MACF (maximum-amplitude-based coarse-to-fine algorithm) will be described in detail.

First, the FRFT applied to the MACF is a linear transformation having a transformation order α∈ [0,4). Mathematically, the signal f (t) is mapped to (F ^α f) (u) by the following equation (1).

The conversion kernel, K (.), Is represented as in Equation (2).

, n은 정수,

는 변환 차수에 대응되는 회전각을 나타내고, exp(.)는 지수 함수이다. Where j =

, n is an integer,

Denotes a rotation angle corresponding to the conversion order, and exp (.) Is an exponential function.

On the other hand, the value of FRFT which concerns on this invention contains the following.

1) Index rule: for all α, β

는 단일 연산자이며, Parseval의 정리(Parseval's theorem)를 따른다.2)

Is a single operator and follows Parseval's theorem.

는 주기 4를 갖는다:

3)

Has cycle 4:

The value according to equation (2) indicates that the energy of the signal does not depend on the conversion order. That is, as shown in equation (3).

의 최대값이 클수록 대응하는 분수차 푸리에 도메인이 더 컴팩트해질 수 있다. 또한,

이 최대값을 갖는 경우, 대응하는

도 최대값이 되어야 한다. 따라서, 신호의 α번째 차수 FRFT의 조밀도를 측정하기 위해 다음 수학식 4와 같은 기준이 적용된다.Thus, if the signal has compact support in the fractional Fourier domain, the corresponding energy spectrum must have a large value for some u and very small for another u. In other words,

The larger the maximum of, the more compact the corresponding fractional Fourier domain can be. Also,

If this has a maximum value, the corresponding

Should also be the maximum value. Therefore, in order to measure the density of the α-th order FRFT of the signal, a criterion as shown in Equation 4 below is applied.

Thus, the optimal transform order is obtained by equation (5).

The problem now is that it is necessary to perform several FRFTs in order to obtain an optimal order. This problem can be solved using FRFT values according to equations (1) and (3). Explain that the latter is α∈ [0,4). Since FRFT is a rotational operation on time frequency variance, the range of α can be further reduced to [0,1] for even symmetric signals. Since we can assume f (t) = f (-t) (t <0), this result applies to most real signals. Thus, α _opt It is not necessary to perform a FRFT on each α in order to calculate the value correctly. Alternatively, you can start at the coarse level and end at the fine level.

3A-3C schematically illustrate the main idea of the coarse-to-fine procedure.

At the starting point, J (α) is calculated from the values of α which are rarely distributed over the interval [0,1]. Α _opt calculated by Equation 5 Better search can be done on small intervals around the value. This procedure is achieved with α _opt The value may be repeated until the value satisfies the predefined accuracy requirements.

The above-described MACF algorithm can be executed by the following five steps (Δα, λ, ε is a positive number less than 1).

Step 1: α _begin = 0, α _end = 1, Δα = Δα ₀

Step 2: α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end } Is called.

Step 3: Perform FRFT of the input signal for each α.

Step 4: Find α _opt using Equations 4 and 5.

Step 5: α _begin = max {0, α _opt 0.5Δα}, α _end = min {1, α _opt + 0.5Δα}, Δα = Δα × λ, Δα ≦ ε and return to step 2.

Step 5 must be performed repeatedly for a number of T loops having T determined by Equation 6 below.

Here, ┌ 천장 means a ceiling function (ceiling function).

The total number of FRFTs required by the MACF is calculated by the following equation.

Here, └ ┘ means floor function.

The computational complexity of a FRFT is O (N _FRFT × NlogN) because the computational complexity of discrete FRFTs can achieve O (NlogN). Where N is the length of the discrete signal, and N _FRFT is determined by _{Δα 0} , λ, ε.

4 is a diagram for describing a signal estimation method according to an exemplary embodiment.

According to FIG. 4, first, a fractional Fourier transform amplitude spectrum of a target signal is measured (S410).

Next, we apply a maximum-amplitude-based coarse-to-fine (MACF) algorithm that uses a coarse to fine technique starting at the maximum and coarse levels of the fractional Fourier amplitude spectrum and ending at the fine level. The optimal transform order of the fractional Fourier transform (FRFT) is determined (S420).

Subsequently, the compact fractional Fourier domain is estimated using the optimal transform order of the determined fractional Fourier transform (FRFT) (S430).

In operation S420, the maximum value of the fractional Fourier amplitude spectrum may be determined by Equation 4 above.

In addition, the optimal conversion order can be determined by the above equation (5).

In addition, the coarse to fine technique applied in step S420 may include the following steps (a) to (e).

(a) α _begin = 0, α _end = 1, DELTA α = DELTA α ₀

(b) α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end };

(c) performing FRFT of the input signal for each α

(d) obtaining α _opt using Equations 4 and 5

(e) α _begin = max {0, α _opt 0.5Δα}, α _end = min {1, α _opt (B) repeating step (b) until the condition of + 0.5Δα}, Δα = Δα × λ, Δα≤ε

Hereinafter, the simulation result of the signal estimation method according to the present invention will be described.

Ozakatas et al (HM Ozaktas et al., “Digital computation of the fractional Fourier transform,” IEEE Trans.Signal Process., Vol. 44, pp. 2141-2150, 1996.). Matlab) was tested for MACF according to the present invention.

Here, the present invention is compared with the GA-EB (GA-based minimum essential-bandwidth) method of Serbes and Durak. This is because it is possible to estimate the optimal fractional Fourier order for a signal whose latter consists of multi-chirp (frequency varying) elements.

It is assumed that t 20 [-20, 20] and the sampling interval is 0.05. The parameters of the MACF are set to Δα ₀ = 0.1, λ = 0.1, ε = 0.01, and it is noted that it is necessary to perform 33 FRFTs to find the optimal conversion order.

First, the synthetic chirp signal generated by the following equation (8) is applied to evaluate the MACF.

Where G _l (t) = A _l (exp-γ _l t ² ) (l = 1, 2) is the envelope for each chirp element, c ₁ and c ₂ are chirp rates, t ₁₀ , t ₂₀ are time shifts (time shifts).

The number of large amplifications is calculated by the following Equation 9 and applied to evaluate the performance of the MACF.

Where M≤N, η _i ∈ {1, 2, ..., N}, (F ^{α opt} f) _{η i} is the η _i th coefficient of the discrete FRFT of the signal, and E is the signal calculated by Equation 3 Is energy.

fix γ ₁ = γ ₂ = 1/60, and in Equation 8 the other parameters A ₁ , A ₂ , c ₁ , c ₂ , t ₁₀ , t ₂₀ are {0.02, 0, 1/3, 0, 0 , 0}, {0.7, 0.3, 1/2, 4/9, 0, 0} and {0.7, 0.3, 1/3, 1/3, 0, 5}, simple chirp signal, two without time shift A signal consisting of two chirp elements, the second signal corresponding to a signal consisting of two chirp elements each having a time shift of 5. The optimum orders obtained by MACF and GA-EB are shown in Table 1 with the number of amplitudes of large magnitude. The amplification spectra in the corresponding compact domains are shown in FIGS. 5A-5C.

5A shows the amplification spectrum in the compact domain of a single chirp signal.

5B shows the amplification spectrum in the compact domain of a multi-chirp signal without time shift.

5C shows an amplification spectrum in the compact domain of a multi-chirp signal with time shift.

The actual population echo location chirp signal with 400 samples is used to demonstrate the efficacy of the MACF. The optimal order is 0.8222 for MACF and 0.782 for GA-EB.

6 shows amplification of the collective signal in the compact domain.

According to FIG. 6, the number of large coefficients is 142 for MACF and 146 for GA-EB.

It can be seen from Table 1 that the optimal order of the single chirp signal estimated for the MACF is 0.6253, which is exactly equal to the theoretical value. These results indicate that, despite the large number of amplitudes, the MACF is more likely to find a compact domain, despite the fact that GA-EB employs a more complex search technique, GA. In addition, this simulation shows that the MACF can find the optimal conversion order for a signal consisting of single chirp elements (with or without time shift) and multi-chirp elements.

As described above, the MACF method for estimating the compact fractional Fourier domain has been proposed in the present invention. The complexity of the calculation is O (N _FRFT x NlogN), and the simulation shows that N _FRFT = 33 is sufficient.

GA-EB (A. Serbes and L. Durak, "Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains," Commun. Nonlin. Sci. And Numer. Simul., Vol. 15, no. 3, pp 675-689, Mar. 2009), although the computational complexity is O (T ₁ × NlogN) (T ₁ is the number of generations of GA), the value of T ₁ is greater than or equal to N _FRFT , which depends heavily on the parameters of GA. May be less. However, since the search algorithm applied to MACF is very simple, MACF is generally faster than GA-EB. In addition, the simulation shows that MACF has better performance in finding the optimal fractional domain on three types of synthesized signals as well as the actual collective echolocation chirp signal.

In addition, although the preferred embodiment of the present invention has been shown and described above, the present invention is not limited to the above-described specific embodiment, the technical field to which the invention belongs without departing from the spirit of the invention claimed in the claims. Of course, various modifications can be made by those skilled in the art, and these modifications should not be individually understood from the technical spirit or the prospect of the present invention.

110: spectrum measurement unit 120: calculation unit
130: signal estimator

Claims (6)

Measuring a fractional Fourier transform amplitude spectrum for the target signal;
Fractions by applying a maximum-amplitude-based coarse-to-fine (MACF) algorithm using a coarse to fine technique starting at the maximum and coarse levels of the fractional Fourier amplitude spectrum and ending at the fine level. Determining an optimal transform order of a fractional Fourier transform (FRFT); And
Estimating a compact fractional Fourier domain using an optimal transform order of the fractional Fourier transform (FRFT). The method of claim 1,
Determining the optimal conversion order,
The maximum value of the fractional Fourier amplitude spectrum is determined by the following equation,

A signal estimating method characterized in that the optimum transform order is determined by the following equation;

Where J (α) is the maximum value of the fractional Fourier amplitude spectrum, F is the fractional Fourier transform, α is the order of the fractional Fourier transform, α _opt is the optimal order of the fractional Fourier transform, f (u) is the target signal Indicates. The method of claim 2,
The coarse to fine technique,
(a) α _begin = 0, α _end = 1, Δα = Δα ₀ ;
(b) α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end };
(c) performing a FRFT of the input signal for each α;
(d) α _opt using Equations 4 and 5 Obtaining (optimal transform order of FRFT); And
(e) α _begin = max {0, α _opt 0.5Δα}, α _end = min {1, α _opt Repeating step (b) until the condition of + 0.5Δα}, Δα = Δα × λ, and Δα ≦ ε are satisfied. A spectrum measuring unit measuring a fractional Fourier transform amplitude spectrum of the target signal;
Fractions by applying a maximum-amplitude-based coarse-to-fine (MACF) algorithm using a coarse to fine technique starting at the maximum and coarse levels of the fractional Fourier amplitude spectrum and ending at the fine level. An arithmetic unit for determining an optimal transform order of a fractional Fourier transform (FRFT); And
And a signal estimator for estimating a compact fractional Fourier domain using the determined optimal transform order. The method of claim 4, wherein
The calculation unit,
The maximum value of the fractional Fourier amplitude spectrum is determined by the following equation,

A signal estimating apparatus for determining the optimum transform order by the following equation;

Where J (α) is the maximum value of the fractional Fourier amplitude spectrum, F is the fractional Fourier transform, α is the order of the fractional Fourier transform, α _opt is the optimal order of the fractional Fourier transform, f (u) is the target signal Indicates. The method of claim 5,
The coarse to fine technique,
(a) α _begin = 0, α _end = 1, Δα = Δα ₀ ;
(b) α = {α _begin , Δα + α _begin , 2 △ α + α _begin , ..., α _end };
(c) performing a FRFT of the input signal for each α;
(d) obtaining α _opt using Equations 4 and 5; And
(e) α _begin = max {0, α _opt 0.5Δα}, α _end = min {1, α _opt Repeating step (b) until the condition of + 0.5Δα}, Δα = Δα × λ, Δα ≦ ε is satisfied.

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