A fast method for estimating chirp signal parameters.

1、A fast method for estimating chirp signal parametersPETER OSHEASchool of Electrical & Electronic Systems Engineering,Queensland University of Technology, GPO Box 2434, Brisbane. 4001.AUSTRALIA..au/people/osheap/Abstract: - This paper presents a fast algorithm for estimating the p

2、arameters of a linearly frequency modulated (chirp) signal. The algorithm is computationally efficient, especially when compared to the maximum likelihood method. The parameter estimates are close to optimal at high signal to noise ratio (SNR). Simulations are provided to verify the results.Key-Word

3、s: - Parameter estimation, Wigner Distribution, Phase interpolation.1 IntroductionAssume that a complex linear frequency modulated (FM) signal is specified by:, . (1)If this signal is immersed in additive complex white Gaussian noise, zw(n), of power, , the observed signal assumes the form:, (2)It i

4、s assumed that N is odd and that the sampling rate is unity. The parameters may be optimally estimated from the observed sequence with the maximum likelihood (ML) method. If the ML estimates of b0, a0, a1 and a2 are denoted respectively by , , and , then these estimates can be found according to 1:(

5、3)(4)(5)The problem with the ML procedure is that it is computationally intensive, with the maximisation in (3) requiring a 2D search. Various alternative parameter estimation methods have been suggested in an effort to reduce the computation of the parameter estimation task. They are reviewedin 2.

6、Some are based on unwrapping the phase and estimating the parameters by a subsequent least-squares fit. Others are based on time-varying parametric spectral estimation. Yet others re-express(3) in a different form, and then take advantage of this form to yield a sub-optimal but computationally simpl

7、er algorithm 3. The method proposed in this paper is similar in spirit to the one in 3, but is computationally simpler.2 Basis for the new algorithmFor the method in this paper, the maximisation in (3) is re-stated as a maximisation involving a line integration in the Wigner domain 4:(6)where Re. de

8、notes the real part and is the one-sided Wigner distribution (WD) defined by:(7)Analogously to the approach in 3, it is possible to abbreviate the line integration in (7), albeit at the expense of global optimality. In particular, the following line integration, which employs two WD slices rather th

9、an N of them, is useful:(8)The particular form for the abbreviated line integration in (8) was chosen for a number of reasons. Two slices only are used in an effort to limit the computation as much as possible. Since the two slices are centred at n=-(N+1)/4 and n=(N+1)/4, they contain no common samp

10、les and are thus independent. It is convenient at this point to define the following:.(9) and correspond respectively to the instantaneous angular frequencies (IAFs) of the signal at times, -(N+1)/4 and (N+1)/4, the centre positions of the two slices. Using (9) one can express (8) equivalently as: (

11、10)or as: (11)where (12)(12) can be further simplified to: (13) (14)Equations (11), (13) and (14) form the basis for the fast algorithm presented in this paper. (13) and (14) clearly involve the estimation of IAFs by maximisation of one-sided WDs. It is useful to enter into some discussion about the

12、 maximisation of these WDs before formally introducing the new method.Maximisation of a one sided WD can be implemented with the aid of an FFT. A search is first performed on the FFT bins to locate the coarse maximum, and then a “fine maximization” is implemented. The fine maximisation can be achiev

13、ed by either a Newton algorithm or with the computationally simpler alternative explained in the following paragraphs.Note that for a given value of n, the product, zr(n+m)zr*(n-m), in the WD transforms a linear FM signal into a sinusoidal signal with zero initial phase. Consequently, the WD of a li

14、near FM signal (at a given n) is essentially the Fourier transform of a sinusoid with known initial phase. Because of this knowledge of the initial phase, one can use the phase near the WD peak to estimate where the peak is. This is explained mathematically below.Assume that the WD of the noiseless

15、linear FM signal in (1) is being evaluated at some time, n, and that the resulting WD slice is M samples long. Assume further that this WD slice is implemented by taking the M-point FFT of zs(n+m)zs*(n-m). The FFT bin closest to the true maximum will correspond to the true IAF, , plus an offset, . T

16、hen at this bin position, the WD phase, , will be: (15)The angular frequency offset, , then, may be calculated from the relation: (16)(16) is the equation which implements the fine maximisation. It is seen to be extremely computationally efficient and very simple to implement. When additive noise is

17、 present in the signal, FFT maximization based on this type of phase interpolation gives rise to an estimate whose variance is only 33 % higher than if obtained with exact maximisations 5. The estimation of an IAF using this approach, then, can be summarised as follows:1) use an FFT of the product,

18、zr(n+m)zr*(n-m),to determine the IAF corresponding to the “coarse maximum”.2) evaluate the phase at the coarse maximum and apply (16) to estimate the offset from the true IAF. Subtract this offset from the coarse IAF estimate to get the refined estimate.3 The parameter estimation algorithmStep 1.Obt

19、ain initial estimates of the IAFs with two K point WDs (K is chosen to be just large enough so that the IAF estimation is above threshold for the minimum required SNR): (17) (18)where is the WD of the K samples of zr which are centered around time, n. “arg maxp” denotes the “argument which maximizes

20、” but with the maximisation comprising 1) a coarse maximization by searching over the bins of an FFT, and 2) a refinement achieved by using the phase interpolation approach described in Section 2.Step 2. Refine the above IAF estimates using an iterative phase interpolation procedure. That is, set Q=

21、ceil(N/2K), L=ceil(K/2), cum1=|zr(-(N+1)/4)|2 and cum2=|zr(N+1)/4)|2, where ceil(.) denotes rounding up to the nearest integer. Then iterate the refinement with the following loop:For q=1:Q; (19); (20); (21); (22)End;Step 3. Compute the final a2 estimate by scaling the IAF estimates from Step 2: (23

22、)Step 4. Obtain the final a1 estimate byaveraging the estimates of the IAF at n= - (N+1)/4 and (N+1)/4 and refining with phase interpolation: (24)Step 5. Find the b0 and a0 estimates: (25) (26)The efficient procedure inherent in the algorithm is well suited to SNRs above -2dB. For lower SNRs, the fi

23、nal parameter estimates have mean-square errors (MSEs) in excess of 3dB above the CR bound, and as such, are not “good” estimators. For SNRs above -2dB, the computation for the algorithm requires a few multiples of N operations and is thus O(N).4 SimulationsA signal with the parameter values, and N=

24、1023 was immersed in complex white Gaussian noise and the parameters were estimated with the new algorithm at various SNRs. 200 simulations were run at each SNR value. Figure 1 shows the MSEs of the a2, a1 and b0 parameters as a function of SNR. The MSEs are marginally above the CR bound (full line)

25、 above a threshold of about -2dB. Theoretical derivations for the MSEs of the parameter estimates have also been done in the appendix and the resultant MSEs are plotted in the graphs (broken line). The measured MSEs are close to the theoretically predicted MSEs. Part of this work was performed at th

26、e School of Electrical and Computer Systems Engineering, RMIT.5 ConclusionA fast algorithm for estimating the parameters of alinear FM (chirp) signal has been presented. The algorithm yields estimates which are close to the Cramer-Rao bound above -2dB.References:1 P. Bello, “Joint estimation of dela

27、y, Doppler and Doppler rate”, IRE Trans. on IT, 1960, pp.3300-341.2 B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal”, Proc. of the IEEE, Vol.8, 1992, pp.520-568.3 S. Peleg and B. Porat, Linear FM signal parameter estimation from discrete-time observations, IEEE Tran

28、s. on AES, V27, 1991, pp.607-615.4 S. Kay and G. Boudreaux-Bartels, “On the optimality of the Wigner distribution for det- ection”, Proc of ICASSP, 1985, pp.1017-1020.5 D. McMahon and R. Barrett, “An efficient method for the estimation of the frequency of a single tone in noise from the phases of di

29、screte Fourier transforms”, Signal Processing, 1986, pp.169-177.6 Appendix: Statistical analysisIt is known that the maximum of the real part of a one-sided M-point WD gives an unbiased estimate of the IAF of a linear FM signal, and that the MSE of the IAF estimate is 2: (27)Now in (11) is obtained by obtaining two IAF estimate