摘要:
本文研究了如下的带噪声中的指数信号模型$Y_j(t)=\sum_{i=1}^n a_{i s} \lambda_i+o_g(t), \quad t=0,1, \cdots, n-1, j=1, N$其中λ1,λ2,…,λq是未知的模为1的复参数,λq+1,…,λp是未知的模小于1的复参数。并假设λ1,λ2,…,λq不相同,p已知,q未知,aky（k=1,p,j=1,N）为未知的复参数。ej（t）(t=0,n-1,j=1,N)为独立同分布的复随机噪声变量,且有其中σ2未知, $E_{\sigma_1}(0), E\left|\epsilon_{\theta_1}(0)\right|^2=\sigma^2, 0<\sigma^2<\infty, E\left|\sigma_1(0)\right|^4<\infty$ 本文给出了 1.q的强相合估计、 2.λ1,λ2,…,λq,σ2及|akj|（k≤q）的强相合估计; 3.上述某些估计的极限分布; 4.λk及akj（k>q）不存在相合估计的证明; 5.N→∞情形的讨论。
Abstract:
This paper studies the model of superimposed exponential sigdals in noise: $Y_f(t)=\sum_{i=1}^n a_{i s} \lambda_i+o_g(t), \quad t=0,1, \cdots, n-1, j=1, N$ whereλ1,…,λq are unknown complex parameters with module 1, λq+1,…,λp. are unknown complex parameters with module 1, λq+1,…,λp are unknown complex parameters with module less than 1, λ1,…,λq are assumed distinct, p assumed known and q unknown. aky, k=1,…p, j=1, …, N are unknown complex parameters. ej（t）,t=0,l ,…, n-1,j=1,…, N, are i.i.d. complex random noise variables such that $E_{\sigma_1}(0), E\left|\epsilon_{\theta_1}(0)\right|^2=\sigma^2, 0<\sigma^2<\infty, E\left|\sigma_1(0)\right|^4<\infty$ and σ2 is unknown. This paper gives: 1. A strong consistent estimate of q; 2. Strong consistent estimates of λ1,…,λq, σ2 and|aky|,k＜q; 3. Limiting distributions for some of these estimates; 4. A proof of non-existence of consistent estimates for λk and aky k＞q. 5. Adiscussion of the case that N→∞
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