对两个频率相近成分作频谱校正的非迭代形式研究

Non-iterative spectrum correction for signals with closely spaced frequency components

双频率模型(DFM)的频谱校正是否存在类似于单频率模型(CSM)的非迭代校正形式、相应的校正性能尚未见文献报道。分析了CSM存在简单校正公式的原因,给出了DFM加矩形窗的非迭代校正式,采用包含幅度相差10倍的DFM对校正式进行仿真考核。研究表明:CSM存在简单校正式的原因在于窗谱函数可以分解为超越函数与有理分式的乘积,前者在相邻的离散谱线上绝对值相等;由这两个特性可建立DFM相对简单频谱校正关系,但是除了DFM加矩形窗的情形外,其它均涉及高于2次的多项式方程。仿真考核表明,给出的校正式对幅度强的成分的误差小于幅度弱者,并且当DFM频率间隔超过两个经典频率分辨率时,最后采用CSM校正式。

For a complex sinusoid model （CSM）, there exists a set of explicit expressions for spectrum correction. However, it is not clear whether there exist corresponding formulas for double-frequency model （DFM）. To determine the applicability of explicit expressions to DFM, the nature of ratio correction for CSM was analyzed. The explicit correction formulas were presented for DFM without windowing, and were examined by using a DFM signal with one strong component whose amplitude is 10 times greater than the weaker one. The study shows： the essence of existence of simple correction expression for CSM is that the spectrum function with common windows can be factorized as a transcendental part multiplying by a rational fraction, and absolute values of the neighbor lines in discrete spectrum of the transcendental part are equal. With above properties, the frequency equations for DFM, only containing the rational fraction, can be deduced. Only for the case of DFM without windowing, can the frequency equation be simplified to a quadratic polynomial, and at least cubic polynomial for other cases, such as Hanning window. In conclusion, it is not worthy or impossible to find the explicit correction expressions except for DFM without windowing. The simulation results show that, the precision of the given correction expression for the strong component is better than that of the weaker one, with the frequency scanning of 0.5 -2.0 resolutions of the fast Fourier transform. But the CSM based correction is preferred if the frequency difference of DFM is greater than 2 canonical resolutions of FFT.