论时域可分解信号的采样分析

【作者】 冯锦；

【导师】 任长学；

【作者基本信息】 湖南大学 ， 电路与系统， 2002， 硕士

【摘要】 信号的数字化一直是信号处理中的基本问题。在这些信号中有一类信号可以分解为一些已知信号的线性和。本文从纯时域角度出发，给出了可分解信号及其采样的基本概念；也指出了信号可离散化的条件；利用线性代数理论给出了对这类信号进行采样的分析理论及相应的推论；并用这些结论对典型的带限信号进行了分析，证明可分解信号采样定理与经典采样理论的一致性；初步探讨了对这类信号的实行采样与恢复的工程实现问题。 本文通过分析认为，当信号集中的任一信号可表示为一系列已知信号的线性代数和时，信号集便构成可分解信号集，已知信号称为基波信号；对可分解信号而言，基波系数构成一序列，当对指定的基波信号集分解唯一确定时，系数序列本身便是信号的一个表示；可分解信号采样定理指出当基波样值矩阵的秩等于基波数时，则由采样点处的采样样构成的样值序列必与基波系数序列一一对应，从而由该样值序列构成的采样必为完全采样；可分解信号采样定理中的推论指出，对信号集进行采样，采样频率必须大于其信号分解的基波数与其对应时长之比；对有限带宽信号，若基波信号为正弦信号时，由可分解信号采样定理推论给出的结论与经典采样定理一致。更多还原

【Abstract】 The problem how to convert analog into digital signal has been playing a major role in the signal theory. Among various analog signal,there is one kind can expressed as linear sum of existing signal. Taking a view of pure time field,this paper presents dissolvable signals and basic concepts of sampling;And also gives the very condition of discrete of signal;This paper uses linear algebra to analysis this kind of signal and then gives some results and relevant deductions;Based on these results,1 made a further step to analysis some typical band-limited signals in order to proof the coherence of my Theory of Dissolvable Signal to the typical sampling theory;I made some preliminary study about the feasibility of sampling and recovering of this kind of signal.After the scientific analysis above,we can make the following conclusion:when one of signals in one signal pool can be denoted as the linear sum of existing signals,then the signal pool can be thought as the dissolvable signal pool. The signal we named it Fundamental wave;according to the Fundamental wave,coefficients of the fundamental wave can be lined in a sequence . when the unique of the dissolve of the fundamental wave can be confirmed,the sequence of the coefficients can be regarded as one of representation forms of the signal itself;Theory of Dissolvable Signal shows that when order of the matrix of Fundamental wave sampling equals to number of fundamental waves,the sequence of the sampling values from sampling points must be matched one by one with the sequence of the coefficients of fundamental waves. The sampling composed by sequences of the sampling values must be full sampling;The relevant deductions of the Theory of Dissolvable Signal shows that when sampling the signal,sampling frequency must be lager than the ratio of the number of fundamental waves to the occupation time of the fundamental waves;To band-limited signals,when the fundamental wave is a sine signal,the results from the relevant deductions of Theory of Dissolvable Signal is coherent to the classic sampling theory.更多还原