信号总结.docVIP

Program/ Transform Fourier Transform Laplace Transform Z-Transform Definition ; ; ; Abbreviate ,, , , , Property of ROC The Sufficient Conditions: The Dirichlet Conditions: 1. be absolutely integrable, that is 2. have a finite number of maxima and minima within any finite interval. 3. have a finite number of discontinuities within any finite interval. Futhermore each of these discontinuities must be finite 对比傅里叶变换和傅里叶级数，有： 。 对比拉普拉斯变换和Z变换，共同点有： 收敛域的形状：平行于jw轴变换为以原点为圆心的圆。 收敛域内均不包含任何极点。 若信号在持续有限区间内有值，收敛域为整个平面。 信号为右边信号时，收敛域为极点的右边或者外边。 信号为左边信号时，收敛域为极点的左边或者里边。 信号为双边信号时，收敛域为两极点的之间的带状区域或者圆环区域。 变换域频谱为有理式时，收敛域为极点所限的区域或者趋向于无穷远处。 变换域频谱为有理式时，若信号为右边信号时，收敛域为最右边（外边）极点的右边（外边），若信号为左边信号时，收敛域为最左边（里边）极点的左边（里边）。 The ROC of consists of strip parallel to the jω-axis in the s-plane. For rational Laplace Transfroms, the ROC doesn’t contain any poles. If is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. If is right sided, and if the line is in the ROC, then all value of s for which will also be in the ROC. If is left sided, and if the line is in the ROC, then all value of s for which will also be in the ROC. If is two sided, and if the line is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line . If the Laplace Transfroms of is rational, then its ROC is bound by poles or extends to infinity. In addition, no poles of are contained in the ROC If the Laplace Transfroms of is rational, if is right sided, the ROC is the region in the s-plane to the right of the rightmost pole, if is left sided, the ROC is the region in the s-plane to the left of the leftmost pole. 对比拉普拉斯变换和Z变换，不同点有： 由于两个变换在定义上的不一样，拉普拉斯变换为无穷积分的形式，而Z变换为无穷级数关于z的幂次项求和的形式，故在考虑Z变换的收敛域时要明确的考虑z=0和z=∞这两点，若X(z)包含z的正幂次项时，收敛域应该不包括z=∞这点，若X(z)包含z的负幂次项时，收敛域应该不包括z=0这点。 拉普拉斯变换可以看成是信号与指数衰减信号相乘以后的连续时间傅里叶变换，而连续时间傅里叶变换可以看成是实部等于零的拉普拉斯变换。 Z变换可以看成是信号与指数衰减信号相乘以后的离散时间傅里叶变换，而离散时间傅里叶变换可以看成是模等于一的Z变换。 The ROC of consists of a ring in the z-plane centered about t