信号与系统课件第9章拉普拉斯变换剖析.ppt

Mason’s Formula (梅森规则): Here is the graph determinant(特征行列式). In a signal flow graph, the transfer value (transfer function) between any source node and sink node or mixed node can be determined by the following equation: is the gain of each loop. is the multiplication of the gains of two loops which have no shared nodes and branches . is the graph determinant of the left graph after removing the k-th forward parth. is the gain of the k-th forward parth between the source node and the sink node. Example 9. 19 Compute the transfer functions between nodes A and B, and nodes A and C in the following signal flow graph. X(s) Y(s) 1/s 2 A 3 1/s 1 5 1 1 –10 C –1 4 1/s W(s) B –1 1/s A to B: After removing G1 , the left graph is: Thus, X(s) Y(s) 1/s 2 A 3 1/s 1 5 1 1 –10 C –1 4 1/s W(s) B A to C: After removing G1 , the left graph is: Thus, Y(s) 1/s 1 5 –10 –1 4 1/s X(s) Y(s) 1/s 2 A 3 1/s 1 5 1 1 –10 C –1 4 1/s W(s) B 9.9 THE UNILATERAL LAPLACE TRANSFORM bilateral Laplace transform: (双边拉普拉斯变换) unilateral Laplace transform: (单边拉普拉斯变换) The lower limit of integration, , signifies that we include in the interval of integration any impulses or higher order singularity functions concentrated at t = 0. (奇异函数) The bilateral transform depends on the entire signal from t = –∞ to t = +∞, whereas the unilateral transform depends only on the signal from to ∞. The bilateral transform and the unilateral transform of a causal signal are identical. The ROC for the unilateral transform is always a right-half plane. Example 9.20 Consider the signal The bilateral transform X(s) for this example can be obtained from Example 9.1 and the time-shifting property: By contrast, the unilateral transform is The evaluation of the inverse unilateral Laplace transforms is also the same as for bilateral transforms, with the constraint that the