Apply the Z-transform to solve the difference equation
- Signal and System Spring 2023 Homework Requirements and Reference Answer Summary
- Signals and Systems 2023 (Spring) Assignment Requirements - Eleventh Assignment
- Signals and Systems 2023 (Spring) Homework Reference Answers - 11th Homework
01 The eleventh homework
1. Introduction to Exercises
Take the z transformation on the left and right of the difference equation, and use the displacement characteristics of the z transformation to get an algebraic equation, where Y(z) is the solution of the equation. By solving the algebraic equation, you can get the z corresponding to the solution of the difference equation transform. In the eleventh assignment, two systems of difference equations are used to practice the method of solving difference equations using the z-transform. Let us discuss the solution ideas below.
2. Problem solving
1. The first sub-question
The first sub-problem is a first-order backward difference equation. Take the z-transformation for the left and right sides of the equation, and bring its physical inspection and initial conditions into the equation. After simplification, the expression of Y(z) can be obtained. The following uses the factor Decomposition method, solve the inverse transformation of Y(z), write out the time domain sequence corresponding to each factor, and the solution of the equation corresponds to the sequence on the right. This is the answer to the first question.
2. Second question
The second subproblem is also a first-order backward difference equation. Take the z-transformation for the left and right sides of the equation at the same time, and substitute the known initial conditions. By simplifying the equation, we can finally contact the expression of Y(z). The expression is further simplified, the expression corresponding to the previous X(z) is known, and the latter item is subjected to z inverse transformation, and the final expression of y[n] can be obtained. This is the answer to this small question. At the end, u[n] should be added to represent the sequence on the right.
※ Summary ※
This paper discusses the application of the z-transform to solve the difference equation, and solves two problems in the eleventh assignment.
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