Signals and Linear Systems Flip Class Notes 18 - Application of z Transform
The Flipped Classroom18 of Signals and Linear Systems
Corresponding textbook: "Signal and Linear System Analysis (Fifth Edition)" Higher Education Press, written by Wu Dazheng
1. Key points
(1, focus) Master the inverse z-transform method corresponding to different convergence domains: including power series expansion method and partial fraction method; (
2) Use z-transform to solve linear constant coefficient difference equations;
(3, focus) Discrete LTI system System function, system solution based on system function;
(4) z-domain block diagram of discrete LTI system, conversion of various discrete system models;
(5) The connection between z-transform and Laplace transform.
2. Questions and Answers
(1)
(*2) For the system differential equation model and initial conditions given in Exercise 6.16 (3), solve its zero input response, zero state response and full response. And compare it with the solution process and results of Exercise 6.17 to explain their differences.
(*3) Solve exercise 6.24 and summarize the methods of mutual conversion between discrete system models such as system functions, z-domain block diagrams, differential equations, and unit responses.
(*4) Solve exercise 6.29 and summarize its solution ideas.
(5)
1. Convergence region and sequence type
(1) For causal sequences, if z transformation exists, the convergence domain is |z|>ρ1 (ρ1 is the convergence radius) outside the circle.
(2) For a reverse causal sequence, its bilateral z transformation may exist, and its convergence domain is |z|<ρ2 (ρ2 is also called the convergence radius).
(3) For a bilateral sequence, the convergence region of its bilateral z-transform is the annular convergence region, while the convergence region of the unilateral z-transform is outside the ρ1 circle. A bilateral sequence with a bilateral z-transformation must also have a unilateral z-transformation, but a bilateral sequence with a unilateral z-transformation does not necessarily have a double-transformation z-transformation (for example, the sequence a^k, -∞<h<∞)
2. The z-transform method solves the response of discrete systems
For the system differential equation model and initial conditions given in Exercise 6.16(3), solve its zero input response, zero state response and full response. And compare it with the solution process and results of Exercise 6.17 to explain their differences.
6.16(3)
6.17
6.16 uses the unilateral left shift property, and 6.17 uses the unilateral right shift property.
When finding the full solution of 6.16, we did not find the zero input response and the zero state response separately, but directly gave the full solution through the inverse z-transform. The initial conditions given in 6.17 and 6.16 are different. When solving the full response of 6.17, they are solved separately. That is, the zero input response and the zero state response are obtained and then added together to obtain the full response.
3. Mutual conversion between discrete system models
Solve exercise 6.24 and summarize the methods of mutual conversion between discrete system models such as system functions, z-domain block diagrams, differential equations, and unit responses.
The relationship between Y(z) and F(z) can be written from the z-domain block diagram. The system function is obtained from Y(z)/F(z). Then the difference equation is obtained from the system function. The system function is inverted in the z-domain. The transformation yields a unit sequence response.
4、
Solve exercise 6.29 and summarize its solution ideas.
5、
It can be seen from the above formula: the left half plane of the s plane (σ<0) is mapped to the inside of the unit circle of the z plane (|z|=ρ<1); the right half plane of the s plane (σ>0) is mapped to z Outside the unit circle of the plane (|z|=ρ>1); the imaginary axis of the s plane (σ=0) is mapped to the unit circle of the z plane (|z|=ρ=1); the real axis of the s plane (w =0) is mapped to the positive real axis of the z-plane (θ=0), while the origin (σ=0, w=0) is mapped to the point z=1 on the z-plane (ρ=1, θ=0).
3. Reflection and Summary
None yet