Use properties to solve z-transform
- 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
In the eleventh assignment, there is an exercise about applying the z-transform property to solve the sequence z-transform. These exercises help us to solve the sequence z-transform results. Let's take a look at how they are solved.
2. Problem solving
1. The first sub-question
The first sub-problem, is an exponential sequence multiplied by n. There are two ways to solve it. The first way to solve it is to apply the sequential linear weighting property. Equal to the z-transform derivative of the sequence, multiplied by negative z. The second applies series exponential weighting, equal to the z-transform of the series for scaling.
The first solution idea starts from the z-transformation of the exponential sequence. Write down the corresponding conversion factor. Then consider independent variable weighting. It is equivalent to deriving the z-transform. After calculation and simplification, the z-transform result of the sequence can be obtained. Next, consider the second solution idea.
Find the z-transform of n sequences first. It can be found from the typical signal z-transform table. Considering exponential weighting, replace z in the result with half of z. Simplify it, and finally get the transformation result. It can be seen that the two results are the same. In the first sub-problem, we reviewed the linear and exponential weighting properties of the z-transform.
2. Second question
The second sub-question actually contains two sequences, the front is the n sequence, and the back is the constant sequence. They are all typical sequence signals. The corresponding z-transforms can be written separately, combined together, and the z-transform of the sequence can be obtained after simplification. This solution is relatively simple.
You can also use the displacement characteristics of the z-transform to solve the z-transform of this small problem. Start with the z-transform of the n-sequence, and use the unilateral z-transform feature to write the corresponding results. By simplifying the result, the z-transform of the sequence can be obtained. A minus sign was missed during the simplification in the previous result. In comparison, it is more convenient to decompose the sequence into two and solve them separately.
3. The third question
The third sub-problem includes a series accumulation part, which is equal to the convolution of negative one index series and u(n). The z-transform of the sequence can be written, and then the convolution theorem of the z-transform is used, and the z-transform of the series accumulation sequence is equal to their product. This results in the z-transform of the accumulated sequence. Consider the previous index series weighting again. The previous z-transformation is subjected to variable substitution, and finally the z-transformation of the sequence after exponential weighting is obtained. This is the z-transform of the x3(n) sequence.
4. The fourth question
For the fourth subproblem, include a rational fraction for n. Now the n+1th sequence starts, and its z-transform can be obtained by looking up the common sequence z-transform table, which corresponds to multiplying z by the log z-1 score z. Then consider the n+2 1/2 sequence. Using the z-transform displacement property, its corresponding z-transform is multiplied by z on the basis of the previous result. Finally, consider exponentially weighting a to the nth power. Corresponding to dividing z by a, after variable substitution, the final transformation result can be obtained. In this process, the z-transformation of the n+1 signal is applied, the displacement characteristic and the exponential weighting characteristic.
5. The fifth question
The fifth sub-question looks cumbersome, but in fact it can be regarded as the result of a cosine sequence weighted by one-fifth proportional sequence. A cosine sequence is a typical signal, and its corresponding z-transform can be written through the common sequence z-transform table. On this basis, considering the influence of exponential weighting, replace z with 5z, and thus obtain the z transformation of the sequence. This is the result of this calculation.
※ Summary ※
This article discusses the application of properties to solve the z-transform of a sequence. The flexible application of these properties can help us to obtain the z-transform of the sequence efficiently and easily.
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