Scale properties of the 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
Scaling operations for discrete sequences have certain particularities. Normally, only integer multiples of snapshot compression and integer multiples of zero-padding stretching can be performed. Given here is a triple compression and stretching of the sequence. Compression is an irreversible operation, but stretching is a reversible operation. In z-transform, if the z-transform of the sequence is known. So what is the z-transform of the sequence after compression and stretching? This is an exercise left in the eleventh homework. Let us discuss its solution ideas below.
2. Problem solving
The solution is firstly solved from the z-transform of zero-padded stretching of the sequence. According to the definition of z-transform, write the z-transform expression of x2[n]. Because x2[n] is not zero only when n is a multiple of scattered integers, so take n equal to 3k and perform variable substitution. The original accumulation expression becomes the accumulation about k. Then according to the relationship between x2 and x[n], the series about x[n] is finally formed. Taking the cubic power of z as a variable, the z transformation of x[n] is finally obtained, but the variable is the cubic power of z. This is the z-transform corresponding to zero-padding the sequence stretched three times. According to this proof process, we can know that if other multiples are stretched, the corresponding z-transformation is equal to the nth power of the z-transformation of the original sequence.
The z-transform for snapshot compression of sequences is discussed below. According to the z-transform formula, write the sequence z-transform expression. Applying variable substitution, let 3n equal k and n equal one third of k. On the basis of this accumulation expression, a constraint condition needs to be added, that is, k is an integer multiple of 3. In order to remove this constraint. A special expression needs to be introduced below. This expression includes three items, which are easy to verify. When n is an integer multiple of 3, it is equal to 1, and when n is not an integer multiple of 3, the expression is 0. In fact, these three terms are the three cubic roots of 1 in the field of complex numbers. Applying this expression, it multiplies x[k] so that the constraint on k is removed. This forms the z-transform of the three sequences. The first is the sequence itself, and the latter two are the complex exponential weighting of the sequence. After sorting, the expression after z transformation can be obtained.
Since the signal is compressed by snapshots, part of the sequence information is actually lost. The corresponding z-transform is actually the original z-transform, after a certain deformation, superimposed in the transform domain. This transforms the z-transform expression to appear aliased. Furthermore, the original z-transform cannot be completely recovered.
※ Summary ※
In this paper, the z-transform after sequence scale transformation is solved. The expression for the corresponding z-transform stretched by zero-padding is relatively simple.
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