Math Problem Statement
(b) Recall and prove any two properties of Z-Transform.
Solution
Properties of Z-Transform and Their Proofs
The Z-Transform is widely used in signal processing and control systems to analyze discrete-time signals. Here, we will recall and prove two fundamental properties: Linearity and Time Shifting.
1. Linearity Property
Statement:
If and have Z-Transforms and , respectively, then: [ \mathcal{Z}{a \cdot x_1[n] + b \cdot x_2[n]} = a \cdot X_1(z) + b \cdot X_2(z), ] where and are constants.
Proof:
The Z-Transform of a sequence is defined as:
For : [ \mathcal{Z}{x[n]} = \sum_{n=-\infty}^\infty \left(a \cdot x_1[n] + b \cdot x_2[n]\right) z^{-n}. ]
Expanding the summation: [ \mathcal{Z}{x[n]} = a \sum_{n=-\infty}^\infty x_1[n] z^{-n} + b \sum_{n=-\infty}^\infty x_2[n] z^{-n}. ]
From the definition of the Z-Transform:
Thus, the property is proven.
2. Time Shifting Property
Statement:
If has a Z-Transform , then the Z-Transform of is given by: where is an integer.
Proof:
The Z-Transform of is:
For the shifted sequence : [ \mathcal{Z}{x[n-k]} = \sum_{n=-\infty}^\infty x[n-k] z^{-n}. ]
Let , so . Substituting this into the summation: [ \mathcal{Z}{x[n-k]} = \sum_{m=-\infty}^\infty x[m] z^{-(m+k)}. ]
Simplify the exponent: [ \mathcal{Z}{x[n-k]} = \sum_{m=-\infty}^\infty x[m] z^{-m} z^{-k}. ]
Factor out (independent of ): [ \mathcal{Z}{x[n-k]} = z^{-k} \sum_{m=-\infty}^\infty x[m] z^{-m}. ]
From the definition of the Z-Transform:
Thus, the property is proven.
Would you like additional details or examples for these properties?
5 Related Questions:
- What are other fundamental properties of the Z-Transform?
- How does the time-shifting property relate to delay in discrete systems?
- Can you explain the region of convergence (ROC) for Z-Transforms in these cases?
- How is the Z-Transform related to the Laplace Transform?
- What is the physical significance of the linearity property in signal processing?
Tip:
Always verify the region of convergence (ROC) when applying Z-Transform properties, as it affects the stability and causality of the system.
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Math Problem Analysis
Mathematical Concepts
Z-Transform
Signal Processing
Discrete-Time Systems
Formulas
Z-Transform: X(z) = Σ x[n] z^(-n)
Linearity: Z{a * x1[n] + b * x2[n]} = a * X1(z) + b * X2(z)
Time Shifting: Z{x[n-k]} = z^(-k) * X(z)
Theorems
Linearity Property of Z-Transform
Time Shifting Property of Z-Transform
Suitable Grade Level
Undergraduate (Electrical Engineering, Signal Processing)
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