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 x1[n]x_1[n] and x2[n]x_2[n] have Z-Transforms X1(z)X_1(z) and X2(z)X_2(z), 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 aa and bb are constants.

Proof:

The Z-Transform of a sequence x[n]x[n] is defined as: X(z)=n=x[n]zn.X(z) = \sum_{n=-\infty}^\infty x[n] z^{-n}.

For x[n]=ax1[n]+bx2[n]x[n] = a \cdot x_1[n] + b \cdot x_2[n]: [ \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: Z{x[n]}=aX1(z)+bX2(z).\mathcal{Z}\{x[n]\} = a \cdot X_1(z) + b \cdot X_2(z).

Thus, the property is proven.


2. Time Shifting Property

Statement:

If x[n]x[n] has a Z-Transform X(z)X(z), then the Z-Transform of x[nk]x[n-k] is given by: Z{x[nk]}=zkX(z),\mathcal{Z}\{x[n-k]\} = z^{-k} X(z), where kk is an integer.

Proof:

The Z-Transform of x[n]x[n] is: X(z)=n=x[n]zn.X(z) = \sum_{n=-\infty}^\infty x[n] z^{-n}.

For the shifted sequence x[nk]x[n-k]: [ \mathcal{Z}{x[n-k]} = \sum_{n=-\infty}^\infty x[n-k] z^{-n}. ]

Let m=nkm = n-k, so n=m+kn = m+k. 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 zkz^{-k} (independent of mm): [ \mathcal{Z}{x[n-k]} = z^{-k} \sum_{m=-\infty}^\infty x[m] z^{-m}. ]

From the definition of the Z-Transform: Z{x[nk]}=zkX(z).\mathcal{Z}\{x[n-k]\} = z^{-k} X(z).

Thus, the property is proven.


Would you like additional details or examples for these properties?

5 Related Questions:

  1. What are other fundamental properties of the Z-Transform?
  2. How does the time-shifting property relate to delay in discrete systems?
  3. Can you explain the region of convergence (ROC) for Z-Transforms in these cases?
  4. How is the Z-Transform related to the Laplace Transform?
  5. 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)