Math Problem Statement

Inspect using long division and determine the inverse z transform of X(z) = z + 1 Z2 – 3z + 2

Solution

To find the inverse Z-transform of X(z)=z+1z23z+2X(z) = \frac{z + 1}{z^2 - 3z + 2} using long division, we will perform the division and identify the resulting sequence in the time domain.

Step 1: Perform Long Division

Given: X(z)=z+1z23z+2X(z) = \frac{z + 1}{z^2 - 3z + 2}

First, let's factor the denominator if possible: z23z+2=(z1)(z2)z^2 - 3z + 2 = (z - 1)(z - 2)

Now, perform the long division:

  1. Divide the highest degree term in the numerator by the highest degree term in the denominator: zz2=1z\frac{z}{z^2} = \frac{1}{z} But we need terms in decreasing powers of zz. This step suggests we'll rewrite it as 00 for this stage of division and continue for lower degree terms.

Since ( \frac{z}{z^

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Math Problem Analysis

Mathematical Concepts

Inverse Z-transform
Long Division
Partial Fraction Decomposition

Formulas

Inverse Z-transform: X(z) = Σ x[n]z^(-n)
Long Division of Polynomials
Partial Fraction Decomposition of Rational Functions

Theorems

Inverse Z-transform Theorem
Long Division Theorem

Suitable Grade Level

Undergraduate level (Engineering/Applied Mathematics)