# 3 Detenmimo the coneclution fas [a] =mu [a) and ag [a] =2" [n-1] by means tranofom noly base on the fomulas and (m+nJa" u(a]Z, 1-az-1
# 3 Detenmimo the coneclution fas [a] =mu [a) and ag [a] =2" [n-1] by means tranofom noly base on the fomulas and (m+nJa" u(a]Z, 1-az-1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exercise #3**
**Objective:**
Determine the convolution of \( a[n] = m \cdot u[n] \) and \( a_2 \cdot u[n-1] \) by means of the Z-transform.
**Steps:**
1. Use the following Z-transform formulas:
\[
a \cdot u[n] \xrightarrow{Z} \frac{1}{1 - a \cdot z^{-1}}, \quad |z| > |a|
\]
2. Apply for \( a[n] = m \cdot u[n] \):
\[
\frac{a}{1 - a \cdot z^{-1}}
\]
3. Apply for \( a_2 \cdot u[n-1] \):
\[
z \cdot \text{(previous result)}
\]
4. Final form:
\[
\left( (1 - a \cdot z^{-1})^{-1} \right) \xrightarrow{Z}
\]
**Graph/Diagram Description:**
No graphs or diagrams are present in this exercise. The text outlines a mathematical process for finding a convolution using the Z-transform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e3992ca-2280-40ba-b65a-68dc98c03d5d%2F502dddcc-8f20-4593-9f07-c90432eff7d9%2F9ahlq0r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise #3**
**Objective:**
Determine the convolution of \( a[n] = m \cdot u[n] \) and \( a_2 \cdot u[n-1] \) by means of the Z-transform.
**Steps:**
1. Use the following Z-transform formulas:
\[
a \cdot u[n] \xrightarrow{Z} \frac{1}{1 - a \cdot z^{-1}}, \quad |z| > |a|
\]
2. Apply for \( a[n] = m \cdot u[n] \):
\[
\frac{a}{1 - a \cdot z^{-1}}
\]
3. Apply for \( a_2 \cdot u[n-1] \):
\[
z \cdot \text{(previous result)}
\]
4. Final form:
\[
\left( (1 - a \cdot z^{-1})^{-1} \right) \xrightarrow{Z}
\]
**Graph/Diagram Description:**
No graphs or diagrams are present in this exercise. The text outlines a mathematical process for finding a convolution using the Z-transform.
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