1) If x(п) —D — 6(п - 2) — 8(п — 4) where N-6 Find a) X(k) b) Y(n) if y(k) = e]tk x(k)
1) If x(п) —D — 6(п - 2) — 8(п — 4) where N-6 Find a) X(k) b) Y(n) if y(k) = e]tk x(k)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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![**Problem Statement:**
Given:
\[ x(n) = -\delta(n - 2) - \delta(n - 4) \]
where \( N = 6 \)
**Tasks:**
1. Find:
a) \( X(k) \)
b) \( Y(n) \) if \( y(k) = e^{j\pi k} x(k) \)
**Detailed Explanation:**
- \( x(n) \) represents a discrete-time signal defined as a combination of shifted Dirac delta functions.
- \( \delta(n - 2) \) is a Dirac delta function shifted by 2 units.
- \( \delta(n - 4) \) is a Dirac delta function shifted by 4 units.
- \( N = 6 \) indicates the period of the discrete-time signal.
**Findings:**
**a) \( X(k) \):**
- \( X(k) \) represents the Discrete Fourier Transform (DFT) of the signal \( x(n) \).
**b) \( Y(n) \):**
- \( Y(n) \) is derived by first computing \( y(k) \) using the relationship given:
\[ y(k) = e^{j\pi k} x(k) \]
- Determine \( Y(n) \) by computing the inverse DFT of \( y(k) \).
**Notation:**
- \( j \) represents the imaginary unit.
- \( \pi \) is the mathematical constant Pi.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8241d13-b770-4dad-946c-6942db4056c7%2F3dd2e7fd-7ca3-4488-a757-a14e92870c7a%2F4e7sf08_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given:
\[ x(n) = -\delta(n - 2) - \delta(n - 4) \]
where \( N = 6 \)
**Tasks:**
1. Find:
a) \( X(k) \)
b) \( Y(n) \) if \( y(k) = e^{j\pi k} x(k) \)
**Detailed Explanation:**
- \( x(n) \) represents a discrete-time signal defined as a combination of shifted Dirac delta functions.
- \( \delta(n - 2) \) is a Dirac delta function shifted by 2 units.
- \( \delta(n - 4) \) is a Dirac delta function shifted by 4 units.
- \( N = 6 \) indicates the period of the discrete-time signal.
**Findings:**
**a) \( X(k) \):**
- \( X(k) \) represents the Discrete Fourier Transform (DFT) of the signal \( x(n) \).
**b) \( Y(n) \):**
- \( Y(n) \) is derived by first computing \( y(k) \) using the relationship given:
\[ y(k) = e^{j\pi k} x(k) \]
- Determine \( Y(n) \) by computing the inverse DFT of \( y(k) \).
**Notation:**
- \( j \) represents the imaginary unit.
- \( \pi \) is the mathematical constant Pi.
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