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Consider a discrete-time LTI system with impulse response h[k] ←+++ -10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9 10 11 12 Suppose the following signal is passed as input to this system: f[k] 3 5 ←+++ + k -10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9 10 11 12 Let y[k] denote the resulting output signal. (a) What is the value of y[k] at index k = 1? (b) What is the value of y[k] at index k = 4? 3

Consider a discrete-time LTI system with impulse response h[k] ←+++ -10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9 10 11 12 Suppose the following signal is passed as input to this system: f[k] 3 5 ←+++ + k -10-9-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8 9 10 11 12 Let y[k] denote the resulting output signal. (a) What is the value of y[k] at index k = 1? (b) What is the value of y[k] at index k = 4? 3

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Discrete-Time LTI System with Impulse Response This example considers a discrete-time Linear Time-Invariant (LTI) system characterized by its impulse response \( h[k] \). #### Impulse Response \( h[k] \): - A sequence represented on a number line (x-axis) from \( k = -10 \) to \( k = 12 \). - Notable non-zero values: - At \( k = 0 \), \( h[0] = 3 \) - At \( k = 1 \), \( h[1] = 5 \) - At \( k = 2 \), \( h[2] = 4 \) #### Input Signal \( f[k] \): Suppose the following signal is used as input to this LTI system, represented by \( f[k] \): - A sequence on the same number line, \( k = -10 \) to \( k = 12 \). - Notable non-zero values: - At \( k = 0 \), \( f[0] = 4 \) - At \( k = 1 \), \( f[1] = 5 \) - At \( k = 3 \), \( f[3] = 6 \) - At \( k = 4 \), \( f[4] = 4 \) The task involves using the convolution of \( f[k] \) and \( h[k] \) to find the resulting output signal \( y[k] \). #### Output Signal \( y[k] \) We need to determine the values of \( y[k] \) at specific indices: (a) **At index \( k = 1 \):** - Calculate using the convolution sum \( y[1] = \sum_{n} f[n] \cdot h[1-n] \). (b) **At index \( k = 4 \):** - Calculate using the convolution sum \( y[4] = \sum_{n} f[n] \cdot h[4-n] \). This process involves systematically applying the convolution formula to find the desired values of the output signal.
Transcribed Image Text:### Discrete-Time LTI System with Impulse Response This example considers a discrete-time Linear Time-Invariant (LTI) system characterized by its impulse response \( h[k] \). #### Impulse Response \( h[k] \): - A sequence represented on a number line (x-axis) from \( k = -10 \) to \( k = 12 \). - Notable non-zero values: - At \( k = 0 \), \( h[0] = 3 \) - At \( k = 1 \), \( h[1] = 5 \) - At \( k = 2 \), \( h[2] = 4 \) #### Input Signal \( f[k] \): Suppose the following signal is used as input to this LTI system, represented by \( f[k] \): - A sequence on the same number line, \( k = -10 \) to \( k = 12 \). - Notable non-zero values: - At \( k = 0 \), \( f[0] = 4 \) - At \( k = 1 \), \( f[1] = 5 \) - At \( k = 3 \), \( f[3] = 6 \) - At \( k = 4 \), \( f[4] = 4 \) The task involves using the convolution of \( f[k] \) and \( h[k] \) to find the resulting output signal \( y[k] \). #### Output Signal \( y[k] \) We need to determine the values of \( y[k] \) at specific indices: (a) **At index \( k = 1 \):** - Calculate using the convolution sum \( y[1] = \sum_{n} f[n] \cdot h[1-n] \). (b) **At index \( k = 4 \):** - Calculate using the convolution sum \( y[4] = \sum_{n} f[n] \cdot h[4-n] \). This process involves systematically applying the convolution formula to find the desired values of the output signal.
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