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Consider the x[n] and h[n] sequences below, defined for all n by x[n] = { (-2)", 3-n, n < 0 n 2 0 and h[n] = 8[n] – 28[n – 1] + 38[n – 2] – 48[n – 3] - Working in the time domain, determine y[n] = (x * h)[n] Specify the four separate cases in n.

Consider the x[n] and h[n] sequences below, defined for all n by x[n] = { (-2)", 3-n, n < 0 n 2 0 and h[n] = 8[n] – 28[n – 1] + 38[n – 2] – 48[n – 3] - Working in the time domain, determine y[n] = (x * h)[n] Specify the four separate cases in n.

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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### Convolution of Discrete-Time Signals Consider the discrete-time sequences \( x[n] \) and \( h[n] \) defined for all \( n \): \[ x[n] = \begin{cases} (-2)^n, & \text{if } n < 0 \\ 3^{-n}, & \text{if } n \geq 0 \end{cases} \] and \[ h[n] = \delta[n] - 2\delta[n - 1] + 3\delta[n - 2] - 4\delta[n - 3] \] where \( \delta[n] \) is the discrete-time unit impulse function. ### Objective Determine the convolution \( y[n] = (x \ast h)[n] \) by working in the time domain. ### Approach 1. **Recall the Convolution Sum:** \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \] 2. **Break into Four Separate Cases:** The convolution \( y[n] \) will be evaluated by considering different ranges of \( n \). These ranges are typically chosen to simplify the evaluation of the convolution sum by taking advantage of the non-zero extents of \( x[n] \) and \( h[n] \). ### Range Considerations 1. **Case 1: \( n < 0 \)** - Both \( x[n] \) and \( h[n] \) primarily interact when \( n \) is less than 0. 2. **Case 2: \( n = 0 \)** - Evaluation at the transition point where \( x[n] \) switches from \( (-2)^n \) to \( 3^{-n} \). 3. **Case 3: \( 0 < n < 3 \)** - For this range, \( x[n] = 3^{-n} \), and \( h[n] \) has impulse responses affecting up to \( n=3 \). 4. **Case 4: \( n \geq 3 \)** - Here, \( x[n] = 3^{-n} \), and \( h[n] \)'s influence can be fully considered within the convolution sum due to the impulse responses up to \( n-3 \). Each case will be specific
Transcribed Image Text:### Convolution of Discrete-Time Signals Consider the discrete-time sequences \( x[n] \) and \( h[n] \) defined for all \( n \): \[ x[n] = \begin{cases} (-2)^n, & \text{if } n < 0 \\ 3^{-n}, & \text{if } n \geq 0 \end{cases} \] and \[ h[n] = \delta[n] - 2\delta[n - 1] + 3\delta[n - 2] - 4\delta[n - 3] \] where \( \delta[n] \) is the discrete-time unit impulse function. ### Objective Determine the convolution \( y[n] = (x \ast h)[n] \) by working in the time domain. ### Approach 1. **Recall the Convolution Sum:** \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \] 2. **Break into Four Separate Cases:** The convolution \( y[n] \) will be evaluated by considering different ranges of \( n \). These ranges are typically chosen to simplify the evaluation of the convolution sum by taking advantage of the non-zero extents of \( x[n] \) and \( h[n] \). ### Range Considerations 1. **Case 1: \( n < 0 \)** - Both \( x[n] \) and \( h[n] \) primarily interact when \( n \) is less than 0. 2. **Case 2: \( n = 0 \)** - Evaluation at the transition point where \( x[n] \) switches from \( (-2)^n \) to \( 3^{-n} \). 3. **Case 3: \( 0 < n < 3 \)** - For this range, \( x[n] = 3^{-n} \), and \( h[n] \) has impulse responses affecting up to \( n=3 \). 4. **Case 4: \( n \geq 3 \)** - Here, \( x[n] = 3^{-n} \), and \( h[n] \)'s influence can be fully considered within the convolution sum due to the impulse responses up to \( n-3 \). Each case will be specific
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