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)
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd07e8233-789e-4760-ba71-e70869af7b54%2F6d166d61-92cf-409b-bb6c-2cf62fd3f413%2Fjwerug_processed.png&w=3840&q=75)
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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