Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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![**Handwrite and Step-by-Step Solutions**
---
**Explanation of the Problem:**
Time-limited signals cannot be bandlimited, but their rate of decay in the frequency domain depends on the sharpness of the transitions in the time domain. For each of the given time-limited signals, we will estimate the rate of decay of \(|\hat{X}(f)|\) as \(|f|\) becomes large. This can be answered without detailed computation of the Fourier transform.
---
**Given Time-Limited Signals:**
(a) \( x_1(t) = (2 - |t|)[I_{[-2,2]}(t)] \).
(b) \( x_2(t) = |t|[I_{[-2,2]}(t)] \).
(c) \( x_3(t) = \cos(\pi t / 4)[I_{[-2,2]}(t)] \).
---
**Method to Estimate Decay Rate:**
- **Understanding the Time-Domain Characteristics:**
- The smoothness and continuity of the time-domain signal influence how rapidly its Fourier transform decays.
- Discontinuities or sharp transitions in the time domain will contribute to slower decay in the frequency domain.
**Analysis:**
(a) **Signal \( x_1(t) \):**
- **Form:**
- Linear piecewise function on the interval \([-2, 2]\).
- Signal transitions smoothly (linear decay towards zero at the edges).
- **Decay Rate:**
- The linear transitions suggest a faster decay rate as smooth changes typically yield a decay rate proportional to \(1/|f|^3\).
(b) **Signal \( x_2(t) \):**
- **Form:**
- Absolute value function peaks at the edges.
- Sharp transition at \(t = 0\).
- **Decay Rate:**
- This sharp transition implies the decay rate is less rapid than for \(x_1(t)\), potentially \(1/|f|^2\).
(c) **Signal \( x_3(t) \):**
- **Form:**
- Cosine function modulated within \([-2, 2]\).
- Oscillatory behavior within the interval.
- **Decay Rate:**
- Oscillations generally slow the decay rate compared to non-oscillatory signals. However, due to the cutoff outside \([-2,](https://content.bartleby.com/qna-images/question/584ffa08-94ad-4eb4-a87d-fd258db9f802/a05621ed-0972-4c8c-a005-61cca58b297e/v6qfvts_processed.png)
Transcribed Image Text:**Handwrite and Step-by-Step Solutions**
---
**Explanation of the Problem:**
Time-limited signals cannot be bandlimited, but their rate of decay in the frequency domain depends on the sharpness of the transitions in the time domain. For each of the given time-limited signals, we will estimate the rate of decay of \(|\hat{X}(f)|\) as \(|f|\) becomes large. This can be answered without detailed computation of the Fourier transform.
---
**Given Time-Limited Signals:**
(a) \( x_1(t) = (2 - |t|)[I_{[-2,2]}(t)] \).
(b) \( x_2(t) = |t|[I_{[-2,2]}(t)] \).
(c) \( x_3(t) = \cos(\pi t / 4)[I_{[-2,2]}(t)] \).
---
**Method to Estimate Decay Rate:**
- **Understanding the Time-Domain Characteristics:**
- The smoothness and continuity of the time-domain signal influence how rapidly its Fourier transform decays.
- Discontinuities or sharp transitions in the time domain will contribute to slower decay in the frequency domain.
**Analysis:**
(a) **Signal \( x_1(t) \):**
- **Form:**
- Linear piecewise function on the interval \([-2, 2]\).
- Signal transitions smoothly (linear decay towards zero at the edges).
- **Decay Rate:**
- The linear transitions suggest a faster decay rate as smooth changes typically yield a decay rate proportional to \(1/|f|^3\).
(b) **Signal \( x_2(t) \):**
- **Form:**
- Absolute value function peaks at the edges.
- Sharp transition at \(t = 0\).
- **Decay Rate:**
- This sharp transition implies the decay rate is less rapid than for \(x_1(t)\), potentially \(1/|f|^2\).
(c) **Signal \( x_3(t) \):**
- **Form:**
- Cosine function modulated within \([-2, 2]\).
- Oscillatory behavior within the interval.
- **Decay Rate:**
- Oscillations generally slow the decay rate compared to non-oscillatory signals. However, due to the cutoff outside \([-2,
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