As soon as possible **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,

Since you have posted multiple , I will solve the first question only as per guidelines. please…

Handwrite and step by step solutions Time-limited signals cannot be bandlimited, but their rate of decay in the fre- quency domain depends on the sharpness of the transitions in the time domain. For each of the time-limited signals below, estimate the rate of decay of |Â(ƒ)| as |ƒ| gets large. You should be able to answer this without detailed computation of the Fourier transform. (a) x₁(t) = (2- |t|)I[_2,2] (t). (b) x₂(t) = |t|I[-2,2] (t). (c) x3 (t) = cos(nt/4) I[-2,2] (t).

Handwrite and step by step solutions Time-limited signals cannot be bandlimited, but their rate of decay in the fre- quency domain depends on the sharpness of the transitions in the time domain. For each of the time-limited signals below, estimate the rate of decay of |Â(ƒ)| as |ƒ| gets large. You should be able to answer this without detailed computation of the Fourier transform. (a) x₁(t) = (2- |t|)I[_2,2] (t). (b) x₂(t) = |t|I[-2,2] (t). (c) x3 (t) = cos(nt/4) I[-2,2] (t).

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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