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6. Explain why the following function generates an impulse function as €→ 0: f(t) = E/TT €²+t² for-∞ ≤t≤ 00

6. Explain why the following function generates an impulse function as €→ 0: f(t) = E/TT €²+t² for-∞ ≤t≤ 00

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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**Problem 6. Impulse Function Generation** **Objective:** Explain why the following function generates an impulse function as \(\epsilon \to 0\): \[ f(t) = \frac{\epsilon/\pi}{\epsilon^2 + t^2} \quad \text{for} \quad -\infty \leq t \leq \infty \] --- **Explanation:** The given function is based on the Lorentzian distribution, which is known to approximate a delta function, or impulse function, in the limit as \(\epsilon\) approaches zero. As \(\epsilon\) becomes very small, the peak of the function becomes much sharper and taller, concentrating the area under the curve around \(t = 0\). The integral of this function over all time remains constant due to its normalization factor \((\epsilon/\pi)\), ensuring the total "area" is always equal to 1, which is characteristic of the delta function. This behavior can be rigorously shown by evaluating the limit and showing that \(f(t)\) converges to the Dirac delta function, \(\delta(t)\), which is fundamentally used in system analysis to model an idealized point impulse.
Transcribed Image Text:**Problem 6. Impulse Function Generation** **Objective:** Explain why the following function generates an impulse function as \(\epsilon \to 0\): \[ f(t) = \frac{\epsilon/\pi}{\epsilon^2 + t^2} \quad \text{for} \quad -\infty \leq t \leq \infty \] --- **Explanation:** The given function is based on the Lorentzian distribution, which is known to approximate a delta function, or impulse function, in the limit as \(\epsilon\) approaches zero. As \(\epsilon\) becomes very small, the peak of the function becomes much sharper and taller, concentrating the area under the curve around \(t = 0\). The integral of this function over all time remains constant due to its normalization factor \((\epsilon/\pi)\), ensuring the total "area" is always equal to 1, which is characteristic of the delta function. This behavior can be rigorously shown by evaluating the limit and showing that \(f(t)\) converges to the Dirac delta function, \(\delta(t)\), which is fundamentally used in system analysis to model an idealized point impulse.
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