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5. We have seen that a differentiator has the Fourier property dx (t) dt ⇒jwX (jw) The derivative of a function can be obtained from the limit = = dx(t) dt [x(t + 7) − x(t− 2)] T ·lim [2(t+3 T-0 Derive the Fourier transform of the term inside the square brackets and show that it converges to the differentiation property. (a) Suppose we designed a simple differentiator which is a discrete approximation: x[n + 1] − x[n − 1] 2 y[n] = Using the delay property of the discrete-time Fourier transform, derive the transfer function H(ejw) of this system. If w=0.9, find the ratio between this transfer function and the transfer function of the "true" differentiator.

5. We have seen that a differentiator has the Fourier property dx (t) dt ⇒jwX (jw) The derivative of a function can be obtained from the limit = = dx(t) dt [x(t + 7) − x(t− 2)] T ·lim [2(t+3 T-0 Derive the Fourier transform of the term inside the square brackets and show that it converges to the differentiation property. (a) Suppose we designed a simple differentiator which is a discrete approximation: x[n + 1] − x[n − 1] 2 y[n] = Using the delay property of the discrete-time Fourier transform, derive the transfer function H(ejw) of this system. If w=0.9, find the ratio between this transfer function and the transfer function of the "true" differentiator.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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5. We have seen that a differentiator has the Fourier property dx(t) dt ⇒jwX (jw) The derivative of a function can be obtained dx(t) lim [2(t+3 dt – = from the limit [x(t + 7) − x(t− 2)] T Derive the Fourier transform of the term inside the square brackets and show that it converges to the differentiation property. (a) Suppose we designed a simple differentiator which is a discrete approximation: x[n + 1] − x[n − 1] 2 y[n] = Using the delay property of the discrete-time Fourier transform, derive the transfer function H(ej) of this system. If w=0.9, find the ratio between this transfer function and the transfer function of the "true" differentiator.
Transcribed Image Text:5. We have seen that a differentiator has the Fourier property dx(t) dt ⇒jwX (jw) The derivative of a function can be obtained dx(t) lim [2(t+3 dt – = from the limit [x(t + 7) − x(t− 2)] T Derive the Fourier transform of the term inside the square brackets and show that it converges to the differentiation property. (a) Suppose we designed a simple differentiator which is a discrete approximation: x[n + 1] − x[n − 1] 2 y[n] = Using the delay property of the discrete-time Fourier transform, derive the transfer function H(ej) of this system. If w=0.9, find the ratio between this transfer function and the transfer function of the "true" differentiator.
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