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9. Prove the following properties of the Fourier convolution: (a) {f (x) * g (x)} = f' (x) * g (x) = f (x) * g' (x), d dx (b) (f * g) (x)] = (f' * g') (x) = (f" * g) (x), d? dx2 %3D (c) (f * g)(m+n) (x) = [f(m) * g(n)] (x),

9. Prove the following properties of the Fourier convolution: (a) {f (x) * g (x)} = f' (x) * g (x) = f (x) * g' (x), d dx (b) (f * g) (x)] = (f' * g') (x) = (f" * g) (x), d? dx2 %3D (c) (f * g)(m+n) (x) = [f(m) * g(n)] (x),

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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9. Prove the following properties of the Fourier convolution: (a) {f (x) *g (x)} = f' (x) * g (x) = f (x) * g' (x), d %3| (b) [(f * 9) (x)] = (f' * g') (x) = (f" * g) (x), d? dx2 (c) (f * g)m+n) (x) = [f(m) * g(n)] (x), (d) (s *9) (2) da = s (u) du g (v) dv. (4) / (e) If g (x) = H (a – x), then 2a (f * g) (x) is the average value of f (x) in [x - a, x + a]. 12.18 Exercises 52 (-), then (G; * G,) (x) = G++s (x). 1 (f) If G; (x) = V4Tkt 4kt
Transcribed Image Text:9. Prove the following properties of the Fourier convolution: (a) {f (x) *g (x)} = f' (x) * g (x) = f (x) * g' (x), d %3| (b) [(f * 9) (x)] = (f' * g') (x) = (f" * g) (x), d? dx2 (c) (f * g)m+n) (x) = [f(m) * g(n)] (x), (d) (s *9) (2) da = s (u) du g (v) dv. (4) / (e) If g (x) = H (a – x), then 2a (f * g) (x) is the average value of f (x) in [x - a, x + a]. 12.18 Exercises 52 (-), then (G; * G,) (x) = G++s (x). 1 (f) If G; (x) = V4Tkt 4kt
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