2) Assume that Y1(t) = § h1(T)X(t –T)dr and Y2(t) = h2(T)X(t – T)dr where h1 (t) andh2(t) are the real-valued impulse responses of two linear time-invariant systems and X(t)is the input to the two systems. Furthermore, X(t) is real-valued and wide-sense-stationaryand has zero mean and the correlation function Kx(T) = E{X(t+T)X(t)}. Prove thefollowing relationships:a) Ky v,(T) = E{Y;(t + 7)Y½(t)} = SS h(T1)h2(T2)Kx(7 – T1 + T2)dr1dr2 whereKy Y2(T) is called the cross-correlation function between Y (t) and Y2(t).b) Sy,,Ya(f) = H;(f)H;(f)Sx(f) where Sy, y,(f), Sx(f), H1(f) and H2(f) are theFourier transforms of Ky, Y (T), Kx(T), h1(t) and h2(t), respectively. Then, answerthe following questions:i) If h1(t) = sinc(t/T) and h2(t) = sinc(t/T – 1), what is Sy, Y, (f)?ii) If h1(t) = sinc(t/T) and h2(t) = sinc(t/T) cos(27t), what is Sy, y,(f)?

Answered: Image /qna-images/answer/192d7520-7a73-4c3e-8d2b-dbdeabd49627.jpg

2) Assume that Y₁(t) = f h₁(7)X(t — 7)dr and Y₂(t) = f h₂(7)X(tr)dr where h₁ (t) and h₂(t) are the real-valued impulse responses of two linear time-invariant systems and X(t) is the input to the two systems. Furthermore, X(t) is real-valued and wide-sense-stationary and has zero mean and the correlation function Kx(7) = E{X(t + 7)X(t)}. Prove the following relationships: a) Ky₁₁₂(T) = {{Y₁(t + 7)Y₂(t)} = S Sh₁(71)h₂(T₂)Kx(7 − T₁ + T₂)d7₁dT2 where Ky₁₂ (7) is called the cross-correlation function between Y₁(t) and Y₂(t). b) Sy₁,₂(f) = H₁(ƒ)H;(ƒ)Sx(ƒ) where Sy₁,₂(f), Sx(ƒ), H₁(f) and H₂(ƒ) are the Fourier transforms of K₁,₂(7), Kx(7), h₁(t) and h₂(t), respectively. Then, answer the following questions: i) If h₁(t) = sinc(t/T) and h₂(t) = sinc(t/T − 1), what is Sy₁,₂(f)? ii) If h₁(t) = sinc(t/T) and h₂(t) = sinc(t/T) cos(2 t), what is Sy₁,₂(f)?

2) Assume that Y₁(t) = f h₁(7)X(t — 7)dr and Y₂(t) = f h₂(7)X(tr)dr where h₁ (t) and h₂(t) are the real-valued impulse responses of two linear time-invariant systems and X(t) is the input to the two systems. Furthermore, X(t) is real-valued and wide-sense-stationary and has zero mean and the correlation function Kx(7) = E{X(t + 7)X(t)}. Prove the following relationships: a) Ky₁₁₂(T) = {{Y₁(t + 7)Y₂(t)} = S Sh₁(71)h₂(T₂)Kx(7 − T₁ + T₂)d7₁dT2 where Ky₁₂ (7) is called the cross-correlation function between Y₁(t) and Y₂(t). b) Sy₁,₂(f) = H₁(ƒ)H;(ƒ)Sx(ƒ) where Sy₁,₂(f), Sx(ƒ), H₁(f) and H₂(ƒ) are the Fourier transforms of K₁,₂(7), Kx(7), h₁(t) and h₂(t), respectively. Then, answer the following questions: i) If h₁(t) = sinc(t/T) and h₂(t) = sinc(t/T − 1), what is Sy₁,₂(f)? ii) If h₁(t) = sinc(t/T) and h₂(t) = sinc(t/T) cos(2 t), what is Sy₁,₂(f)?

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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