**Auto-Correlation Function and CTFT Explanation**The auto-correlation function \( a(t) \) associated with the signal \( x(t) \) is given as:\[ y(t) = x(t) \ast x^*(-t) \]Using Continuous-Time Fourier Transform (CTFT) properties, derive an expression for \( Y(j\omega) \), the CTFT of \( y(t) \), in terms of \( X(j\omega) \) and other relevant variables. **Explanation**:- **Auto-correlation Function**: This mathematical operation measures the similarity of a signal with a time-shifted version of itself. For a signal \( x(t) \), the auto-correlation is expressed as \( y(t) = x(t) \ast x^*(-t) \), where \( x^*(-t) \) is the complex conjugate and time-reversal of \( x(t) \). - **CTFT and its Use**: The Continuous-Time Fourier Transform (CTFT) is a tool used to analyze the frequency content of continuous signals. Deriving \( Y(j\omega) \) involves utilizing CTFT properties to express the frequency representation of the auto-correlation function \( y(t) \).

Given, Auto correlation of signal x(t) is y(t). i.e y(t) = x(t) * x*(-t) We need to determine the relationship between Y(jw) and X(jw). Where, X(jw) = CTFT{x(t)} Y(jw) = CTFT {y(t)} We know that, FT{x*(t)} = X(-jw) = X1(jw) Let x1(t) = x*(t) then let x2(t) = x1(-t) = x*(-t) Apply CTFT on both sides, X2(jw) = X1(-jw) , [ since time reverse property] Therefore, X2(jw) = X(-(-jw)) = X(jw) Given, y(t) = x(t) * x*(-t) Apply CTFT on both sides Y(jw) = X(jw). CTFT{x*(-t)} = X(jw). X(jw) Therefore, Y(jw) = [X(jw)]2

Engineering

Electrical Engineering

The auto-correlation function a(t) associated with the signal x(t) is given as y(t) x(t) * x*(-t). - Using CTFT properties, derive an expression for Y(jw), the CTFT of y(t), in terms of X(jw) and other relevant variables.

The auto-correlation function a(t) associated with the signal x(t) is given as y(t) x(t) * x*(-t). - Using CTFT properties, derive an expression for Y(jw), the CTFT of y(t), in terms of X(jw) and other relevant variables.

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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