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Composite Systems Consider the following impulse responses of two LTI systems: h(t) = 8(t-2), g(t) = eat, a € R+ (a) Find H(w) and G(w), the frequency responses (CTFT) of the systems. (b) Consider interconnected LTI systems with impulse responses h(t) and g(t): System 1, with overall impulse response of s₁(t): x(t) x(t) h(t) System 2, with overall impulse response of s₂(t): (c) Now suppose g(t) h(t) Y₁ (t) + Y2(t) g(t) Find the impulse responses s₁(t) and s₂(t). Also find their respective fre- quency responses S₁ (w) and S₂(w). x(t) = eiwot Calculate the output y(t) of the two systems S₁ and S₂ when given x(t) as an input.

Composite Systems Consider the following impulse responses of two LTI systems: h(t) = 8(t-2), g(t) = eat, a € R+ (a) Find H(w) and G(w), the frequency responses (CTFT) of the systems. (b) Consider interconnected LTI systems with impulse responses h(t) and g(t): System 1, with overall impulse response of s₁(t): x(t) x(t) h(t) System 2, with overall impulse response of s₂(t): (c) Now suppose g(t) h(t) Y₁ (t) + Y2(t) g(t) Find the impulse responses s₁(t) and s₂(t). Also find their respective fre- quency responses S₁ (w) and S₂(w). x(t) = eiwot Calculate the output y(t) of the two systems S₁ and S₂ when given x(t) as an input.

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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Composite Systems Consider the following impulse responses of two LTI systems: h(t) = 8(t - 2), g(t) = e-alt, a € R+ (a) Find H(w) and G(w), the frequency responses (CTFT) of the systems. (b) Consider interconnected LTI systems with impulse responses h(t) and g(t): System 1, with overall impulse response of $₁(t): x(t) System 2, with overall impulse response of s₂(t): x(t) (c) Now h(t) suppose g(t) h(t) y₁ (t) g(t) Find the impulse responses s₁(t) and s₂(t). Also find their respective fre- quency responses S₁ (w) and S₂(w). x(t) = eiwot + Y₂(t) Calculate the output y(t) of the two systems S₁ and S₂ when given x(t) as an input. Hint: Convert x(t) to the frequency domain, and feed it through the com- posite system before taking the inverse transform of the result. As a further hint, note that convolution in the time domain is equivalent to multiplication in the frequency domain.
Transcribed Image Text:Composite Systems Consider the following impulse responses of two LTI systems: h(t) = 8(t - 2), g(t) = e-alt, a € R+ (a) Find H(w) and G(w), the frequency responses (CTFT) of the systems. (b) Consider interconnected LTI systems with impulse responses h(t) and g(t): System 1, with overall impulse response of $₁(t): x(t) System 2, with overall impulse response of s₂(t): x(t) (c) Now h(t) suppose g(t) h(t) y₁ (t) g(t) Find the impulse responses s₁(t) and s₂(t). Also find their respective fre- quency responses S₁ (w) and S₂(w). x(t) = eiwot + Y₂(t) Calculate the output y(t) of the two systems S₁ and S₂ when given x(t) as an input. Hint: Convert x(t) to the frequency domain, and feed it through the com- posite system before taking the inverse transform of the result. As a further hint, note that convolution in the time domain is equivalent to multiplication in the frequency domain.
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