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.
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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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