What is the effect of cascading two LTI systems on their transfer functions?

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Multiple Choice

What is the effect of cascading two LTI systems on their transfer functions?

Explanation:
Cascading two LTI systems multiplies their transfer functions. When the input goes through the first system, the output is Y1(s) = H1(s) X(s). Feeding that into the second system gives Y(s) = H2(s) Y1(s) = H2(s) H1(s) X(s). Therefore the overall transfer function is H_total(s) = Y(s)/X(s) = H2(s) H1(s). Since multiplication of transfer functions is commutative for these systems, this is also equal to H1(s) H2(s). In other words, the cascade corresponds to convolving the impulse responses, and taking the Laplace transform turns that into a product of the two transfer functions.

Cascading two LTI systems multiplies their transfer functions. When the input goes through the first system, the output is Y1(s) = H1(s) X(s). Feeding that into the second system gives Y(s) = H2(s) Y1(s) = H2(s) H1(s) X(s). Therefore the overall transfer function is H_total(s) = Y(s)/X(s) = H2(s) H1(s). Since multiplication of transfer functions is commutative for these systems, this is also equal to H1(s) H2(s). In other words, the cascade corresponds to convolving the impulse responses, and taking the Laplace transform turns that into a product of the two transfer functions.

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