Cascade a time delay and a time scaling

Consider the following two systems:

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow y(t)=x(t+2) $

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t)=x(5t) $

Obtain a simple expression for the output of the following cascade:

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t) $

(Sorry, I don't know how to make a real "box" to represent a system. If somebody knows, please help. -pm)

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Answer 1

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t) = x(5(t + 2)) = x(5t + 10) $ --Cmcmican 16:05, 15 January 2011 (UTC)

Instructors comments: Unfortunately, the answer is not correct. (I actually expected that mistake to happen. Almost everybody does it the first time.) Try to carefully write the output after each step of the cascade, and change the variable explicitely. -pm

Answer 2

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t) = x(5t + {2 \over {5}}) $

--Rgieseck 11:47, 18 January 2011 (UTC)

Draw an example signal and apply the systems graphically to logically deduce this answer.

Instructors comments: Unfortunately, this answer is not correct either. I know the graphical approach is tempting, but there is a point where problems become too complicated to be handled that way. I think ECE301 is such a point. Let me suggest another approach below. -pm

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow z(t)=x(t+2) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow w(t)= z(5t) = x(5t + 2 ) $

Does everybody understand why z(5t)=x(5t+2)? -pm

No, I am confused as to why this is true. Could you please explain in more detail? -RG

Answer 3

write it here.

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