Response of LTI System A to x(t): y(t)=x(t)-x(t-2a)

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The discussion focuses on determining the response y(t) of LTI system A to the input x(t), expressed as y(t) = x(t) - x(t-2a). The user seeks clarification on how to derive this relationship, particularly for the interval 2a < t < 4a. The equations for x(t) and y(t) are provided, highlighting their behavior in specified time intervals. The user asks for agreement on the derived equations of the lines, indicating a need for validation of their calculations. Understanding the relationship between y(t) and x(t) is crucial for analyzing the system's response.
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Hi,
y(t) is the response of LTI system A to x(t). How could I have figured out that y(t) = x(t) - x(t-2a) (please see attachment). By looking at the graphs this isn't apparent to me :S.
 

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Write out the equations of the lines. You know that the equations are equal for 0<t<2a and 4a<t<6a.

Write out the equation of y(t) for 2a<t<4a. Relate that equation to the equation for x(t).
 
x(t) = 2t - 2(t-a) + 2(t-3a) - 2(t-4a)
y(t) = 2t - 2(t-a) - 4(t-2a) + 4(t-3a) + 2(t-4a) - 2(t-5a) + 2(t-6a)
These are the equations of the lines. Would you agree?
 

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