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This method of observing the pattern and determining what the system is doing appears to work in this situation. However, it is not very rigorous, and does not explain how linearity plays a role. It may work for simple problems; however, it may also prove difficult/impossible to utilize correctly with more complex systems. Actually, from HW2, problem E clearly demonstrates this issue. A more rigorous method would be to write the input <math>\cos(2t)</math> as a linear combination of the given input functions, and then use the definition of linearity to write the response as a linear combination of the given response functions. --Jeff Kubascik | This method of observing the pattern and determining what the system is doing appears to work in this situation. However, it is not very rigorous, and does not explain how linearity plays a role. It may work for simple problems; however, it may also prove difficult/impossible to utilize correctly with more complex systems. Actually, from HW2, problem E clearly demonstrates this issue. A more rigorous method would be to write the input <math>\cos(2t)</math> as a linear combination of the given input functions, and then use the definition of linearity to write the response as a linear combination of the given response functions. --Jeff Kubascik | ||
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| + | I 100% agree with Jeff, it appears that your method just happened to get you the correct answer and doesn't use linearity at all in the solution (which for a more complicated input would be required). Euler's formula is the key to this problem, split up the cosine and you'll see the pattern. -Travis Safford | ||
Revision as of 18:11, 17 September 2008
This method of observing the pattern and determining what the system is doing appears to work in this situation. However, it is not very rigorous, and does not explain how linearity plays a role. It may work for simple problems; however, it may also prove difficult/impossible to utilize correctly with more complex systems. Actually, from HW2, problem E clearly demonstrates this issue. A more rigorous method would be to write the input $ \cos(2t) $ as a linear combination of the given input functions, and then use the definition of linearity to write the response as a linear combination of the given response functions. --Jeff Kubascik
I 100% agree with Jeff, it appears that your method just happened to get you the correct answer and doesn't use linearity at all in the solution (which for a more complicated input would be required). Euler's formula is the key to this problem, split up the cosine and you'll see the pattern. -Travis Safford
