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This question already has an answer here:

i need to find the frequency response of the following equations:

$$h(t) = \delta(t) +e^{-2(t-1)} u(t-1)$$ $$x(t) = \cos\left(\frac{pi}{4}t\right)+2\cos\left(\frac{pi}{2}t\right)$$

find $y(t)$

However i am quite confused on how to do this, please explain and show every step so i can learn

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marked as duplicate by nispio, Peter K. Nov 21 '13 at 15:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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This looks like a homework problem, so I'm not going to give you the answer, rather I will try to point you in the right direction.

H(t) is the impulse response of the filter. X(t) is the input. The response of a filter in the time domain (i.e. y(t)) can be calculated by convolving h(t) and x(t). Convolution is a linear process so you can do it by convolving all of h(t) with all of x(t), or you can break h(t) up into two pieces- $\delta(t)$ and $e^{-2(t-1)}u(t-1)$- and/or break x(t) into two pieces- $cos(\frac{\pi}{4}t)$ and $2cos(\frac{\pi}{2}t)$- then convolve the individual pieces and add up the results to get the total result. Breaking the input and filter into simpler pieces is usually the way to go because it is easier.

An even easier way is to do it in the frequency domain. Convolution in the time domain is equivalent to multiplication in the frequency domain, so the process is to 1) calculate the Fourier transform of the input and the filter impulse response, 2) multiply the transforms together, and 3) inverse transform the result back to the time domain. All of these steps are linear operations, so once again you can break up the filter impulse response and the input into pieces and run the process on the individual pieces. You will find that that is a much easier way to do things, particularly because you can make good use of the Fourier identities/properties.

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