# Frequency response using fourier transform [duplicate]

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

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

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.