Hey all I have the following question and I am having trouble getting to the correct answer. Here is what I have: (sorry I do not understand how to format the question)
Consider a linear time-invariant system such that
$$H(e^{j\omega}) = \frac{1}{(1-\frac{1}{2}e^{j\omega})^2}$$
If the input x ̃[n] is periodic with period N0 = 8, then determine the output Fourier series coefficient y4 if x4 = 9.
I have this so far:
$$y_4 = x_4 H(e^{jk\frac{2\pi}{N_0}})$$
$$= 9 H(e^{j\pi})$$
Goes back into the given function:
$$= \frac{9}{(1-\frac{1}{2}e^{-j\omega})^2}$$
I am stuck here. How would I get a solid answer from this. I know that in the end the answer is 4.
Thanks.
Best Answer
I wasn't really able to follow your reasoning, but I can tell you how you can simplify your second equation:
$$9 H(e^{j\pi})$$
Note that $$e^{j\pi} = -1$$ (Euler's formula).
Inserting that into the equation above gives:
$$ \frac{9}{(1-\frac{1}{2}(-1))^2} = \frac{9}{1.5^2}= 4$$ I hope this helps ;)