Linear time-invariant system

Question

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.

Answer 1

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 ;)