Frequency Response of FIR Filters

Throughout this problem consider the length-three moving average FIR filter

$$y[n] = \frac{1}{3} \sum_{k=0}^2 x[n-k].$$

1.

Compute the output of this filter if the input is $x_1[n]=\exp(j 2 \pi 0 n)$. (Hint: sketch x₁[n] first.)
Provide your answer both as a sequence of numbers and in the form $y_1[n]=H_1 \exp(j 2 \pi 0 n)$, where H₁ is a constant.

2.

Compute the output of this filter if the input is $x_2[n]=\exp(j 2 \pi \frac{1}{2} n)$. (Hint: sketch x₂[n] first.)
Provide your answer both as a sequence of numbers and in the form $y_2[n]=H_2 \exp(j 2 \pi \frac{1}{2} n)$, where H₂ is a constant.

3.

Compute the output of this filter if the input is x[n]=x₁[n]+3x₂[n].
Provide your answer both as a sequence of numbers and in the form $y[n]=H_1 \exp(j 2 \pi 0 n) + H_2 \exp(j 2 \pi \frac{1}{2} n)$, where H₁ and H₂ are constants.

4.

Compute the frequency response of this filter.

5.

Bonus: For which frequency is the frequency response equal to 0?