Frequency Response Homework: Impulse & Cosine Output

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Homework Statement

3. (12 pts) Consider the causal FIR filter with {bk } = {1, 4, 5, 4, 1}.
(a) What is the impulse response of this filter (in terms of delta functions)?
(b) What is the frequency response of this filter? Simplify using Euler’s inverse formula.
(c) What is the output y[n] of this system when the input is x[n] = cos(0.5πn)?

Homework Equations

The Attempt at a Solution

(a) h[n]= δ[n]+ 4δ[n-1]+ 5δ[n-2]+ 4δ[n-3]+δ[n-4]

(b)

$$H(e^{j\hat{\omega}}) = 1 + 4e^{-j\hat{\omega}}+5e^{-j2\hat{\omega}}+4e^{-j3\hat{\omega}}+e^{-j4\hat{\omega}}$$

$$H(e^{j\hat{\omega}}) = e^{-j2\hat{\omega}} [e^{j2\hat{\omega}} + e^{-j2\hat{\omega}}+4e^{j\hat{\omega}}+e^{-j\hat{\omega}}+5]$$

$$H(e^{j\hat{\omega}}) = e^{-j2\hat{\omega}} [2cos(2\hat{\omega})+8cos(\hat{\omega})+5]$$

(c) Does 0.5pi get substituted for omega hat to solve part c?

 

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Hopefully someone can give me a hint on this... It seems to me that I need an fs to complete this problem. omega hat = omega*Ts. If you are not given fs, do you just use the Nyquist rate?