# Frequency Response

1. Nov 16, 2013

### freezer

1. The problem statement, all variables and given/known data

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)?

2. Relevant equations

3. 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?

2. Nov 17, 2013

### freezer

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?

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