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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)
[tex]
H(e^{j\hat{\omega}}) = 1 + 4e^{-j\hat{\omega}}+5e^{-j2\hat{\omega}}+4e^{-j3\hat{\omega}}+e^{-j4\hat{\omega}}
[/tex]
[tex]
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]
[/tex]
[tex]
H(e^{j\hat{\omega}}) = e^{-j2\hat{\omega}} [2cos(2\hat{\omega})+8cos(\hat{\omega})+5]
[/tex]
(c) Does 0.5pi get substituted for omega hat to solve part c?