Question

PROBLEM 4A (15 pts.) The frequency response of a FIR filter is given by $H(e^{j\omega}) = e^{-j2\omega}(3\cos(\omega) + a\cos(2\omega))$ (i) (4 pts.) If the input x[n] = (-1)$^n$ (n ? Z) results in the (same) output y[n] = (-1)$^n$ (n ? Z) what is the value of a? (ii) (7 pts.) Using the value of a found in (i), determine the output y[n] when the input is given by $x[n] = 1 + \cos(\frac{\pi n}{3} - \frac{\pi}{4})$ Simplify your answer. (iii) (4 pts.) Derive the input-output relationship of this FIR filter, in the standard form $y[n] = \sum_{k=0}^{M} b_k x[n-k]$

          PROBLEM 4A (15 pts.)
The frequency response of a FIR filter is given by
$H(e^{j\omega}) = e^{-j2\omega}(3\cos(\omega) + a\cos(2\omega))$
(i) (4 pts.) If the input
x[n] = (-1)$^n$  (n ? Z)
results in the (same) output
y[n] = (-1)$^n$ (n ? Z)
what is the value of a?
(ii) (7 pts.) Using the value of a found in (i), determine the output y[n] when the input is given by
$x[n] = 1 + \cos(\frac{\pi n}{3} - \frac{\pi}{4})$
Simplify your answer.
(iii) (4 pts.) Derive the input-output relationship of this FIR filter, in the standard form
$y[n] = \sum_{k=0}^{M} b_k x[n-k]$

PROBLEM 4A (15 pts.)
The frequency response of a FIR filter is given by
H(e^jω) = e^-j2ω(3cos(ω) + acos(2ω))
(i) (4 pts.) If the input
x[n] = (-1)^n (n ? Z)
results in the (same) output
y[n] = (-1)^n (n ? Z)
what is the value of a?
(ii) (7 pts.) Using the value of a found in (i), determine the output y[n] when the input is given by
x[n] = 1 + cos((π n)/(3) - (π)/(4))
Simplify your answer.
(iii) (4 pts.) Derive the input-output relationship of this FIR filter, in the standard form
y[n] = ∑k=0^M bk x[n-k]

Added by Paul M.

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