Let h[n] be the impulse response of a discrete time linear time invariant (LTI) filter. The impulse response is given by h(0)= $${1 \over 3};h\left[ 1 \right] = {1 \over 3};h\left[ 2 \right] = {1 \over 3};\,and\,h\,\left[ n \right]$$ =0 for n < 0 and n > 2. Let H ($$\omega $$) be the Discrete- time Fourier transform (DTFT) of h[n], where $$\omega $$ is the normalized angular frequency in radians. Given that ($${\omega _o}$$) = 0 and 0 < $${\omega _0}$$ < $$\pi $$, the value of $${\omega _o}$$ (in ratians ) is equal to ____________.

Consider the signal $$x\left[ n \right] = 6\delta \left[ {n + 2} \right] + 3\delta \left[ {n + 1} \right] + 8\delta \left[ n \right] + 7\delta \left[ {n - 1} \right] + 4\delta \left[ {n - 2} \right]$$.

If X$$({e^{t\omega }})$$is the discrete-time Fourier transform of x[n],

then $${1 \over \pi }\int\limits_{ - \pi }^\pi X ({e^{j\omega }}){\sin ^2}(2\omega )d\omega $$ is equal to ____________.

Let $$\widetilde x\left[ n \right]\, = \,1 + \cos \left[ {{{\pi n} \over 8}} \right]$$ be a periodic signal with period 16. Its DFS coefficients are defined by
$${a_k}$$ = $${1 \over {16}}\sum\limits_{n = 0}^{15} {\widetilde x} \left[ n \right]\exp \left( { - j{\pi \over 8}kn} \right)$$ for all k. The value of the coeffcients $${a_{31}}$$ is _____________________.

A 5-point sequence x [n] is given as x$$\left[ { - 3} \right]$$ =1, x$$\left[ { - 2} \right]$$ =1, x$$\left[ { - 1} \right]$$ =0, x$$\left[ { - 0} \right]$$ = 5, x$$\left[ { - 1} \right]$$ = 1. Let X$$({e^{j\omega }})\,$$ denote the discrete - time Fourier transform of x(n). The value of $$\int\limits_{ - \pi }^\pi x $$ ($$({e^{j\omega }})\,$$ d$$\omega $$ is