Consider the signals $x[n]=2^{n-1} u[-n+2]$ and $y[n]=2^{-n+2} u[n+1]$, where $u[n]$ is the unit step sequence. Let $X\left(e^{j \omega}\right)$ and $Y\left(e^{j \omega}\right)$ be the discrete-time Fourier transform of $x[n]$ and $y[n]$, respectively. The value of the integral

$$ \frac{1}{2 \pi} \int_o^{2 \pi} X\left(e^{j \omega}\right) Y\left(e^{-j \omega}\right) d \omega $$

(round off to one decimal place) is $\_\_\_\_$

Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n],

where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}} $$.

The value (correct to two decimal places) of $$\sum\limits_{n = 0}^3 {x\left[ {2n} \right]} $$ is ___________.

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 ____________.

Two discrete-time signals x [n] and h [n] are both non-zero only for n = 0, 1, 2 and are zero otherwise. It is given that x(0)=1, x[1] = 2, x [2] =1, h[0] = 1, let y [n] be the linear convolution of x[n] and h [n]. Given that y[1]= 3 and y [2] = 4, the value of the expression (10y[3] +y[4]) is _____________________.