signals and systems mid

2

odd signal of $x [n]$ is ${x [n] - x [- n], n >= 0, n < 0$

3

b

odd signal $x [n] = - x [- n]$
even signal $x [n] = x [- n]$
$x_{o} [n] = {x [n] - x [- n], n >= 0, n < 0$

$x_{e} [n] {x [n] x [- n], n >= 0, n < 0$

product of $x_{o} [n]$ and $x_{e} [n]$ is ${x [n]^{2} - x [- n]^{2}, n >= 0, n < 0$ still are odd signal since $x [n] = - x [- n]$

$x = x_{o} + x_{e} x (t)^{2} = x_{o} (t)^{2} + 2 x_{o} (t) x_{e} (t) + x_{e} (t)^{2} \int_{- \infty}^{\infty} x^{2} (t) d t \int_{- \infty}^{\infty} x^{2} (t) d t \int_{- \infty}^{\infty} x^{2} (t) d t = \int_{- \infty}^{\infty} x_{o}^{2} (t) d t + \int_{- \infty}^{\infty} 2 x_{o} (t) x_{e} (t) d t + \int_{- \infty}^{\infty} x_{e}^{2} (t) d t = \int_{- \infty}^{\infty} x_{o}^{2} (t) d t + 0 + \int_{- \infty}^{\infty} x_{e}^{2} (t) d t = \int_{- \infty}^{\infty} x_{e}^{2} (t) d t$

4

	linear	time invariant
$y (t) = t^{2} x (t - 1)$	no	no
$y [n] = x [n + 2] - x [n - 3]$	yes	yes
$y [n] = O_{d} {x [n]}$	yes	no

5

	period
$x (t) = 3 cos (4 t + \frac{π}{3})$	$\frac{π}{2}$
$x (t) = O_{d} {sin (4 π t) u (t)}$	$0.5$
$x [n] = cos (\frac{n}{8} - π)$	$None$

6

	memoryless	time invariant	linear	causal	stable
$y [n] = E_{v} {x [n - 1]}$	❌	✅	✅	✅	✅
$y [n] = x [n - 2] - 2 x [n - 3]$	❌	✅	✅	✅	✅
$y (t) = cos (3 t + 2) x (t)$	✅	✅	❌	✅	✅
$y [n] = {0, x (t) + x (t - 2), if t < 0 otherwise$	❌	✅	❌	✅	✅

something