a) Encontrar \(y[n]\)
\[
x[n] = [1, 0, 2, -3]
\]
\[
h[n] = [3, -3]
\]
\[
y[n] = ?
\]
b) Encontrar \(n_y [n]\) sabiendo que:
\[
n_x[n] = [0, 1, 2, 3]
\]
\[
n_h[n] = [0, 1]
\]
Desarrollo
2) Intervalo 1
Cuando \(n < 0\)
\[
y[n] = 0
\]
3) Intervalo 2
Cuando \(n = 0\)
\[
y[0] =
\color{orange} 3 \color{black} \cdot \color{purple} 1 \color{black}
=
3
\]
3) Intervalo 3
Cuando \(n = 1\)
\[
y[0] =
\color{orange} 3 \color{black} \cdot \color{purple} 0 \color{black}
+ \color{orange} (-3) \color{black} \cdot \color{purple} 1 \color{black}
=
-3
\]
3) Intervalo 4
Cuando \(n = 2\)
\[
y[0] =
\color{orange} 3 \color{black} \cdot \color{purple} 2 \color{black}
\color{orange} -3 \color{black} \cdot \color{purple} 0 \color{black}
=
6
\]
3) Intervalo 5
Cuando \(n = 3\)
\[
y[0] =
\color{orange} 3 \color{black} \cdot \color{purple} (-3) \color{black}
\color{orange} -3 \color{black} \cdot \color{purple} 2 \color{black}
=
-15
\]
3) Intervalo 6
Cuando \(n = 4\)
\[
y[0] =
\color{orange} (-3) \color{black} \cdot \color{purple} (-3) \color{black}
=
9
\]
4) Finalmente escribimos la salida \(y[n]\)
\[
\color{limegreen} \boxed{ \color{black}
y[n] = [3, -3, 6, -15, 9]
}
\]
\[
\color{limegreen} \boxed{ \color{black}
n_y[n] = [0, 1, 2, 3, 4]
}
\]