I am trying to find a general way to find system stability.I have applied these methods and struck in confusion
For eg:
If
$$y(t)= \int_{-\infty}^{t} x(\tau) \sin(4\tau) d\tau$$
then find whether the system is stable or unstable?
Process (1):
If $$y(t) = \int_{-\infty}^{t} x(\tau)\sin(4\tau) d\tau$$
then
$$h(t) = \int_{-\infty}^{t} \delta(\tau) \sin(4\tau) d\tau$$
$$\implies h(t) = \int_{-\infty}^{\infty} \delta(\tau)\sin(4\tau) u(t-\tau) d\tau$$
$$\implies h(t) = [\sin(4\tau)u(t-\tau)]|_{\tau=0}$$
$$\implies h(t)=0$$.
so system is stable as $$\int_{-\infty}^{\infty} |h(t)| dt= 0 < \infty$$ ;
Process (2):
If $$y(t)= \int_{-\infty}^{t} x(\tau)\sin(4\tau) d\tau$$
then when $x(t)=\sin(4t)$
$$y(t)= \int_{-\infty}^{t} \sin(4\tau)\sin(4\tau) d\tau = \infty $$
so the system produces unbounded output $y(t)$ for bounded input $x(t)= \sin(4t)$
Hence the system is unstable
Thus actually what will be the system stability and which process is right and which is wrong and how it is wrong?? please explain....