Memory
A system has no memory if the output at time t=t0 is a function of the input at time t=t0 , and if the output at time t=t0 is not a function of the input for\(\ t\neq t_0\) .
Memoryless system :
A system will be said to be memoryless if its output at time t only depends on the input at the same time.
For example,
y(t)=Rx(t)
This equation can represent an output that corresponds to the voltage across a resistor with the current x(t) as the input. Current does not influence voltage before or after a given moment; it is the product of current and resistance.
For example (Counter-example),
\(y\left(t_0\right)=\frac{1}{c}\int_{-\infty}^{t_0}x\left(z\right)dz\)
This equation represents an output, which is the voltage across a capacitor with a current x\(\left(t\right)\) as the input. The voltage is a function of the previous current, i.e., \(x\left(t\right) for {t<t}_0\) .