Memoryless and Memory Systems

Memoryless system: The output at time $t$ only depends on the input at time $t$.

Memory system: The output at time $t$ depends on past inputs as well.

• Finite memory: Output is determined by the input in the last $T$ seconds
• Infinite memory: Outputs are dependent to the entire past $(-\infty ,t)$

Example:

• Memoryless:

$$y(t)=(t-1)x(t)$$

• Finite with memory $T=3$:

$$y(t)=(t-1)x(t)+x(t-3)$$

• Infinite memory (accumulator/summer in this case):

$$y[n]=y[n-1]+x[n]=\sum _{k=-\infty }^{n}x[k]$$