What is a System

Basically a function \(f(t)\) that takes some signal.

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Classification

Memory

If the output changes based on inputs from current and previous times, it has memory.

In other words, a system is memoryless if the output \(y\) only depends on the input at a given time \(t\), and \(y\) is always the same for the same \(t\).

Invertibility

If the system has an inverse function, then the system is invertible.

• Inverse function: for each \(t\) there exists a unique \(x(t)\), so that an inverse function exists \(w(x(t)) = t\)

Causality

If a system output depends only on inputs from current and past time, the system is causal. Ex: \(y = x(t) + x(t-T)\)

In other words, if an output considers values from future time to the input, then it is non-causal. Ex: \(y = x(t) + x(t+T)\)

Stability

If a system output does not change aggressively on small input changes, it is stable.

In other words, if a small change in inputs causes big changes in outputs, it is not stable.

Formal def here

Linearity

A system is linear if inputs and outputs follow superposition:

\[x(t) = y,\quad x(t) = x_1(t) + x_2(t)\] \[x_1(t) + x_2(t) = y_1 + y_2 = y\] \[ax(t) = ay(t)\]

Some tips: \(x(t) + C,\quad x(t), y(t)\) are multiplied with derivatives/integrals, or scaled with non-integer exponents, or there are abs(x,y) then it is not linear. Imagine a similar criteria to linear ODEs.

Time Invariance

If a time shift on x causes the same time shift in y, then it is time-invariant.

\[x(t) => y(t), \quad x(t + T) = y(t + T)\]