Time-Invariant Systems

A time-invariant system with shifted input and output

A system is time-invariant if shifting the input sequence on time axis leads to an equivalent shift of the output sequence along the time axis, with no other changes. So if an input s[n] generates an output r[n], then

    \begin{align*}       \textmd{input}\quad s[n-n_0] ~& \xrightarrow{\hspace*{2cm}}~ r[n-n_0] \quad \textmd{output}     \end{align*}

In other words, if an input signal is delayed by some amount n_0, output also just gets delayed by n_0 and remains exactly the same otherwise. Whether a system is time-invariant or not can be determined by the following steps.

  1. Delay the input signal s[n] by n_0 samples to make it s[n-n_0]. Find the output r_1[n] for this input.
  2. Delay the output signal r[n] by n_0 samples to make it r[n-n_0]. Call the new output r_2[n].
  3. If r_1[n] = r_2[n], the system is time-invariant. Otherwise, it is time-variant.
Example


Consider a system

    \begin{equation*}       r[n] = s[n] + s[n-3]     \end{equation*}

We will follow the steps above to determine whether this system is time-invariant or not.

  1. Delaying the input signal by n_0 samples, the output in response to s[n-n_0] is

        \begin{equation*}       r_1[n] = s[n-n_0] + s[n-n_0-3]     \end{equation*}

  2. Delaying the output signal by n_0 samples, we get

        \begin{equation*}       r_2[n] = s[n-n_0] + s[n-3-n_0]     \end{equation*}

  3. Since r_1[n] = r_2[n], this system is time-invariant.

Now consider another system

    \begin{equation*}       r[n] = n s[n]     \end{equation*}

and apply the same steps.

  1. Delaying the input signal by n_0 samples, the output in response to s[n-n_0] is

        \begin{equation*}       r_1[n] = n s[n-n_0]     \end{equation*}

  2. Delaying the output signal by n_0 samples, we get

        \begin{equation*}       r_2[n] = (n-n_0)s[n-n_0]     \end{equation*}

  3. Since r_1[n] \neq r_2[n], this system is time-variant.

In other words, a system is time-invariant if the order of delaying does not matter, i.e.,

    \begin{align*}   s[n] ~~ \rightarrow ~~ \textmd{Delay} ~~ \rightarrow &~~ \textmd{System} ~~ \rightarrow ~~ \textmd{Output} ~r_1[n] \end{align*}

is the same as

    \begin{align*}   s[n] ~~ \rightarrow ~~  \textmd{System} ~~ \rightarrow &~~ \textmd{Delay} ~~ \rightarrow ~~ \textmd{Output} ~r_2[n] \end{align*}

Linear Time-Invariant (LTI) System


A system that is both linear and time-invariant is, not surprisingly, called Linear Time-Invariant (LTI) system. It is a particularly useful class of systems that not only truly represents many real-world systems but also possesses an invaluable benefit of having a rich set of tools available for its design and analysis.

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