Objectives_template

Signals in Natural Domain

Chapter 3 : Discrete-Time Systems

Time invariance :

A system is said to be time invariant if the behavior and characteristics of the system do not change with time.Thus a system is said to be time invariant if a time delay or time advance in the input signal leads to identical delay or advance in the output signal. Mathematically if

{y[n]} = T ({x[n]})

then {y[n-n₀]} = T({x[n-n₀]}) for any n₀

Let us consider the accumulator system

$\displaystyle y[n]=\sum\limits_{k=- \infty}^{n}x[k]$

If the input is now {x₁[n]} = {x[n-n₀]} then the corresponding output is

$\displaystyle y_1[n]=\sum\limits_{k=-\infty}^{n}x_1[k]$

$\displaystyle =\sum\limits_{k=-\infty}^{n}x[k]$

The shifted output signal is given by

$\displaystyle y[n-n_0]==\sum\limits_{k=-\infty}^{n-n_0}x[k]$

The two expression look different, but infact they are equal. Let us change the index of summation by l = k - n₀ in the first sum then we see that

$\displaystyle y_1[n]==\sum\limits_{l=-\infty}^{n-n_0}x[l]$

$\displaystyle =y[n-n_0]$

Hence, {y[n]} = {y[n-n₀]} and the system is time-invariant. As a second example consider the system defined by y[n] = nx[n]

if $\displaystyle \{x_1[n]\}=\{x[n-n_0]\}$

$\displaystyle y_1[n]=nx_1[n]=nx[n-n_0]$

while $\displaystyle y[n-n_0]=(n-n_0)x[n-n_0]$

and so the system is not time-invariant. It is time varying. We can also see this by giving a counter example. Suppose input is $\{x[n]\}=\{\delta[n]\}$ then output is all zero sequence. If the input is $\{\delta [n-1]\}$ then output is $\{\delta [n-1]\}$ which is definitely not a shifted version version of all zero sequence.