Recall that a discrete-time system is a mathematical entity that takes an input signal (usually denoted $x$) and produces an output signal (usually denoted $y$). In the study of signal processing, systems that have certain characteristics are of particular interest. Among these characteristics is
time-invariance .
There is a very basic intuition behind the notion of time-invariance. In many cases, we would like a system to behave a certain way, no matter when an input may be given. To use a practical analogy, people expect their toasters to operate the same way on Tuesdays as they do on Mondays.
A system is time-invariant if a time delay for an input results in the same time delay on the output. Expressing this idea of system time-invariance mathematically is straightforward. Consider a system $H$; let $x[n]$ be some arbitrary input, and call the system's output for that input $y[n]$. $H$ is time-invariant if, for some arbitrary integer value $q$, $H[x[n-q]]=y[n-q]$:
Example: moving average system
Consider, for example, the moving average system $y[n]= \frac{1}{2}(x[n]+x[n-1])$. Is this system time invariant? To find out, let's delay the input by some value $q$, and see what the output is. We'll call the delayed input signal $x'[n]$ (=$x[n-q]$) and the new output $y'$:$y'[n]=\frac{1}{2}(x'[n]+x'[n-1])=\frac{1}{2}(x[n-q]+x[(n-1)-q])$Now the important question, is $y'[n]=y[n-q]$? That will be easy to determine, simply replace every $n$ with $n-q$ in that original $y[n]$ equation and see if it is the same as $y'[n]$:
$y[n-q]=\frac{1}{2}(x[n-q]+x[n-q-1])=\frac{1}{2}(x[n-q]+x[(n-1)-q])=y'[n]\checkmark$
Example: decimation
Now let's consider a decimation system, $y[n]=x[2n]$. Is this system time-invariant? First, create a new signal that is a delayed version of $x[n]$: $x'[n]=x[n-q]$. Next, find the output corresponding to this new signal: $y'[n]=x'[2n]$. Now express this output in terms of the original input: $y'[n]=x'[2n]=x[2n-q]$. Finally, check to see if this is the same as $y[n-q]$. As $y[n-q]=x[2(n-q)]$ is not equivalent to $y'[n]$ (for $2n-1\neq 2(n-q)$ for all $n$ and $q$), the system is not time-invariant.
Now practice on the system examples below; which of them are time-invariant?
All of the discussion above has considered systems which take infinite-length signals as inputs. A system which takes finite-length signals as inputs and produces them as outputs can also exhibit the characteristic of time-invariance, defined slightly differently. Such systems are time-invariant if any
circular shift on the input results in a corresponding
circular shift on the output:
Questions & Answers
discuss how the following factors such as predation risk, competition and habitat structure influence animal's foraging behavior in essay form
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
