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The output y(n) of a system T(.) to the input x(n) is given as y(n) = T(x(n)) = x(a0n). Where a0 is a positive integer not equal to 1. This system is

The output y(n) of a system T(.) to the input x(n) is given as y(n) = T(x(n)) = x(a0n). Where a0 is a positive integer not equal to 1. This system is Correct Answer Only Linear but not Time-invariant

Concept:

A system is called linear if it satisfies two mathematical properties:

  • Additive 
  • Homogeneity

This is as explained as shown:

​[ alt="F1 R.D. N.J 26.09.2019 D 16" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2016.png" style="width: 344px; height: 39px;">

To check for time invariance, we first shift the input in time and observe the output.

Next, we shift the original output in time by the same amount. We then compare both the outputs. If they are equal, then the system is time-invariant

Calculation:

Given, a system can be defined as:

[ alt="F1 R.D. N.J 26.09.2019 D 17" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2017.png" style="width: 256px; height: 28px;">

Checking for linearity first:

[ alt="F1 R.D. N.J 26.09.2019 D 18" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2018.png" style="width: 285px; height: 28px;"> 

[ alt="F1 R.D. N.J 26.09.2019 D 19" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2019.png" style="width: 284px; height: 28px;">

When both the input is simultaneously applied:

[ alt="F1 R.D. N.J 26.09.2019 D 20" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2020.png" style="width: 406px; height: 46px;">

Therefore, the system is Linear.

Checking for time invariance now:

Shifting the input by no,

[ alt="F1 R.D. N.J 26.09.2019 D 21" src="//storage.googleapis.com/tb-img/production/19/10/F1_R.D._N.J_26.09.2019_D%2021.png" style="width: 335px; height: 28px;">

Shifting the original output by the same amount,

i.e. y(n - no) ⇒ x(ao(n - no)) = x(aon - aono)

Clearly, x(aon - aono) ≠ x(aon - no)

The system is time-variant.

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