**Practice Question on "Signals and Systems"**

Topic: System Properties

## Question

The input x[n] and the output y[n] of a system are related by the equation

$ y[n]=x[n-1]+x[1-n]. $

Is the system time invariant (yes/no)? Justify your answer.

You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!

### Answer 1

No, this system is time variant. $ x[n] \to \Bigg[ time\ delay\ n_0 \Bigg] \to y[n]=x[n-n_0] \to \Bigg[ system \Bigg] \to z[n]=y[n-1]+y[1-n]=x[(n-1)-n_0]+x[(1-n)-n_0] $

$ x[n] \to \Bigg[ system \Bigg] \to y[n]=x[n-1]+x[1-n] \to \Bigg[ time\ delay\ n_0 \Bigg] \to z[n]=y[n-n_0]=x[(n-n_0)-1]+x[1-(n-n_0)] $

$ =x[n-1-n_0]+x[1-n+n_0]\, $

The second term in the last equation has a factor of $ +n_0 $, so the two are not equal, therefore this system is time variant.

--Cmcmican 19:07, 26 January 2011 (UTC)

- TA's comment: Correct. This system is a time-varying system. Good job!

--Ahmadi 17:22, 27 January 2011 (UTC)

### Answer 2

Write it here.

### Answer 3

Write it here.