# Are these Linear and Time Invariant Systems?

We were given the above systems and we were asked to tell whether they are linear (or not) and time invariant(or not).

1. $y(m,n)=x(m,n)+c,\quad c>0$

2. $y(m,n)=x(m,n)+x(m,-n)$

3. $y(m,n)=x(m,n+m)$

4. $y(m,n)=x(m,n) + y(m-a,n-b)$

Although I know the theory for the signal to be linear and time invariant I have trouble applying it. Can any of you explain one of the above so I can continue on my own for the rest of them? Thank you in advance

• Hi, welcome to DSP SE. What did you do so far? Where are you stuck? Did you do some searching on the website? There is already many answers to such questions. – jojek Apr 13 '16 at 23:22
• Thanks for editing my question, actually i have problem proving all these without having the analytic form of the signal x. – johny Apr 13 '16 at 23:38

## 1 Answer

Well, the system described in problem 1 is non-linear and time invariant. A linear system cannot have a constant term(or a term independent of input/output). Here is the proof of it's non-linearity: Let y1 and x1 be a set of the o/p and i/p respectively y1(m,n)=x1(m,n)+c Similarly, let y2 and x2 be another such set y2(m,n)=x2(m,n)+c Now, let x(m,n)=x1(m,n)+x2(m,n)......(1) For the system to be linear, when x(m,n)[which is a sum of two signals] is given as input, the output should be a sum of the individual outputs. Let's check: y(m,n)=x(m,n)+c from (1), we can write y(m,n)=x1(m,n)+x2(m,n)+c which is not equal to y1(m,n)+y2(m,n). Hence proved. The time invariance property is quite straight forward to prove.