# linear and time-invariant system

#### cupcake

Joined Sep 20, 2010
73
consider these systems is linear or time invariant?
a) Y[n]=aX[n]+b
b) Y[n]=X[n] X[n-5]
c) Y[n]=5X[n]+9X[n-5]

I know that time-invariant system means if a time shift of the input sequence causes a coresponding shift in the output sequence.

here are the attempt at a solution

a) Y[n]=aX[n]+b
input X[n]
X1[n]= X[n-no]
Y1[n]= aX[n-no]+b
Y[n-no]= aX[n-no]+b
so,this system is time-invariant

b)Y[n]=X[n] X[n-5]
input X[n]
Y1[n]=X[n-no] X[n-no-5]
Y[n-no]=X[n-no] X[n-no-5] <- here I'm not sure..
but, I think this system is time-invariant

c)Y[n]=5X[n]+9X[n-5]
input X[n]=X[n-no]
Y1[n]=5X[n-no]+9X[n-no-5]
Y[n-no]=5X[n]-5X[no]+9X[n]-9X[no]+9X[-5]
since the output wasn't as we expect, so the system is not time-invariant

can, someone check my works, am I correct or wrong? and also, how to check whether the system is linear, I know that theoretically the linear system should satisfy the superposition and proportionality, but I'm not sure how to work out with it.. thanks

#### Georacer

Joined Nov 25, 2009
5,182
I 'm not very good with S&S myself, but I think b is variant and c is invariant.

I base my claim to the fact that b. is of the form y(n)=a(n)x(n-k), where a is a time-related coefficient.
As for the system c., I think it is time-invariable, since it is a linear combination of the inputs.

Someone should explain it in more words, though.

#### cupcake

Joined Sep 20, 2010
73
but, I have to prove whether the system time-invariant or not step by step, that's why I need to show some workings...

do you know how to prove whether the system is linear or not? I know the condition theoretically as I mentioned before, but I don't know how to work it out step by step..can someone help me with this?

#### cupcake

Joined Sep 20, 2010
73

#### narasimhan

Joined Dec 3, 2009
72
I think you missed the subscripts in the question so I assumed that the questions were
b) Y[n]=X1[n] X2[n-5]
c)Y[n]=5X1[n]+9X2[n-5]

a and b are non-linear and time-invariant
c is linear and time-invariant

Refer Signals and systems by Oppenheim the very 1st unit contains lots of such problems
I have a national entrance exam(entire ece syllabus) next sunday. So I will provide full answers later.

#### narasimhan

Joined Dec 3, 2009
72
On second thought I guess the questions are valid without the subscript.
But either way the answer that I posted previously holds.

#### cupcake

Joined Sep 20, 2010
73
thanks..
hmm... how did you prove that the systems are linear or non-linear system?
and what made you think that c is time-invariant?

#### narasimhan

Joined Dec 3, 2009
72
For linearity
let input be x1(n) and output be y1(n)
and input be x2(n) and output be y2(n)
Find the output for ax1(n)+bx2(n). Let the output be y3(n).
If y3(n)=ay1(n)+by2(n) then the system is linear. This is clearly given in oppenheim 1st unit. Refer it for further details.

By looking at c itself one can guess that the system is time-invarient, since it doesn't have any terms with n outside the brackets nor does it have any -n within the brackets of any term.
But a rigorous and fool proof way to find out is to use the method given in the same textbook.
let input be x(n) and output be y(n). Find y(n-T) and name it as y1(n) i.e. y1(n)=y(n-T)
let input be x(n-T) and output be y2(n)
where T can be any integer

Now if y1(n)=y2(n) then the system is time-invarient

#### cupcake

Joined Sep 20, 2010
73
ok, thanks I will look for that book and read more..

another questions, what is the impulse response of these systems (a) (b) and (c)?? maybe you can give me the brief explanation and I will try working out the answer..thanks

#### narasimhan

Joined Dec 3, 2009
72
Only LTI systems are completely specified by impulse response. Which means only c has impulse response. Just substitute delta(n) instead of x(n) and you'll get impulse response.

#### cupcake

Joined Sep 20, 2010
73
Thanks a lot!

#### narasimhan

Joined Dec 3, 2009
72
Thanks a lot!
You don't need to thank everytime. We are here to help.

Anyway there is an easier way to thank someone(right bottom corner of any post)