A system has the inputoutput relation given by yn rxn nxn

A system has the input-output relation given by y[n] = r{x[n]} = nx[n] Determine whether the system is (a) linear, (b) time-invariant. Determine if the following system is linear and/or time-invariant y(t) = cos(3t) x(t)

Solution

Time Variant or Time Invariant Systems

Definition:

A system is said to be Time Invariant if its input output characteristics do not change with time. Otherwise it is said to be Time Variant system.

Explanation:

As already mentioned time invariant systems are those systems whose input output characteristics do not change with time shifting. Let us consider x(n) be the input to the system which produces output y(n)

Now delay input by k samples, it means our new input will become x(n-k). Now apply this delayed input x(n-k) to the same system as shown in figure below.

Now if the output of this system also delayed by k samples (i.e. if output is equal to y(n-k)) then this system is said to be Time invariant (or shift invariant) system.

If we observe carefully, x(n) is the initial input to the system which gives output y(n), if we delayed input by k samples output is also delayed by same (k) samples. Thus we can say that input output characteristics of the system do not change with time. Hence it is Time invariant system.

y(t)= x(t) cos (3t)
we need to determine whether or not this system is linear
1) first I shift the input by t0    ----> x(t-t0) cos (3(t-t0))
2) second I shift the output by t0 and inputaccordinglyt:   y(t-t0) = x(t-t0) cos (3(t-t0)

1 and 2 match up, therefore the system is time invariant