A linear system, in continuous time or discrete time.is a system that possesses the property of superposition.If an input consists of the weighted sum of several signals,then output is the superposition-that is the weighted sum-of the responses of the system to each of those signals.
Thus the system is considered to be linear when a) The response to x1(t)+x2(t) is y1(t)+y2(t) b) The response to ax1(t) is ay1(t),where a is any complex constant.
Example: Consider a system S whose input x(t) and output y(t) are related by y(t) = tx(t),we choose two arbitrary inputs x1(t) and x2(t) x1(t) -> y1(t)=t x1(t) x2(t) -> y2(t) = t x2(t) and let x3 be a linear combination of x1(t) and x2(t),ie x3(t) = ax1(t)+bx2(t)
If x3 is input to S then corresponding output may be expressed as: y3(t) = tx3(t) = ay1(t)+by2(t)
where as a non-linear system is a system that doesn't possess the property of superposition. Example:
y[n] = 2x[n] + 3 the system is non linear as it violates the additive property: if x1[n] = 2 and x2[n] = 3 then x1[n] -> y1[n] = 2x1[n]+3 = 7 x2[n] -> y2[n] = 2x2[n]+3 = 9
However, response to x3[n] = x1[n]+x2[n] is y3[n]= 2{x1[n]+x2[n]} + 3 = 13 which is not equal to y1[n]+y2[n] = 16
A system is called time invariant if for any input signal x(t)(x[n]) and for any t0 belongs to R, the response to the shifted inputX(t-t0) is the shifted output y(t-t0).
. Example: X(t) ->SYSTEM -> y(t) = 10 x(t) is time invariant because X(t) -> t0 -> y(t) = X(t-t0) -> SYSTEM -> z(t) = 10 y(t) = 10 x(t-t0)where as a system is called time variant when we find an input signal for which the condition of time invariance is violated.
. Example:
y[n] = nx[n] Proof: consider an input signal x1[n] = d[n] which yields an output y1[n] that is identically 0.However the input x2[n] = d[n-1] yields the output y2[n] = nd[n-1] = d[n-1].
