discrete signals - Initial conditions for the LTI systems described as a difference equations

Why do we need the initial conditions to be zero for the LTI systems described as a difference equations?

1. 
   

   First question is why do we need it for linearity? I can't think of any example of the non linear system described as a difference equation with constant coefficients.

   
   


2. 
   

   The definition of TI system is that it does not depend on particular time the input is applied. I can't understand how does this definition relate to the initial conditions? Those initial conditions will be the same whenever we apply the input so why they have to be zero?

   
   

Thanx

Answer

The problem is that non-zero initial conditions cause a term in the output signal that does not depend on the input signal. This explains why a system with non-zero initial conditions can neither be linear nor time-invariant. A linear system must have a zero output for zero input. With non-zero initial conditions the output will generally be non-zero, even for a zero input signal. Alternatively, think of scaling a given input signal. A linear system will have a response that is scaled in the same way. However, the part of the output signal caused by non-zero initial conditions will not scale accordingly, because it's independent of the input signal.

The same is true concerning time-invariance. For a time-invariant system, a shifted version of the input signal must result in an output signal with the same shift. However, the output term caused by non-zero initial conditions will not shift accordingly, as it is independent of the input signal.

Consequently, a system described by a linear difference equation with constant coefficients plus initial conditions is only linear and time-invariant if the initial conditions are zero.