Can Any Formal System Prove Its Own Consistency?

Prove wheter a discrete-time system is LTI (linear time-invariant)?

• This is an assignment question which make me so puzzled. This question says, For the following discrete-time systems with input x(n) and output y(n), prove whether it is LTI. Show your working. y[n] = | cos ( 2π x(n) ) | Please help, many thanks!

• Answer:

  1) Test for linearity: The summation of the output for two different inputs should be the same as the output for the summed inputs.Hence the system is linear if: y1(n)=y2(n) where y1(n) = cos[2π x1(n)]+cos[2π x2(n)] and y2(n) = cos{2π[x1(n)+x2(n)]} we have the formula cos(a+b)=cos(a)cos(b)-sin(a)sin(b) and hence when this formula is used on y2(n) it does not equal y1(n). Hence the system is non linear. 2) Test for time invariance: Identical shifts of time in the input should produce identical shifts of time in the output. Hence when both the input and output are time shifted by 'm' units: y(n-m)=cos(2π x(n-m)). This equation is not true. Hence the system is time variant. Therefore it can be declared that the system is not LTI. Hope that helped you.