Q10Digital Signal Processing

Nonlinearity: for a system to be linear, it must satisfy both additivity and homogeneity (scaling), and specifically the zero-input, zero-output property (a linear system must produce y(n)=0 for x(n)=0). Setting x(n)=0 in y(n+2)+2y(n) = x(n+1)+2 gives y(n+2)+2y(n) = 2 ≠ 0, meaning a zero input does not produce a zero output. This directly violates the homogeneity property of linearity, since scaling the input x(n) by zero should scale the output by zero for a linear system, but here the constant term '+2' remains regardless of the input scaling — hence the system is nonlinear.

Time-invariance: to check time-invariance, replace n with n-n0 in the original equation: y(n-n0+2) + 2y(n-n0) = x(n-n0+1) + 2. Comparing this with the response to a shifted input x(n-n0), the output is correctly the shifted version of the original output y(n-n0) satisfying the same difference equation form with the same constant term, since the coefficients (1, 2, and the constant 2) do not explicitly depend on n. Therefore, shifting the input by n0 produces the correspondingly shifted output, confirming the system is time-invariant.