Question

Consider a discrete-time system with input z[n] and output y[n], where y[n] = nx[n-3].

(a) Explain whether the system is linear. (b) Explain whether the system is shift-invariant.

Answer 1

The given system is not linear because it does not satisfy the additivity property. It is also not shift-invariant as shifting the input does not correspond to a shift in the output.

Step-by-step explanation:

(a) A linear system satisfies two properties: additivity and homogeneity. Additivity means that if the input is the sum of two signals, the output will be the sum of their individual outputs.

Homogeneity means that if the input is multiplied by a constant, the output is also multiplied by that constant. In this case, the system y[n] = nx[n-3] is not linear because it does not meet the additivity property.

For example, if we have x1[n] and x2[n] as inputs, the output would be y1[n] = nx1[n-3] and y2[n] = nx2[n-3], but y[n] = y1[n] + y2[n] would not be equal to nx[n-3].

(b) A shift-invariant system means that if the input is shifted, the output is also shifted by the same amount. In this case, the system y[n] = nx[n-3] is not shift-invariant because shifting the input by 3 units does not result in a corresponding shift of the output by 3 units. For example, if we shift x[n] by 3 units to get x[n-3], the output y[n] would still have the term n multiplying the shifted input x[n-3].