Digital Signal Processing Basics
Which of the following systems with input x[n] and output y[n] is both linear and time-invariant (LTI)?
- Ay[n] = x[n] + 2
- By[n] = x[−n]
- Cy[n] = 3x[n] − x[n − 2]
Show answer and explanation
Answer: C. y[n] = 3x[n] − x[n − 2]
Linearity means that a weighted sum of inputs gives the same weighted sum of outputs, and time invariance means that delaying the input by k samples only delays the output by k samples. y[n] = 3x[n] − x[n − 2] only multiplies the present and past inputs by constants and adds them, so it satisfies both. y[n] = x[n] + 2 outputs 2 even for a zero input, so it is not linear, and y[n] = x[−n] reverses time, so delaying the input makes the output earlier and it is not time-invariant.
- A. Adding the constant 2 gives an output of 2 even for a zero input, so superposition fails and the system is not linear.
- B. It is linear, but because it reverses time, delaying the input makes the output earlier, so it is not time-invariant.
- C. Correct. It adds the present and past inputs with constant coefficients, so it is linear and time-invariant.