English
Menu

Digital Signal Processing Basics

Which of the following correctly describes the system y[n] = n·x[n]?

  1. AIt is linear and also time-invariant
  2. BIt is linear but not time-invariant
  3. CIt is not linear but it is time-invariant
Show answer and explanation

Answer: B. It is linear but not time-invariant

The output for the input a·x₁[n] + b·x₂[n] is n(a·x₁[n] + b·x₂[n]) = a·y₁[n] + b·y₂[n], so the system is linear. However, the output for the delayed input x[n − k] is n·x[n − k], while the delayed output is (n − k)·x[n − k], and the two differ. For example, the input δ[n − 1] gives δ[n − 1], but δ[n − 2] gives 2δ[n − 2], so the response is not simply the same shape delayed. A system whose multiplier changes with the time index n is linear but not time-invariant.

  • A. It is linear, but the multiplier n changes with time, so it is not time-invariant.
  • B. Correct. Superposition holds, but delaying the input does not simply delay the same output shape.
  • C. A sum of inputs gives the sum of outputs (superposition holds), so it is linear. It is also not time-invariant.
Practice this topic Back to topic

Books on Digital Signal Processing Basics Ad

As an Amazon Associate I earn from qualifying purchases.

More Digital Signal Processing Basics questions

  1. Which of the following systems with input x[n] and output y[n] is causal (its output depends only on inputs up to the present time)?
  2. Let u[n] be the unit step. Which of the following LTI systems, given by their impulse responses, is BIBO stable (every bounded input gives a bounded output)?
  3. The sequence x[n] = {1, 2, 3} (n = 0, 1, 2) is applied to an LTI system with impulse response h[n] = {1, −1} (n = 0, 1). Which of the following are the outputs y[0] to y[3]?
  4. A unit step (x[n] = 1 for n ≥ 0) is applied from n = 0 to the system described by the difference equation y[n] = x[n] + 0.5y[n − 1]. With y[−1] = 0, which of the following is y[2]?
  5. A signal sampled at 48 kHz is analyzed with a 1024-point discrete Fourier transform (DFT). Which of the following is the spacing of the frequency bins?