Signals and LTI systems · Definition

Linear time-invariant systems

signal processingsignals and lti systems

Before you start you need to know

LTI systems are the central model in signal processing because they let us analyze any input by decomposing it into shifted impulses. This property reduces the computation of the output to a convolution operation.

Definition

A linear time-invariant (LTI) system is a system that satisfies superposition and whose behavior does not change when the input is shifted. It is determined by its impulse response \(h[n]\).

\[y[n]=\sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]

A system is linear if it satisfies superposition: the response to a sum of inputs is the sum of the responses, so the response to \(a\,x_1[n]+b\,x_2[n]\) is \(a\,y_1[n]+b\,y_2[n]\). It is time-invariant if a shift of the input produces the same shift in the output. Together these properties imply that the output is the convolution of the input with \(h[n]\).

Consider a system with impulse response \(h[n]=\left(\tfrac{1}{2}\right)^n u[n]\) and input \(x[n]=\delta[n]+\delta[n-1]\). The output is \(y[n]=h[n]+h[n-1]\), that is, the sum of two responses shifted by one sample. This decomposition shows how each input impulse contributes to the output.

A typical mistake is assuming that any system with constant coefficients is LTI, but if the system is nonlinear, such as \(y[n]=x[n]^2\), superposition fails even though the system does not depend on time. Another common error is forgetting that convolution requires summing over all indices \(k\) where \(x[k]\) is nonzero.

Think about it

Can a system be time-invariant without being linear?

Yes. For example, \(y[n]=x[n]^2\) does not depend on time, but it does not satisfy superposition.

Exercises 10

  1. Which of these properties is required for a system to be LTI?
  2. A system is time-invariant if a shift of the input produces:
  3. What is the impulse response of an identity system, where \(y[n]=x[n]\)?
  4. A system with \(y[n]=x[n]^2\) is linear.
  5. Which operation is used to compute the output of an LTI system from its input and its impulse response?
  6. If \(h[n]=\delta[n]+\delta[n-2]\) and \(x[n]=\delta[n-1]\), what is \(y[n]\)?
  7. If \(h[n]=\left(\tfrac{1}{2}\right)^n u[n]\) and \(x[n]=\delta[n]\), what is \(y[2]\)?
  8. If \(x[n]=u[n]\) and \(h[n]=\delta[n]-\delta[n-1]\), what is \(y[n]\)?
  9. Let \(h_1[n]=\delta[n]+\delta[n-1]\) and \(h_2[n]=\delta[n-2]\) be connected in cascade. If \(x[n]=\delta[n]\), what is the output \(y[n]\) of the overall system?
  10. An LTI system has \(h[n]=\left(\tfrac{1}{2}\right)^n u[n]\). For the input \(x[n]=\delta[n]+\delta[n-1]\), what is \(y[1]\)?

References

  • Alan V. Oppenheim, Alan S. Willsky, Signals and Systems, 2nd edition, Pearson, 1996.
  • Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing, 3rd edition, Pearson, 2010.