Signals and LTI systems · Definition

Impulse response

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If you know a system's response to an impulse, you can predict its response to any signal. This idea is the core of the analysis of linear time-invariant systems.

Definition

The impulse response \(h[n]\) is the output of an LTI system when its input is the unit impulse \(\delta[n]\).

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

By linearity and time invariance, the response to \(\delta[n-k]\) is \(h[n-k]\), and decomposing \(x[n]=\sum_k x[k]\delta[n-k]\) yields the convolution sum. In causal systems \(h[n]=0\) for \(n<0\), and in stable systems \(\sum_n |h[n]|<\infty\). These properties can be read directly from \(h[n]\).

Consider a two-sample averager, \(y[n]=\tfrac12(x[n]+x[n-1])\). Applying the input \(\delta[n]\) gives \(h[0]=\tfrac12\) and \(h[1]=\tfrac12\), with all other values zero, so \(h\) has finite length. This finite-duration response defines an FIR system.

A typical mistake is confusing \(h[n]\) with the input, or forgetting that convolution uses \(h[n-k]\) rather than \(h[k-n]\). Another common error is assuming every system has a finite-duration impulse response, when recursive systems such as \(y[n]=x[n]+\tfrac12 y[n-1]\) produce infinite responses.

Think about it

Why is the impulse response enough to know the output for any input in an LTI system, and what would fail if the system were not time-invariant?

Linearity lets any input be decomposed into shifted impulses, and time invariance makes each one produce a shifted copy of h. Without time invariance, each shift would need its own response, so h[n] alone would not suffice.

Exercises 10

  1. What is the output of an LTI system called when the input is \(\delta[n]\)?
  2. What value does \(\delta[n]\) take at \(n=0\)?
  3. If \(h[n]=0\) for \(n<0\), what property does the system have?
  4. If \(h[n]\) is nonzero only for \(n=0,1,2\), what is the system called?
  5. Which operation combines the input \(x[n]\) with the response \(h[n]\) to produce the output?
  6. A system has \(h[n]=(0.5)^n u[n]\). What is \(h[2]\)?
  7. For \(h[n]=\delta[n]+2\delta[n-1]\) and input \(x[n]=\delta[n]\), what is \(y[1]\)?
  8. A system has \(h[n]=u[n]-u[n-3]\). What is \(h[1]\)?
  9. If \(h[n]=\delta[n-2]\), what output does the input \(x[n]=\delta[n]+\delta[n-1]\) produce at \(n=2\)?
  10. Let \(h[n]=(0.5)^n u[n]\) and \(x[n]=u[n]\). Compute \(y[0]\) and \(y[1]\) by convolution and pick the correct option.

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.