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What is Impulse response?

The output h[n] of a system when the unit impulse δ[n] (1 only at n = 0 and 0 elsewhere) is applied. For a linear time-invariant (LTI) system, the output for any input is the convolution of the input with h[n], y[n] = Σx[k]h[n − k], so h[n] fully describes the system. A filter whose h[n] ends after a finite length is an FIR filter, and one whose h[n] continues forever is an IIR filter. BIBO stability requires Σ|h[n]| to be finite, and the z-transform of h[n] is the transfer function H(z).

Japanese name: インパルス応答

Impulse response appears in 5 practice questions across 1 topic, including Digital Signal Processing Basics.

Field: Digital Signal Processing Basics

Try it: questions about Impulse response

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)?

  1. Ah[n] = (−0.9)ⁿu[n]
  2. Bh[n] = u[n]
  3. Ch[n] = (1.1)ⁿu[n]
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Answer: A. h[n] = (−0.9)ⁿu[n]

An LTI system is BIBO stable if and only if the sum of the absolute values of its impulse response, Σ|h[n]|, is finite. For h[n] = (−0.9)ⁿu[n], Σ|h[n]| = Σ0.9ⁿ = 1/(1 − 0.9) = 10, which is finite, so the system is stable. h[n] = u[n] is an accumulator that keeps adding its input; the sum is infinite, and a constant input makes the output grow without limit. (1.1)ⁿ grows with time, so that system is unstable.

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]?

  1. A−1, −1, −1, 3
  2. B1, 3, 5, 3
  3. C1, 1, 1, −3
Show answer and explanation

Answer: C. 1, 1, 1, −3

For h[n] = {1, −1}, the convolution y[n] = Σx[k]h[n − k] becomes y[n] = x[n] − x[n − 1], the difference from the previous input. y[0] = 1 − 0 = 1, y[1] = 2 − 1 = 1, y[2] = 3 − 2 = 1 and y[3] = 0 − 3 = −3. The output length is 3 + 2 − 1 = 4, from the input length 3 and the impulse-response length 2.

Impulse response in the explanations

Questions that mention Impulse response (5)

  1. 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)?
  2. 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]?
  3. Which of the following is a characteristic of an FIR (finite impulse response) filter?
  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. Which of the following correctly describes the bilinear transform used to design a digital filter from the transfer function H(s) of an analog filter?

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Related terms

IIR filter Convolution FIR filter Linear phase Difference equation Aliasing Cutoff frequency Decimation Discrete Fourier transform Fast Fourier transform (FFT)

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