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What is IIR filter?

A digital filter that feeds past outputs back to the input side and whose impulse response continues forever (infinite impulse response filter). It is computed with a difference equation such as y[n] = 0.2x[n] + 0.8y[n − 1] (exponential smoothing), and its transfer function has a denominator. It achieves a steep response with few coefficients, little computation and small delay, but its phase is not linear. It becomes unstable unless all its poles lie inside the unit circle, and high-order filters are sensitive to coefficient quantization, so they are often implemented as a cascade of second-order sections (biquads). Designing from an analog filter with the bilinear transform is a common method.

Japanese name: IIRフィルタ

IIR filter appears in 4 practice questions across 1 topic, including Digital Signal Processing Basics.

Field: Digital Signal Processing Basics

Try it: questions about IIR filter

Which of the following gives the gains at DC and at the Nyquist frequency of the first-order IIR low-pass filter given by the exponential smoothing y[n] = 0.2x[n] + 0.8y[n − 1]?

  1. A1 at DC and 0 at the Nyquist frequency
  2. B5 at DC and about 0.56 at the Nyquist frequency
  3. C1 at DC and about 0.11 at the Nyquist frequency
Show answer and explanation

Answer: C. 1 at DC and about 0.11 at the Nyquist frequency

The transfer function is H(z) = 0.2/(1 − 0.8z⁻¹); at DC (z = 1) the gain is 0.2/(1 − 0.8) = 1, and at the Nyquist frequency (z = −1) it is 0.2/(1 + 0.8) = 1/9 ≈ 0.11 (about −19 dB). The new input is weighted by 0.2 and the previous output by 0.8; the weights sum to 1, so a constant input passes unchanged. Remembering only the previous output gives an average of past inputs with exponentially decreasing weights. The pole at z = 0.8 lies inside the unit circle, so the filter is stable, and the closer the feedback coefficient is to 1, the stronger the smoothing and the slower the response.

Which of the following is a correct precaution when implementing a high-order IIR filter in fixed point with coefficients of limited bit width?

  1. AThe poles of an IIR filter are not affected by coefficient quantization, so the coefficient bit width only needs to be chosen for passband gain accuracy
  2. BLike an FIR filter, it always stays stable after coefficient quantization, so even at high order it is best implemented as a single difference equation
  3. CImplementing the high-order denominator as a single expression lets coefficient errors move the poles and can make it unstable, so it is split into a cascade of second-order sections
Show answer and explanation

Answer: C. Implementing the high-order denominator as a single expression lets coefficient errors move the poles and can make it unstable, so it is split into a cascade of second-order sections

The roots (poles) of a high-order polynomial are sensitive to tiny changes in its coefficients; especially in steep filters whose poles cluster near the unit circle, merely rounding the coefficients can push a pole outside the unit circle. Splitting the filter into second-order sections (biquads), each handling two poles, and connecting them in cascade keeps each section's coefficient errors from affecting the other poles, making it robust to quantization. An FIR filter stays stable after coefficient quantization, although its frequency response changes.

IIR filter in the explanations

Questions that mention IIR filter (4)

  1. Which of the following gives the gains at DC and at the Nyquist frequency of the first-order IIR low-pass filter given by the exponential smoothing y[n] = 0.2x[n] + 0.8y[n − 1]?
  2. Which of the following is a correct precaution when implementing a high-order IIR filter in fixed point with coefficients of limited bit width?
  3. 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]?
  4. Which of the following is a characteristic of an FIR (finite impulse response) filter?

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Related terms

FIR filter Impulse response Nyquist frequency Fixed point Difference equation Linear phase Convolution Decimation Discrete Fourier transform Fast Fourier transform (FFT)

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