What is Linear phase?

The property of a filter whose phase response is a straight line proportional to frequency. Every frequency component is then delayed by the same time (the group delay is constant), so the waveform keeps its shape. An FIR filter has exactly linear phase when its coefficients are symmetric (or antisymmetric) about the center, with a delay of (N − 1)/2 samples for N taps. A stable, causal IIR filter cannot achieve exactly linear phase, so FIR filters are widely used where waveforms must be preserved, as in audio, communication receivers and measurement.

Japanese name: 直線位相

Linear phase appears in 3 practice questions across 1 topic, including Digital Signal Processing Basics.

Field: Digital Signal Processing Basics

Try it: questions about Linear phase

Which of the following 5-tap FIR filters has linear phase?

Ah = {1, 2, 3, 2, 1}
Bh = {1, 2, 3, 4, 5}
Ch = {5, 4, 3, 2, 1}

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Answer: A. h = {1, 2, 3, 2, 1}

If the coefficients of an FIR filter are symmetric about the center (h[n] = h[N − 1 − n]), the phase is a straight line proportional to frequency, and every frequency component is delayed by the same (N − 1)/2 samples. {1, 2, 3, 2, 1} is symmetric about the central 3, so it is a linear-phase filter with a constant delay of 2 samples. {1, 2, 3, 4, 5} and {5, 4, 3, 2, 1} are neither symmetric nor antisymmetric, so their delay changes with frequency.

Which of the following is the group delay of a 101-tap linear-phase FIR filter (with symmetric coefficients)?

A50 samples
B50.5 samples
C101 samples

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Answer: A. 50 samples

The group delay of an N-tap FIR filter with symmetric coefficients is (N − 1)/2 samples at every frequency. For N = 101 this is (101 − 1)/2 = 50 samples, the position of the center coefficient (the 51st coefficient, h[50]). At a sampling frequency of 48 kHz this is a delay of 50 ÷ 48000 ≈ 1.04 ms, and in real-time processing it must be kept in mind that more taps mean more delay.

Linear phase in the explanations

{1, 2, 3, 2, 1} is symmetric about the central 3, so it is a linear-phase filter with a constant delay of 2 samples.
Which of the following 5-tap FIR filters has linear phase?
With symmetric coefficients it has linear phase, delaying every frequency component by the same time and preserving the shape of the waveform.
Which of the following is a characteristic of an FIR (finit…

Questions that mention Linear phase (3)

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