FIR Filter (Digital Filter)

The signal is sampled, runs through a delay line and is summed with weighting coefficients. The coefficients can be dragged freely with the mouse.

Explanation Tasks

What is happening?

A filter as a recipe

An analogue filter shapes the sound with components – inductor, capacitor or, as in the spring–mass model, spring and mass. A digital filter does the same with numbers only: it remembers the last few samples, weights them and adds them up.

An everyday analogy: if you average the temperature of the last seven days, the quick day-to-day jumps disappear and the slow trend remains. That is already a low-pass. An FIR filter (finite impulse response) generalises this idea: each of the last N values gets its own weight, the coefficient bₖ (k counts back from the newest value: b₀ belongs to the newest, b₁ to the one before, and so on) – and negative weights are allowed too.

Sampling and delay line

On the left of the animation the continuous “Input u(t)” arrives (t stands for time). The switch closes briefly fs times per second (sample rate fs) and measures a sample. Each new value lands in the first cell of “Sampling and delay line x[n−k]” and moves on by one cell (z⁻¹, “one clock of delay”) at every tick. So the last N values x[n], x[n−1] … x[n−N+1] are always available: x[n] is the newest sample, x[n−k] the value from k ticks ago.

Try it: press “Pause” and then “Step” a few times (or the → key). At each step all samples move exactly one cell to the right and the oldest drops out at the end. Because 8 kHz would be far too fast for the eye, everything runs in slow motion; the factor is shown in the “Slow motion” row.

Weighting and adding

Below each cell is a bar of the “Coefficients bₖ”. Each sample is multiplied by its coefficient (“Products bₖ · x[n−k]”), Σ adds all products, and out comes the output value y[n]: y[n] = b₀·x[n] + b₁·x[n−1] + … + b₍N−1₎·x[n−N+1]. This value is held until the next tick – which is why the output looks like a staircase in the oscilloscope; the dots mark the samples.

Worked example with “Custom” and the coefficients 0.5 and 0.5: the filter averages two consecutive values. At 0 Hz both are equal and the output is fully there. At fs/2 = 4 kHz (fs = 8 kHz) consecutive values have opposite signs and cancel. In general, the magnitude of the frequency response |H(f)|, i.e. the ratio of output to input amplitude at the frequency f, is |H(f)| = |cos(π·f/fs)| here, so 0.71 (−3 dB) at 2 kHz. Drag b₁ to −0.5 with the mouse: now the filter takes half the difference, 0 Hz disappears and 4 kHz passes fully – a single sign turns the low-pass into a high-pass.

The “Moving average” design averages N values with equal weights. With N = 4 a sine at fs/N = 2 kHz disappears completely, because exactly one period then fits into the four values and their sum cancels.

The impulse response is the coefficient list

Choose the “Impulse” waveform: once per period of the input signal exactly one sample of height 1 arrives, all others are 0. Advance tick by tick with “Step”. The impulse travels through the line and meets b₀, b₁, b₂ … one after another – exactly the coefficients appear at the output. After N ticks it has left the line and the output is exactly zero again.

Hence the name “finite impulse response”. Unlike the spring–mass resonator, which in theory rings forever, an FIR filter is done after N ticks, and it can never become unstable because nothing is fed back. At f = 250 Hz and fs = 8 kHz an impulse arrives only every 32 ticks; with N = 15 the response is long over before the next one comes. A common misconception: the bars are some kind of frequency control. In fact they are a piece of time signal – the frequency response only results from how they work together.

A low-pass made to measure

The “Low-pass” design computes the coefficients from a sinc function, the impulse response of an ideal low-pass. That, however, is infinitely long; the model cuts it to N values and fades it out towards the edges with a “Window”. More coefficients give a steeper slope. The window trades steepness for stop-band attenuation: with “Rectangular” (simply cut off) the slope is steepest, but the pass band ripples and the stop band is weak. Hann and Hamming attenuate much better.

Numbers for fs = 8 kHz, “Cutoff frequency fc” = 1 kHz (where the pass band turns into the stop band) and N = 15: with Hann every tone from 2 kHz upwards is at least 44 dB lower, with Rectangular only about 28 dB, and Rectangular even boosts 500 Hz by 1.8 dB. Watch out for a common mistake: in this design fc is not the −3 dB point but roughly the −6 dB point. With Hann and N = 15, −3 dB is already reached at about 770 Hz.

Try it: the square-wave input at 250 Hz has overtones at 750, 1250, 1750 Hz … In the “Frequency response” you see them as pale bars (before the filter) and strong bars (after the filter). The 3rd harmonic loses 2.8 dB, the 5th about 12 dB, the 7th almost 40 dB – in the oscilloscope the square becomes round. The “High-pass”, by the way, is made from a low-pass by flipping the sign of every second coefficient; this mirrors the frequency response about fs/4.

Comb filter: delay becomes timbre

“Comb +” has only two non-zero coefficients: b₀ = 0.5 and, at the end, b₍N−1₎ = 0.5. So the filter adds the signal to a copy delayed by D = (N−1)/fs. Where the copy arrives exactly half a period later, it cancels the original. The cancellations lie at f = 1/(2D), 3/(2D), 5/(2D) … at equal spacing – a comb in the frequency response.

Worked example: N = 5 at 8 kHz gives D = 4/8000 s = 0.5 ms; cancellation at 1 kHz and 3 kHz, full gain at 0, 2 and 4 kHz. “Comb −” subtracts the copy and swaps peaks and notches: cancellation at 0, 2 and 4 kHz.

The same happens without any plug-in whenever two time-shifted copies of a signal are mixed: direct sound and table reflection at a speaker's microphone, two microphones at different distances from a guitar, or a parallel bus with uncompensated latency. The typical hollow, “phasey” sound is always this comb.

Linear phase – and what it costs

If the coefficients are symmetric (or antisymmetric, as with “Comb −”), the filter is linear-phase: all frequencies are delayed by exactly the same time, namely (N−1)/2 ticks. The waveform is preserved, it just arrives later. The “Delay” row shows this value; with asymmetric custom coefficients it reads “not linear-phase”.

Try it: raise the “Number of coefficients N” from 15 to 31. The slope in the frequency response gets steeper, but the delay rises from 7 to 15 ticks, i.e. from 0.875 ms to 1.875 ms at 8 kHz. This is the basic conflict of digital filters: steepness costs coefficients, and with linear phase coefficients cost latency. A steep linear-phase EQ with 4096 coefficients delays by about 2048 ticks ≈ 43 ms at 48 kHz. In mastering that does not matter, the DAW compensates it. In in-ear monitoring, however, the singer hears themself like an echo – there short or minimum-phase filters are used. So more coefficients are not automatically “better”.

Nyquist, aliasing and limits of the model

A digital system can only represent frequencies below half the sample rate unambiguously, the Nyquist frequency fs/2. In the “Frequency response” the range above it is grey. Overtones of the input above fs/2 are mirrored to lower frequencies when sampling (aliasing): f_alias = |f − k·fs|, where f is the frequency of the overtone and k the whole number that brings the result below fs/2. A 1100 Hz square at 8 kHz has a 7th harmonic at 7700 Hz, which reappears as 300 Hz – below the fundamental and not harmonic. In the oscilloscope the sampling points then fall differently relative to the edges in every period, and the output looks irregular. A higher “Sample rate fs” makes the problem smaller.

Limits of the model: the input is sampled without an anti-aliasing filter so that aliasing stays visible; real converters filter beforehand. The harmonic bars in the frequency response only show components up to fs/2; mirrored components do not appear there. The output is held until the next tick, without a reconstruction filter. The filter has at most 31 coefficients, dragged values lie between −1 and +1, and fc stays below 0.45·fs. All frequencies are real quantities; only the animation runs in slow motion.

Tasks

Practice questions

Questions from studio, stage and mastering – to think about, discuss and check with the module.

1. A linear-phase EQ with 4096 coefficients at 48 kHz is to run in the singer's in-ear monitoring. What is the problem?

Show answer

The delay is (N − 1)/2 samples, here about 2048 – i.e. around 43 ms. The singer hears themself clearly late, like an echo, and sings against it. During playback the DAW compensates such latencies automatically; live this is not possible. For monitoring, minimum-phase or short filters are used.
2. A parallel bus carries a linear-phase filter with 32 samples of delay (48 kHz), and delay compensation is off. What happens when it is mixed with the dry signal?

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A comb filter appears, just like the comb filter design in the module. The delay is 32/48,000 s ≈ 0.67 ms. Cancellations lie at 750 Hz, 2250 Hz, 3750 Hz and so on – wherever the filtered signal still passes. With delay compensation the effect is gone.
3. A speaker sits at a table; the microphone picks up the direct sound and a reflection from the tabletop with a 0.5 m longer path. Where are the dips in the frequency response, and what can you do?

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The detour corresponds to 0.5/343 s ≈ 1.46 ms of delay. As with the comb filter in the module, dips appear at 343 Hz, 1029 Hz, 1715 Hz … (odd multiples of 1/(2 · 1.46 ms)) – not quite down to zero, because the reflection is quieter than the direct sound, but clearly audible as the typical “hollow” sound. Remedies: move the microphone closer to the mouth, damp the reflection (cloth, absorber) or place a boundary microphone directly on the table – then the detour is practically zero.

Tasks

Predict first, then try it out, then open the solution. “Load starting point” sets the matching initial values.

1. The impulse response

Pause and advance clock by clock with “Step”. Which values appear one after another at the output y[n]? When is the output zero again?

Load starting point

Show solution

Exactly the coefficients b₀, b₁ … b₁₄ appear at the output: the impulse response of an FIR filter is its list of coefficients. After N = 15 clocks the impulse has left the delay line and the output is zero again. Hence the name Finite Impulse Response.
2. When is the average zero?

The “Average” design with N = 4 takes the mean of the last four samples. At which frequency does a sine disappear completely? Hint: when do four consecutive samples cancel exactly?

Load starting point

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At 2 kHz = fs/N. Then one period lasts exactly four samples and their sum is zero. Another zero lies at 4 kHz (fs/2). At 1 kHz, by contrast, the sine is attenuated by only 3.7 dB.
3. The comb filter

“Comb filter +” with N = 5 adds the signal to a copy delayed by four clocks. At which frequencies do the two cancel? Check with the sine. What changes with “Comb filter −”?

Load starting point

Show solution

At 1 kHz and 3 kHz. The delay is 4/8000 s = 0.5 ms. At 1 kHz that is exactly half a period: original and copy are in antiphase and cancel. At 2 kHz it is a full period and both add up. With “Comb filter −” it is the other way round: cancellation at 0, 2 and 4 kHz.
4. One sign

The filter has two custom coefficients, both 0.5. Test with low and high frequencies. Then drag b₁ to −0.5. What kind of filter was it before, what is it now?

Load starting point

Show solution

With 0.5 / 0.5 it averages two samples: a low-pass that fully passes 0 Hz and cancels fs/2 = 4 kHz. With 0.5 / −0.5 it takes half the difference: a high-pass that cancels 0 Hz and fully passes 4 kHz. A single sign decides the filter type.
5. Steepness costs time

Increase N from 15 to 31. What happens to the slope in the frequency response and to the “Delay”? Why does this matter in live applications?

Load starting point

Show solution

The slope gets steeper, but the delay rises from 7 to 15 samples (0.875 → 1.875 ms at 8 kHz). A linear-phase FIR delays by (N − 1)/2 clocks. Steep linear-phase filters need many coefficients and therefore a lot of latency: at 48 kHz and 4096 coefficients it is already more than 40 ms. On-stage monitoring suffers from that; mastering does not.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

FIR Filter (Digital Filter)