Forward-backward filtering (filtfilt)

A minimum-phase filter applied once backwards and once forwards: the phase shifts cancel and the result is symmetric – with pre-ringing. Step by step.

Explanation Tasks

What is happening?

A filter without phase shift

In the “Pre-ringing” module a linear-phase filter rings before the edge. This module shows an intuitive way to arrive at such symmetric behaviour – starting from an ordinary minimum-phase filter, like any analogue EQ. The trick: send the signal through the same filter once backwards and once forwards. After the function of the same name in MATLAB and SciPy, this is often simply called “filtfilt”.

The result has the magnitude response of both passes combined, but no phase shift at all: every frequency arrives exactly at the same time as in the original. This is called a zero-phase filter. A linear-phase FIR filter behaves in the same way, only shifted by its latency. Both ring equally before and after every event.

How to read the display

The large chart shows the signal around an event at t = 0: blue the input, orange the filtered signals. “Step” selects one of the five stages. “Play” runs through them one after another; each step is animated. When a filter runs, its curve grows from left to right with the label “filter runs →”. When the signal is reversed (“⟲ reverse”), the previous curve turns over about t = 0 like the page of a book. The signal from the step before stays visible in a faint colour. Below the chart you can read what happens in the selected step. ← and → move one step back or forward.

In steps 2 and 3 the time axis runs backwards; it is then labelled −t. Shaded regions mark post-ringing and pre-ringing. In step 5 the dashed blue-green curve “2× forward (minimum phase)” shows, for comparison, the same filter applied twice in the normal forward direction. The charts below show magnitude and phase response, the readouts are on the right.

The five steps

Step 1 is the input x(t), a click or a step. In step 2 the signal is reversed, as if a tape were played backwards: x(t) becomes x(−t). The rising step becomes a falling one.

In step 3 the minimum-phase filter runs as usual, sample by sample from left to right. It can only ring afterwards – in the reversed signal, that is to the right of the event. Step 4 reverses the result again. Now all ringing lies before the event and nothing comes after it. Such a filter is called anti-causal or maximum-phase. In real time it is impossible, because it reacts to something that has not happened yet. Offline, with the whole file in memory, it is no problem.

In step 5 the same filter runs forwards once more. It adds post-ringing, exactly as much as pre-ringing was created before. The result is mirror-symmetric about t = 0. With a click as input you see the impulse response of the zero-phase filter itself: its maximum lies exactly on the event, and it rings equally to the left and to the right.

Why the phase cancels

A filter with frequency response H = |H|·e^(jφ) amplifies each frequency by |H| and shifts its phase by φ. Reversing time keeps the magnitude but flips the sign of the phase: the reversed filter has |H|·e^(−jφ). That is exactly the backward pass of steps 3 and 4. In series the two frequency responses multiply: |H|·e^(−jφ) · |H|·e^(+jφ) = |H|². The phases add up to −φ + φ = 0, the magnitudes multiply.

The “Phase response” chart shows this directly. The solid grey line is the phase of a forward pass (+φ), the dashed grey one that of the backward pass (−φ). They mirror each other about the 0° line. Their sum, the orange “filtfilt” line, lies exactly at 0° at all frequencies. The reference “2× forward”, in contrast, shifts twice as far as one pass, because there +φ and +φ add up.

By the way, the order does not matter: first forwards, then backwards gives the same result, since linear filters may be swapped.

Half the gain per pass

Because the filter acts twice, its magnitude is squared. In decibels: the gain doubles at every frequency. For the bell, the module therefore sets half the gain G/2 per pass, so that both together reach exactly G at the centre. The readout “Gain at f (both passes)” then shows G again; in the magnitude chart the dashed grey curve “one pass” lies at half the height of the orange “filtfilt” curve. The total curve is not exactly the same as a single bell with full gain and the same Q, but very similar.

For the low-pass each pass is a second-order filter with −3 dB at f (Butterworth at Q = 0.707). Together that gives −6 dB at f and a slope twice as steep, 24 dB per octave. This is familiar from the Linkwitz-Riley crossover, which also consists of two identical Butterworth filters – but both running forwards.

This is exactly the comparison the reference “2× forward” shows: the same filter twice in series, but both times forwards. Its magnitude response is exactly the same; the dashed blue-green line lies on the orange one. The only difference is the phase.

Same magnitude, different ringing

Set the input to “Step” and go to step 5. For the bell in its default setting (1 kHz, Q = 2, +12 dB), the readouts for filtfilt show a pre-ringing of about 23 % of the step height and just as much post-ringing. The reference has 0 % pre-ringing, but about 44 % post-ringing. For the low-pass with Q = 0.707 it is 3.4 % before and after versus 0 % and 6.3 %. Symmetry spreads the ringing across both sides, and each side rings less than the minimum-phase filter alone.

How long the ringing lasts depends on the filter, not on the method: the higher Q and the lower f, the longer. At 100 Hz everything takes ten times as long as at 1 kHz, the height stays the same. A narrow boost in the bass is therefore particularly noticeable with zero phase, because before a loud event the ear masks only about 10 to 20 ms (pre-masking), but 100 to 200 ms afterwards (post-masking).

Where it is used

filtfilt is standard in measurement and data analysis, for example as scipy.signal.filtfilt or in MATLAB: measured curves are smoothed without shifting peaks in time. Audio editors that process a whole file offline can work this way, too.

In real time this is impossible, because the backward pass starts at the end of the signal. Linear-phase EQs in a DAW solve this differently: they compute a symmetric FIR impulse response with the desired magnitude and truncate it to a finite length. The signal is delayed by half that length – the latency the plug-in reports to the DAW, which compensates for it. The behaviour in time is the same as here: symmetric, with pre-ringing. The same rule of thumb as in the “Pre-ringing” module therefore applies: zero or linear phase for sums, band splitting and measurements, minimum phase for percussive material and anything that needs low latency.

Limits of the model

The computation runs at 44.1 kHz in a buffer of just over 32,000 samples (about 0.74 s) with the event in the middle. Each pass is a biquad after Robert Bristow-Johnson's “Audio EQ Cookbook”. So that the reversed step does not start with a switch-on click, the filters start in their settled state, as if the first value had always been there. At very low frequencies and high Q the ringing has not fully died away at the buffer ends; in the middle, which the chart shows, this does not matter.

Pre- and post-ringing measure the largest deviation from 0 before the event and from the final value after the rise, each in percent of the step height. Half a period of f around the edge counts as the rise itself and is not included. No sound is played, because filtfilt needs the whole signal in advance.

Tasks

Practice questions

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

1. Measurement software and editors that process whole files like to use “filtfilt”. Why can a plug-in running live on a mixing desk channel not do the same?

Show answer

The backward pass needs the signal up to its end – it starts at the last sample. Live, the future is unknown. Linear-phase plug-ins therefore approximate the result with an FIR filter whose impulse response is truncated, and delay the signal by half its length. This latency is the price of filtering without phase shift.
2. A monitor mixer on stage only offers minimum-phase EQs. Is that a drawback?

Show answer

Hardly. For in-ear and wedge mixes low latency matters most, and minimum-phase EQs add no extra delay. They shift the phase depending on frequency, but never ring before a transient. A linear-phase EQ would have to delay the signal by several milliseconds – disturbing on stage, because musicians would hear themselves late compared with the direct sound.
3. In mastering you boost the kick at 60 Hz by 10 dB with a narrow linear-phase EQ. Afterwards the transients sound softer, almost “sucked in”. Why, and what helps?

Show answer

A linear-phase EQ behaves like filtfilt: it spreads the ringing of the bell symmetrically before and after each hit. At 60 Hz and high Q this ringing lasts very long, often longer than the ear's pre-masking. Remedies: a wider Q (shorter ringing), less gain, or a minimum-phase EQ for this move, which only rings after the hit.

Tasks

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

1. Which way does it ring?

A low-pass, step 3. On which side of the click does the filter ring? Then switch to step 4: where is the ringing now, and what is left after the click?

Load starting point

Show solution

In step 3 the ringing lies to the right of the click, because the filter runs from left to right and only rings afterwards. As time is reversed here, “right” means “earlier”. Step 4 shows this: after reversing, all ringing lies before the click and the output is 0 afterwards. This is an anti-causal (maximum-phase) filter – impossible in real time, no problem offline.
2. Evenly spread

Low-pass with Q = 2, step input, step 5. Compare the pre- and post-ringing readouts of filtfilt with those of the reference “2× forward”. Both have exactly the same magnitude response.

Load starting point

Show solution

filtfilt rings equally before and after, about 28 % of the step height each. The reference does not ring before at all (0 %), but about 64 % afterwards. Same magnitude, different phase: filtfilt spreads the ringing symmetrically, the minimum-phase filter puts all of it behind the edge – and rings harder there.
3. Phases that cancel

Bell at 1 kHz with Q = 4. Read from the phase response how far one pass shifts the phase at 900 Hz and at 1.1 kHz. How far does the backward pass shift it, how far both together – and how far the reference?

Load starting point

Show solution

Forwards, one pass shifts by about +19° at 900 Hz and about −19° at 1.1 kHz; exactly at the centre the phase is 0°. Backwards the values are the same with opposite sign. Together that gives 0° everywhere – the horizontal “filtfilt” line. The reference runs forwards twice and shifts twice as far, about ±38°.
4. Half the gain per pass

Bell with G = +12 dB. How much gain does a single pass have at 1 kHz? Does the doubling in dB also hold away from the centre, say at 707 Hz? Compare the curves “one pass” and “filtfilt” in the magnitude response.

Load starting point

Show solution

One pass has +6 dB, both together +12 dB. This holds at every frequency: in series the magnitudes multiply, in dB they add. At 707 Hz one pass has about +2.03 dB, filtfilt +4.05 dB. That is why the module sets G/2 per pass. The shape of the total curve is not exactly that of a single 12 dB bell with the same Q, only very similar.
5. Low frequencies

The same bell (Q = 4, +12 dB), but at 100 Hz instead of 1 kHz. Watch the time axis: how long before the step does the ringing start now? How large is the pre-ringing compared with 1 kHz?

Load starting point

Show solution

The height stays the same (about 17.5 % of the step height, as at 1 kHz), but everything takes ten times as long: the window now spans about ±50 ms, and the pre-ringing is visible over tens of milliseconds. The ringing duration grows with Q/f. Before a loud event the ear masks only about 10 to 20 ms – a narrow, low zero-phase boost can therefore audibly ring in advance.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

Forward-backward filtering (filtfilt)