# filtfilt

Zero-phase digital filtering

## Syntax

`y = filtfilt(b,a,x)y = filtfilt(SOS,G,x)y = filtfilt(d,x)`

## Description

`y = filtfilt(b,a,x)` performs zero-phase digital filtering by processing the input data, `x`, in both the forward and reverse directions [1]. `filtfilt` operates along the first nonsingleton dimension of `x`. The vector `b` provides the numerator coefficients of the filter and the vector `a` provides the denominator coefficients. If you use an all-pole filter, enter `1` for `b`. If you use an all-zero filter (FIR), enter `1` for `a`. After filtering the data in the forward direction, `filtfilt` reverses the filtered sequence and runs it back through the filter. The result has the following characteristics:

• Zero-phase distortion

• A filter transfer function, which equals the squared magnitude of the original filter transfer function

• A filter order that is double the order of the filter specified by `b` and `a`

`filtfilt` minimizes start-up and ending transients by matching initial conditions, and you can use it for both real and complex inputs. Do not use `filtfilt` with differentiator and Hilbert FIR filters, because the operation of these filters depends heavily on their phase response.

 Note:   The length of the input `x` must be more than three times the filter order, defined as `max(length(b)-1,length(a)-1)`.

`y = filtfilt(SOS,G,x)` zero-phase filters the data, `x`, using the second-order section (biquad) filter represented by the matrix `SOS` and scale values `G`. `SOS` is an L-by-6 matrix containing the L second-order sections. `SOS` must be of the form

$\left(\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& {a}_{01}& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& {a}_{02}& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& {a}_{0L}& {a}_{1L}& {a}_{2L}\end{array}\right)$

where each row are the coefficients of a biquad filter. The vector of filter scale values, `G`, must have a length between 1 and L + 1.

 Note:   When implementing zero-phase filtering using a second-order section filter, the length of the input, `x`, must be more than three times the filter order. You can use `filtord` to obtain the order of the filter.

`y = filtfilt(d,x)` zero-phase filters the input data, `x`, using a digital filter, `d`. Use `designfilt` to generate `d` based on frequency-response specifications.

## Examples

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### Zero-Phase Filtering of an Electrocardiogram Waveform

Zero-phase filtering helps preserve features in a filtered time waveform exactly where they occur in the unfiltered signal.

To illustrate the use of `filtfilt` for zero-phase filtering, consider an electrocardiogram waveform.

```wform = ecg(500); plot(wform) axis([0 500 -1.25 1.25]) text(155,-0.4,'Q') text(180,1.1,'R') text(205,-1,'S') ```

The QRS complex is an important feature in the ECG. Here it begins around time point 160.

Corrupt the ECG with additive noise. Reset the random number generator for reproducible results. Construct a lowpass FIR equiripple filter and filter the noisy waveform using both zero-phase and conventional filtering.

```rng default x = wform' + 0.25*randn(500,1); d = designfilt('lowpassfir', ... 'PassbandFrequency',0.15,'StopbandFrequency',0.2, ... 'PassbandRipple',1,'StopbandAttenuation',60, ... 'DesignMethod','equiripple'); y = filtfilt(d,x); y1 = filter(d,x); subplot(2,1,1) plot([y y1]) title('Filtered Waveforms') legend('Zero-phase Filtering','Conventional Filtering') subplot(2,1,2) plot(wform) title('Original Waveform') ```

Zero-phase filtering reduces noise in the signal and preserves the QRS complex at the same time it occurs in the original. Conventional filtering reduces noise in the signal, but delays the QRS complex.

Repeat the above using a Butterworth second-order section filter.

```d1 = designfilt('lowpassiir','FilterOrder',12, ... 'HalfPowerFrequency',0.15,'DesignMethod','butter'); y = filtfilt(d1,x); subplot(1,1,1) plot(x) hold on plot(y,'LineWidth',3) legend('Noisy ECG','Zero-Phase Filtering') ```

## References

[1] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

[2] Mitra, Sanjit K. Digital Signal Processing. 2nd Ed. New York: McGraw-Hill, 2001, secs. 4.4.2 and 8.2.5.

[3] Gustafsson, F. "Determining the initial states in forward-backward filtering." IEEE® Transactions on Signal Processing. Vol. 44, April 1996, pp. 988–992.