Zero-phase digital filtering
y = filtfilt(SOS,G,x)
y = filtfilt(d,x)
y performs zero-phase digital filtering by processing the
x, in both the forward and reverse
along the first nonsingleton dimension of
b provides the numerator coefficients of
the filter and the vector
a provides the denominator
coefficients. If you use an all-pole filter, enter
If you use an all-zero filter (FIR), enter
After filtering the data in the forward direction,
the filtered sequence and runs it back through the filter. The result
has the following characteristics:
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
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
differentiator and Hilbert FIR filters, because the operation of these
filters depends heavily on their phase response.
The length of the input
y = filtfilt(SOS,G,x) zero-phase filters
x, using the second-order section (biquad)
filter represented by the matrix
SOS and scale
SOS is an L-by-6
matrix containing the L second-order sections.
be of the form
where each row are the coefficients of a biquad
filter. The vector of filter scale values,
have a length between 1 and L + 1.
When implementing zero-phase filtering using a second-order
section filter, the length of the input,
y = filtfilt(d,x) zero-phase filters the
x, using a digital filter,
designfilt to generate
on frequency-response specifications.
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')
 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.
 Mitra, Sanjit K. Digital Signal Processing. 2nd Ed. New York: McGraw-Hill, 2001, secs. 4.4.2 and 8.2.5.
 Gustafsson, F. "Determining the initial states in forward-backward filtering." IEEE® Transactions on Signal Processing. Vol. 44, April 1996, pp. 988–992.