In many instances we need to filter a signal to remove unwanted frequencies. If we use classical filters such as Butterworth, Chebyshev or even Bessel then a phase delay is introduced. This phase delay is itself a function of frequency so that the signal content at one frequency is delayed a different amount to that at another frequency. Why does this matter?
Fig. 1 : Filtered and unfiltered signals
Fig. 2 : Both signals overlaid. The “wine glass” graph !
Well, of course, we all remember from our Fourier analysis that a signal may be represented as a set of sines and cosines at different frequencies. Now at a peak in the signal then most of the constituent sines are in phase with each other. Applying a classical filter will cause some components to be delayed with respect to others and this may affect the size and position of the peak. It could be higher or lower ! Using phaseless filters eliminates the possibility of this distortion.
Other situations occur where we are interested in the actual time difference of an event between two signals. If both signals have had the same filtering applied and also the same general frequency content then probably all is well. If the signals are filtered differently then there could be a problem.
Fig. 3 : Original and filtered (Standard Filter)
Fig. 4 : Original and filtered (Phaseless Filter)
One method of producing phaseless filters is to use what are called Finite Impulse Response (FIR) filters. These typically have to be designed and then make use of a convolution scheme to apply to the original signal. Some FIR filters suffer from ringing and passband ripple and they do not have the nice amplitude characteristics of the classical filters.
Fig. 5 : Startup of signal using standard filter
Also FIR filters typically have a large number of multiply / add operations for each data point and this increases quite dramatically as the filter cutoff becomes steeper.There is a simpler and faster technique to produce phaseless filtering which can use classical filters. With this technique we have the advantage of using normal amplitude shapes and minimising the arithmetic. The technique is elegant in its simplicity. First filter the signal as normal, reverse the sequence, filter the reversed sequence and then do a final reversal. The result is a phaseless filter!
Fig. 6 : Startup of signal using phaseless filter
Obviously this cannot be applied in real time, but then most signal analysis is actually post processing. Because we filter twice then the total cutoff rate is twice the expected and the corner frequencies will need to be adjusted but these are trivial to compensate for. A little thought shows that we can, with a small effort, even ‘undo’ the phase effects caused by anti-aliasing filters if we have the equivalent software filter. This is rarely necessary, but…..
Dr Colin Mercer
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