Filtering practice

We often read that digital filtering is an operation that matches one sequence of numbers x_n with another sequence of numbers y_n. This definition, while accurate, obscures the essential fact, in our opinion: the important thing is not the sequence of numbers but the signal, most often continuous, which is the image of the physical phenomenon to which the experimenter is attached.

Let's define digital filtering as follows: let's say we have a signal $x\left(t\right)$ to which we want to match a signal $y\left(t\right)$ resulting from $x\left(t\right)$ through linear frequency filtering. Digital filtering is an operation on the samples x_n of the signal $x\left(t\right)$ which will lead to samples y_n, which should make it possible to reconstitute a signal as close as possible to the desired signal $y\left(t\right)$.

In our opinion, a good knowledge of analog (continuous-time) filtering is a prerequisite for approaching digital filtering, since physicists most often reason in terms of continuous signals, and what they want is a graph rather than a sequence of numbers (cf. [R 1 102]).

It should not be forgotten that prior to any sampling, it is essential to perform a low-pass filtering (see [R 1 102]) on the signal (see § 1.4.2) to avoid spectrum aliasing. The more precise the digital filter, the more elaborate the anti-aliasing filter must be. However, the power of today's digital circuits is such that the analog head filter is increasingly dispensed with. All that's needed is to sample the raw signal at the sensor's output fast enough to roughly comply with Shannon's theorem (recalled in § 1.1), and then perform digital low-pass filtering.

After a short section on the necessary mathematical tools, we will discuss transverse filters with finite impulse response (§ 2), which have the following characteristics:

• they have the advantage of always being stable;

• their phase shift can be made proportional to frequency (constant group delay);

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