--- title: "Smoothing Out Noisy Signals Using Discrete Mollifier Transforms" url: "https://www.bekk.christmas/post/2022/12/smoothing-out-noisy-signals-using-discrete-mollifier-transforms" authors: ["Simon Foldvik"] published: "2022-12-12" updated: "2024-11-07T13:05:28Z" language: "en-US" type: "article" tags: ["Algoritmer"] keywords: ["smoothing", "noise", "reduction", "transform", "operator", "time series", "signal", "processing", "algorithm"] --- # Smoothing Out Noisy Signals Using Discrete Mollifier Transforms We present a family of discrete regularization operators, suitably adapted to remove noise from time series data. The operators are discrete analogues of their continuous [mollifier](https://en.wikipedia.org/wiki/Mollifier) counterparts, which are well-known in the literature on partial differential equations for producing smooth approximations of irregular functions. In this article we demonstrate the utility of the noise reduction filters on electrocardiographic data, after which a technical discussion is presented in [part two](https://www.bekk.christmas/post/2022/14/smoothing-out-noisy-signals-using-discrete-mollifier-transforms-part-2) of this two-part series. ## In short Discrete mollifier transforms smooth noisy time series by approximating a clean underlying signal while preserving the structure needed for downstream analysis. The article uses ECG data and QRS detection to show how noise reduction can help avoid false or premature detections, and sets up the mathematical and technical details for a later part. - **For:** Developers and data scientists working with noisy time series or ECG signals - **Good to know beforehand:** Basic programming experience _AI-generated_ ### Time series Time-dependent data may only be sampled at discrete points in time, thus producing a discrete sequence of observations. Such a time-sampled sequence is usually referred to as a _time series_. Examples of time series include electrocardiograms (ECG) from hospital patients and name popularity trends over time, the first of which is discussed in more detail below. Other examples include digital audio signals, the evolution of stock prices, et cetera. ### Noise Sampling continuous phenomena over time is not always smooth sailing. The famous [Shannon–Nyquist theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) is a classical example providing sufficient conditions on the _sampling rate_ of a signal, that is, the number of samples per second, to avoid such effects as aliasing; it tells us that sampling at a rate twice that of the highest frequency occuring in the signal will be sufficient. Another challenge presented to us is _noise_. Even if the underlying phenomenon being sampled is of a continuous, or even smooth, nature, the procedure of sampling a signal generally introduces noise as an artefact on top of the true signal which one wants to capture – measured data which is not really there. It is therefore natural to ask if it is possible to remove the noise and recover the true, underlying signal. Many such techniques do indeed exist, and the noise reduction filters presented here were developed by the author when he was working as a summer intern on the [Pacertool](https://www.simula.no/research/projects/pacertool-patient-specific-biofeedback-guiding-tool-improve-cardiac) project at [Simula Research Laboratory](https://www.simula.no/) in the summer of 2019. ### Removing noise from electrocardiographic (ECG) signals During the summer of 2019 at Simula Research Laboratory, the author was tasked with the creation of tools for automated analysis of ECG signals. This entailed in part the automatic detection of QRS complexes in such signals. An ECG signal is produced by placing electrodes on the patient's limbs and chest to record the electrical activity of their heart, thereby producing a waveform of the type which is here exemplified by public data from [WFDB](https://wfdb.readthedocs.io/en/latest/): ![ECG waveform with QRS complex annotations](https://cdn.sanity.io/images/ah2n1vfr/bekk-blogg-prod/8b8b67f55715857af0eca33535b579d709f51086-793x684.png) _ECG waveform with two QRS complexes. Source: WFDB._ An algorithm for automatic QRS detection in ECG signals must be able to precisely pinpoint the onsets (Q) and offsets (S) of the complex, a task which is readily seen to be complicated by the presence of noise. Indeed, signal spikes produced by the noise may fool the algorithm into premature onset or offset detection, or even into thinking there is a QRS complex where there is none. A common remedy is to preprocess the signal by smoothing out noise before pinpointing the complexes, here achieved by the discrete mollifier transform to be studied in more detail in part two of this series: ![ECG waveforms before and after smoothing](https://cdn.sanity.io/images/ah2n1vfr/bekk-blogg-prod/44e85cf8b68b20a570d02fcf7c8619cf352caf28-1059x569.png) _Smoothing the waveform before QRS complex detection reduces false positives in onset and offset detection._ ### Conclusion There are plenty of situations where noise reduction of time series data is desirable, and the techniques for doing so are well within reach for anyone with minimal programming experience. Here we had a brief look at situations where noise reduction is beneficial. In the [next article](https://www.bekk.christmas/post/2022/14/smoothing-out-noisy-signals-using-discrete-mollifier-transforms-part-2), we take a mathematical and technical look at how the particular class of discrete mollifier transforms used above are defined, implemented, and how they relate to other common regularization operators. See you there! ## Relevant resources recommended by the author - [Smoothing Out Noisy Signals Using Discrete Mollifier Transforms – Part 2](https://www.bekk.christmas/post/2022/14/smoothing-out-noisy-signals-using-discrete-mollifier-transforms-part-2): An analysis of the mathematical and technical aspects of the noise reduction filters used in this article.