Radar signal processing is a promising tool for vital sign monitoring. For contactless observation of breathing and heart rate a precise measurement of the distance between radar antenna and the patient's skin is required. This results in the need to detect small movements in the range of 0.5 mm and below. Such small changes in distance are hard to be measured with a limited radar bandwidth when relying on the frequency based range detection alone. In order to enhance the relative distance resolution a precise measurement of the observed signal's phase is required. Due to radar reflections from surfaces in close proximity to the main area of interest the desired signal of the radar reflection can get superposed. For superposing signals with little separation in frequency domain the main lobes of their discrete Fourier transform (DFT) merge into a single lobe, so that their peaks cannot be differentiated. This paper evaluates a method for reconstructing the phase and amplitude of such superimposed signals.

Radar appliances are a promising platform for contactless vital sign monitoring.
Due to regulations the usable bandwidth and hence the distance resolution is limited.
With signal analysis based on the Fourier transform this results in limitations of the maximum achievable frequency and distance resolution.
Algorithms like estimation of signal parameters via rotational invariance techniques (ESPRIT) or MUltiple SIgnal Classification (MUSIC) allow detection of signal frequencies with much closer limits.
They provide frequency estimates or a pseudospectrum, that can be used for peak detection.
While those methods provide good signal separation in frequency domain, they do not provide values for signal phase or amplitude.
Figure

Superposition of signals with similar signal frequencies
and their reconstruction in time domain:

Superposition of two signals

The corresponding power spectral densities (PSD) displayed in Fig.

Section

For a successful phase and amplitude reconstruction an accurate signal model is required.
Using a matrix representation of superimposed harmonic signals as described in Sect.

A sampled harmonic oscillation

For the analysis in this paper frequency estimation is done using ESPRIT.
Since the method of frequency estimation is exchangeable and not the main scope of this paper, only a rough outline of the ESPRIT algorithm is given in this section

Based on the harmonic signal model the signal vectors

Because the actual signals

The signal model presented in Sect.

Similarly Eq. (

By using a pseudo inverse of

The methods described in Sect.

Testing parameters.

With the parameters listed in Table

For further understanding of the limitations connected to the provided method of phase determination first some limitations of the ESPRIT algorithm are presented in Sect.

When analyzing the resulting measurements, the limits ESPRIT must be examined.
For this purpose a range of randomly distributed signals according to Table

Rate of successful signal separation based on ESPRIT:
The number of successfully detected distinct oscillations in relation to the total number of measurements

As shown in Fig.

Notable is the increase of erroneous detections close to 0 Hz and close to the Nyquist frequency 64 Hz.
Below the minimum frequency of 1 Hz and above of 1 Hz below the Nyquist frequency the rate of correctly identified frequencies drops to less than

The set of successfully separated signals is evaluated further.
The system's accuracy is analyzed regarding the accuracy in frequency estimation and phase measurement.
Table

RMSE of the signal frequencies estimated by the ESPRIT method for all successfully separated signals.

Summarized results.

The distribution of the estimated frequencies root mean square error (RMSE) displayed in Fig.

The resulting RMSE of the measured signal phase based on the actual signal frequencies is displayed in Fig.

RMSE of the signal phase derived from the measured signal using the true signal frequency for the successfully separated signals.

RMSE of the signal phase derived from the measured signal using the signal frequency estimated by the ESPRIT method for the successfully separated signals.

In comparison to the phase measurement based on the true frequency value, the phase error is two orders of magnitude worse, when it's based on imperfect frequency measurements.
Across the complete set of measurements the root mean square error is

In conclusion the method described in Sect.

Code will be uploaded to:

All data used is generated using the matlab random number generator.

ARD conceived the parent idea of decomposing overlapping signals, that are close in frequency domain. CS implemented, tested and refined the reconstruction algorithm.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Kleinheubacher Berichte 2020”.

This paper was edited by Madhu Chandra and reviewed by two anonymous referees.