Periodic average magnitude difference function for remote heart rate monitoring
Chi Zhang1, Shaoming Wei2, Ge Dong1, and Yajun Zeng2
1 School of Aerospace Engineering, Tsinghua University, Beijing 100080, China
2 School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
Email: dongge@tsinghua.edu.cn.
With the increasing attention on remote monitoring of human heart rate by radar, there is a need to develop a method that can estimate heart rate quickly and reliably. In this study, a new estimation method using a periodic average magnitude difference function (PAMDF) is proposed to estimate the heart rate from the radar signal. PAMDF advances the classical average magnitude difference function (AMDF) with the help of maximum likelihood (ML) theory. It operates in the time domain and estimates the heart rate by calculating the signal magnitude difference between all heartbeat periods. The proposed technique is more accurate than AMDF and allows rounding interpolation to improve resolution, while maintaining the low complexity advantage of AMDF. The algorithm was validated using radar data from a publicly available dataset.
Introduction: Radar has been shown to be able to monitor vital signals such as respiratory and heartbeat by detecting human chest fluctuations [1]. Motions caused by respiratory and heartbeat modulate radar echoes, enabling estimation of the respiratory and heartbeat rate. The performance of radar in remote respiratory monitoring is well recognized, but its effectiveness in heartbeat monitoring remains inconclusive, due to the small amplitude of the heartbeat induced motion, which is difficult to detect [2].
Many studies focus on using frequency-domain methods to estimate heart rate. The most straightforward approach is to identify the peaks of the spectrum within the heartbeat frequency bands. However, this method is less effective when processing heart beat signals that consist of many harmonics, even though it performs well in estimating respiratory rate [1, 3].
Estimation techniques can be also performed in the time domain, using methods such as the classical autocorrelation function (ACF) and its simplified version, the average magnitude difference function (AMDF) [4]. The AMDF algorithm is particularly suited for real-time systems, as it is multiplication-free, which is desirable for heartbeat rate monitoring. Estimating heart rate is equivalent to estimating heartbeat period, and both ACF and AMDF have long been used to estimate signal periods in radar [5]. However, the ACF and AMDF have much room for improvement in period estimation. While many adjustments have been proposed, they often lack a solid theoretical basis and offer only limited improvement [6].
A novel method named the periodic average magnitude difference function (PAMDF) is proposed in this paper. It combines the maximum likelihood (ML) estimation with the AMDF to achieve lower complexity for real-time monitoring and improved performance than classical AMDF. The proposed method utilizes the signal comprehensively through the difference between all observed heartbeat periods, thus counters the noise and interference in remote monitoring. Specifically, the goals of this study are to:
  1. derive a heart rate estimation method named PAMDF,
  2. reveal the relationship between the proposed PAMDF and the classical AMDF, and
  3. validate the proposed PAMDF using measured data.
Preliminary: A continuous-wave radar with carrier wavelengthobserves a scatter. The sample of the received signal is
where is the radar cross section (RCS) of the scatter and is distance of the scatter to the radar at the sampling time.
In remote vital signal monitoring, the target is a person sitting or lying down. Echoes from the scatters on the mannequin is picked up by the radar [2]. A periodic change in caused by heartbeat would modulate the phase of the . In practice, the target has an unknown number of scatters. The received signal is the superimposition of many ’s that have different waveforms but a shared period depending on heart rate. So, the observed signal is periodic and is difficult to further describe with a detailed model.
Therefore, the heart rate estimation problem is directly modeled as a heartbeat period length estimation problem with unknown waveform repetition in the time domain. The period of the received signal is samples, and the signal is composed by repetition of the real or complex samples. The observed signal contains samples of periods with white Gaussian noise :
Generally, the observed signal length is much longer than the period length, so the effect of an incomplete period is ignored. The and the waveform determined by samples are unknown. We assume that the samples in one period are independent and identically distributed.
Additionally, respiratory related echoes are also received by radar unavoidably. However, low pass filtering is sufficient to eliminate most received respiratory components in most cases. The effects of respiration are not considered in this paper.
Proposed method : The classical AMDF was originally designed for audio signal processing and defined on real data [4]. Here we define a new AMDF that is suitable for complex radar signals:
For complex input , the differences of the real and imaginary parts are calculated separately to avoid a complex modulo. The signal period length can be estimated by finding the that minimize . However, only uses the information between adjacent periods, while the relationship between non-adjacent periods with interval of , is not exploited. Therefore, the AMDF can be used for heart rate estimation but is unsatisfactory.
An improvement of AMDF with a solid theoretical basis for signal with multiple periods is desired. Starting with the derivation of the likelihood function of the signal model in , the ML estimation of heartbeat period is given by
The may vary slightly, depending on how the incomplete period is treated.
It is too cumbersome to solve directly for real-time heart beat estimation. Reformulate in the form of the sum of the square amplitude differences of the signal:
Solving finds the that minimize the square differences between samples. If happens to be the actual period, signal components cancel each other out and the differences only produce noise in . Conversely, some or all of the differences will contain signal components, making the expectation of their square value larger.
The periodicity of the signal is characterized by signal square difference in and may be measured in other ways. To avoid the large number of multiplications in or , we propose to change the sum of squares to absolute values. By removing the 0 terms and exchanging the order of summations in , a new function is defined as the PAMDF:
The relation between the classical AMDF and the proposed PAMDF is revealed. Substitute into ,
The AMDF is advanced by averaging with a certain interval. The magnitude difference between non-adjacent periods represented by is included in the new function. Ideally, it still has the minimum value when is the actual period, like the classical AMDF.
In addition, although the resolution of the AMDF is limited by the sampling rate, the PAMDF allows non-integer values by interpolating the input signal by rounding. The simplest interpolation method introduces no multiplication, which is consistent with PAMDF. It could improve the estimation resolution and reduce the “picket fence” effect. For non-integer with signal rounding, and become
where [] means rounding. The non-integer samples of input signal are replaced by the nearest samples. As shown in , it is equivalent to round the AMDF. The AMDF defined on integers can be used to calculate non-integer PAMDF.
It is preferred to calculate the PAMDF via AMDF by using rather than by using . It is to avoid duplicate difference operations in when there are many candidate , especially with interpolation. In this way, the PAMDF based method’s complexity is mainly determined by the addition operations in the AMDF. Only real additions and one real multiplication are needed to calculate from AMDF for each candidate in the possible heartbeat period range. It is a small amount of calculation compared with the real additions (subtractions) required by AMDF for complex radar signal. The PAMDF, which requires fewer multiplications, retains a significant complexity advantage over the original ML estimation and other frequency domain techniques.
The algorithm is given as ALGORITHM 1.
ALGORITHM 1: PAMDF period estimation
Input: Discrete periodic signal sequence ,
Possible period range samples,
Interpolated estimation resolution samples,
1: Calculate the of by using .
2: Calculate of each possible period value by using .
3: Find the that minimizes as the estimated period. Convert the period length to heart rate in beats per minute.
Output: Estimated heart rate.
Estimation by PAMDF suffers from period ambiguity like the classical ACF and AMDF. Specifically, has similar near-zero troughs at integral multiples of , which will lead to period ambiguity when the multiple of is included in the possible period range. Fortunately, when estimating the heart rate, there is sufficient prior knowledge to constrain the period range.
Experiments and analysis: To validate the algorithm, open datasets that consist of data from nine healthy subjects were used [7]. The datasets included 10-GHz and 24-GHz Doppler radar data, as well as electrocardiogram data for reference. Nine groups of 600-second data were used for heart rate estimation. A high-pass filter with a cutoff frequency of 0.8 Hz was applied to eliminate respiratory component. Each data was divided into 57 adjacent windows with a length of 10 seconds, with the first 30 seconds discarded. The possible heart rate range was set between 50 to 92 beats/minute. The equivalent heartbeat period range was 0.65 seconds to 1.2 seconds.
The evaluation was based on the proportion of windows with accurate estimation of the heart rate, which was measured in beats per minute. An estimation was considered accurate only if the difference between the estimated heart rate and the electrocardiogram reference was less than one per minute.
The proposed algorithm was compared with the ML estimation by , the AMDF by and the FFT-based method commonly used for radar heart rate estimation. The MATLAB program for the FFT-based method was provided by the dataset authors [7]. The first experiment was conducted directly on the original data with a sampling rate of 1000Hz.
For the second experiment, the radar signals in the dataset were down-sampled to 20 Hz. The PAMDF interpolation resolution was set to one tenth of the sampling interval, while other methods remained unchanged. The results are presented in Table 1.