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:
- derive a heart rate estimation method named PAMDF,
- reveal the relationship between the proposed PAMDF and the classical
AMDF, and
- 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.