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Chapter 3 Improved Single-Target Respiration Rate Estimation
Chapter 2 demonstrates the capability of UWB signals to detect the respiration rate of a human target. Theoretically, the FFT-based approach is able to estimate accurate respiration rate even in very low SNR , as the simulation results show in Figure 2.12. However, from the experimental results, large error in respiration rate estimation occasionally occurs, e.g., 30% error in respiration rate in Experiment 10 of Table 2.1. Based on the observation of the slow-time variation of the received signals, we found that the large error was not caused by noise, but rathor the large-scale body motion of the target during the measurement. We noticed that although the target is required to be still during the measurement, it is quite common in the received data that the motion components not only include the chest-wall displacement, but also contain slower variations which are mainly caused by large-scale body motion of the target. This is reflected in the target trajectory as mean position drifting or an abrupt shift of the periodic variation along the slow-time domain, as shown in Figure 3.1. In the frequency domain, it exhibit as some peaks within low frequency range. Sometimes, the relatively large-scale motion will cause strong frequency components in the lower frequency spectra and interfere with the detection of the true respiration rate.
It is intuitive to use a high-pass filter to remove the low-frequency components in the sig-nals. However, such an approach is unreliable, mainly because it is diﬃcult to predict the appropriate cut-oﬀ frequency to eﬀectively block unwanted frequency components caused by the relative large-scale body motion, yet pass the respiratory signal, especially when the patient is in critical condition with a very low breathing rate. In Figure 3.2, although we have already applied a BPF with a passband frequency of 0.1-5Hz, we still observe a strong low frequency component just below 0.1Hz that was not eliminated by filtering. The signal in Figures 3.1 and 3.2 is the target trajectory from Experiment 10 in Table 2.1. This ex-plains the large error in this experiment. Thus, our goal is to find a robust mean position compensation method which will be adaptive to diﬀerent measurement environments.
Mean Position Tracking Algorithms
For the sake of clarity, since the vector of slow-time variation along the range bin contain-ing the maximum energy reflects the characteristic frequency of both the respiration and heartbeat activities, we denote it as the observed trajectory.
Moving Average Filter
The simplest way to compute local first order statistics is to use the empirical average of the observation samples in an appropriate window around each time instance. As the time instant of interest proceeds, the window moves at the same speed. Assuming the window size L to be even for simplicity of notation, the typical method of calculating the moving average over a fixed window is formulated as, The benefit of the moving average filter is its computational eﬃciency. However, this esti-mator is eﬀective only if the window size closely matches the signal period which is unknown in our case. We applied this filter with diﬀerent window length to the same observed trajec-tory. As shown in Figure 3.3 where the window size is close to the respiration period, the estimated mean position is smooth and reflects the true variation caused by body motion. On the other hand, when the window size L is smaller than the respiration period as illus-trated in Figure 3.4, the estimated mean value exhibits undesirable oscillations following the original trajectory. In this case, once we subtract the obtained mean trajectory from the original trajectory to compensate the drifting mean position, the desired periodic signal may be attenuated or canceled which would impact the estimation. The reason for the oscillations is straightforward since the moving average filter does not take into consideration the nature of the original signal and it only averages the data within the window. With shrinking of the window size, the mean trajectory will gradually converge to the original trajectory. On the other hand, if the window size increases, the filter will become less sensitive to the mean variation and even loose tracking.
Additionally, the actual respiration motion is semi-sinusoidal and usually modeled as a mod-ified cosine function , which is defined as, with n = 2 and where fr is the respiration rate and A is the maximum displacement. We simulated a set of data based on this waveform to verify the performance of the moving average filter which is illustrated in Figure 3.5. In Figure 3.5(a), we set the window size equal to the signal period. Although the estimated mean trajectory is a smooth and stable curve, it doesn’t reflect the true mean position which is zero. Further in Figure 3.5(b) where the window size is mismatched to the signal period, the estimated mean fluctuates away from the true mean. Therefore, it is obvious that the mean estimation for the ideal respiratory model using a moving average filter is not reliable and very sensitive to the window size.
Ellipse Fitting Algorithm
In order to circumvent the direct filtering approach and exploit the semi-sinusoidal feature of the respiratory and heartbeat motion, we adopt another method known as the ellipse fitting algorithm introduced in , . This approach uses a relatively low degree-of-freedom ellipse model to fit a cluster of 2-dimensional data obtained from the self-delayed version of the sinusoidal trajectory, to an ellipse. The advantage of this method is its robustness to noise and unreliable observation data.
Given a set of the one-dimensional discrete data xi = x(ti) sampled from a sinusoidal wave-form x(t), we can obtain a second set of data from a delayed version of the original waveform yi = x(ti − σ) = xj , where (i − j)Δt = σ. The particular choice of the delay length is not too crucial, and here we use τ = 0.5 seconds. From these two data sets, we have a two-dimensional cluster consisting of (xi, yi) with i = 1, 2, …n. The x − y plot of the signals results in an elliptical shape. In order to retrieve the parameters (i.e., the instantaneous mean value) of the original sinusoidal model, we need to first compute the ellipse parameters that best fit the signals. Here, we apply the “direct ellipse fitting ” algorithm introduced by .
The model of an ellipse can be represented by a general conic equation where x = [x2 xy y2 x y 1]T , and α = [a b c d e f ]T , and superscript T denotes the trans-pose operation. Note that the discriminant must be negative for an ellipse, i.e., b2 − 4ac < 0. F (α, xi) is called the “algebraic distance” of a point (xi, yi) to the curve F (α, x) = 0. The fitting of the ellipse conic can by approached by minimizing the sum of squared algebraic distances of the conic to the N data points xi.
In order to avoid the trivial solution α = 06, and recognizing that any multiple of a solution α represents the same conic, the parameter vector α is constrained so that the conic that it represents is forced to be an ellipse. The appropriate constraint is that the discriminant b2 − 4ac be negative. However, the imposition of the inequality constraint is diﬃcult in general. In this case we have the freedom to arbitrarily scale the parameters, so we may simply incorporate the scaling into the constraint and impose the equality constraint b2 − 4ac = −1. This is a quadratic constraint which may be expressed in the matrix form by Bookstein , if a quadratic constraint is set on the parameters, the minimization of (3.4) can be solved by considering a rank deficient generalized eigenvalue system. Therefore, this constrained ellipse fitting problem becomes equivalent to minimizing where X = [x1, x2, …, xL]T , is defined as the design matrix. It is subject to the constraint that αT Cα = 1.
Introducing the Lagrangian multiplier λ and diﬀerentiating, we arrive at the system of si-multaneous equations which can be written as This may be rewritten as the system where S is the scatter matrix equal to XT X. This system is readily solved by considering the generalized eigenvectors of (3.8). Based on Theorem 1 in , there is exactly one positive generalized eigenvalue which corresponds to the unique local minimum of the Lagrangian. Let us assume that (λ, u) solves equation (3.7), then αˆ = u is the optimal solution to the ellipse parameters in equation (3.6), where is equal to Therefore, as the ellipse equation is obtained with optimal vector αˆ, the ellipse parameters can be extracted from the equation in terms of the principal axes, rotation angle, and center of ellipse.
Further, we want to transform the ellipse parameters to the original signal parameters. Assuming the two-dimensional cluster obtained from the the “observed trajectory”is modeled .
Therefore, the center of the cluster (x0, y0) in equation (3.14) which corresponds to the mean value of the original observed trajectory which can be obtained from the vector αˆ.
Note that we define S(t) = X(t)T X(t) = t+L/2 xixi T with L indicating the window size.
As mentioned in Section 3.2, the window size L also determines the compromise between response speed (tracking eﬃciency) and robustness (tracking stability) for this ellipse fitting algorithm, which means it is helpful to choose a window size close to its optimal value which should be identical with the signal period. However, since the mean value is gained based on the sinusoidal nature of the trajectory, it is less sensitive to the window size as compared to the moving average filter. A simulation result based on a sinusoidal trajectory with noise illustrates the capability of the ellipse fitting algorithm for accurate mean position estimation even when the window length disagrees with the signal period. In Figure 3.6, we plot the fitted ellipse for the 2-D cluster when the window is identical to the signal period (middle) and much less than the signal period (bottom). It is clear from the fitted ellipse which is overlaid on the observation cluster, the technique outputs an accurate mean position estimate in bose cases. The bottom plot in Figure 3.6 illustrates the situation where the window size is less than the signal period, which can be regarded as the case where the observation data is only available for part of the period (the arc shape of the cluster). We noticed that although only partial arc data is available for fitting, the fitted ellipse still provides good a approximation for the cluster and an accurate mean position estimate. This empirical study justifies our confidence in ellipse fitting method as a smooth mean estimator.
To further verify this method for our experiment, we first apply this method to the simulated modified cosine function which is regarded as the ideal respiratory trajectory and compare it with the moving average filter discussed in Section 3.2.1. Figure 3.7(a) shows that the ellipse fitting algorithm are not only smooth but also able to provide the true mean position (zero). In Figure 3.7(b), it is obvious that the mean position estimated from the this approach is pleasantly stable when the window size mismatches the signal period.
Next, we test this approach using data from real experiments. Since the normal breathing rate of human-beings is in the range of [4-30] breaths/min, we choose the window size to be consistent with the average of 15 breaths/min for both moving average filter and the ellipse fitting algorithm. Figures 3.8 and 3.9 demonstrate the mean tracking ability of two methods applied to experimental data. They present similar performance when the period of the signal is close to the window size as shown in Figure 3.8, but the ellipse fitting method is more stable and closer to the true mean position when the breathing period has a relatively large diﬀerence compared to the window size, as shown in Figure 3.9. (Note that for more complicated respiration-motion-related monitoring using UWB in the future, the mean position may have a more significant influence ,)
Finally, after subtracting the estimated drifting mean value from the original observed tra-jectory, the slow variation caused by large-scale body motion has been removed and the respiratory signal becomes much easier to detect. We used the same set of experimental data shown in Figure 3.1, and applied the ellipse fitting algorithm to track the mean po-sition. The result is shown in Figure 3.10. The FFT spectrum after the mean tracking is removed is shown in Figure 3.11. Compared with the spectrum of the original trajectory
in Figure 3.2, the respiratory frequency is now the largest magnitude response. Therefore, compensation of the drifting mean value provides some resistance to false detection due to body motion.
The same experimental data given in Table 2.1 was processed using the ellipse fitting algo-rithm. The experimental results are summarized in Table 3.1. As we can see, in most cases the accuracy of the respiration rate estimation is nearly identical to the simple FFT-based approach. One exception to that trend is Experiment 10. In this particular experiment, the subject was moving in a more pronounced manner, and thus the mean tracking filter provided substantial improvement. The advanced average error was reduced from approxi-mately 9% to roughly 3%, and the error in Experiment 10 was redueced from 24% to 3%.
In this chapter, we identified that large-scale body motion is a major interference for accu-rate respiration frequency estimation. The motion is manifested as mean position drifting of the observed trajectory in the time domain and exhibits some peaks in the low frequency range. A mean position tracking filter was proposed to estimate the drifting mean position and remove it from the observed trajectory. Instead of the common moving average filter, we apply the ellipse fitting algorithm for the mean estimation due to its robustness and accu-racy for the sinusoidal-like trajectory. Experimental results demonstrate that the proposed method eﬀectively avoids false detection due to body motion.
1.2 System Concept
1.3 Previous Work
1.4 Ultra-Wideband Technology
1.5 Thesis Organization
2 FFT-based Signal Processing Techniques for Vital-signs Estimation
2.2 Measurement Setup
2.3 Mathematical Formulation
2.4 Simulation Results
2.5 Experimental Results
3 Improved Single-Target Respiration Rate Estimation 40
3.2 Mean Position Tracking Algorithms
3.3 Experimental Results
4 Improved Single-Target Heartbeat Rate Estimation
4.2 Respiratory Interference Cancellation
4.3 Simulation Results
4.4 Frequency Estimation for Non stationary Signals
4.5 Experimental Results
5 Multi-Target Vital-Signs Estimation 85
5.2 Multi-Target Respiration Rate Estimation
5.3 Multi-target Heartbeat Rate Estimation
6 Concluding Remarks 104
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