Can Vital Signs be Measured Using mmWave FMCW Radar?

TL;DR

Yes! Theoretically, it’s possible to measure a subject’s heart rate, respiration rate, and even blood pressure using mmWave FMCW radar. In reality, though, the numerous sources of noise and target obfuscation make doing so a very complicated problem. To date, many solutions for extracting vital signs information from radar devices’ output signals have been proposed by the research community. The most successful of these implementations are those that utilize advanced signal processing techniques and/or artificial intelligence to obtain reasonable vital signs estimates despite real-world measurement conditions.

Introduction

Vital signs provide valuable insight into the current state of a patient’s health and well-being. In particular; heart rate, respiration rate, and blood pressure are all instrumental for the detection of certain pulmonary and cardiovascular diseases, for assessing the fitness of athletes and non-athletes alike, and for gauging both the physical and psychological stress of individuals. Measurements of these vital signs are most often acquired using traditional, contact-based methods such as electrocardiogram (ECG) systems, respiration belts, and blood pressure cuffs. While these methods are straightforward to understand and provide accurate results, they are obtrusive and place little priority on the comfort of the patient. This makes them unsuitable for applications requiring long-term, non-invasive monitoring; including monitoring of special patients (e.g., newborns, burn victims, infectious disease patients), sleep quality monitoring, emergency detection for home healthcare patients, detection of driver fatigue, etc. For these and other applications, new contactless vital signs detection methods are currently being developed, proposed, and implemented by the research community. To date, mmWave radar is utilized most frequently as the enabling technology.

Note: Body temperature is also an important vital sign. However, it is not applicable to this discussion because 1) it cannot be measured by radar systems and 2) contactless body temperature measurement using infrared sensing technology is already well-established and widely available at low cost. Therefore, any reference to vital signs in this article only refers to heart rate, respiration rate, and blood pressure.

There are many varieties of mmWave radar. Earlier research tended to favor continuous wave (CW) radar for vital signs detection applications due to its simple construction, low cost, and straightforward operating principle. It is commonly referred to as Doppler radar as it relies on the Doppler effect to detect targets and determine their radial velocity. However, an iteration of CW radar called frequency modulated continuous wave (FMCW) radar is now frequently being employed by state-of-the-art research as it can determine both the range and velocity of the target(s). Though the benefits of CW radar are somewhat eroded by the FMCW scheme, the added ranging capability is far more advantageous for target detection, tracking, and separation from background clutter. Together with the current availability of compact, low-power radar modules still capable of 10+ meter ranging and clothing penetration, the benefits have arguably made FMCW radar the top contender for contactless monitoring of vital signs.

The core concept behind using mmWave radar to measure vital signs is a simple one: use it to detect and monitor the displacements that occur on the surface of the body resulting from cardiopulmonary activity. The most discernible of these displacements are caused by respiration. As the diaphragm contracts and relaxes, it causes the abdomen and chest cavity to visibly expand and contract. Less prominent are the tiny displacements (often referred to as “micro-motions”) caused by the beating heart muscle and the resulting changes in blood flow. These micro-motions are typically most prominent in the chest and abdomen, though, they may also be detected at pulse points such as those at the neck or wrists. Most papers point to [1] and [2] as sources for the values given in Table 1 providing typical motion parameters of the chest wall related to respiration and heartbeat activity. Notice that the displacements caused by heartbeats are indeed quite subtle; well under a millimeter in magnitude. Herein lies the key to answering the title question of this article. If the radar sensor being employed can detect the sub-millimeter (sub-mm) movements of a subject, then it can theoretically be used to measure the vital signs of the subject! To reveal if this is the case, we must first understand the operating principle of FMCW radar.

Table 1: Typical motion parameters of chest wall displacements resulting from cardiopulmonary activity.

Activity Frequency (Hz) Displacement (mm)
Respiration 0.1–0.3 4–12
Heartbeat 1–2 0.2–0.5

FMCW Radar Operating Principle

A simplified block diagram of a typical FMCW radar system is shown in Figure 1, the top half of which illustrates that an electromagnetic wave is generated and transmitted by amplifying the signal from a voltage-controlled oscillator. In a CW radar system, the transmitted signal is simply a continuous sinusoid of constant frequency. In an FMCW radar system, however, the instantaneous frequency of the signal is constantly changing as defined by a modulation function over a fixed duration. This signal is repeated, resulting in a series of identical frequency modulated signals being transmitted in quick succession.

The function most frequently used for modulation is a ramp with positive slope, yielding a series of up-chirp signals similar to that shown in Figure 2. The first of these transmitted chirps can be defined as follows.

s_T(t) = A_T\cos{\left(2\pi{}f_0t + \pi\frac{B}{T_c}t^2 + \phi(t)\right)} \quad \text{for $0 \leq t \leq T_c$} \tag{1}

where the bandwidth B is the difference between the ending frequency and the starting frequency (f_0), T_c is the chirp sweep duration, and \phi(t) is the phase noise. The subsequent chirp waveforms are identical to the first, though shifted in time by multiples of the chirp period T. Assuming these chirps repeat indefinitely, the equation for the entire transmitted waveform can be written as:

\begin{multline*} s_T(t,m) = A_T\cos{\left(2\pi{}f_0(t-mT) + \pi\frac{B}{T_c}(t-mT)^2 + \phi(t)\right)} \\ \text{for $0 \leq (t-mT) \leq T_c$} \tag{2} \end{multline*}

where m \in \{0, 1, 2, ...\} is the chirp index. Note that the idle time between chirps (the span from T_c to T) is technically undefined. The implementation of this interval is defined by the radar device’s hardware design and its duration depends on how said device is configured. Figure 3 provides another view of these chirp parameters by plotting the frequency of the transmitted signal (the solid line) over time. Here, the undefined region of the signal is explicitly grayed out and only included for illustration purposes.

Figures 1 and 3 further depict that a portion of the transmitted signal’s energy is reflected by the target and collected by the receiving antenna. The round-trip time of this reflected signal (the dashed line in Figure 3) is given by \tau = \frac{2R}{c}, where R is the distance of the target from the radar sensor and c is the speed of light. If the target is not stationary, the frequency of the received signal will also be shifted by some f_D due to the Doppler effect. This Doppler shift can be calculated as f_D = \frac{2v}{\lambda}, where v is the velocity of the target and \lambda is the wavelength of the emitted radar signal.

For the application of vital signs detection, the subject is generally instructed to remain stationary and any minor movements that they inadvertently make will have considerably low velocity. Therefore, with \lambda on the order of millimeters, the value of f_D is unlikely to far exceed 1 kHz. In relation to f_0, which for mmWave radar will be tens of gigahertz in magnitude, this shift in frequency is insignificant and can be ignored. By doing so, it becomes straightforward to define a single received chirp signal as a transmitted signal (1) delayed by \tau.

\begin{multline*} s_R(t) = \alpha{}A_T\cos\left(2\pi{}f_0(t-\tau) + \pi\frac{B}{T_c}(t-\tau)^2 + \phi(t-\tau)\right) \\ \text{for $\tau \leq t \leq (T_c + \tau)$} \tag{3} \end{multline*}

where \alpha is simply an attenuation factor which summarizes the path losses.

It should be noted that this definition of s_R(t) is admittedly incomplete. In reality, the received waveform will be composed of multiple reflections from multiple objects in the radar’s field of view. Even if only one target is positioned in front of the sensor, its various surfaces (e.g., the head, chest, legs, etc. of a human subject) will each reflect energy from the transmitted waveform. This is also true of any background clutter. The following equation captures the effect of these L reflected signals while also extending the definition to describe the complete received chirp signal corresponding with (2).

\begin{multline*} s_R(t, m, l) = A_T\sum_{l=1}^L \alpha_l\cos\bigg( 2\pi{}f_0((t-mT)-\tau_l) + \pi\frac{B}{T_c}((t-mT)-\tau_l)^2 \\ + \phi((t-mT)-\tau_l)\bigg) \quad \text{for $\tau_l \leq (t-mT) \leq (T_c + \tau_l)$} \tag{4} \end{multline*}

While this expression is important to understand and will be referenced later in this article, the additional information is not strictly necessary for addressing the possibility of detecting sub-mm movements. In that regard, it can be assumed without loss of generality that only one reflection is captured from a single target in front of the sensor (i.e., L=1). This simplifies the equation as follows.

\begin{multline*} s_R(t, m) = \alpha A_T \cos\bigg( 2\pi{}f_0((t-mT)-\tau) + \pi\frac{B}{T_c}((t-mT)-\tau)^2 \\ + \phi((t-mT)-\tau)\bigg) \quad \text{for $\tau \leq (t-mT) \leq (T_c + \tau)$} \tag{5} \end{multline*}

It is clear by observing Figure 3 that the delay value \tau (and to a marginal extent, the Doppler shift f_D) leads to a difference between the instantaneous frequencies of the transmitted and received signals (shown in red). This frequency difference, referred to as the intermediate frequency (IF) or the beat frequency, is proportional to the range of the target from which the received chirp signal was reflected. Thus, by producing a new sinusoidal signal with a frequency of f_{b}, said frequency can be directly measured and the range of the target determined.

The bottom half of Figure 1 illustrates that the IF signal is obtained by first mixing the transmitted and received signals, effectively multiplying them. We know that the result of multiplying two sinusoids is as follows: \cos{(\alpha)}\cos{(\beta)} = \frac{1}{2}\left(\cos{(\alpha + \beta)} + \cos{(\alpha - \beta)}\right). So, by providing the output of the mixer as input to a low-pass filter with an appropriate cutoff frequency, the high-frequency component \cos{(\alpha + \beta)} will be virtually removed from the signal whereas the low-frequency component \cos{(\alpha - \beta)} will be all but unaffected. This allows a single IF signal using (1) and (3) to be calculated as:

\begin{multline*} x_{IF}(t) = (A_T)(\alpha A_T)\dfrac{1}{2}\cos\bigg( \left[ 2\pi{}f_0t + \pi\frac{B}{T_c}t^2 + \phi(t) \right] \\ \shoveright{- \left[ 2\pi{}f_0(t-\tau) + \pi\frac{B}{T_c}(t-\tau)^2 + \phi(t-\tau) \right] \bigg)} \\ \shoveleft{x_{IF}(t) = \dfrac{\alpha{}A_T^2}{2}\cos\bigg( 2\pi{}f_0t + \pi\frac{B}{T_c}t^2 + \phi(t) - 2\pi{}f_0t} \\ \shoveright{+ 2\pi{}f_0\tau - \pi\frac{B}{T_c}t^2 + 2\pi\frac{B}{T_c}t\tau - \pi\frac{B}{T_c}\tau^2 - \phi(t-\tau) \bigg)} \\ \shoveleft{x_{IF}(t) = \dfrac{\alpha{}A_T^2}{2}\cos\left( 2\pi{}f_0\tau + 2\pi\frac{B}{T_c}t\tau - \pi\frac{B}{T_c}\tau^2 + \Delta\phi(t) \right) \quad \text{for $\tau \leq t \leq T_c$} \tag{6}} \end{multline*}

Once this signal is amplified, it is then digitally sampled and the resulting data points are made available to the host for further processing. A common responsibility of the host is to calculate the discrete Fourier transform of the signal using the FFT function to ascertain its frequency, followed then by a simple calculation to obtain the range of the target. While this is indeed useful, we still haven’t shown whether these range measurements have the sub-mm resolution required for vital signs monitoring.

Sub-mm Detection

As previously mentioned, we are assuming for this application of vital signs detection that the subject in view of the radar is attempting to remain motionless at a fixed range R from the device. If they are successful in doing so, then the only movements observable by the sensor will be the small cardiopulmonary displacements discussed in the introduction. Therefore, we can redefine R as R = R_0 + x(t), where R_0 is the range of the subject and x(t) is the small, time-varying displacement we are trying to detect. Using this model and recalling the fact that \tau = \frac{2R}{c}, x_{IF}(t) can be expanded as shown.

\begin{multline*} x_{IF}(t) = \dfrac{\alpha{}A_T^2}{2}\cos\Bigg( 2\pi{}f_0\left(\dfrac{2(R_0+x(t))}{c}\right) + 2\pi\frac{B}{T_c}t\left(\dfrac{2(R_0+x(t))}{c}\right) \\ \shoveright{ - \pi\frac{B} {T_c}\left(\dfrac{2(R_0+x(t))}{c}\right)^2 + \Delta\phi(t) \Bigg)} \\ \shoveleft{x_{IF}(t) = \dfrac{\alpha{}A_T^2}{2}\cos\Bigg( \frac{4\pi{}f_0(R_0+x(t))}{c} + \frac{4\pi{}B(R_0+x(t))t}{cT_c}} \\ - \frac{4\pi{}B(R_0+x(t))^2}{c^2T_c} + \Delta\phi(t) \Bigg) \quad \text{for $\tau \leq t \leq T_c$} \tag{7} \end{multline*}

The above equation may look complicated, but some major simplifications can be made by considering the typical values of these variables. Table 2 provides some concrete examples of operating parameters for a reference vital signs monitoring application to give a sense of their proportion. Of course, these should in no way be construed as the ideal values since they will change depending on the chosen radar device and vital signs detection algorithm.

Table 2: Representative operating parameters for FMCW radar devices configured for vital sign detection.

Parameter Value
f_0 58 GHz
B 5 GHz
T_c 64 us
T 520 us
R_0 0.5 m

Because we are assuming that the subject is located relatively close to the radar sensor, we can ignore the phase noise \Delta\phi(t) due to the range correlation effect [3]. This change leaves three remaining terms which comprise the phase of x_{IF}(t). Another simplification that can be made comes by observing that the third of these terms is being divided by c^2 (a very large number), rendering it negligibly small and allowing us to ignore it. At this point, we are approximating the IF signal as:

x_{IF}(t) \approx \dfrac{\alpha{}A_T^2}{2}\cos{\left( \frac{4\pi{}f_0(R_0+x(t))}{c} + \frac{4\pi{}B(R_0+x(t))t}{cT_c} \right)} \quad \text{for $\tau \leq t \leq T_c$} \tag{8}

A third simplification comes from rewriting the second phase term as 4\pi{}B(R_0t+x(t)t)/cT_c and examining the quantity x(t)t. Since for FMCW radar, T_c is on the order of microseconds, the time t will be on the order of microseconds as well. Also, by our definition of R, x(t) is on the order of millimeters. Thus, the product of these two will not be large enough to have a significant effect on x_{IF}(t) and can be omitted. By removing this quantity from the equation, the IF signal approximation can be re-arranged into a more familiar form.

x_{IF}(t) \approx \frac{\alpha{}A_T^2}{2}\cos{\left( 2\pi{}f_bt + \phi_b(t) \right)} \quad \text{for $\tau \leq t \leq T_c$} \tag{9}

where

\begin{align*} f_b &= \frac{2BR_0}{cT_c} \\[2ex] \phi_b(t) &= \frac{4\pi{}(R_0+x(t))}{\lambda} \end{align*}

That is, the frequency of the IF signal will be approximately f_b and the phase of the IF signal will be approximately \phi_b(t). Notice that this equation confirms the aforementioned assertion that the frequency f_b is determined by the distance of the subject from the radar sensor. Since B and T_c are known quantities, it’s trivial to calculate R_0 after the host processor computes f_b. Notice also that f_b is a constant value whereas \phi_b(t) is time varying. To see the full implication of this, we must consider the sampled representation of the IF signal for all chirps sent and received by the radar device. Specifically, the equation x_{IF}(n, m) defining the samples provided to the host processor by the sensor.

The first step in obtaining said equation is to derive its continuous-time counterpart x_{IF}(t, m). Doing so is simply a matter of repeating the derivation of x_{IF}(t) using the repeating sequence of chirp signals defined by (2) and (5) rather than the solitary chirp definitions given by (1) and (3). Applying the same assumptions and approximations presented in this section of the article, we arrive at the following.

x_{IF}(t,m) \approx \frac{\alpha{}A_T^2}{2}\cos{\left( 2\pi{}f_b\cdot(t-mT) + \phi_b(t) \right)} \quad \text{for $\tau \leq (t-mT) \leq T_c$} \tag{10}

Next, the discretized IF signal can easily be attained by replacing every instance of t in (10) with n\frac{T_c}{N} + mT, where n \in \{0, 1, 2, ..., N\} is the sample index.

x_{IF}(n,m) \approx \frac{\alpha{}A_T^2}{2}\cos\Bigg( 2\pi{}f_b\cdot \left( \left( n\frac{T_c}{N}+mT \right)-mT \right) + \phi_b\left( n\frac{T_c}{N}+mT \right) \Bigg)
\begin{multline*} \shoveleft{x_{IF}(n,m) \approx \frac{\alpha{}A_T^2}{2}\cos\Bigg( 2\pi{}f_bn\frac{T_c}{N} + \phi_b\left( n\frac{T_c}{N}+mT \right) \Bigg) \quad \text{for $n \leq N$} \tag{11}} \end{multline*}

Finally, referring again to Table 2, we see that the chirp duration T_c is significantly smaller than the chirp repetition period T in a typical vital signs detection system. Thus, by assuming \phi_b\left( n\frac{T_c}{N}+mT \right) \approx \phi_b(mT), we realize our final approximation of the sampled IF signal.

x_{IF}(n,m) \approx \frac{\alpha{}A_T^2}{2}\cos\left( 2\pi{}f_bn\frac{T_c}{N} + \phi_b\left( mT \right) \right) \quad \text{for $n \leq N$} \tag{12}

It can now clearly be seen that the frequency of the approximated IF signal will not change from chirp to chirp because the frequency term (2\pi{}f_b n\frac{T_c}{N}) does not contain the chirp index m. Only the phase of the signal will change as consecutive chirp signals are sent and received by the radar device. Therefore, detection of sub-mm changes in the range of the subject (\Delta x) can only be achieved by observing the corresponding change in phase of the IF signal (\Delta\phi_b). Of course, this change in \phi_b must be substantial enough to be measurable by the digital system. Comparing the phase from subsequent chirps, we can see how \Delta\phi_b relates to \Delta x.

\begin{align*} \Delta\phi_b &= \phi_b((m+1)T) - \phi_b(mT) \\ \Delta\phi_b &= \frac{4\pi{}(R_0+x((m+1)T))}{\lambda} - \frac{4\pi{}(R_0+x(mT))}{\lambda} \\ \Delta\phi_b &= \frac{4\pi(R_0 + x((m+1)T) - R_0 - x(mT))}{\lambda} \\ \Delta\phi_b &= \frac{4\pi(x((m+1)T) - x(mT))}{\lambda} \\ \Delta\phi_b & = \frac{4\pi\Delta{}x}{\lambda} \tag{13} \end{align*}

Note that for mmWave radar, \lambda will by definition be on the order of millimeters. Thus, the sub-mm movements of the chest cavity caused by heartbeats and respiration will indeed cause a significant change in the phase! As an example of this, consider a 60 GHz radar system (i.e., \lambda = 5mm). In order to reliably detect the heartbeats of a human subject, said system must be sensitive to at least a 0.2mm displacement of the chest wall (see Table 1).

\Delta\phi_b = \frac{4\pi(0.0002)}{.005} = \frac{4\pi}{25}

Remarkably, by (13), we expect such a small displacement to result in a sizable 4\pi/25 radian change in \Delta\phi_b (i.e., 28.8°). Even on limited hardware, detecting such a change is perfectly feasible using relatively simple digital signal processing methods. Notice that the sensitivity of this phase change is entirely dependent on \lambda. The smaller \lambda is, the larger \Delta \phi_b will be for a given \Delta x. Thus, higher frequency radars are more capable of detecting sub-mm movements made by the target. For this reason, contactless vital signs detection solutions are typically based on 60GHz or 77GHz (or higher) radar systems rather than those operating at 24GHz.

It’s important to be aware that changes in phase larger than \pm\pi (i.e., \lvert\Delta\phi_b\rvert > \pi) will alias into the range -\pi < \Delta \phi_b \le\pi, meaning there is a maximum \Delta x that can be detected. For instance, we can easily use (13) to determine that a 60GHz radar system cannot detect a small change in displacement greater than 1.25mm. Referring once again to Table 1, this clearly isn’t a problem for heart rate monitoring as the displacements caused by heartbeats typically don’t exceed 0.5mm. By that same token, though, it may appear to present an issue for the monitoring of respiration rate. Indeed, if T were chosen such that chirps were transmitted only at the beginning and end of every inhalation/exhalation, the processor would not be able to calculate the coterminous 4–12mm chest displacements. However, we know this will not be the case as T is typically set orders of magnitude lower than the respiration period (see Table 2). Thus, \Delta x will correspond to only a small fraction of the respiration cycle displacements and remain comfortably below its maximum value.

Theory vs. Reality

Having established the theoretical feasibility of measuring a human subject’s vital signs using FMCW radar, it’s now worth taking a moment to reflect on the journey we took to get to this point. Those who encounter the final equation derived in the previous section (i.e., equation (13)) without comprehending its origins may infer from its simplicity that acquiring vital sings measurements from radar output is a straightforward endeavor. This isn’t entirely unreasonable considering that by itself, (13) suggests micro-motions of the chest wall can be precisely measured given the IF signal’s phase variations. We must keep in mind, though, that numerous assumptions and simplifications were made en route to deriving this final equation. Namely,

  1. Ignore the Doppler shift f_D when deriving s_R(t)
  2. Neglect many sources of noise (e.g., thermal noise)
  3. Disregard the phase noise \Delta\phi(t)
  4. Assume the subject is remaining perfectly stationary (i.e., no random body movements such as swaying or fidgeting)
  5. Disregard several phase terms in the IF signal x_{IF}(t)
  6. Omit additional chirp reflections from background clutter
  7. Assume the phase of the IF signal does not change during a chirp (i.e., \phi_b\left( n\frac{T_c}{N}+mT \right) \approx \phi_b(mT)).

Beginning with an already idealized model of an FMCW radar system, there were at least seven liberties we had to take to distill a highly complex problem into a very simple equation. Each item in this list (and undoubtedly others as well) erodes at the validity of (13), making it unsuitable for accurately and precisely calculating sub-mm displacements of the target. That is to say, (13) is merely an estimate and just how credible it is varies over the entire measurement interval as environmental conditions change.

Perhaps most detrimental to the quality of the vital signs signal obtained from \Delta\phi_b are random body movements and background clutter (items 4 and 6 in the above enumeration). Acting in concert with each other, these sources of noise can potentially have a greater contribution to \Delta\phi_b than the micro-motions associated with respiration and the cardiac cycle. For example, the inadvertent swaying motions of a subject standing before the radar sensor can easily interfere with respiration monitoring. Even smaller movements at different areas of the body; such as twitching, speaking, and blinking; will have an effect on the IF signal and likely overshadow the heartbeat component. While it is convenient to ignore these issues from a theoretical perspective, it cannot be done from a practical one as contactless monitoring solutions in the real-world must be able to contend with random body movements and background clutter.

Many solutions have thus far been proposed by researchers attempting to extract an accurate vital signs signal from the output of FMCW radar sensors. As one may expect, the problem is not terribly difficult when the subject is seated or lying down in an uncluttered environment and instructed to remain as motionless as possible. However, the noise floor is rapidly elevated once these restrictions are relaxed and the subject is allowed to perform simple actions like standing in place or calmly speaking. Thus, advanced signal processing techniques are regularly employed to suppress unwanted motion artifacts and isolate only those corresponding to vital signs. These include various adaptive filtering, spectral analysis, and/or decomposition algorithms. Recent works have even begun to feature artificial intelligence models trained to produce more robust vital signs estimates when the subject is given more freedom to move naturally. Future solutions may even be able to produce reasonably accurate data when subjects are walking, running, cycling, strength training, etc.

Conclusion

For the time being, FMCW radar is the favored technology for contactless vital signs detection. By transmitting a sequence of frequency modulated waveforms and down-converting the received reflections, the sensor produces a series of output signals in which the frequency components correlate to the range of the target(s). We were able to establish that under controlled conditions and by making sensible simplifications, these outputs may be modeled as simple sinusoids with phase differences proportional to the sub-mm displacements of the target. Thus, observing this change in phase over time enables the detection of displacements on the surface of the body caused by respiration and the beating heart muscle. Unfortunately, the realities of the physical world make it very difficult to derive a faultless vital signs signal from radar output without restricting the movements of the subject and confining them to an uncluttered environment. To avoid these limitations and better realize the potential benefits of contactless vital signs detection, researchers continue to develop more advanced solutions which promise to yield accurate measurements without sacrificing the comfort and mobility of the subject.

References

[1] A. De Groote, M. Wantier, G. Cheron, M. Estenne and M. Paiva, “Chest wall motion during tidal breathing,” Journal of Applied Physiology, vol. 83, no. 5, pp. 1531-1537, 1997.

[2] G. Ramachandran and M. Singh, “Three-dimensional reconstruction of cardiac displacement patterns on the chest wall during the P, QRS and T-segments of the ECG by laser speckle inteferometry,” Medical and Biological Engineering and Computing, vol. 27, pp. 525-530, 1989.

[3] M. C. Budge and M. P. Burt, “Range correlation effects in radars,” The Record of the 1993 IEEE National Radar Conference, pp. 212-216, 1993.