By restricting ourselves to small objects of interest, which are sufficiently far from the antennas, the set of differential equations (i.e., Maxwell equations) describing the scenario may be approximated by a simple transmission model. It will be finally based on the Friis transmission formula and the radar equation, which we extend to time-domain conditions here. In order to make that approximation already valid for short distances, the involved antennas should be quite small.

In what follows, we first introduce this transmission model for the simple case of free space propagation and summarize some methods of localizing/imaging of small, invariant objects without the need of an inversion of Maxwell’s equations. We then discuss variant objects under free space conditions and some methods are introduced to detect them under noisy conditions. For the sake of brevity, we will only refer in this connection to the time variance of those objects. But, this approach will also include the “targeted object modification” (which is typically time-dependent) and the “difference between two scenarios”-method, if we regard the related measurements as sequentially done in (observation) time $T$. Finally, we will extend the scenario model to strong multipath conditions (i.e., the case will be considered in which a weak time-variant object is embedded in a strong multipath environment).

#### 2.1. Invariant Object in Free Space and Its Localization

The generic situation is depicted in

Figure 2. The antennas and target are referred to as points in space assigned by their position vectors

${r}_{i}$ and

${r}_{q}$. We call these points radiation and scattering centers, respectively. The interaction of the involved objects—the antennas and target—with the electric field, we describe by impulse response functions (IRF)

${T}_{i}\left(t\right)$,

${R}_{i}\left(t\right)$ and

${\Gamma}_{i}\left(t\right)$,

$\Lambda \left(t\right)$. Herein,

$t$ refers to the propagation time (often also called fast time).

${T}_{i}\left(t\right)$ and

${R}_{i}\left(t\right)$ are the transmission IRFs of antenna

$i$ if it works in transmitter and receiver mode, respectively. Both are linked via the reciprocity relation

${T}_{i}\left(t\right)=\frac{1}{2\pi c}\frac{\mathrm{d}}{\mathrm{d}t}{R}_{i}\left(t\right)$, where

$c$ is the speed of light in the propagation medium.

${\Gamma}_{i}\left(t\right)$ and

$\Lambda \left(t\right)$ represent reflection IRFs of the antenna feeding and the target. For an introduction into the concept of impulse response functions see [

1]. Any angular dependencies and polarimetric issues of

${T}_{i},\text{\hspace{0.17em}}{R}_{i}$ and

$\Lambda $, as well as the aperture reflection of the antenna, we will omit here for the sake of brevity. For such a scenario, the individual response functions in (1) can be expressed by the time domain Friis-formula and radar equation by (

$\delta \left(t\right)$—Dirac delta function):

Herein, the left term of ${S}_{ii}$ represents the feed point reflections of the antenna, and the right term is a mono-static radar equation, while ${S}_{ji}$ is derived from the Friis-formula (left term; often called crosstalk) and a bi-static radar equation (right term). Multiple reflections between antenna–antenna and antennas–target have been omitted in Equation (2). The delay times ${\tau}_{a},\text{\hspace{0.17em}}{\tau}_{ji},\text{\hspace{0.17em}}{\tau}_{iq}$ refer to the propagation delay between the measurement plan and the radiation center of the antennas, the propagation time between two antennas, and the propagation time between the antenna and the target, respectively.

Following (2), the response matrix

$S$ can be decomposed into two parts as follows:

where

${S}_{0}$ only involves antenna effects (i.e., feed point reflection and crosstalk) and

${S}_{\Lambda}$ is called the multi-static response matrix, which covers all transmission paths including the scattering object.

Figure 3 illustrates typical signals for an idealized scenario with electrically short antennas and a point scatterer. Note that the antenna IRF in transmission and receiving mode is simply the derivative

${T}_{i}\sim \frac{\mathrm{d}}{\mathrm{d}t}\cdots $ and a delta function

${R}_{j}\left(t\right)\sim \delta \left(t\right)$, respectively. The IRF of a point scatterer leads to a second derivative

$\Lambda \left(t\right)\sim \frac{{\mathrm{d}}^{2}}{\mathrm{d}{t}^{2}}\cdots $ [

1]. The signal

${b}_{ji}$ refers to the received signal of antenna

$j$, if antenna

$i$ is stimulated by the pulse.

For the reconstruction of the target locations (i.e., the image of the scenario) from the measurements ${b}_{ji}$, one has several options.

Method 1: The volume to be observed is subdivided into a grid of voxels. In order to get the intensity value

${I}_{p}$ of the voxel located at position

${r}_{v}$, one superimposes the signal components received by all antennas with propagation times corresponding to the related voxel-antenna distance. Note, for that purpose, the wave speed is supposed to be known. In a mathematically generalized form, inspired from the p-norm, this may be expressed in different ways, as for example by (

$p=0,1,2,3,\cdots \text{\hspace{0.17em}}$):

If the voxel position coincides with the target position

${r}_{v}={r}_{q}$, the signals are coherently superimposed, leading to a large intensity value. In the opposite case (i.e.,

${r}_{v}\ne {r}_{q}$), the signals are incoherently added, so the voxel intensity tends to small values. For the suppression of measurement errors, sidelobes, or other image defects, it may be meaningful to modify every signal

${b}_{ji}\left(t\right)$ by an individual weighting function

${h}_{ji}\left(t\right)$ before summing (see e.g., [

11,

12,

13,

14,

15] for examples). Furthermore, in Equation (4),

$w\left(t\right)$ represents a gating function whose duration is on the order of the width of the stimulus pulse. This ensures that only the desired signal sections are added up. In Equation (4), for

$p$, typically 1 or 2 are also assigned as a delay-and-sum approach, which is widely used in the literature.

Figure 4A illustrates an intensity plot based on the delay-and-sum approach for a simple two-dimensional (2D)-scenario with a five-element linear antenna array. Obviously, the method provokes many sidelobes, which limit the contrast of the image. To improve the contrast, the number of antennas has to be increased or other methods of signal superposition have to be applied. Equations (5) and (6) depict two examples. The first method, we call the delay-and-multiply approach. A related result is demonstrated in

Figure 4B. Obviously, the contrast is dramatically increased, because the sidelobes have disappeared. Nevertheless, the method needs some care, because an unwanted zero in the data, which actually should be superimposed, will “destroy” the related target.

Finally, the routine Equation (6) superimposes a set of cross-correlation functions—also assigned as delay-multiply-and-sum—determined from the captured response signals [

16,

17,

18,

19,

20].

Method 2: Instead of superimposing the measured signals as demonstrated in Equations (4)–(6), one may also try to numerically solve the problem of target localization. If we consider a single measurement

${b}_{ji}\left(t\right)$ (compare

Figure 3) or even a cross-correlation function between signals

${b}_{ji}\left(t\right)$ and

${b}_{lk}\left(t\right)$, we can identify the propagation time of arrival (ToA) and the propagation time difference of arrival (TDoA), respectively, between the involved antennas and the target. Based on the known wave speed, the propagation distance may be estimated, so a quadric surface (sphere, ellipsoid, elliptic hyperboloid) can be calculated on which the target is localized. By repeating this for different antenna positions, the intersections of all quadric surfaces finally gives the actual target position.

In what follows, the localization of a single target shall be illustrated. For the sake of demonstration, we restrict ourselves to mono-static measurements

${b}_{ii}\left(t\right)$ solely, which provide us the target range via roundtrip time measurement

${r}_{iq}={\tau}_{iq}c/2$ (refer to

Figure 3). Corresponding to the measurement setup in

Figure 2, one finds:

where

${r}_{i}$ and

${r}_{q}$ represent

$\left[3,1\right]$ position vectors. In order to remove

${r}_{q}^{2}$ from the equation, the results from two different antennas are subtracted. For an antenna array of

$K\ge 4$ antennas, this leads to a set of

$P=K!/\left(2\left(K-2\right)!\right)\ge 6$ combinations, which are arranged in following manner:

Here,

$R$ represents a

$\left[P,3\right]$ matrix of the known antenna position vectors,

${r}_{q}$ is the

$\left[3,1\right]$ column vector of the wanted target position, and

$Q$ is a

$\left[P,1\right]$ column vector built from known antenna distances and the measured target distances. The minimum least square solution of the overdetermined Equation (8) for the target position is given by the following:

This type of solution is optimal if the measurement errors of the involved quantities are Gaussian distributed. The squaring of ${r}_{iq}$ in Equation (8) leads, however, to a non-Gaussian error distribution, and Equation (9) will then provide a biased solution.

In order to avoid the bias, a maximum likelihood approach can be applied, which aims to maximize the probability density function (PDF) in Equation (10) with respect to the target position

${r}_{q}$:

Here again, one subdivides the volume to be observed in voxels assigned by their position ${r}_{v}$. The distances of a voxel to all antenna positions are summarized in the $\left[1,K\right]$ column vector ${D}_{v}={\left[\begin{array}{cccc}\left|{r}_{1}-{r}_{v}\right|& \left|{r}_{2}-{r}_{v}\right|& \cdots & \left|{r}_{K}-{r}_{v}\right|\end{array}\right]}^{T}$. The actual distances between the antennas and the target gained from roundtrip time measurements, we have arranged in the $\left[1,K\right]$ column vector $D={\left[\begin{array}{cccc}{r}_{1q}& {r}_{2q}& \text{\hspace{0.17em}}\cdots & {r}_{Kq}\end{array}\right]}^{T}$, and the uncertainties of the measurement are represented by the $\left[K,K\right]$ covariance matrix $\Sigma \approx {\sigma}^{2}I$, which is a diagonal matrix if the range measurements are mutually independent. $I$ is the identity matrix, and ${\sigma}^{2}$ gives the variance of the range estimation, which can be taken often as equal for all measurement channels.

Figure 5 depicts a simple 2D-example of the PDF illustrating the impact of the measurement uncertainty

$\sigma $ and the array structure onto the target localization. In this context, it should not go unmentioned that the measurement uncertainty is not only due to random measurement errors, but also caused by the ambiguity of the distance definition of spatially extended objects [

21,

22]. Furthermore, the method is only able to estimate the position of a single target, so in multi-target scenarios, the roundtrip times extracted from the measurement data must be correctly assigned to the related targets beforehand.

Method 3: The last imaging approach, at least that we will mention here, exploits the reciprocity of the transmission paths characterized by the multi-static response matrix

${S}_{\Lambda}$. Further it is limited to a small number of point scatterers involved in the test scenario; hence, the method is also assigned as sparse scene imaging. For details of the method, the interested reader is referred to [

1,

23,

24,

25,

26,

27]. For the first trial, the reciprocity properties of

${S}_{\Lambda}$ were used by the DORT (Decomposition de l'Operateur de Retournement Temporel) approach [

28] with the aim of concentrating wave energy at the position of the strongest scatterer under the multi-path condition. Related ideas may also be adapted for imaging purposes, which exploit Eigen-value or singular value decompositions and the MUSIC concept for target localization. It should be noted that this method assumes knowledge of the wave propagation speed within the scenario of the test.

#### 2.2. Time-Variant Objects and Their Emphasis from Noise

Restricting ourselves only to the multi-static transmission matrix, its components may be expressed as follows in the case of a time-variant scenario (compare also Equation (2)):

Here, $T$ symbolizes the observation (slow) time. Assuming antennas at fixed positions, the time variance can only affect the target reflectivity $\Lambda \left(t,T\right)$ and the target ranges ${r}_{iq}\left(T\right),\text{\hspace{0.17em}}{r}_{jq}\left(T\right)$. Observation time-independent components in (11) are summarized by $\Theta \left(t\right)={T}_{j}\left(t\right)\ast {R}_{i}\left(t\right)\ast \delta \left(t-2{\tau}_{a}\right)$.

In order to capture the time variance of the scenario, it is measured in typically regular time intervals

$\Delta T$, and the measurements are displayed as functions of the observation time

$T$ (i.e.,

$b\left(t,T\right)=S\left(t,T\right)\ast a\left(t\right)$). The related representation is called a radargram, as exemplified in

Figure 6A for a walking person, which moves toward the radar antennas and back. From these measurements, the roundtrip time

${\tau}_{2}$ has to be estimated. This is not a task of an unambiguous result, because the time shape of the back scattered signal permanently changes (refer

Figure 6C). As a consequence, the precision of the target position does not only depend on the range resolution of the UWB radar, but also on the variability of the target response, which is a matter of geometric shape variations during walking. In this paper, as a kind of compromise, the energetic center

${\tau}_{2}$ is used to estimate the target distance as follows:

By removing the range influence from the radar data via Equation (13) below, one actually may observe the variability of the backscattering while walking (

Figure 6C).

In order to image the target motion in space, one can follow one of the approaches discussed in

Section 2.1 for every timepoint of the observation time.

In many applications, the targets are small, weakly reflecting and moving only a little. This issue, we will address below. A small-sized target leads to Rayleigh-scattering, which provokes a twofold differentiation of the incident field. Hence, in the case of a time variance, such a target may only vary the strength of the backscattering caused by variations of its permittivity, permeability, or volume. This, we will express by:

Here,

${\Lambda}_{0}$ represents the average reflection strength of the target, and

$\Delta \Lambda $ symbolizes its variable part. If the target finally also moves a little by

$\Delta r$, the receiver signals can be modeled by (15). Without loss of generality, we only refer to the mono-static components here. Joining the observation time-independent part

$\Theta \left(t\right)$ with the twofold derivative in Equation (14)

$\Phi \left(t\right)={\mathrm{d}}^{2}\left(\Theta \left(t\right)\ast a\left(t\right)\right)/\mathrm{d}{t}^{2}$ and insertion of Equation (14) in Equation (11), the receiving signal becomes as follows:

To shorten the notation, we include the delay term in the argument of

$\Phi $ and substitute

$\xi =t-2{r}_{iq,0}/c$, where

${r}_{iq,0}={r}_{iq}\left(T=0\right)$ is the range of the target at the beginning of the observation time. In order to approach realistic conditions, we have also added receiver noise now, which is modeled by the random process

${\underset{\u02dc}{\nu}}_{ii}\left(t,T\right)$. Developing Equation (15) in a Taylor series and omitting higher terms and cross-terms yields the following (using

$\dot{\Phi}=\mathrm{d}\Phi /\mathrm{d}t$):

As seen from Equation (16), the radar response

${b}_{ii}\left(t,T\right)$ is composed from several components. Term A represents the observation time-independent part (i.e., the response of the target in a static state). Term B is an amplitude modulation of the channel response, where the second term

$\Delta {r}_{iq}/{r}_{iq,0}$ is usually negligible compared to the effect in terms C and D. Terms C and D are caused from time-delay modulations. The modulation term C is dominant at the signal edges (large first derivative) of the received signals. It is proportional to the target motion

$\Delta {r}_{iq}\left(T\right)$. The modulation term D mainly appears in the region of peak values of the receiving signal, because there the magnitude of the second derivative is maximum. This term is usually of minor interest, because it is less sensitive than term C and it provides only the squared target motion.

Figure 7 illustrates both terms for a weak sinusoidal target motion.

Figure 7A picks out the signal section, which is located around the target reflection (compare

Figure 3 for the complete signal). In practice, the received signal is sampled. Hence, we know it only at discrete timepoints. Five such sampling points

${t}_{1}\cdots {t}_{5}$ are selected, and their variation in observation time is emphasized. Two points at

${t}_{1},\text{\hspace{0.17em}}{t}_{5}$ are placed on a falling signal edge, one point at

${t}_{3}$ is located at the rising edge, and two points at

${t}_{2},\text{\hspace{0.17em}}{t}_{4}$ are to be found close to signal peaks.

Figure 7B plots the observation time variation of the different samples. One can observe the following:

the largest modulation is provided by the sample located at the steepest part of the received signal;

the modulations at rising and falling edges are inverted; and

the modulation at the signal peaks has double frequency (due to the squaring) and is quite weak.

For motionless targets, where only its reflectivity $\Delta \Lambda $ is time variant, the modulation term B is responsible. It provides the largest contribution at the signal peaks.

In the case of properly designed measurement receivers, which are typically based on sub-sampling, the receiver noise

${\underset{\u02dc}{\nu}}_{ij}\left(t,T\right)$ of the different measurements may be considered as Gaussian distributed with a white spectrum in both

$t$ and

$T$, as well as uncorrelated between different response functions and measurement channels. Moreover, as will be seen in the next section, the strength of the noise will depend on the received signal itself, and its variance becomes dependent on propagation time:

(i.e., the noise increases at steep signal edges). Here,

${\sigma}_{n}^{2}$ characterizes the variance of the additive random effect as thermal and quantization noise, while

${\phi}_{j}^{2}$ refers to the variance of the sampling jitter, which can be considered as “time noise”. These properties of the raw (i.e., unfiltered) data for mono- and bi-static channels are summarized in the following (Gaussian distributed; white; uncorrelated):

Because in the case of weak targets, the modulation effects will also be quite weak, the major challenge will be to detect the time-variable targets under noise conditions. Many methods are investigated with that goal [

29,

30,

31,

32,

33,

34,

35,

36,

37]. Here, for the sake of brevity, we will only consider the most effective method, which is based on a matched filter concept. For that purpose, we pick out the data samples captured at an arbitrary time point

${t}_{0}$ and call the related signal

$x\left(T\right)$. Removing the DC-value

$\overline{{b}_{ii}\left({t}_{0}\right)}$,

$x\left(T\right)$ may be decomposed into an observation time-dependent modulation function

$\chi \left(T\right)$ and a propagation time-dependent value

${x}_{0}\left({t}_{0}\right)$ that determines the strength of the modulation dependent on the sample position as follows:

In case 1, the selected time sample is located within a flat part of the receiving signal, so we get only noise (

${x}_{0}=0$). In case 2, the time sample is located close to a signal peak, and the time variance is caused from a variation of the target reflectivity. Cases 3 and 4, refer to a weakly moving target if the sampling time is placed on a signal edge (case 3; i.e., one of the blue signals in

Figure 7B) or close to a signal peak (case 4; i.e., one of the red signals in

Figure 7B). Case 4 is mostly out of interest and will not be considered further.

In order to suppress the noise, the deterministic part

${x}_{0}\left({t}_{0}\right)\text{\hspace{0.17em}}\chi \left(T\right)$ in Equation (19) has to be emphasized. Assuming we know approximately the time shape of the modulation function, except for an unknown time delay

$\zeta \left(T\right)\approx \chi \left(T-{\tau}_{0}\right)$, this is the case when the test scenario may be externally modulated by the operator or, in the case of heartrate and breathing detection, where one can assume an approximately sinusoidal modulation. From the assumed modulation function and the measurement data, one can establish a summation over the observation time period

${T}_{0}=N\text{\hspace{0.17em}}\Delta T$ (

$\Delta T$ = repetition interval between consecutive measurements, see

Figure 7;

$N$ = number of repetitions), which finally represents a cross-correlation function as follows:

Using Equation (18), its expected value and variance result in the following:

In the case of a perfect match between modulation and reference function

$\zeta \left(T\right)=\chi \left(T-{\tau}_{0}\right)$, the signal-to-noise ratio at a given time sample

${t}_{0}$ yields the following:

As seen from Equation (22), the detection performance can be arbitrarily increased by extending the integration time

${T}_{0}$ (i.e., increasing

$N$). However, for vital data detection, the modulation is not stable over time. Hence,

${T}_{0}$ should not be selected too long in order to avoid de-correlation [

38].

In the simplest case of sinusoidal modulation, the correlation function (20) is implemented by a Fourier-transform in observation time direction (

$\varphi $—“observation time” frequency) as follows:

Figure 8 depicts an example. Obviously it is hardly possible to identify the moving target in the original radar data if it is too noisy. However, already the correlation over the short duration

${T}_{0}$ makes the target visible, and further enlargement of the integration time increasingly suppresses the noise.

In many cases, however, the target modulation is not predictable in advance. Hence, the question arise from where to take the reference modulation in Equation (20). One possible option is to illuminate the target from a second antenna at a slightly different position. The resulting radar data is

${b}_{jj}\left(t,T\right)$. According to Equation (19), we again pick out data samples captured at

${t}_{1}$ and call the related signal

$y\left(T\right)$ as follows:

Because the antennas observe the same object, the modulation function

$\chi \left(t\right)$ is identical in

$x\left(t\right)$ and

$y\left(t\right)$. Taking the cross energy from both signals, we get the following:

Assuming the target modulation at both signals is largest at the sampling points

${t}_{0},\text{\hspace{0.17em}}{t}_{1}$ and both receivers have identical noise behavior, we can state that

${y}_{0}\left({t}_{1}\right)\approx {x}_{0}\left({t}_{0}\right)={x}_{0}$ and

${\sigma}_{x}^{2}\left({t}_{0}\right)\approx {\sigma}_{y}^{2}\left({t}_{1}\right)={\sigma}^{2}$, so the signal-to-noise ratio becomes the following:

which gives a result of SNR only twice as poor as in the ideal case for Equation (22).

Because the optimum sampling positions

${t}_{0},\text{\hspace{0.17em}}{t}_{1}$ are not known prior, one has to run through all possible combinations leading to the cross-energy matrix, as illustrated in

Figure 9. To calculate the

$\left[M,M\right]$ cross-energy matrix

${E}_{ij}$, we assume that the radar data are given by two

$\left[M,N\right]$ matrices

${B}_{ii}$ and

${B}_{jj}$ (

$M$ = number of samples in propagation time;

$N$ = number of impulse responses measured during the observation interval

${T}_{0}$). In a first step, the DC-value of every row is removed from both matrices leading to

${\stackrel{\u2323}{B}}_{ii}$ and

${\stackrel{\u2323}{B}}_{jj}$. Finally, from this, the cross-energy is calculated from the following:

The maximum position of that matrix gives the roundtrip times from both antennas. The idea behind the cross-energy is the noise independency of the merged signals, so with increasing integration time, the noise will mutually cancel out. Such noise independency, we also observe in single antenna measurements as long as the signals are not joined with themselves (i.e., the diagonal elements of the cross-energy matrix have to be set zero).

Singular value decomposition is also often proposed for noise reduction purposes. It will, however, not help if the signal to be detected is already buried beneath noise.

#### 2.3. Time-Variant Objects in Multi-Path Environment

The experimental situation is roughly illustrated in

Figure 10 by two examples. One of them is with mostly free space propagation, real static objects (wall, furniture), and a target, which moves over distances much larger than the radar range resolution. The second example deals with minor motion detection (much smaller than the radar range resolution) and a wave propagation in a lossy environment, which is not very stable, because a person is not able to keep limbs truly motionless.

We first restrict ourselves to a stable propagation environment, as depicted in

Figure 10A. If threefold and higher order reflections are omitted, we can identify four different types of propagation paths. The first type ① refers to all paths, which only involve static objects. The second ② is linked with the object of interest, and the third ③ contains a twofold scattering, one by the target and one by a static object. The forth transmission path ④ symbolizes the transmission behavior of the wall, which affects the antenna signal (e.g., by multiple wall reflections). Merging all this together, the received signal for a single antenna arrangement may be modeled in simplified form as:

Here, we neither respect any angular dependency of the impulse response function nor the range influence (i.e., wave spreading and attenuation) of the signal magnitude or the polarization of the electric fields. Without loss in general, we only involve one static and one time-variable target. The different symbols stand for antenna impulse responses $T\left(t\right),\text{\hspace{0.17em}}R\left(t\right)$, wall transmission $\Xi \left(t\right)$, scattering behavior of the static object ${\Lambda}_{S}\left(t\right)$, and scattering of time-variable object ${\Lambda}_{T}\left(t,T\right)$. The symbol $\tau $ indicates the related path propagation time. In Equation (2), the path delay was respected by convolution with Dirac functions. Here, it is part of the arguments of the function to shorten the notation. In the bottom line of Equation (29), the different components of the transmission paths are merged into different functions. ${b}_{S}\left(t\right)$ symbolizes all transmission paths (including multiple reflections), which do not change in observation time. This part represents the strongest component of the measured signal. It is often orders of magnitude larger than the other components. ${b}_{T}\left(t,T\right)$ refers to the time-variable target, in which we are actually interested, and ${b}_{ST}\left(t,T\right)$ represents multipath components, which involve the time-variant target. ${b}_{S}\left(t\right)$ and ${b}_{ST}\left(t,T\right)$ are often assigned as clutter. They must be removed from the measured signal.

Before we do that, the noise term in Equation (29) must be accounted for more seriously, because this will be important for the estimation of the clutter reduction. As already mentioned in Equation (17), the randomness of the measurement is caused by amplitude noise (additive noise)

$\underset{\u02dc}{n}\sim N\left(0,{\sigma}_{n}^{2}\right)$ and “time” noise (i.e., sampling jitter)

$\Delta {\underset{\u02dc}{\tau}}_{j}\sim N\left(0,{\phi}_{j}^{2}\right)$. In well-designed receiver electronics, both can be assumed to be Gaussian distributed, white, independent, and ergodic. By considering both noise terms separately, Equation (29) has to be modified to the following:

The propagation delay is omitted in the signal components for the sake of a shorter notation. Due to the ergodicity of the additive noise, the randomness of the sampling procedure does not influence its statistical properties, where the jitter $\Delta {\underset{\u02dc}{\tau}}_{j}$ is omitted in the noise term $\underset{\u02dc}{n}\left(t,T\right)$.

In order to suppress the static paths, one needs to know

${b}_{S}\left(t\right)$. It can either be determined from measurements where the target is still absent or it is estimated by averaging over the captured data. In both cases, one performs an integration in observation time in order to reduce the noise influence. So, we can approximately write the following:

Equation (31) assumes that the variable target parts are canceled out due to a sufficiently long integration. Furthermore, it shows that the time shape of the static signal is slightly modified by “a low pass filter” whose IRF is given by the PDF ${p}_{\Delta \tau}\left(t\right)$ of the jitter. However, this becomes only remarkable if the standard deviation ${\phi}_{j}$ of the jitter is on the order of the rise time of the signals. The additive noise in Equation (31) is omitted, because it is largely suppressed by the averaging. In practical implementation, the integration in Equation (31) may be performed over the whole captured data set or even by slighting averaging or low pass filtering the observation time. The signal $\overline{{b}_{S}\left(t\right)}$ is often assigned as background.

By subtracting

$\overline{{b}_{S}\left(t\right)}$ from the measured data, the result after some manipulation is as follows:

The second line in Equation (32) comes from a Taylor series expansion of Equation (30) and the assumption of a not significantly large jitter, so higher terms of the series may be neglected. Further on, the third line ignores the jitter-affected time-variant signals, because they are of very low magnitude, and ${p}_{\Delta \tau}\left(t\right)$ is approximated by the first element of its Taylor series.

As we can observe from Equation (32), we get after background removal the wanted signal

${b}_{T}\left(t,T\right)$, but it is still bothered by other components. One of them is the multipath component

${b}_{ST}\left(t,T\right)$. It is illustrated in

Figure 11.

Figure 11A is based on the same scenario as already discussed in

Figure 6, but now it refers to the completely captured data set. Because the person was walking in a room, it created a shadow on the wall opposite to the antenna, which may have the same strength as the wanted signal. However, the signal from the shadow has a larger propagation time compared to the direct target reflection. This gives us the opportunity to single out the signal from the shadow and the direct target reflection. This will work better with a larger bandwidth of radar, because due to the better range resolution, targets close to the wall may be separated from their shadows more easily.

Then, we still have the additive noise

$\underset{\u02dc}{n}$ in Equation (32) affecting the detection performance. This noise has been treated in many previous papers and will hence not be considered here in detail. A more serious effect comes from the jitter, which leads to the bias term

${\dot{b}}_{S}\left(t\right)\text{\hspace{0.17em}}{\phi}_{j}$ and the random term

${\dot{b}}_{S}\left(t\right)\Delta {\underset{\u02dc}{\tau}}_{j}$. The bias term is independent of the observation time and thus will be less critical for the target detection (note that

${\phi}_{i}$ is independent of

$t$ and

$T$). The opposite is valid for the random term.

$\Delta {\underset{\u02dc}{\tau}}_{j}$ is random in

$t$ and

$T$. Furthermore, its random effect on

$c\left(t,T\right)$ is weighted by the first derivative

${\dot{b}}_{S}\left(t\right)$ of the very strong signal scattered from the static objects. Under strong multipath conditions, these signals are typically spread over the whole radar range.

Figure 11B gives an example of signal spreading even under simple conditions. To get a better impression of the process of dying out, the data are logarithmically scaled by keeping the signal sign. The scaling function is given by

${b}_{\mathrm{log}}=\mathrm{sign}\left(b\right)\cdot \left(\mathrm{max}\left[20\mathrm{lg}\left(b/{b}_{\mathrm{max}}\right),-D\right]+D\right)$.

$D\text{\hspace{0.17em}}\left[\mathrm{dB}\right]$ is the dynamic range over which the data are depicted.

The strength of the background is often 2–3 orders of magnitude larger than the target reflections and with increasing bandwidth of the radar, the signal derivation leads to an additional emphasis of their influence. Hence, jitter may seriously affect the detection performance of a time-variable target under strong multipath conditions (especially if it is at the same range as a strong target—compare

Figure 10—person, and file cabinet). UWB radar experiments for motion detection under (nearly) free space conditions are therefore less trustworthy.

So far, we have assumed that the clutter objects are static. In a situation as depicted in

Figure 10B, this cannot be strictly presumed, because a living organism can never suppress completely the motion of its limbs. Hence, by referring to the signal model in Equation (30), we also have to take into account a minor time variance of the signal component

${b}_{s}\left(t\right)\to {b}_{s}\left(t,T\right)$. Under the condition that all motion effects are small (i.e., time-delay modulations are smaller than the rise time of the sounding signal), we can follow the approach introduced with respect to Equation (19). That is, we pick out data samples at different propagation timepoints

${t}_{k}$ and observe their magnitude dependent on the observation time

$b\left({t}_{k},T\right)$ or

$c\left({t}_{k},T\right)$. The modulation of the majority of data samples will follow the global motion of the scenario under testing conditions (e.g., the arm), and only few data samples will also contain a modulation caused by the target of interest. The goal is to separate both modulations and to extract the target.

Under the assumption that both modulations are independent or orthogonal, respectively, this can be done by principal component analysis (PCA), which exploits singular value decomposition (SVD) [

40].

Figure 12 gives an example. Assuming, we want to measure the artery motion in the arm. It is approximated in our example by the sinusoidal modulation

$\chi \left(T\right)$ (

Figure 12A). Due to the vital motion of the body, we get an additional, irregular modulation

$\zeta \left(T\right)$, which is even stronger than the signal of interest. Both modulations overlap in the radar signal, but they are connected with slightly different roundtrip times. These two signals affect the radar signal in the same way, as depicted in

Figure 7. Because we are interested in a periodic motion, FFT (Fast Fourier Transform) is performed in observation time with the hope of finding the wanted signal (

Figure 12B). However, we do not succeed, because the perturbing modulation

$\zeta \left(T\right)$ suppresses the modulation

$\chi \left(T\right)$ of interest. Exploiting PCA on the radargram leads to two separable principal components (

Figure 12C). Because the second principal component meets best our expectation about the wanted signal, the related signal parts are extracted from the radar signal. Now, the spectrum shows the wanted sinusoidal modulation signal (

Figure 12D).