An Algorithm Based Wavelet Entropy for ... - Semantic Scholar

sensors Article

An Algorithm Based Wavelet Entropy for Shadowing Effect of Human Detection Using Ultra-Wideband Bio-Radar Huijun Xue 1,† ID , Miao Liu 1,† , Yang Zhang 2 , Fulai Liang 1 , Fugui Qi 1 , Fuming Chen 3 Hao Lv 1 , Jianqi Wang 1, * and Yang Zhang 1, * 1

2 3

* †

ID

,

Department of Biomedical Engineering, Fourth Military Medical University, Xi’an 710032, China; [email protected] (H.X.); [email protected] (M.L.); [email protected] (F.L.); [email protected] (F.Q.); [email protected] (H.L.) Center for Disease Control and Prevention of Guangzhou Military Region, Guangzhou 510507, China; [email protected] Department of Medical Engineering, Lanzhou General Hospital of Lanzhou Military Area Command of PLA, Lanzhou 730050, China; [email protected] Correspondence: [email protected] (J.W.); [email protected] (Y.Z.); Tel.:+86-29-8477-4843 (J.W.); Fax: +86-29-8477-9259 (J.W.) These authors contributed equally to this work and should be regarded as co-first author.

Received: 9 August 2017; Accepted: 24 September 2017; Published: 30 September 2017

Abstract: Ultra-wide band (UWB) radar for short-range human target detection is widely used to find and locate survivors in some rescue missions after a disaster. The results of the application of bistatic UWB radar for detecting multi-stationary human targets have shown that human targets close to the radar antennas are very often visible, while those farther from radar antennas are detected with less reliability. In this paper, on account of the significant difference of frequency content between the echo signal of the human target and that of noise in the shadowing region, an algorithm based on wavelet entropy is proposed to detect multiple targets. Our findings indicate that the entropy value of human targets was much lower than that of noise. Compared with the method of adaptive filtering and the energy spectrum, wavelet entropy can accurately detect the person farther from the radar antennas, and it can be employed as a useful tool in detecting multiple targets by bistatic UWB radar. Keywords: ultra-wide band (UWB); multiple target detection; wavelet entropy

1. Introduction In recent years, the application of ultra-wideband (UWB) radar for remote sensing of human beings at short ranges through walls and ruins has led to a range of intensive studies [1–3]. Different from thermal imaging, magnetic induction, X-ray, and some other success-restrictive methods [4], UWB radar determines human objects by detecting his/her chest subtle motion due to respiration, without the influence of temperature, material of non-metal wall and target clothes [2,5]. Furthermore, because the low-frequency pulse signals occupy an extremely broad bandwidth, the electromagnetic wave generated by UWB radar is beneficial to penetrate walls and various nonmetal materials. Moreover, compared with the continuous-wave radar system, UWB radar can locate human targets with the required accuracy. Accordingly, the UWB radar is a good solution for detection and localization purposes in rescue missions after disasters such as earthquakes, mine accidents, and collapse [2,6–9]. Short-range human targets detection using UWB radar systems through walls and ruins has been studied, e.g., in [10–13]. Up to now, the bistatic UWB radar (with one transmitting antenna and one receiving antenna) has resolved detection of a single stationary human target, yet the problem of multi-stationary target detection has been less well addressed. Several experiments for detecting Sensors 2017, 17, 2255; doi:10.3390/s17102255

www.mdpi.com/journal/sensors

Sensors 2017, 17, 2255

2 of 13

multiple stationary human targets through walls have been carried out and the detection results have shown that bistatic UWB radar is able to accurately detect targets who are closest to the radar antennas only, whereas other farther targets are not able to be detected accurately. This is mainly attributed to three factors. Firstly, because of the low reflectivity of human respiration and the strong backscattered responses of obstacles, the signal to noise and clutter ratio (SNCR) of the echo signal is low. Secondly, as the transmission distance increases, the energy of the electromagnetic wave is attenuated, hence, the energy of electromagnetic waves reaching farther targets is inevitably weaker than that reaching the closest target. Thirdly, the closest target reflects the partial energy of the electromagnetic wave and, as a result of the decrease of the radar electromagnetic illumination, a shadowing region is formed and the respiration signal of farther targets located in the shadowing region are too weak to be detected [14]. Some methods were applied for detecting the weak signal of human targets by bistatic UWB radar. For example, mean subtraction was applied to remove the stationary clutter to enhance the SNCR of the echo signal in a real scenario, but the disadvantage of mean subtraction is that this algorithm affects the scattering information of a specific human target [13,15]. Singular value decomposition (SVD) was also used for removing the interference of non-stationary clutter, but this method is not efficient with a low SNCR [16]. The most common algorithm is adaptive filter [11,17], which needs two channels of input signals, but this algorithm ordinarily initiates an echo signal over-fitting. These signal processing algorithms above can suppress interference in a certain sense, however, for multiple stationary human targets, especially the farther targets located in the shadowing region, they do not work well. As in previous studies, during radar detection of the respiration of stationary human targets, the distance between the radar and targets does not change. This means that the respiration signal always appears at a constant moment in the propagation time. Considering that the physical size of the human body is a certain thickness, the respiration signal of the human target always appears in several neighboring cells; thus, the signal in the area of the human target location will present a characteristic of regularity. This is different from the characteristic of noise and clutter, which are more likely to be random and disordered. According to the theory of information, entropy is a natural approach to measure the degree of order of a signal through its entropy spectrum [18]. It is supposed that the entropy amplitude of human respiration and noise will have a certain difference. Furthermore, as a result of the radar echo signal of human respiration being non-stationary, wavelet transform (WT) is often used to analyze the signal of respiration in time-frequency decomposition. If the algorithm of entropy and WT are in mutual cooperation, wavelet entropy (WE) is expected to quantify the time dynamics with the order/disorder of the radar echo signal. In [19], since humans and animals have different characteristic radar echos, the wavelet entropy method was used to distinguish human targets and dogs, and experimental results showed that the entropy value of the human target is much lower than a dog’s. It also indicated that the signals of human targets are more in order than that of dogs. Fugui Qi et al. also employed wavelet entropy to detect apnea in the respiratory signal of obstructive sleep apnea (OSA) by bio-radar, with surveillance results showing that the entropy value of apnea is higher than that of normal respiration [20]. In all of the above, wavelet entropy has been applied to distinguish human targets and dogs, the normal respiration and apnea with the assistance of the time-frequency characteristic of human respiration. However, analyzing the echo signal with WE analysis for detecting multiple targets, especially in the case of the shadowing effect, has not been reported [21,22]. Considering that entropy is closely related to the characteristic of the time-frequency instead of its energy, WE is proposed in this paper for detecting multiple stationary human targets. The five sections are discussed as follows: In Section 2, the IR-UWB radar system is briefly described. Section 3 states three experimental scenarios of multiple target detection. The method for detecting multiple human targets in the shadowing effect is proposed in Section 4, and some data are analyzed. Then, the experiment results are presented and analyzed in Section 5. Finally, the conclusion and some proposals for future work are provided in Section 6.

2017, 2017,17, 17,2255 2255

33ofof1313

analyzed. Then, analyzed. Then, the experiment experiment results results are are presented presented and and analyzed analyzed inin Section Section 5.5. Finally, Finally, the Sensors 2017, 17, 2255 the 3 ofthe 13 conclusion conclusionand andsome someproposals proposalsfor forfuture futurework workare areprovided providedininSection Section6.6. 2.2.UWB UWBRadar RadarSystem System The UWB radar setupisisshown showninin inFigure Figure The UWB radar is stuck onwall the with wall with two 1.1.1. The UWB radar isisstuck on TheUWB UWBradar radarsetup shown Figure The UWB radar stuck onthe the wall withtwo twobowbowbow-tie dipole antennas, which employed transmitting and receivingthe theelectromagnetic electromagnetic wave, wave, tie antennas, which are employed for transmitting and receiving tiedipole dipole antennas, which areare employed forfor transmitting and receiving the electromagnetic wave, respectively. repetition 128 kHz. respectively. AApulse respectively.A pulsegenerator generatorproduces producestrigger triggersignals signalswith withaapulse pulserepetition repetitionfrequency frequencyofof128 128kHz. kHz. The The Thepulse pulsesignals signalsare aresent senttotothe thetransmitter transmitterand andshaped shapedinto intobipolar bipolarpulses pulsesfor forexciting excitingthe thetransmitter transmitter antenna. antenna.The Theecho echopulses pulsesare arereceived receivedby bythe thereceiver receiverantenna, antenna,which whichisisidentical identicaltotothe thetransmitter transmitter antenna. antenna.Then, Then,the thereceived receivedpulses pulsesare aresampled sampledby bymeans meansofofequivalent-time-sampling equivalent-time-samplingwith withdifferent different time delay, thus, the signals ofofdifferent different ranges away from radar can be timedelay, delay,thus, thus,the thesignals signalsof differentranges rangesaway awayfrom fromradar radarcan canbe beobtained. obtained.These Thesesignals signalswere were integrated and amplified, then stored in the form of waveforms through the digital signal processor. integrated and amplified, then stored in theform formofofwaveforms waveformsthrough throughthe thedigital digitalsignal signalprocessor. processor. The waveforms are sampled into 2048 points. Thewaveforms waveformsare aresampled sampledinto into2048 2048points. points.The Therecorded recordedduration durationisis60 60ns, ns,and andthe thedetection detectionrange range was wasup uptoto99m. m.Considering Consideringthat thatthe theUWB UWBradar radarwill willbe beused usedfor forlife lifedetection detectionand andidentification identificationafter after an penetrability and spatial resolution. anearthquake, earthquake,ititdemands demandsthat thatthe theradar radarguarantee guaranteeadequate adequatepenetrability penetrabilityand andspatial spatialresolution. resolution. Thus, Thus,the thebistatic bistaticUWB UWBradar radarwith withaacenter centerfrequency frequencyand andbandwidth bandwidthofof500 500MHz, MHz,respectively, respectively,was was employed. waveform and its spectrum are shown in Figure 2a,b, respectively. A received are shown in Figure 2a,b, respectively. employed. A received waveform and its spectrum are shown in Figure 2a,b, respectively.

Figure 1.1.Bistatic (UWB) bio-radar setup. Figure1. Bistaticultra-wideband ultra-wideband(UWB) (UWB)bio-radar bio-radarsetup. setup. Figure Bistatic ultra-wideband

Figure and the spectrum ofofthe the received waveform. Figure2. (a)One Onereceived receivedwaveform; waveform;(b) (b)and andthe thespectrum spectrumof thereceived receivedwaveform. waveform. Figure 2.2.(a) (a) One received waveform; (b)

3.3.Experimental ExperimentalStatement Statement 3. Experimental Statement In order illustrate fundamental problem ofof multi-stationary target Inorder order illustrate the fundamental problem multi-stationary human target detection, detection, In tototo illustrate thethe fundamental problem of multi-stationary humanhuman target detection, several several males aged between 25 and 30 years were involved, and three cases of experiments were several males aged between 25 and 30 years were involved, and three cases of experiments were males aged between 25 and 30 years were involved, and three cases of experiments were carried out in carried out in this study. Measurements took place in the Bio-Radar Laboratory, and experiments carried out in this study.took Measurements took placeLaboratory, in the Bio-Radar Laboratory, andperformed experiments this study. Measurements place in the Bio-Radar and experiments were to were totodetect human targets brick wall. first wereperformed performed detect human targetsthrough through a24-cm-thick 24-cm-thick brick wall.For Forthe the firstexperiment, detect human targets through a 24-cm-thick brick awall. For the first experiment, which isexperiment, shown in Figure 3a, target A who was closer to the radar antennas stood at 3 m in the front of the antennas, and

2017, 17, 2255

4 of 13

Sensors 2017, 17, 2255

4 of 13

which is shown in Figure 3a, target A who was closer to the radar antennas stood at 3 m in the front of the antennas, and target B who was father away from the antennas stood at 6 m in the shadowing target Bof who was A. father from the antennas stood at 6 in m in the shadowing of target The region target Theaway second experiment was shown Figure 3b; target region A retained theA.same second experiment was shown in Figure 3b; target A retained the same position as the first scenario, position as the first scenario, and target B stood at 6 m out of the shadowing region. The third and target Bwas stood at 6 minout of the3c; shadowing The third experiment in Figure 3c; experiment shown Figure there wasregion. only one person located at 3was m inshown front of the radar there was only one person located at 3 m in front of the radar antennas. antennas.

(a)

(b)

(c)

Figure 3. (a) (a)Target TargetA A stood in front the antennas, whileB target B 6stood at 6shadowing m in the Figure 3. stood at 3atm3inmfront of theof antennas, while target stood at m in the shadowing region of target A; (b) target A stood at 3 m in front of the antennas, while target B out stood region of target A; (b) target A stood at 3 m in front of the antennas, while target B stood at 6 m of at 6 m out of the shadowing region; and (c) target A stood at 3 m in front of the radar antennas. the shadowing region; and (c) target A stood at 3 m in front of the radar antennas.

4. Signal Processing and Analysis 4. Signal Processing and Analysis 4.1. 4.1. Signal Signal Pre-Processing Pre-Processing and and Analysis Analysis The The above-mentioned above-mentioned received received waveforms waveforms were werestored storedin inaatwo-dimensional two-dimensionalmatrix matrix RR with with , where M indicates the detection distance, and it was sampled at discrete instants in fast M × N M × N, where M indicates the detection distance, and it was sampled at discrete instants in fast time τ = mT ( m = 1, 2,..., M ) time ; each fast time corresponded with a specific distance. N indicated the f 1, 2, . . . , M ); each fast time corresponded with a specific distance. N indicated the time τ = mT (m = f

t =s (nT ( n 1, = 1, N )) .. In order to better time signal of each fast time, it was sampled in slow signal of each fast time, and and it was sampled in slow timetime t = nT n s= 2,2,..., ...,N targets, the the signal pre-processing was applied in threein steps: Distance estimate the thestationary stationaryhuman human targets, signal pre-processing was applied three(i)steps: (i) accumulation was usedwas for compressing the sample along thealong fast time dimension to shorten Distance accumulation used for compressing thepoint sample point the fast time dimension to the length of the data, and the sample point was compressed from 2048 to 200; (ii) The clutter caused shorten the length of the data, and the sample point was compressed from 2048 to 200; (ii) The clutter by stationary objects,objects, such assuch walls, leads to a baseline drift in those signals alongalong the slow caused by stationary as usually walls, usually leads to a baseline drift in those signals the time dimension. For removing the clutter of stationary objects,objects, the average of all waveforms’ values slow time dimension. For removing the clutter of stationary the average of all waveforms’ with slow time were subtracted by a slidebywindow, and the width of the slideofwindow 100; (iii) A values with slow time were subtracted a slide window, and the width the slidewas window was low pass to remove random noise. As the respiration signal is a kind of 100; (iii) Afilter lowwas passapplied filter was appliedsubstantial to remove substantial random noise. As the respiration signal low-frequency narrowband narrowband signal, a finitesignal, impulse response (FIR)response filter was(FIR) adopted slow is a kind of low-frequency a finite impulse filteralong was the adopted time dimension for suppressing high-frequency Then, a new matrix W could expressed along the slow time dimension the for suppressing thenoise. high-frequency noise. Then, a newbematrix W as W = w1 , w2 , wi .as . . wW200=)(, wwhere row vector w contained 64t (t was sampling time in slow time, , w , w ... w ) , where row vector w contained 64t ( was sampling could be(expressed t i 1 2 i 200 i the unit was time, second) time in slow thesampling unit was point. second) sampling point. After pre-processing, pre-processing, the spectrum of the data data in in Figure Figure 3a 3a was was calculated; calculated; the the result result was was After the power power spectrum of the shown in Figure 4a. Meanwhile, experimental data of Figure 3b were processed by the same method shown in Figure 4a. Meanwhile, experimental data of Figure 3b were processed by the same method and the the power In previous previous studies, studies, the the magnitude magnitude of of the the power power and power spectrum spectrum is is shown shown in in Figure Figure 4b. 4b. In spectrum in the range dimension represents the degree of the signal power at different ranges [19]. spectrum in the range dimension represents the degree of the signal power at different ranges [19]. In Figure 4a, the maximum peak appears at a distance of 3 m, which corresponds to the location of In Figure 4a, the maximum peak appears at a distance of 3 m, which corresponds to the location of target A. A. However, Figure 4b 4b target However, target target B B cannot cannot be be detected, detected, since since he he stood stood in in the the shadowing shadowing region. region. Figure shows that there are two peaks that appear at distances of 3 m and 6 m, respectively; the magnitude shows that there are two peaks that appear at distances of 3 m and 6 m, respectively; the magnitude of the the peak peak at at aadistance distanceofof66mmisissmaller smallerthan thanthat that 3 m. This is because of the attenuation of of at at 3 m. This is because of the attenuation of the the electromagnetic wave of the radar thedistance distanceincreases. increases. Therefore, Therefore, the the traditional traditional power power electromagnetic wave of the radar asasthe spectrum method is not credible for detecting multiple human targets, especially in the case with the spectrum method is not credible for detecting multiple human targets, especially in the case with the shadowing effect. effect. shadowing

2017, 17, 2255 2017, 17, 2255 Sensors 2017, 17, 2255

5 of 13 5 5ofof1313

(a) (a)

(b) (b)

Figure 4. (a) The power spectrum of two targets with a shadowing effect; and (b) the power spectrum Figure (a)The The power spectrumofofeffect. twotargets targetswith withaashadowing shadowingeffect; effect;and and(b) (b)the thepower powerspectrum spectrum Figure 4.4.targets (a) power two of two with nospectrum shadowing twotargets targetswith withno noshadowing shadowingeffect. effect. ofoftwo

In Figure 3a, as a result of target A partially reflecting the energy of the electromagnetic wave, InFigure Figure 3a,asasa result afrom result target A partially reflecting the energy of from the electromagnetic wave, 3a, of of target partially reflecting the energy of the electromagnetic wave, the theIn energy reflected target B isAreduced compared with the energy the same position of a the energy reflected from target B is reduced compared with the energy from the same position of a energy from one target B is reduced compared with the energy from the position of aof single singlereflected person (only person standing at 6 m). Fortunately, although thesame power spectrum target single person (only one person standing at 6 m). Fortunately, although the power spectrum of target person (only personin standing at the 6 m). Fortunately, although thereceived power spectrum target BFigure cannot5a B cannot beone observed Figure 4, signals of target B are also by radarof antenna. Bshows cannot befrequency observed inthe Figure 4, signals of target BThe are signals also by radar antenna. Figure 5a be observed in Figure 4,spectrum signals of target B Figure are also3a. received byreceived radar Figure 5acomponents; shows the the of the target B in haveantenna. many frequency shows thespectrum frequency ofFigure target3a. B inThe Figure 3a. The have many frequency components; frequency ofspectrum target B in signals havesignals many frequency components; a principal a principal frequency of 0.25 corresponds to human respiration. In order to find the features of the a principal frequency of 0.25 corresponds to human respiration. In order to find the features the frequency 0.25 B, corresponds to human respiration.region, In order find the features the signals of signals ofoftarget who is located in the shadowing thetosignals of the same of position in of Figure signals of target B, who is located in the shadowing region, the signals of the same position in Figure target B, who is located in the shadowing region, the signals of the same position in Figure 3c were 3c were processed, and the result is shown in Figure 5b. There are wider bands with more complex 3c were processed, and the result is Figure shown5b. in Figure 5b.wider Therebands are wider with more complex processed, and the result is shown in There are withbands more complex frequency frequency components than Figure 5a. frequency components than Figure 5a. components than Figure 5a.

Figure The frequency spectrum target Figure and frequency spectrum the Figure 5. 5. (a)(a) The frequency spectrum ofof target BB inin Figure 3a;3a; and (b)(b) thethe frequency spectrum at at the Figure 5. (a) The frequency spectrum target B3c. in Figure 3a; and (b) the frequency spectrum at the same distance target B of Figure in Figure same distance asas target B of Figure 3a3a inof Figure 3c. same distance as target B of Figure 3a in Figure 3c.

4.2.Auto-Correlation Auto-Correlation Algorithm 4.2. Algorithm 4.2. Auto-Correlation Algorithm previously known, correlation is closely to the characteristic of of the period of AsAs previously known, correlation is closely related related to the characteristic of the period respiration As previously known, correlation isimprove closely related thethe characteristic ofpre-processing, the period of respiration rather that ofIn energy. to improve thetoSNCR of the signal after pre-processing, rather than that ofthan energy. order In toorder the SNCR of signal after respiration rather than that ofisenergy. In order toisimprove the SNCR of wavelet the signal afterThe pre-processing, an auto-correlation algorithm is proposed, which is fundamental entropy. The equation an auto-correlation algorithm proposed, which fundamental for for wavelet entropy. equation is an auto-correlation algorithm is proposed, which is fundamental for wavelet entropy. The equation is expressed as follows: expressed as follows: is expressed as follows: R (t ) = E [ x (t ) x (t − Δt )] = E {[ s(t ) + n (t )][ s(t − Δt ) + n (t − Δt )]} R x (t) R=x(E [ x (t) x (t − ∆t)] = E{[s(t) + n(t)][s(t − ∆t) + n(t − ∆t)]} x t ) = E [ x ( t ) x ( t − Δt )] = E {[ s ( t ) + n ( t )][ s ( t − Δt ) + n ( t − Δt )]} t ) s∆t (t − (t()tn− (t −∆tΔ)] t )]++ E E[[ss((tt))n n ((tt −−Δ∆t t )])] ++ E [E n ([tn)(st()t s−(tΔ− t )]∆t)] (1) (1) = E[s(=t)Es[(st(− )]Δ+t )]E+[nE([tn)n = E [ s (t ) s (t − Δt )] + E [n (t )n (t − Δt )] + E [ s (t )n (t − Δt )] + E [n (t ) s (t − Δt )] (1) t )n+(tR)n + (t )R+snR(snt()t + ) +RRns = Rs (=t)R+s (R (tt)) ns ( = Rs (t ) + Rn (t ) + Rsn (t ) + Rns (t )

2017, 17, 2255 Sensors 2017, 17, 2255

6 of 13 6 of 13

Where s(t ) is the wanted signal in time dimension after the step of pre-processing, n(t ) is noise, Where signal in time dimension afterdimension. the step of pre-processing, n(t) is noise, x (t) =is x(t ) = ss((t) +isnthe (t ) wanted is a mixed signal in the time An auto-correlation algorithm simplemented (t) + n(t) is a for mixed signal in the time dimension. An auto-correlation algorithm is implemented each row vector wi . Since the human life signal is not correlated with noise in for the each row vector wi . Since the human life signal is not correlated with noise in the time dimension, time dimension, Rsn (t ) and Rns (t ) are approximately zero, and as Δt increases, the major energy Rsn (t) and Rns (t) are approximately zero, and as ∆t increases, the major energy of noise focuses on of noise focuses on the location of Δt = 0 , R (t ) is equal to Rs (t ) , and the noise is removed. the location of ∆t = 0, R x (t) is equal to Rs (t), xand the noise is removed. Figure 6a 6a shows shows the the results data of of Figure 5a. 5a. As Figure results of of the the auto-correlation auto-correlationalgorithm algorithmforforthethe data Figure mentioned above, the respiration signal appears in several cells and they have a high correlation with As mentioned above, the respiration signal appears in several cells and they have a high correlation eacheach other. Thus, in Figure 6a,6a, thethe stationary and the theprincipal principal with other. Thus, in Figure stationaryclutter clutterand andnoise noise are are eliminated, eliminated, and respiration frequency of 0.25 is reserved. Figure 6b shows the frequency spectrum of Figure 5b, where respiration frequency of 0.25 is reserved. Figure 6b shows the frequency spectrum of Figure 5b, where there is only one human target at a position of 3 m. Compared to Figure 5b, the frequency components there is only one human target at a position of 3 m. Compared to Figure 5b, the frequency components havebeen beenreduced, reduced,but butare arericher richerthan thanininFigure Figure6a. 6a. have

Figure6.6.(a) (a)The Thefrequency frequencyspectrum spectrumof ofFigure Figure5a 5aprocessed processedby byauto-correlation; auto-correlation;and and(b) (b)the thefrequency frequency Figure spectrumofofFigure Figure5b 5bprocessed processedby byauto-correlation. auto-correlation. spectrum

4.3.Wavelet WaveletTransform Transform 4.3. Wavelettransform transformisisaauseful usefultool toolfor foranalyzing analyzingaanon-stationary non-stationary signal, signal, such such as as that that of of radar. radar. Wavelet Continuous wavelet transform (CWT) decomposes the continuous signal into a family of functions Continuous wavelet transform (CWT) decomposes the continuous signal into a family of functions localizedin intime timeand andfrequency, frequency,respectively. respectively.The Thedecomposition decompositionisisconstituted constitutedby byaaset setof offunctions, functions, localized ψ ( t ) ψ ( t ) , which are constructed by dilation and translation of a function , called the mother a , τ ψa,τ (t), which are constructed by dilation and translation of a function ψ(t), called the mother wavelet.

a factor wavelet. In this formulation, the parameter is a dilation factor which measures the degree of In this formulation, the parameter a is a dilation which measures the degree of compression, τ factor compression, andτthe parameter is the which translation factor the which time location and the parameter is the translation determines timedetermines location of the wavelet. ψa,τ (tof ) wavelet. isthe defined as: ψ a ,τ (t ) is defined as: 1 t−τ ψa,τ (t) = √ ψ( ), a, τ ∈ R, a 6= 0 (2) at − τ a1 ψ a ,τ (t ) = ψ( ), a ,τ ∈ R, a ≠ 0 (2) a a Compared with CWT, the discrete wavelet transform (DWT) is more computationally efficient. The main difference is CWT, that the factor a and the translation τ ofcomputationally the DWT can only have Compared with thedilation discrete wavelet transform (DWT)factor is more efficient. power of two values, i.e., The main difference is that the dilation factor a and the translation factor τ of the DWT can only a = 2j (3) have power of two values, i.e., j τ = ka a ==2 jk2

(4) (3)

where j is the level of DWT analysis performed and k is the parameter, k ∈ Z. (4) ka = k 2 j into approximation coefficients and detail The coefficients C ( a, τ ) of the DWT can τbe= divided coefficients. coefficients are theand high-scale, components of signal the approximation level of DWT analysis performed k is thelow-frequency parameter, k ∈ Z. where j isThe f (n), whereas the detail coefficients are the low-scale, high-frequency components. The coefficients C (a,τ ) of the DWT can be divided into approximation coefficients and detail The discrete wavelet transform approximation coefficients of signal f (n) at level j are expressed coefficients. The approximation coefficients are the high-scale, low-frequency components of signal as follows: f ( n ) , whereas the detail coefficients are the low-scale, high-frequency components.

2017, 17, 2255

7 of 13

SensorsThe 2017,discrete 17, 2255

wavelet transform approximation coefficients of signal f ( n ) at level j7 ofare 13

expressed as follows: ∞ ∞ 1 n − k2 j ∞ f (n) Aj =∞ f (n )φ j ,k (n ) =  (5) 1j φ ( n −j k2)j A j = ∑0 f (n)φj,k (n) = ∑ f (n) √ (5) 0 2 φ( 2 j ) 2 2j 0 0 where φ (n) is the scaling function associated with the wavelet function ψ (n) . Similarly, the where φj,kj ,k(n) is the scaling function associated with the wavelet function ψj,k (n).j ,kSimilarly, the detail j are expressed as follows: detail coefficients coefficients of levelofj level are expressed as follows: ∞ 1 n − k2 j ∞ Dj =  f (n ) 1 ψ (n − jk2 j ) (6) Dj = ∑ f (n) √2 j ψ( 2 j ) (6) 0 j 2 2 0 Based on Equations (5) and (6), the signal f ( n ) is iteratively decomposed as the number of Based on Equations (5) and (6), the signal f (n) is iteratively decomposed as the number of levels levels increases. Thus, a hierarchical set of approximations and details can be obtained through increases. Thus, a hierarchical set of approximations and details can be obtained through various various decomposition levels. In this paper, the example of the three-level wavelet decomposition of decomposition levels. In this paper, the example of the three-level wavelet decomposition of signal signal f ( n ) is shown in Figure 7. Firstly, f ( n ) is split into an approximation A1 and a detail D1. f (n) is shown in Figure 7. Firstly, f (n) is split into an approximation A1 and a detail D1 . Secondly, Secondly, the approximation A1 isinto alsoa split into a second-level approximation A2 and a detail the D2. the approximation A1 is also split second-level approximation A2 and a detail D2 . Thirdly, Thirdly, the approximation A 2 is also split into a third-level approximation A3 and a detail D3. approximation A2 is also split into a third-level approximation A3 and a detail D3 .

Figure7.7.An Anexample exampleofofaathree-level three-levelwavelet waveletdecomposition decompositionof ofsignal signal ff((nn)). . Figure

4.4. Wavelet Entropy Algorithm In order to better explain the flow of signal processing by wavelet entropy algorithm, the signal vector ininthe timetime dimension processed by auto-correlation is considered to be Xτ (tto )(τ be = theslow slow dimension processed by auto-correlation is considered 1,X2, . . . , 200). As shown in the following, the DWT of signal Xτ (t) is defined X τ (t ) isas: defined as: τ (t )(τ = 1, 2,..., 200) . As shown in the following, the DWT of signal ∞ ∞

DWTs s((j,jk, k) )==  t t)) DWT ,k (( ∑ XXττ((tt)ψ)ψj j,k

(7) (7)

00

The notonly onlyprovides providesrelevant relevantinformation informationin in aa simple simple way, way, but but also also The wavelet wavelet coefficient coefficient CCj j((kk)) not energies at different scales. It is supposed that the sample value provides an anestimation estimationofofthe thelocation location energies at different scales. It is supposed that the sample corresponds to a uniform time grid. If the discrete dyadic wavelet X = {xX value =), n{ x=01,..., ( n ), N n }= 1, . . . , N } corresponds to a uniform time grid. If the discrete dyadic wavelet 0 (n decomposition is implemented all resolution resolution levels, levels, the the wavelet wavelet expansion expansion of of Xττ ((tt)) can be can be decomposition is implemented over over all expressed as: expressed as: −1

Xτ (t) = ∑ −1∑ Cj (k) ψj,k (t) j = − J, . . . , −1 ( J = log2 ( N )) X τ (t )j=− =  j = − J ,..., −1 ( J = log 2 ( N )) J k C j ( k )ψ j ,k (t ) j =− J

k

(8) (8)

where C1 (n), C2 (nn), . . . , Co J ( n ) are the wavelet coefficients of the wavelet sequence. where C1 ( n ), C2 ( n ),..., CJ ( n ) are the wavelet coefficients of the wavelet sequence. If the family ψj,k (t) is an orthonormal based on the Hilbert space L2 ( R), the concept of energy the familylevel an orthonormal based on the Hilbert space L2 ( R) , the concept of {ψ jcan at theIfresolution be is defined as: ,k (t )}

energy at the resolution level can be defined as: Eτ,j =

2

∑ Cj (k )

k

(9)

2017, 17, 2255

8 of 13

Sensors 2017, 17, 2255

8 of 13

Eτ , j =  C j ( k )

2

(9)

k

The radar signal is divided into non-overlapping temporal windows of length L to follow the Theevolution radar signal is divided intoFor non-overlapping to followthe the L samples, ) with N temporal of the quantifiers. each interval i, temporal (i = 1, . . . windows , NT , NT =ofNLlength = 1,..., N T ,window. N T = N )The temporal evolution of theare quantifiers. For interval , (i the with N samples, appropriate signal values assigned to theeach central pointiof time concept of the L average energy of resolution level j can be defined as: the appropriate signal values are assigned to the central point of the time window. The concept of

the average energy of resolution level j can be defined as: 2 iL 1 Eτ,j = (i) ∑ iL Cj (n) ,2 i = 1, . . . , NT 1 (i −1) L+ 1 EN C j ( n ) , i = 1,..., N T τ , jj = N (j i ) ( i −1) L +1

(10) (10)

(i )

i) where the number of wavelet coefficients at resolution j included the timein window i. j in represents the number of wavelet coefficients at resolution included the time whereNj N (represents j The total energy of the wavelet coefficients at this time window can then be obtained by: window i . The total energy of the wavelet coefficients at this time window can then be obtained by:

Eτ,total Eτ ,total==∑ EEτ,jτ , j jj

(11) (11)

Therelative relativewavelet waveletenergy energythat thatrepresent representthe theenergy’s energy’s probability distribution The probability distribution is:is: Eτ , j Eτ,j (12) Eτ ,total (12) Eτ,total Clearly,  Pτ , j = 1 . According to the Shannon entropy, which gives a measure of the information Clearly, ∑ Pτ,j = 1. According to the Shannon entropy, which gives a measure of the information j Pτ , j = Pτ,j =

j

on any distribution for analyzing and comparing the probability distribution, the wavelet entropy is on any distribution for analyzing and comparing the probability distribution, the wavelet entropy is defined as: defined as: Hτ (HPτ )( P=) =−−∑ ] (13) PPτ,jτ , jlnln[[PPτ,j τ,j] (13) j

j

InInthe human target target produces produces a thefrequency frequencyspectrum spectrumanalysis analysisof of Figure Figure 6, 6, the the respiration respiration of the human a narrow-band narrow-band spectrum; it indicates that the respiration respiration signal signal isisregular. regular.However, However,the thesignal signalofofthe the noise produces a very wide-band spectrum, which indicates that noise is disordered. According to noise produces a very wide-band spectrum, which indicates that noise is disordered. According to entropy entropytheory, theory,a aregular regularsignal signalisisinclined inclinedtotoproduce producelower-value lower-valueentropy. entropy.Conversely, Conversely,a acompletely completely random such asas clutter and noise, is inclined to atohigher entropy value. In order to eliminate the randomsignal, signal, such clutter and noise, is inclined a higher entropy value. In order to eliminate effect of wavelet entropy, two experimental scenarios with no human target and a single human target the effect of wavelet entropy, two experimental scenarios with no human target and a single human located 3 m are The wavelet entropy spectrum these twoof scenarios is exhibited target at located atimplemented. 3 m are implemented. The wavelet entropyofspectrum these two scenariosinis Figure 8. Figure 8a shows that8a the entropy of the experiment no human fluctuated exhibited in Figure 8. Figure shows thatvalue the entropy value of thewith experiment withtarget no human target asfluctuated the rangeas was processed. Excluding the influence of the reflection of the wall from 0 m to 1 m, the range was processed. Excluding the influence of the reflection of the wall fromthe 0m entropy of the value remaining is from about 1.85. 1.79 In Figure 8b, the target located to 1 m, value the entropy of theregion remaining region is 1.79 fromtoabout to 1.85. Inas Figure 8b, asisthe target atis3located m, the entropy value of the target the lowest, owing to the trailing effect of the target, and the at 3 m, the entropy value of is the target is the lowest, owing to the trailing effect of the target, entropy at 3.8 m isatlower the other region. and thevalue entropy value 3.8 mthan is lower than noise the other noise region.

Figure Figure8.8.(a) (a)The Thevalue valueofofwavelet waveletentropy entropywith withno nohuman humantarget; target;and and(b) (b)the thevalue valueofofwavelet waveletentropy entropy with atat 3 m. withone onehuman humantarget targetlocated located 3 m.

Sensors 2017, 17, 2255

9 of 13

5. Experiment Results and Discussion In this section, in order to demonstrate the capability of the wavelet entropy algorithm to detect multi-stationary human targets, the adaptive filter algorithm and power spectrum were also used as references. Two cases of experiments imitating Figure 2 were conducted and one variable was assigned in each experimental case. For the first experimental case, the variable is the distance of target B to radar antennas. Three experiment scenarios were described in the following: For experiment I, target A stood 3 m in front of the antennas, and target B stood 5 m in the shadowing region of target A. For experiment II and experiment III, target A remained in the same position as in experiment I, but the distance of target B to the radar antennas was 6 m and 7 m, respectively. For the second experimental case, the variable is the intersection angle of target A and target B. There were five experiment scenarios. Target A stood at 2.5 m in front of radar antennas and the distance of target B to the radar antennas was 4 m, but their intersection angles were 10◦ , 15◦ , 20◦ , 25◦ , and 30◦ . Real experimental data were processed by the wavelet entropy algorithm and two reference algorithms based on the adaptive filter and power spectrum. The experiment results of the first and the second experimental case are shown in Figures 9 and 10. In Figure 9, only one periodically-varied strip that corresponds to the respiratory-motion response is obvious at the range of 3 m in Figure 9a,d,g. The results of the power spectrum in Figure 9b,e,h are similar to the above pseudo-color, with only one peak located at 3 m away from the radar in each Figure, respectively, which are correlated with target A. The energy of target B (at 5 m, 6 m, and 7 m in the first experimental case), who was located in the shadowing region, is negligible compared to that of target A. Thus, the method based on an adaptive filter and energy detection could easily induce a false negative result. According to the result of Figure 8, the entropy value displays a significant difference in the condition with a human target or without a human target. In this section, the proposed algorithm is also applied for locating the distance of multi-stationary human targets. Figure 9c,f,i show the results of the first case of experimental data processed by wavelet entropy. In these three Figures, the entropy value at 3 m is the lowest, which indicates that the position of target A is accurately estimated. Likewise, another lower entropy value—5 m in Figure 9c, 6 m in Figure 9f, and 7 m in Figure 9i—indicates the position of target B in the three experiments. However, it is noteworthy that the entropy values within 1 m behind target A are lower than the rest of the remaining region in the two Figures; it is easy to incorrectly deduce that one human target is located in this region. According to the theory of electromagnetic wave propagation, as a result of multipath effects and tailing effects, the ground reflects the respiration signals of target A, and the signals behind target A are regular and have the same frequency as the respiration of target A. Furthermore, the respiration signals appear in several neighboring cells and the physical size of the human body is not ignorable. In Figure 9c, the entropy pit of target A is from 2.835 to 3.24 and the entropy pit of target B is from 4.905 to 5.31. In Figure 9f, the entropy pit of target A is from 2.61 to 3.15 and the entropy pit of target B is from 5.76 to 6.25. In Figure 9i, they are from 2.61 to 3.10 and from 6.93 to 7.33; the six ranges of pits correspond to the thickness of the human body. For the lower entropy value that is behind target A, the range of the pits does not obviously agree with the thickness of the human target. Thus, the algorithm of the wavelet entropy can detect two stationary human targets accurately despite one person being in the shadowing region.

Sensors 2017, 17, 2255 2017, 17, 2255

10 of 13 10 of 13

Figure 9. 9. (a) (a) The The data data of of experiment experiment II processed processed by by an an adaptive adaptive filter; filter; (b) (b) the the power power spectrum spectrum of of the the Figure data of experiment I; (c) the entropy spectrum of the data of experiment I; (d) the data of experiment data of experiment I; (c) the entropy spectrum of the data of experiment I; (d) the data of experiment II II processed adaptive filter; thepower powerspectrum spectrumofofthe thedata dataofofexperiment experimentII; II;(f) (f)the the entropy entropy processed byby anan adaptive filter; (e)(e) the spectrum of the data of experiment II; (g) the data of experiment III processed by an adaptive filter; spectrum of the data of experiment II; (g) the data of experiment III processed by an adaptive filter; (h) the the power power spectrum spectrum of of the the data data of of experiment experiment III; III; and and (i) (i) the the entropy entropy spectrum spectrum of of the the data data of of (h) experiment III. experiment III.

The results of the second experimental case are shown in Figure 10. From Figure 10a–i, the The results of the second experimental case are shown in Figure 10. From Figure 10a–i, the intersection angle of target A and target B is 10°, 15°, and 20°, and target B is located in the shadowing intersection angle of target A and target B is 10◦ , 15◦ , and 20◦ , and target B is located in the shadowing region of target A. The results of the experiment processed by the proposed and referenced region of target A. The results of the experiment processed by the proposed and referenced algorithms algorithms complied with the expected results. Even though the distance of the human target to the complied with the expected results. Even though the distance of the human target to the radar antennas radar antennas is decreased, due to the shadowing effect of target A, target B cannot be detected by is decreased, due to the shadowing effect of target A, target B cannot be detected by an adaptive filter an adaptive filter and the power spectrum. Under the condition that the intersection angle of two and the power spectrum. Under the condition that the intersection angle of two targets reduces to 10◦ , targets reduces to 10°, target A reflects almost all of the energy of the electromagnetic wave to target target A reflects almost all of the energy of the electromagnetic wave to target B, and target B can still B, and target B can still be detected by the wavelet entropy algorithm. As their intersection angle be detected by the wavelet entropy algorithm. As their intersection angle increases, the energy of the increases, the energy of the echo signal of target B is stronger, thus, when the intersection angle is 25° echo signal of target B is stronger, thus, when the intersection angle is 25◦ and 30◦ , in Figure 10j,m, and 30°, in Figure 10j,m, the periodically-varied strip corresponding to the respiratory-motion the periodically-varied strip corresponding to the respiratory-motion response of target B appears response of target B appears at the range of 3.7 m and 3.8 m, respectively. There are also small peaks at the range of 3.7 m and 3.8 m, respectively. There are also small peaks of target B located at 3.7 m of target B located at 3.7 m and 3.8 m away from the radar in each power spectrum of Figure 10k,n, and 3.8 m away from the radar in each power spectrum of Figure 10k,n, respectively. Thus, when the respectively. Thus, when the intersection angle is less than 20°, the wavelet entropy is more effective intersection angle is less than 20◦ , the wavelet entropy is more effective for detecting multi-stationary for detecting multi-stationary human targets. human targets.

Sensors 2017, 17, 2255 2017, 17, 2255

11 of 13 11 of 13

(m)

(n)

(o)

◦ intersection Figure10. 10.(a) (a)The Theexperimental experimentaldata dataofof intersection angle processed by adative an adative filter; (b) Figure thethe 1010° angle processed by an filter; (b) the the power spectrum the experimental data awith a 10° intersection the entropy spectrum power spectrum of theofexperimental data with 10◦ intersection angle;angle; (c) the(c) entropy spectrum of the ◦ intersection of the experimental data with a 10° intersection angle; (d) the experimental a 15° intersection experimental data with a 10 angle; (d) the experimental data of data a 15◦of intersection angle angle processed by an adative filter; (e) power spectrum of the experimental data with a 15°

Sensors 2017, 17, 2255

12 of 13

processed by an adative filter; (e) power spectrum of the experimental data with a 15◦ intersection angle; (f) the entropy spectrum of the experimental data with a 15◦ intersection angle; (g) the experimental data of a 20◦ intersection angle processed by an adative filter; (h) the power spectrum of the experimental data with a 20◦ intersection angle; (i) entropy spectrum of the experimental data with a 20◦ intersection angle; (j) the experimental data of a 25◦ intersection angle processed by an adative filter; (k) the power spectrum of the experimental data with a 25◦ intersection angle; (l) entropy spectrum of the experimental data with a 25◦ intersection angle; (m) the experimental data of a 30◦ intersection angle processed by an adative filter; (n) the power spectrum of the experimental data with a 30◦ intersection angle; and (o) the entropy spectrum of the experimental data with a 30◦ intersection angle.

6. Conclusions In the case of multi-stationary human target detection through a brick wall using bistatic UWB radar, since the person closest to radar antennas partially reflects the energy of the electromagnetic wave, the person farther from the radar antennas is not detected accurately, especially when in the shadowing region of the closest person. Considering that the signals of the farther person are contained in the echo signal, and its frequency spectrum is different from the characteristic of noise, wavelet entropy is proposed as a new quantitative algorithm to detect human targets in the shadowing region. This algorithm utilizes the difference between periodic respiration and random noise in the time-frequency domain and calculates the entropy based on the complexity of energy after wavelet transform. In this paper, as the respiration of the human target exhibits more rhythmic behavior compared with noise, the entropy of the human target was significantly lower than in other regions. Through analyzing the two cases of experimental results, a conclusion is drawn: the reference algorithms based on adaptive filtering and the power spectrum can only detect the closest person; they are incapable of detecting the person in the shadowing region. The wavelet entropy algorithm not only detects the closest person and locates their distance, but also detects the farther human target in the shadowing region and gives an accurate distance. This is a breakthrough in detecting multi-stationary human targets. Yet, it must be admitted that the bistatic radar with one transmitter antenna and one receiver antenna restricts the expansion of the multi-stationary human target’s detection. Therefore, in future research, we will consider applying an UWB radar array for detecting multi-stationary targets. We think that information fusion of multiple channels may essentially improve the subtle signal of the remaining shadowed human targets in this kind of situation. Acknowledgments: This research is supported by the National Natural Science Foundation of China (no. 61327805) and the FMMU Scholar Program (4139Z3B8DA). Author Contributions: Each author contributed extensively to the preparation of this manuscript. Huijun Xue and Miao Liu developed the algorithms; Yang Zhang, Fulai Liang, and Fugui Qi designed and performed the experiments; Fuming Chen and Hao Lv analyzed the data. Jianqi Wang, Yang Zhang, Huijun Xue, and Miao Liu revised and improved the paper. All authors participated in the discussion about the proposal and contributed to the analysis of the results. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3.

Chen, K.-M.; Devendra, M.; Wang, H.; Postow, E. An X-banc microwave life-detection system. IEEE Trans. Biomed. Eng. 1986, 33, 697–701. [CrossRef] [PubMed] Chernyak, V. Signal processing in multislte UWB radar devices for searching survivors in rubble. In Proceedings of the 3rd European Radar Conference, Manchester, UK, 13–15 September 2006; pp. 190–193. BitaSobhani, M.M.; Enrico, P.; Marco, C. Multiple target detection and localization in uwbmultistatic radars. In Proceedings of the 2014 IEEE International Conference on Ultra-WideBand (ICUWB), Paris, France, 1–3 September 2014; pp. 135–140.

Sensors 2017, 17, 2255

4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

15.

16. 17. 18. 19. 20. 21. 22.

13 of 13

Stanley, E.B.; Bernard, J.C.; Peter, J.C. Through-the-wall surveillance for homeland security and law enforcement. Proc. SPIE 2015, 5778, 175–185. Sharafi, A.; Baboli, M.; Ahmadian, A. Respiration-rate estimation of a moving target using impulse-based ultra wideband radars. Australas. Phys. Eng. Sci. Med. 2012, 35, 31–39. [CrossRef] [PubMed] Rivera, N.V.; Anderson, C.; Buehrer, R.M. Multi-target estimation of heart and respiration rates using ultra wideband sensors. Eur. Signal Process. Conf. 2006, 27, 1–6. Immoreev, I.Y. Short-distance ultra wideband radars. IEEE Aerosp. Electr. Syst. Mag. 2005, 20, 9–14. [CrossRef] Fontana, R.J. Recent system applications of short-pulse ultra-wideband (UWB) technology. IEEE Trans. Microw. Theory Tech. 2004, 52, 2087–2104. [CrossRef] Lv, H.; Zhang, Y.; Wang, J.Q. An adaptive-mssa-based algorithm for detection of trapped victims using uwb radar. IEEE Geosci. Remote Sens. Lett. 2015, 12, 1808–1812. [CrossRef] Lv, H.; Jing, X.J.; Wang, J.Q. A new ultra-wideband radar for detecting survivors buried under earthquake rubbles. Microw. Opt. Technol. Lett. 2010, 52, 2621–2624. [CrossRef] Li, Z.; Jing, X.J.; Wang, J.Q. A novel method for respiration-like clutter cancellation in life detection by dual-frequency IR-UWB radar. IEEE Trans. Microw. Theory Tech. 2013, 61, 2086–2092. [CrossRef] Zhang, Z.; Lv, H.; Wang, J.Q. Human-target detection and surrounding structure estimation under a simulated rubble via UWB radar. IEEE Geosci. Remote Sens. Lett. 2013, 10, 328–331. [CrossRef] Zetik, R.; Crabbe, S.; Thomä, R. Detection and localization of persons behind obstacles using M-sequence through-the-wall radar. In Proceedings of the SPIE 6201 Sensors, and Command, Control, Communications, and Intelligence (C3I) Technologies for Homeland Security and Homeland Defense V, 62010I, Kissimmee, FL, USA, 10 May 2006. Kocur, D.; Rovnáková, ˇ J.; Urdzík, J.D. Mutual shadowing effect of people tracked by the short-range uwb radar. In Proceedings of the 34th International Conference on Telecommunications and Signal Processing (TSP), Budapest, Hungary, 18–20 August 2011; pp. 302–306. Venkatesh, S.; Rivera, N.V.; Buehrer, R.M. Implementation and analysis of respiration-rate estimation using impulse-based UWB. In Proceedings of the Military Communications Conference (MILCOM), 2005, Atlantic City, NJ, USA, 17–21 October 2005; pp. 17–20. Nezirovic, A.; Yarovoy, A.G.; Ligthart, L.P. Signal processing for improved detection of trapped victims using UWB radar. IEEE Trans. Geosci. Remote Sens. 2010, 48, 2005–2014. [CrossRef] Zhang, Y.; Li, C.Z.; Wang, J.Q. An interference suppression technique for life detection using 5.75- and 35-ghz dual-frequency continuous-wave radar. IEEE Trans. Geosci. Remote Sens. 2015, 12, 482–486. [CrossRef] Shannon, C.E. A mathematical theory of communication. Bell. Syst. Tech. J. 1948, 27, 379–423. [CrossRef] Wang, Y.; Yu, X.; Wang, J.Q. Using wavelet entropy to distinguish between humans and dogs detected by uwb radar. Prog. Electromagn. Res. 2013, 139, 335–352. [CrossRef] Qi, F.; Li, C.; Lu, G. Contact-free detection of obstructive sleep apnea based on wavelet information entropy spectrum using bio-radar. Entropy 2016, 18, 306. [CrossRef] Renyi, A. On measures of entropy and information. Math. Inst. Hung. Acad. Sci. 1961, 1, 547–561. Coifman, R.R.; Wickerhauser, M.V. Entropy-based algorithms for best basis selection. IEEE Trans. Inf. Theory 1992, 38, 713–718. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).