### I. Introduction

### II. Signal Model

*L*represents the number of FMCW chirp signals,

*T*

*defines the total duration of the chirp symbol (e.g.,*

_{F}*T*

*=*

_{F}*T*+

*T*

*),*

_{i}*T*is the chirp symbol’s duration, and

*T*

*is the guard period’s duration. The chirp symbol for vital FMCW radar*

_{i}*s*

_{0}(

*t*) is composed of

*f*

_{0}is the start frequency of the bandwidth,

*μ*=

*2πf*

*/*

_{BW}*T*is the FMCW chirp symbol’s frequency slope as time progresses, and

*f*

*is the FMCW signal’s bandwidth.*

_{BW}*M*targets, the definition of body movement is needed to determine the target’s fixed distance

*d*

_{0,}

*and the time-varying distance*

_{m}*x*

*(*

_{m}*t*) of the

*m*-th target, such as chest displacements caused by respiration and the heartbeat. The time-varying distance between the radar and the human is represented by

*x*

*(*

_{m}*t*) =

*x*

_{m,}_{h}(

*t*) +

*x*

_{m,}_{r}(

*t*), and

*x*

_{m,}_{r}(

*t*) and

*x*

_{m,}_{h}(

*t*) define the

*m*-th human’s body motions by respiration and heartbeat, respectively. The time-varying distance movement

*x*

*(*

_{m}*t*) is composed of

*K*antennae, the

*m*-th human’s received signal is obtained with a time delay

*τ*

*. The received signal*

_{m}*y*

*(*

_{k,l}*t*) in the

*l*-th chirp symbol and the

*k*th antenna array is represented [5] in (5), where

*a*

_{m,}_{h}and

*a*

_{m,}_{r}define the amplitude of the heartbeat and respiration, respectively, and

*f*

_{m,}_{h}and

*f*

_{m,}_{r}denote the frequency of the heartbeat and respiration, respectively.

*ã*

*is the complex amplitude of the*

_{m}*m*-th target,

*x*

_{m,}_{r,}

*(*

_{p}*t*) is the

*p*-th respiration harmonic components of the

*m*-th human, and ζ denotes the phase residual. Further, λ is the wavelength of center frequency,

*d*is the distance between the adjacent antennae,

*θ*

*is the angle of the*

_{m}*m*-th human, and

*ω*(

*t*) is the additive white Gaussian noise (AWGN) signal. In the RX component, the de-chirping method achieves the low complexity of the FMCW radar. In the de-chirping method, as the beat signal, the conjugation multiplication method of the FMCW TX signals

*y*

*(*

_{k,l}*t*) are performed. The beat signal is represented such that

*p*

*(*

_{k,l}*t*) is determined by a Nyquist sampling frequency

*f*

*= 1/*

_{S}*T*

*as the digital signal*

_{S}*p*

*[*

_{k,l}*n*] where

*q*= 1, 2, …,

*Q*. In the case

*q*= 1, the main respiration signal is yielded; in the case of

*q*= 2, 3, …,

*Q*denotes the respiration harmonic components.

##### (7)

*f*

*= 1/*

_{S}*T*

*and the digital data is represented in the vector*

_{S}

*p*

_{k}_{,}

*= [*

_{l}*p*

_{k}_{,}

*[0],*

_{l}*p*

_{k}_{,}

*[1],...,*

_{l}*p*

_{k}_{,}

*[*

_{l}*N*−1]]

^{T}in (7).

*p*

_{k}_{,}

*is expressed by the distance, the vital Doppler, and the angle terms, respectively, such that*

_{l}**and**

*α***are the amplitude and noise vectors, respectively. In order to explain further,**

*ω***= [**

*α**a*

_{0},

*a*

_{1}, ...,

*a*

_{N}_{−1}]

^{T}and

**= [**

*ω**ω*

_{0},

*ω*

_{1}, ...,

*ω*

_{1 }

_{N}_{ −1}]

^{T}, the range vector is denoted by

**, i.e.,**

*R***= [**

*R**R*(

*τ*

_{0}),

*R*(

*τ*

_{1}), ...,

*R*(

*τ*

*)], where*

_{M}*R*(

*τ*

*) is the FMCW beat signal of delay*

_{m}*τ*

*, i.e.,*

_{m}*R*(

*τ*

*) = exp(*

_{m}*j*2

*πμ τ*

_{m}*T*

_{s}*n*). The velocity vector

*v**is shown such that*

_{l}*m-*th element of

*v**is denoted by*

_{l}*v*

*(*

_{l}*m*) and it is represented such that

### III. Proposed Super-resolution DOA Estimation of Vital FMCW Radar

*d*=

*λ*/2, the angle resolution of conventional FMCW radar is low. Because this paper concentrates on angle detection, the Doppler information has been omitted.

### 1. Distance Parameter Estimation by FFT

*P*

_{k,}_{1}= [

*P*

_{k,}_{1}[0],

*P*

_{k,}_{1}[1], …,

*P*

_{k,}_{1}[

*N*-1]]

^{T}at the 1st chirp symbol-based distance information are represented such that

*W**is the matrix of FFT, which consists of*

_{N}*N*column vectors with

*N*×1 elements, i.e.,

*W**=[*

_{N}*W*

_{0},

*W*

_{1}, ...,

*W*

_{N}_{−1}]. Further, the

*u-*th column vector is represented by

*W*

*such that*

_{u}**= [**

*I**I*

_{1},

*I*

_{2}, …,

*I*

*] are obtained. When the distance 1D-FFT’s peak value is utilized, the super-resolution DOA results with a wide array distance are accomplished in the next section.*

_{M}### 2. Proposed Method for the Angle Information

*d*= λ/2 is swept from −1 to 1. When the array distance

*d*is set to λ, the total FOV angle changes from −30° to 30° while the angle range is swept from −1 to 1 using (11). Since conventional studies of vital radar consider a narrow FOV [5], this paper focuses on a narrow FOV. Under a narrow FOV, there is a margin to increase the spacing between arrays. Increasing the spacing between arrays can maintain resolution even if the number of arrays is reduced. By reducing the total number of arrays, the proposed scheme reduces system complexity while maintaining the same resolution.

*θ*such that [13]:

*d*= λ/2, we obtain the ideal angular resolution. When the array distance

*d*is changed to λ, we achieve twice the improvement in resolution. In order to obtain the super-resolution angle results with

*d*= λ, the proposed algorithm with a wide array distance and extrapolation method virtually generates data between antenna spacing to improve the resolution. The extrapolation-based DOA technique depends on modeling the extracted results as the output of a linear system of a rational system form, whereby

*g*and

*h*in (15). Among these three linear models, the AR model with

*h*= 0 is the most widely used because the AR model results in very simple linear equations for the AR parameters. The typical method to estimate an AR parameter for estimating DOA information is the covariance method by [14]. Using extracted AR parameters, we can extrapolate the results of the extracted

*i*th target

*K′*is the number of extrapolated signals. Finally, the extracted 1D-extrapolations are fed into the super-resolution algorithms. Of all the available super-resolution algorithms, this study selected the MUSIC algorithm. When the data extracted by only the distance index is processed by MUSIC [15, 16], the proposed algorithm obtains high-resolution DOA results. When obtaining the DOA, the received signal’s correlation matrix is needed to check a full rank. However, when the angular components of each target are similar to each other, that is, a coherent signal exists, the rank of the matrix can be reduced. The coherent signal refers to multiple targets in the same range cell. This coherent signal correlation matrix is meant as a drawback. This problem is solved by a smoothing method in the DOA direction [13]. The auto-correlation matrix

*R*

_{X}_{,}

*of the extrapolation results of the extracted*

_{i}*i-*th target

*U*≤

*K′*, is represented such that

**is the exchange matrix.**

*J*

*R**is performed [15] such that*

_{fb,i}**= [**

*S*

*s*_{0}, …,

*s*

_{P}_{-1}] is matched with the signal subspace of the forward–backward auto-correlation matrix. The noise eigenvector matrix

**= [**

*N*

*n*_{0,}…,

*n*

_{T}_{-}

_{P}_{-1}] represents the noise subspace of the auto-correlation matrix, and λ

*denotes the*

_{n}*n-*th eigenvalues of

*R**. The total number of human beings at the same distance is denoted by*

_{fb,i}*Q*. The largest

*Q*eigenvalues of

*λ*

_{0}, …,

*λ*

_{Q}_{-1}are matched with the

*Q*signal eigenvectors of

**. The other eigenvalues λ**

*S**, …,*

_{Q}*λ*

_{U}_{-1}are matched with the noise eigenvectors of

**such that λ**

*N**= … = λ*

_{Q}

_{U}_{-1}=

*σ*

^{2}. The eigenvalue is represented such that

*P*signal eigenvectors

**are the signal subspace and the**

*S**U*-

*Q*-1 noise eigenvectors

**represent the noise subspace. The MUSIC algorithm has the characteristics that the steering DOA vector**

*N**a*(

*θ*) and the noise eigenvectors

**are orthogonal such that**

*N**a*(

*θ*) is defined as

*i-*th target among

*M*humans can be estimated such that

### 3. Summary of the Proposed Algorithm

**Step 1:**The distance between multiple human beings is obtained using the FFT outputs of the received beat signal, i.e., the range bins are obtained.**Step 2:**Among FFT outputs, the clutters are removed using characteristics of the vital signals. The phase information of clutters does not change according to time, whereas the phase of respiration and heartbeat have a change over time. This property was used to efficiently eliminate clutter.**Step 3:**Among the distance FFT outputs in the which the clutters are removed, the maximum peaks among*M*targets are obtained to effectively detect angle—that is, the range bins in which targets exist are selected among all the range bins.**Step 4:**Since the conventional studies of vital radar consider narrow a FOV [5], this paper also focused on narrow FOV. Under a narrow FOV, the margin to increase the spacing between arrays exists. After the array distance*d*is set to λ, we obtain the total FOV from −30° to 30° and achieve twice the improvement in DOA resolution.**Step 5:**The array data virtually increase using the extrapolation method.**Step 6:**By performing the MUSIC algorithm on the extrapolated array signal, improved DOA information is obtained.

### IV. Simulations

### 1. Simulation Environment

*C*times, where

*C*is the number of simulations. The RMSE is calculated by

*C*is set to 10

^{3}, where

*θ*

_{m}_{,}

*and*

_{n}*θ̄*

_{m}_{,}

*are the actual DOA and the estimated DOA, respectively, of the*

_{n}*m*-th target at the

*n*-th Monte Carlo simulation.

### 2. Simulation Results

*d*=

*λ*/2 and a number of arrays

*K*= 4, the range of total FOV is from −90° to 90° and the DOA resolution is about 25.4° [14]. However, when the array distance

*d*is set to λ, we expected to obtain a range of the total FOV from −30° to 30°, and the DOA resolution to be about 12.7° through simulation. The SNR parameter was configured to 20 dB. Fig. 3 represents the two targets’ simulation environment. In Fig. 2,

*R*

_{1}and

*R*

_{2}are the ranges of the two targets, respectively. In this simulation,

*R*

_{1}and

*R*

_{2}are set to 3 m in order to confirm the angle resolution of each algorithm.

*K*= 4. The proposed method sets the MUSIC-based array at distance =

*λ*and

*K*= 4. In Fig. 4(a), the simulation results of MUSIC with

*K*= 8 and an array distance = λ/2, MUSIC with

*K*= 4 and an array distance = λ, and the proposed algorithm achieved two signal peaks in terms of the angle and they are represented with the reference data. When the two targets are located at

*θ*

_{1}= 5° and

*θ*

_{2}= 25°, respectively, MUSIC with

*K*= 8, MUSIC with

*K*= 4, and the proposed algorithm can distinguish between the two targets appropriately. When the two targets are located very close together (

*θ*

_{1}= 5° and

*θ*

_{2}= 22°), the spectrum of MUSIC with

*K*= 4 in Fig. 4(b) has only a single peak in terms of the angle; whereas, the conventional MUSIC with

*K*= 8 and the proposed algorithm can still obtain the two peaks at

*θ*

_{1}= 5° and

*θ*

_{2}= 22° with the reference data. In Fig. 4(c), with

*θ*

_{1}= 5° and

*θ*

_{2}= 20°, while conventional MUSIC with

*K*= 4 cannot separate the two targets adequately, the conventional MUSIC with

*K*= 8 and the proposed algorithm can obtain the two peaks compared with the reference data. From these results, we can conclude that the conventional MUSIC with

*K*= 4 cannot discern appropriately; however, the proposed algorithm can distinguish the two targets. Therefore, the proposed method is suitable as a super-resolution algorithm.

*K*= 4. We will concentrate on the RMSE results of the first target when the second target exists. The angle differences are set at 20°, 17°, and 15°, respectively in Fig. 5. In Fig. 5(a), with an angle difference of 20°, the proposed and the conventional algorithms’ RMSEs are similar. When the two persons’ angle difference is closer to the resolution limitation, with an angle difference of 17° as in Fig. 5(b), the proposed algorithm performs better than the MUSIC algorithm with

*K*= 4 under all SNRs. In Fig. 5(c), where the angle difference is 15°, the proposed algorithm and the MUSIC with

*K*= 8 have a low RMSE. The conventional MUSIC with

*K*= 4’s RMSE performance is poor. Through these results, this section concludes that the proposed algorithm with

*K*= 4’s estimation performance is similar to the conventional MUSIC with

*K*= 8, while the proposed one has better estimation results than the MUSIC with

*K*= 4. According to Fig. 5, we set the size of the spectrum to 1,024, thus we showed that angle estimation error reached about 1°.

### V. Experiments

### 1. Experimental Environments

### 2. Experimental Results

*R*

_{1}= 2.9 m and

*R*

_{2}= 4.4, 3.9, 4.9, and 5.4 m, respectively. In terms of case I, II, III, and IV, when the two humans were located at certain distinct distances, the proposed result could classify the two humans from stationary clutter, simultaneously, as shown in Fig. 7. The proposed method effectively removes the clutter terms with a simple algorithm using the characteristics of vital signals. Cases I and II, which have a large distance, distinguished between the two humans from the conventional structure and the proposed structure. In the distance interval of case III, the conventional FFT algorithm did not distinguish the two humans, while the proposed structure could resolve them. Using the distances in case IV, the conventional FFT and the proposed algorithm did not distinguish two humans.

*d*of the radar is set to λ. Finally, the data of the array virtually increases using extrapolation; then, the MUSIC algorithm performs on the extrapolated array signal to obtain the improved DOA information (Fig. 8(c)).

*K*= 4 and 8 in the RF system represented in [13]. In Fig. 9(a), when

*K*= 4, the conventional MUSIC scheme cannot classify the two targets; whereas, when

*K*= 8, the proposed scheme and the conventional MUSIC can resolve the targets, even though the two targets are close. That is, the results of the proposed algorithm with

*K*= 4 is similar to the results obtained by the MUSIC algorithm with

*K*= 8. As the results of the experiment in Fig. 9(b) and (c), the angular resolution of the proposed structure shown in Fig. 9(a) is improved compared to the conventional structure. These results show that the proposed scheme achieves an improvement in the resolution due to the increased number of channels.

### VI. Conclusion

*K*= 4 cannot identify the same two targets as distinct. The RMSE of the proposed algorithm validates the results of the simulations. We analyzed experimentally that the proposed method can separate adjacent targets, while the MUSIC algorithm with

*K*= 4 cannot distinguish between them. Therefore, the proposed method is applicable to FMCW radar due to its high performance in parameter estimation. For further study, the MUSIC algorithm for antenna array processing needs band equalization due to channel mismatch. This issue will be the subject of future improvement of radar prototypes to enhance the milestone described herein.