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2 × 2 = 4

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9 × 9

14.5mm× 14.5mm160◦ 145◦

(1, 1), (1, 2), · · · , (9, 9) (1, 1)

(5, 6)

Δt =(5, 6) (5, 5) (4, 5)

(4, 4)

s1(t) s2(t) s3(t) s4(t)

(5, 6) (5, 5) (4, 5)(4, 4)

0.81λλ

9×9

0.41λ

x z

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−50 0 50

−60

−40

−20

0

20

40

60

Ch1Ch2Ch3

Ch4

x [mm]

z [m

m]

s1(t) s2(t)s3(t) s4(t)

si(t) = s(raw)i (t)− s

(DC)i

(i = 1, 2, 3, 4) s(raw)i (t) s

(DC)i

s(DC)i = 1/T0

∫ T0

0s(raw)i (t)dt

PBF(θ, φ)

PBF(θ, φ) = 1T0

∫ T0

0|a(θ, φ)Hs(t)|2dt,

= a(θ, φ)HRssa(θ, φ).

s(t) = [s1(t), s2(t), s3(t), s4(t)]T

Rss

Rss =1

T0

∫ T0

0

s(t)sH(t)dt,

a(θ, φ)

a(θ, φ) = [k · r1,k · r2,k · r3,k · r4]T ,

k r1 r2 r3 r4

kθ φ

k(θ, φ) = k[sin θ cosφ, sin θ sinφ, cos θ]T,

k = 2π/λ λ = 3.6

y(t) = w∗1s1(t) + w

∗2s2(t) + w

∗3s3(t) + w

∗4s4(t) = wHs(t)

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∗

w = [w1, w2, w3, w4]T

|y(t)|2 = |wHs(t)|2= wHs(t)(wHs(t))H

= wHs(t)sH(t)w,

E[|y(t)|2] = wHRssw

E[]Rss = E[s(t)sH(t)]s(t)

PC(θ, φ)

PC(θ, φ) = minwwHRssw,subject to wHa(θ, φ) = 1.

wE[|y(t)|2] = wHRssw

(θ, φ)

(θ, φ)(θ, φ)

E[|y(t)|2]

wPC(θ, φ)

PC(θ, φ) =1

aH(θ, φ)(Rss + εI)−1a(θ, φ),

Iε

ε = ε0tr(Rss) ·PC(θ, φ)

θ0 φ0

(θ0, φ0)

wopt = argminwwHRssw,subject to wHa0 = 1,

w a0 a0 =a(θ0, φ0) θ0 φ0

argmaxθ,φ PC(θ, φ)

L

L(w, λ0) = 12w

HRssw +Re[λ0(wHa0 − 1)]

= 12w

HRssw + 12λ0(w

Ha0 − 1)

+ 12λ

∗0(w

Ta∗0 − 1),

λ0

L(w, λ0) w

∇wL(w, λ0) = Rssw + λ0a0 = 0.

w

w = −λ0R−1ss a0,

Rss

λ0 = − 1

aH0 R

−1ss a0

.

w =R−1

ss a0

aH0 R

−1ss a0

.

wRss Rss

R−1ss

wopt =(Rss + ε′I)−1a0

aH0 (Rss + ε′I)−1a0

.

sDCMP(t)

sDCMP(t) = wHopts(t).

ε′ = ε′0tr(Rss)

ε

wHs(t)sd(t) (θ0, φ0)

si(t)(θi, φi)

G(θ0, φ0) = 1 G(θi, φi) = |α(w)|2w α(w)

(θi, φi)wHs(t) = sd(t) + α(w)si(t)E[|sd(t) + α(w)si(t)|2] = E[|sd(t)|2] + |α(w)|2E[|si(t)|2]

|sd(t)|2

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w |α(w)|si(t)

sDCMP(t)(θ0, φ0)

sDCMP(t)

� sDCMP(t)

x z(x, y) = (−0.3m, 0.5m) (x, y) =

(0.3m, 0.5m)

sA(t)sB(t)

sA(t) + sB(t)

sA(t) sB(t)

λ = 35.7λ

λ62.7◦

123.9◦

x zy = 0.5

sA(t)+sB(t)

ε0 = 10−5

(x, z) = (28.9cm,−0.8cm)

(θ0, φ0) a0

sB(t)−1 − j 1 − j −1 + j 1 + j

sB(t)

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x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−30

−25

−20

−15

−10

−5

0

x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−30

−25

−20

−15

−10

−5

0

Rss sB(t)Rss

wopt

a0

PDCMP(θ, φ) = wHopta(θ, φ)

ε′0 = 0.1

x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−30

−25

−20

−15

−10

−5

0

x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−30

−25

−20

−15

−10

−5

0

sDCMP(t) = wHoptsB(t)

P ∝ r(t)−4 P

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−2 −1 0 1 2−2

−1

0

1

2

Real

Imag

inar

y

Ch1 Ch2

Ch3 Ch4

sB(t)

x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−10

−8

−6

−4

−2

0

2

r

s1(t) sDCMP(t)dDCMP(t)

dDCMP(t) =λ

4πunwrap(� sDCMP(t)),

unwrap(·)

unwrap(θ(t)) =

⎧⎨⎩

θ(t) (|θ(t)− θ(t−Δt)| ≤ π),θ(t) + 2π (θ(t)− θ(t−Δt) < −π),θ(t)− 2π (θ(t)− θ(t−Δt) > π).

Δt

-1 -0.5 0 0.5 1Real

-1

-0.5

0

0.5

1

Imag

inar

y

sDCMP(t)

0 10 20 30 40 50−60

−50

−40

−30

−20

−10

0

10

20

30

Time [s]

Dis

plac

emen

t [m

m]

dDCMP(t)sDCMP(t) sB(t)

d1(t) s1(t)

dDCMP(t)� sDCMP(t)

sA(t) + sB(t)

s1(t)sA(t)+sB(t)

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JETCAS.2018.2809582, IEEE Journalon Emerging and Selected Topics in Circuits and Systems

−2 −1 0 1 2−2

−1

0

1

2

Real

Imag

inar

y

Ch1 Ch2

Ch3 Ch4

sA(t)+sB(t)

0 10 20 30 40 50

−80

−70

−60

−50

−40

−30

−20

−10

0

Time [s]

Dis

plac

emen

t [m

m]

s1(t)

sA(t) + sB(t)

Rss θ0 φ0

sDCMP(t)

x [cm]

z [c

m]

−100 −50 0 50 100

−100

−50

0

50

[dB]

−10

−8

−6

−4

−2

0

2

-1 -0.5 0 0.5 1Real

-1

-0.5

0

0.5

1

Imag

inar

y

sDCMP

sA(t)+sB(t)t = 10

0 10 20 30 40 50−60

−50

−40

−30

−20

−10

0

10

20

30

Time [s]

Dis

plac

emen

t [m

m]

dDCMP(t)sDCMP(t) sA(t)+sB(t)

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0 10 20 30 40 50−300

−200

−100

0

100

200

300

Time [s]

Dis

plac

emen

t [µm

]

sDCMP(t)sB(t)

15 s ≤ t ≤ 45 sμ

≤ t ≤

0 10 20 30 40 50−300

−200

−100

0

100

200

300

Time [s]

Dis

plac

emen

t [µm

]

s1(t) sA(t) + sB(t)

0 10 20 30 40 50−300

−200

−100

0

100

200

300

Time [s]

Dis

plac

emen

t [µm

]

sDCMP(t) sA(t) + sB(t)

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sB(t)

s1(t)sDCMP(t)

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7

Time [s]

Inte

rbea

t Int

erva

l [s]

sB(t)

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7

Time [s]

Inte

rbea

t Int

erva

l [s]

sA(t) + sB(t)

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JETCAS.2018.2809582, IEEE Journalon Emerging and Selected Topics in Circuits and Systems

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7

Time [s]

Inte

rbea

t Int

erva

l [s]

sA(t) +sB(t)

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7

Time [s]

Inte

rbea

t Int

erva

l [s]

sA(t) +sB(t)

Rs

Δt′ τECGn τ radarn

n = 1, 2, · · · , N Δt′ = 0.5

|τECGn − τ radarn | < τ0 τ0 = 50

N0

Rs = N0/NRs

Rs

0 10 20 30 40 50Time [sec]

80

90

100

110

120

Hea

rt R

ate

[bpm

]

sB(t)

0 10 20 30 40 50Time [sec]

80

90

100

110

120

Hea

rt R

ate

[bpm

]

sA(t) + sB(t)

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0 10 20 30 40 50Time [sec]

80

90

100

110

120H

eart

Rat

e [b

pm]

sA(t) + sB(t)

-20 -10 0 10 20Heart Rate Error [bpm]

0

0.1

0.2

0.3

0.4

0.5

Nor

mal

ized

Fre

quen

cy

DCMP for One ParticipantDCMP for Two ParticipantsSingle Channel for Two Participants

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JETCAS.2018.2809582, IEEE Journalon Emerging and Selected Topics in Circuits and Systems

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JETCAS.2018.2809582, IEEE Journalon Emerging and Selected Topics in Circuits and Systems