components R
1,k
(t) and R
1,k
(t) can be expressed as
∠R
1,k
(t) − ∠R
1,k
(t) ≈ 2π(f
c2
− f
c1
)
2d
k
C
= 4πf
c
·
d
k
C
(4)
where f
c
is the carrier frequency difference. The relation
d
k
≈ d
i,k
is applied in the formula (4) since the radar
transmitter and receiver antennas are closely located.
As a result, the DOA and range of the k
th
target can be
estimated as
θ
k
= sin
−1
∠R
2,k
(t) − ∠R
1,k
(t)
2πf
c1
· d
· C
(5)
d
k
=
∠R
1,k
(t) − ∠R
1,k
(t)
4πf
c
· C
The localization method, which is based on the direct
phase calculation as shown in formula (5), has been
proved very sensitive to noise interference. To improve the
stability of the DF-CW TWR detection, a modified
method, which is based on the frequency integration
process, is usually applied in practical applications as [22]
θ
k
= sin
−1
2π ·
t
(f
2,k
− f
1,k
)dt + φ
θ
2πf
c1
· d
· C
(6)
d
i,k
=
2π ·
t
(f
1,k
− f
1,k
)dt + φ
d
4πf
c
· C
where f
i,k
is the k
th
target instantaneous frequency at the
receiver R
xi
corresponding to the carrier frequency f
c1
,
while f
i,k
is the counterpart corresponding to the carrier
frequency f
c2
. φ
θ
and φ
d
are the initial phases for the target
DOA and range estimation, respectively. As a result, with
the application of formula (6), the target coordinate can be
estimated from its instantaneous frequency and the
DF-CW TWR localization has been converted into the
frequency estimation issue of the interested targets.
B. Hough Separation Approach
The capability of Hough transfer to accumulate signal
energy along a preset time-frequency trajectory provides
an effective approach to separate multiple components
[24, 25]. Take the separation process of the k
th
target
component as an example and the main steps can be
summarized as follows. First, a proper model of the target
instantaneous frequency is constructed as f
k
(X,t),which
determines the accumulation trajectory of the processing
signal r(t). X is a set of undecided model parameters by
which the accumulation trajectory can be flexibly
adjusted. Next, based on the constructed model, the signal
energy is accumulated as
E
k
(X) =
r(t) · e
−j2π
t
0
f
k
(X,t)dt
2
dt (7)
and the most proper parameter set X for the target, which
is denoted as X
k
, can be determined from the peak
coordinates of the function E
k
(X). Then, a low pass filter
L(f) is applied to extract the baseband demodulation
component from the frequency domain and the
reconstructed target component can be expressed as
r
k
(t) = IFFT
FFT
r(t) · e
−j2π
t
0
f (X
k
,t)dt
· L(f )
· e
j2π
t
0
f (X
k
,t)dt
(8)
where FFT(.) and IFFT(.) are the Fourier and inverse
Fourier operators, respectively. Finally, subtract the target
component from the signal r(t) and calculate the residual
energy rate as
R
residual
=
t
r(t) −
k
i=1
r
i
(t)
2
dt
t
|r(t)|
2
dt
(9)
Update the parameter k = k + 1 and repeat the above steps
until the residual energy rate falls into the noise level.
C. HAF-based Estimation Method
The high-order instantaneous moment (HIM) of a
signal r(t) is defined as [23]
HIM
1
[r(t); τ ] = r(t)
HIM
2
[r(t); τ ] = r(t) · r
∗
(t − τ )
HIM
3
[r(t); τ ] = HIM
2
[HIM
2
[r(t); τ ]; τ ]
(10)
= r(t) · [r
∗
(t − τ )]
2
· r(t − 2τ )
.
.
.
HIM
N
[r(t); τ ] = HIM
2
[HIM
N−1
[r(t); τ ]; τ ]
where τ is the time delay parameter and r*(t) is the
conjugation signal of r(t). As a result, the symbol
HIM
i
[r(t), τ ]isthei
th
order instantaneous moment of the
signal r(t) with the time delay parameter τ and its Fourier
result is called the i
th
HAF as
HAF
i
[r(t); τ ] =
HIM
i
[r(t); τ ] · e
−j2πf t
dt (11)
According to the earlier research, if the signal r(t) is a
single-component P-order polynomial phase signal, its
P-order HIM is a complex sinusoidal signal which can be
expressed as [23]
r(t) = A · e
j
P
k=0
a
k
·t
k
(12)
HIM
P
[r(t); τ ] = e
j
P !τ
P −1
a
P
t+ψ
where is a constant phase term. As a result, the highest
order phase coefficient α
P
of the signal r(t) can be
estimated from its HAF as
a
P
=
2πf
P !τ
P −1
(13)
where f
is the peak frequency coordinate of the signal
HAF.
Demodulate the signal r(t) with the estimated phase
coefficient α
P
as
r
(t) = r(t) · e
−ja
P
·t
P
(14)
Update the signal r(t) = r
(t) and the parameter P = P-1.
Repeat the formula (12) to (14) to sequentially estimate
DING ET AL.: HOUGH-MHAF LOCALIZATION ALGORITHM FOR DF-CW TWR 113