Method and apparatus for processing joint multiple chirp sequences
A non-transitory computer-readable medium stores machine instructions that cause a processor to obtain a range-sample-antenna data cube for a received radar signal comprising reflections of a plurality of interleaved chirp sequences. The processor generates, for each chirp sequence, a Hankel matrix based on subset of range bins for the particular chirp sequence, and generates a block Hankel matrix based on the Hankel matrices for the plurality of interleaved chirp sequences. The processor performs truncated singular value decomposition to estimate subspaces, and object detection to identify selection matrices. The processor calculates least-squares approximations for the selection matrices and the block Hankel matrix to obtain, for each chirp sequence, a first phase shift matrix θ 11 and a second set of phase shift matrices θ 1l , l=2, . . . , L. The processor performs joint diagonalization and Doppler division multiplexing compensation on θ 11 and θ 1l , l=2, . . . , L, then resolves ambiguities in determined velocities based on comparison of θ 11 and θ 1l , l=2, . . . , L.
1 . A system, comprising:
at least one processor; and
at least one non-transitory computer-readable medium storing machine instructions which, when executed by the at least one processor, cause the at least one processor to:
obtain a range-sample-antenna data cube for a received radar signal comprising reflections of a plurality of interleaved chirp sequences off of objects in an environment;
generate, for each chirp sequence of the plurality of interleaved chirp sequences, a Hankel matrix based on a subset of range bins of the range-sample-antenna data cube corresponding to the particular chirp sequence;
generate a block Hankel matrix based on the Hankel matrices corresponding to the plurality of interleaved chirp sequences;
perform truncated singular value decomposition on the block Hankel matrix to estimate subspaces;
perform object detection to identify selection matrices in the estimated subspaces;
calculate least-squares approximations for the selection matrices and the block Hankel matrix to obtain, for each chirp sequence in the plurality of interleaved chirp sequences, a first phase shift matrix θ 11 and a second set of phase shift matrices θ 1l , l=2, . . . , L, wherein the first matrix θ 11 represents a phase shift over a repetition interval between chirps in a same chirp sequence, and wherein the second set of matrices θ 1l , l=2, . . . , L represent a phase shift between chirps of a same sequence index in the plurality of interleaved chirp sequences;
perform joint diagonalization of the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L;
perform Doppler division multiplexing compensation on the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L; and
resolve ambiguities in determined velocities based on a comparison of the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L.
2 . The system of claim 1 , wherein a Hankel matrix corresponding to an lth chirp sequence is represented as:
H
l
=
M
‵
θ
1
l
X
+
N
l
where M′ is an B×(PK) sub-matrix of a Vandermonde matrix M, B is a number greater than a maximum expected number P of objects in a range bin, K is the number of transmitters, θ 1l is a diagonal matrix that represents phase shifts introduced by velocities of the objects in the environment between the first chirp sequence and the lth chirp sequence, X is a matrix representing shifted combination of complex amplitudes of the received radar signal reflected off of objects in the environment with object velocity induced phase changes, and N l is a noise matrix for the lth chirp sequence.
3 . The system of claim 1 , wherein a number of possible selection matrix pairs is represented as two times a number G, where G is represented as:
G
=
L
!
2
!
(
L
-
2
)
!
where “!” indicates the factorial of a number.
4 . The system of claim 1 , wherein the plurality of interleaved chirp sequences includes a number L of interleaved chirp sequences, wherein the least-squares approximation for an lth chirp sequence is represented as:
Ψ
11
=
U
Φ
1
†
U
Φ
2
=
T
θ
11
T
-
1
Ψ
1
l
=
U
θ
1
†
U
θ
l
=
T
θ
1
l
T
-
1
,
l
=
2
,
…
,
L
where Ψ 11 represents a least-squares model for the first matrix θ 11 , Ψ 1l represents a least-squares model for the second set of matrices θ 1l , l=2, . . . , L, and T is a non-singular matrix that jointly diagonalizes Ψ 11 and Ψ 1l , wherein the matrix T has dimensions (PK)×(PK) where P is an expected maximum number of objects in a range bin and K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
5 . The system of claim 1 , wherein the Doppler division multiplexing compensation for the first matrix θ 11 causes a phase shift arg(θ 11,p ) for a particular object p in a number P of objects in the environment over the repetition interval between chirps in the same chirp sequence to be represented as:
θ
11
,
p
=
θ
11
,
p
(
1
)
+
θ
11
,
p
(
2
)
e
-
j
2
π
K
+
…
+
θ
11
,
p
(
K
)
e
-
j
2
π
(
K
-
1
)
K
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
6 . The system of claim 1 , wherein the Doppler division multiplexing compensation for the θ 1l , l=2, . . . , L causes a second set of phase shifts arg(θ 1l,p ) for a particular object p in a number P of objects in the environment phase shift between chirps of the same index in the plurality of interleaved chirp sequences to be represented as:
θ
1
l
,
p
=
θ
1
l
,
p
(
1
)
+
θ
1
l
,
p
(
2
)
+
…
+
θ
1
l
,
p
(
K
)
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
7 . The system of claim 1 , wherein the at least one non-transitory computer-readable medium further stores machine instructions which, when executed by the at least one processor, cause the at least one processor to:
obtain the received radar signal;
perform a first fast Fourier transform (FFT) on the received radar signal to obtain the range-sample-antenna data cube;
for each particular range bin in the range-sample-antenna data cube:
determine whether a characteristic of the range bin satisfies a criterion;
in response to the characteristic of the range bin satisfying the criterion, perform a second FFT on a sample-antenna array corresponding to the particular range bin; and
in response to the characteristic of the range bin not satisfying the criterion, include the sample-antenna array corresponding to the particular range bin in the subset of range bins.
8 . A non-transitory computer-readable medium storing machine instructions which, when executed by at least one processor, cause the at least one processor to:
obtain a range-sample-antenna data cube for a received radar signal comprising reflections of a plurality of interleaved chirp sequences off of objects in an environment;
generate, for each chirp sequence of the plurality of interleaved chirp sequences, a Hankel matrix based on a subset of range bins of the range-sample-antenna data cube corresponding to the particular chirp sequence;
generate a block Hankel matrix based on the Hankel matrices corresponding to the plurality of interleaved chirp sequences;
perform truncated singular value decomposition on the block Hankel matrix to estimate subspaces;
perform object detection to identify selection matrices in the estimated subspaces;
calculate least-squares approximations for the selection matrices and the block Hankel matrix to obtain, for each chirp sequence in the plurality of interleaved chirp sequences, a first phase shift matrix θ 11 and a second set of phase shift matrices θ 1l , l=2, . . . , L, wherein the first phase shift matrix θ 11 represents a phase shift over a repetition interval between chirps in a same chirp sequence, and wherein the second set of matrices θ 1l , l=2, . . . , L represent a phase shift between chirps of a same sequence index in the plurality of interleaved chirp sequences;
perform joint diagonalization of the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L;
perform Doppler division multiplexing compensation on phase shifts in the first matrix θ 11 and the phase shifts collected in second set of matrices θ 1l , l=2, . . . , L; and
resolve ambiguities in determined velocities based on a comparison of the phase shifts from the first matrix θ 11 and the phase shifts from the second set of matrices θ 1l , l=2, . . . , L.
9 . The non-transitory computer-readable medium of claim 8 , wherein a Hankel matrix corresponding to an lth chirp sequence is represented as:
H
l
=
M
‵
θ
1
l
X
+
N
l
where M′ is an B×(PK) sub-matrix of a Vandermonde matrix M, B is a number greater than a maximum expected number P of objects in a range bin, K is the number of transmitters, θ 1l is a diagonal matrix that represents phase shifts introduced by velocities of the objects in the environment between the first chirp sequence and the lth chirp sequence, X is a matrix representing shifted combination of complex amplitudes of the received radar signal reflected off of objects in the environment with object velocity induced phase changes, and N l is a noise matrix for the lth chirp sequence.
10 . The non-transitory computer-readable medium of claim 8 , wherein a number of possible selection matrix pairs is represented as two times a number G, where G is represented as:
G
=
L
!
2
!
(
L
-
2
)
!
where “!” indicates the factorial of a number.
11 . The non-transitory computer-readable medium of claim 8 , wherein the plurality of interleaved chirp sequences includes a number L of interleaved chirp sequences, wherein the least-squares approximation for an lth chirp sequence is represented as:
Ψ
11
=
U
Φ
1
†
U
Φ
2
=
T
θ
11
T
-
1
Ψ
1
l
=
U
θ
1
†
U
θ
l
=
T
θ
1
l
T
-
1
,
l
=
2
,
…
,
L
where Ψ 11 represents a least-squares model for the first phase shift matrix θ 11 , Ψ 1l represent a least-squares models for the second set of matrices θ 1l , l=2, . . . , L, and T is a non-singular matrix that jointly diagonalizes Ψ 11 and Ψ 1l , wherein the matrix T has dimensions (PK)×(PK) where P is an expected maximum number of objects in a range bin and K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
12 . The non-transitory computer-readable medium of claim 8 , wherein the Doppler division multiplexing compensation for the first matrix θ 11 causes a phase shift arg(θ 11,p ) for a particular object p in a number P of objects in the environment over the repetition interval between chirps in the same chirp sequence to be represented as:
θ
11
,
p
=
θ
11
,
p
(
1
)
+
θ
11
,
p
(
2
)
e
-
j
2
π
K
+
…
+
θ
11
,
p
(
K
)
e
-
j
2
π
(
K
-
1
)
K
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
13 . The non-transitory computer-readable medium of claim 8 , wherein the Doppler division multiplexing compensation for the second set of matrices θ 1l , l=2, . . . , L causes a second phase shift arg(θ 1l,p ) for a particular object p in a number P of objects in the environment phase shift between chirps of the same index in the plurality of interleaved chirp sequences to be represented as:
θ
1
l
,
p
=
θ
1
l
,
p
=
θ
1
l
,
p
(
1
)
+
θ
1
l
,
p
(
2
)
+
…
+
θ
1
l
,
p
(
K
)
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
14 . The non-transitory computer-readable medium of claim 8 , further storing machine instructions which, when executed by the at least one processor, cause the at least one processor to:
obtain the received radar signal;
perform a first fast Fourier transform (FFT) on the received radar signal to obtain the range-sample-antenna data cube;
for each particular range bin in the range-sample-antenna data cube:
determine whether a characteristic of the range bin satisfies a criterion;
in response to the characteristic of the range bin satisfying the criterion, perform a second FFT on a sample-antenna array corresponding to the particular range bin; and
in response to the characteristic of the range bin not satisfying the criterion, include the sample-antenna array corresponding to the particular range bin in the subset of range bins.
15 . A method, comprising:
obtaining a range-sample-antenna data cube for a received radar signal comprising reflections of a plurality of interleaved chirp sequences off of objects in an environment;
generating, for each chirp sequence of the plurality of interleaved chirp sequences, a Hankel matrix based on a subset of range bins of the range-sample-antenna data cube corresponding to the particular chirp sequence;
generating a block Hankel matrix based on the Hankel matrices corresponding to the plurality of interleaved chirp sequences;
performing truncated singular value decomposition on the block Hankel matrix to estimate subspaces;
performing object detection to identify selection matrices in the estimated subspaces;
calculating least-squares approximations for the selection matrices and the block Hankel matrix to obtain, for each chirp sequence in the plurality of interleaved chirp sequences, a first phase shift matrix θ 11 and a second set of phase shift matrices θ 1l , l=2, . . . , L, wherein the first phase shift matrix θ 11 represents a phase shift over a repetition interval between chirps in a same chirp sequence, and wherein the second set of matrices θ 1l , l=2, . . . , L represent a phase shift between chirps of a same sequence index in the plurality of interleaved chirp sequences;
performing joint diagonalization of the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L;
performing Doppler division multiplexing compensation on the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L; and
resolving ambiguities in determined velocities based on a comparison of the first matrix θ 11 and the second set of matrices θ 1l , l=2, . . . , L.
16 . The method of claim 15 , wherein a Hankel matrix corresponding to an lth chirp sequence is represented as:
H
l
=
M
‵
θ
1
l
X
+
N
l
where M′ is an B×(PK) sub-matrix of a Vandermonde matrix M, B is a number greater than a maximum expected number P of objects in a range bin, K is the number of transmitters, θ 1l is a diagonal matrix that represents phase shifts introduced by velocities of the objects in the environment between the first chirp sequence and the lth chirp sequence, X is a matrix representing shifted combination of complex amplitudes of the received radar signal reflected off of objects in the environment with object velocity induced phase changes, and N l is a noise matrix for the lth chirp sequence.
17 . The method of claim 15 , wherein a number of possible selection matrix pairs is represented as two times a number G, where G is represented as:
G
=
L
!
2
!
(
L
-
2
)
!
where “!” indicates the factorial of a number.
18 . The method of claim 15 , wherein the plurality of interleaved chirp sequences includes a number L of interleaved chirp sequences, wherein the least-squares approximation for an lth chirp sequence is represented as:
Ψ
11
=
U
Φ
1
†
U
Φ
2
=
T
θ
11
T
-
1
Ψ
1
l
=
U
θ
1
†
U
θ
l
=
T
θ
1
l
T
-
1
,
l
=
2
,
…
,
L
where Ψ 11 represents a least-squares model for the first matrix θ 11 , Ψ 1l represents a least-squares model for the second set of matrices θ 1l , l=2, . . . , L, and T is a non-singular matrix that jointly diagonalizes Ψ 11 and Ψ 1l , wherein the matrix T has dimensions (PK)×(PK) where P is an expected maximum number of objects in a range bin and K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
19 . The method of claim 15 , wherein the Doppler division multiplexing compensation for the first matrix θ 11 causes a phase shift arg(θ 11,p ) for a particular object p in a number P of objects in the environment over the repetition interval between chirps in the same chirp sequence to be represented as:
θ
11
,
p
=
θ
11
,
p
(
1
)
+
θ
11
,
p
(
2
)
e
-
j
2
π
K
+
…
+
θ
11
,
p
(
K
)
e
-
j
2
π
(
K
-
1
)
K
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
20 . The method of claim 15 , wherein the Doppler division multiplexing compensation for the second set of matrices θ 1l , l=2, . . . , L causes a second phase shift arg(θ p ) for a particular object p in a number P of objects in the environment phase shift between chirps of the same index in the plurality of interleaved chirp sequences to be represented as:
θ
1
l
,
p
=
θ
1
l
,
p
(
1
)
+
θ
1
l
,
p
(
2
)
+
…
+
θ
1
l
,
p
(
K
)
,
where K is a number of transmitters that transmitted the plurality of interleaved chirp sequences.
21 . The method of claim 15 , further comprising:
obtaining the received radar signal;
performing a first fast Fourier transform (FFT) on the received radar signal to obtain the range-sample-antenna data cube;
for each particular range bin in the range-sample-antenna data cube:
determining whether a characteristic of the range bin satisfies a criterion;
in response to the characteristic of the range bin satisfying the criterion, performing a second FFT on a sample-antenna array corresponding to the particular range bin; and
in response to the characteristic of the range bin not satisfying the criterion, including the sample-antenna array corresponding to the particular range bin in the subset of range bins.