Signal Processing - March 2017 - 40

decoupled frequency estimation, where the best results are
achieved when it is selected according to the largest frequency separation.
As mentioned in the section "Practical Aspects," high-resolution processing in automotive radar is not only constrained
by limited computational resources but also by memory. When
applied in the original domain the complete data cube has to be
stored, whereas Fourier-domain processing enables a memoryefficient implementation by storing only a reduced number of
processing cells around power detections. Decoupled frequency estimation has been applied in [43] and [44] in the original domain. We extend this approach by applying decoupled
frequency estimation in the Fourier domain. Note that this is
particularly adapted to automotive radar use cases, in which
the number of well-separated targets is typically very large,
e.g., several hundreds. Fourier-domain processing allows a
subdivision into smaller frequency estimation problems with
significantly fewer targets. In most cases, only a single target is
presented in a Fourier-domain processing cell.
In the sequel, we consider a framework for decoupled highresolution frequency estimation in the Fourier domain, which
allows a flexible selection of the resolution dimension and distinguishes a single target from multiple target processing cells.
A special case for high-resolution frequency estimation in the
spatial domain is also presented, corresponding to the considered DOA estimation problem in [25]. This approach has to
be applied when the array elements are not uniformly spaced
and can be advantageous when the number of array elements is
very small so that element-space algorithms are computationally more efficient.

Decoupled models
In automotive radar, a memory-efficient realization of decoupled high-resolution frequency estimation is based on a Fourier-domain model. After the 3-D finite DTFT calculation in
(2), the model in (1) becomes
K

/ ak W

m

(m - m k)

k =1

# W n (n - n k) W o (o - o k) + N (m, n, o),

(3)

where W m (m), W n (n) , and W o (o) are the 1-D finite DTFT of
window functions w m (l s), w n (m s) , and w o (n s) , respectively,
and N (m, n, o) is the 3-D finite DTFT of p (l s, m s, n s) . Note
that N (m, n, o) is colored, circular complex Gaussian noise.
However, for the purpose of frequency estimation, it can be
assumed to be approximately white [42].
For decoupled high-resolution processing, the resolution
dimension can be either the range dimension, the velocity
dimension, or the angular dimension of the model in (3). A
vectorization in the resolution dimensions enables three local
vector models
z m (n, o) =

fk (m, o) w (n k) + noise

(5)

fk (m, n) w (o k) + noise.

(6)

fk (n, o) = a k W n (n - n k) W o (o - o k)

(7)

fk (m, o) = a k W m (m - m k) W o (o - o k)

(8)

fk (m, n) = a k W m (m - m k) W n (n - n k)

(9)

k=1

z o (m, n) =

Kr

/

k=1

Herein,

gather the model terms in the respective remaining dimensions, and
w ^m k h = ;W m c 2r l a - m k m, f, W m c 2r l b - m k mE
L
L

T

(10)

w (n k) = ;W n c 2r m a - n k m, f, W n c 2r m b - n k mE
M
M

(11)

w (o k) = ;W o c 2r n a - o k m, f, W o c 2r n b - o k mE
N
N

(12)

T

T

are the model vectors in the respective resolution dimension,
where l a, l b, m a, m b, n a, and n b are chosen such that indices
l = l a, f, l b, m = m a, f , m b, and n = n a, f, n b, respectively,
contain the support for K r local targets of interest. Typically,
in an unresolved situation, this corresponds to one or two
samples around the detected local maximum. Note that for
the local models, we consider a reduced number of frequencies K r % K in the vicinity of detections. In particular, for
most automotive radar scenarios, we use either K r = 1 or
K r = 2 [25].

Kr

/

fk (n, o) w (m k) + noise

(4)

A calculation similar to (2), but without the angular finite
DTFT is
Y (m, n; n s) =

L s -1 M s -1

/ /

ls = 0 ms = 0

w m (l s) w n (m s) x (l s, m s, n s) e -j (ml s +nms)
(13)

with m ! [0, 2r), n ! [0, 2r), and n s = 0, f, N s - 1. This
form is required when the high-resolution processing is
applied to the spatial domain and has been used in [25].
The signal model in (1) after 2-D finite DTFT calculation in
(13) can be obtained similarly to (3). The corresponding local
vector model is
y (m, n) =

Kr

/

fk (m, n) v (o k) + noise,

(14)

k=1
T
where fk (m, n) is given in (9), and v (o k) =61, e jo k, f, e j (N s -1) ok@
is a ULA steering vector. Note that the model in (14) is in the
original domain, in which the vector elements correspond to
spatial array elements, whereas the model in (6) is in the Fourier domain, in which the vector elements correspond to samples of the angular spectrum.

k=1

40

Kr

/

Spatial domain

Fourier domain

X (m, n, o) =

z n (m, o) =

IEEE SIgnal ProcESSIng MagazInE

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March 2017

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Table of Contents for the Digital Edition of Signal Processing - March 2017

Signal Processing - March 2017 - Cover1
Signal Processing - March 2017 - Cover2
Signal Processing - March 2017 - 1
Signal Processing - March 2017 - 2
Signal Processing - March 2017 - 3
Signal Processing - March 2017 - 4
Signal Processing - March 2017 - 5
Signal Processing - March 2017 - 6
Signal Processing - March 2017 - 7
Signal Processing - March 2017 - 8
Signal Processing - March 2017 - 9
Signal Processing - March 2017 - 10
Signal Processing - March 2017 - 11
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Signal Processing - March 2017 - 17
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Signal Processing - March 2017 - 21
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Signal Processing - March 2017 - 29
Signal Processing - March 2017 - 30
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Signal Processing - March 2017 - 33
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Signal Processing - March 2017 - 37
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Signal Processing - March 2017 - 40
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Signal Processing - March 2017 - 42
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Signal Processing - March 2017 - 101
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Signal Processing - March 2017 - 105
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Signal Processing - March 2017 - 116
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Signal Processing - March 2017 - 118
Signal Processing - March 2017 - 119
Signal Processing - March 2017 - 120
Signal Processing - March 2017 - 121
Signal Processing - March 2017 - 122
Signal Processing - March 2017 - 123
Signal Processing - March 2017 - 124
Signal Processing - March 2017 - Cover3
Signal Processing - March 2017 - Cover4
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