1 Cramer-Rao Lower Bounds for Estimation of Doppler Frequency in Emitter Location Systems J. Andrew Johnson and Mark L. Fowler Department of Electrical and Computer Engineering State University of New York at Binghamton Binghamton, NY Abstract: This paper derives Cramer-Rao bounds on estimates of the Doppler-shifted frequency of a coherent pulse-train intercepted at a single moving antenna. Such estimates are used to locate the emitter that transmitted the pulse train. Coherency from pulse to pulse allows much better frequency accuracy and is considered to be necessary to support accurate emitter location. Although algorithms for estimating the Doppler-shifted fre- quency of a coherent pulse train have been proposed, previously no results were available for the Cramer-Rao lower bound (CRLB) for frequency estimation from a coherent pulse train. This paper derives the CRLB for estimating the Doppler-shifted frequency of a coherent pulse train as well as for a non-coherent pulse train; a comparison of these two cases is made and the bound is compared to previously published accuracy results. It is shown that a general rule of thumb is that the frequency CRLB for coherent pulse trains depends inversely on pulse on-time, number of pulses, variance of pulse times, and the product of signal-to-noise-ratio and sampling frequency SNR×F s ; pulse shape and modulation have virtually no impact on the frequency accuracy. For the case that the K intercepted pulses are equally spaced by the pulse repetition interval (PRI), then the CRLB de- creases as 1/PRI 2 and as 1/K 3 . It is also shown that roughly the gain in coherent accuracy vs. the non-coherent case is K times the ratio of pulse on-time to PRI; since PRI is typically much larger than pulse on-time, the co- herent scenario allows much better frequency estimation accuracy. Index Terms: Emitter Location, Frequency Estimation, Doppler Shift, Cramer-Rao Bound Corresponding Author: Mark Fowler Department of Electrical and Computer Engineering State University of New York at Binghamton P. O. Box 6000 Binghamton, NY 13902-6000 Phone: 607-777-6973 Fax: 607-777-4464 E-mail: [email protected]
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Cramer-Rao Lower Bounds for Estimation of Doppler Frequency in Emitter Location Systems
J. Andrew Johnson and Mark L. Fowler
Department of Electrical and Computer Engineering State University of New York at Binghamton
Binghamton, NY Abstract: This paper derives Cramer-Rao bounds on estimates of the Doppler-shifted frequency of a coherent pulse-train intercepted at a single moving antenna. Such estimates are used to locate the emitter that transmitted the pulse train. Coherency from pulse to pulse allows much better frequency accuracy and is considered to be necessary to support accurate emitter location. Although algorithms for estimating the Doppler-shifted fre-quency of a coherent pulse train have been proposed, previously no results were available for the Cramer-Rao lower bound (CRLB) for frequency estimation from a coherent pulse train. This paper derives the CRLB for estimating the Doppler-shifted frequency of a coherent pulse train as well as for a non-coherent pulse train; a comparison of these two cases is made and the bound is compared to previously published accuracy results. It is shown that a general rule of thumb is that the frequency CRLB for coherent pulse trains depends inversely on pulse on-time, number of pulses, variance of pulse times, and the product of signal-to-noise-ratio and sampling frequency SNR×Fs; pulse shape and modulation have virtually no impact on the frequency accuracy. For the case that the K intercepted pulses are equally spaced by the pulse repetition interval (PRI), then the CRLB de-creases as 1/PRI2 and as 1/K3. It is also shown that roughly the gain in coherent accuracy vs. the non-coherent case is K times the ratio of pulse on-time to PRI; since PRI is typically much larger than pulse on-time, the co-herent scenario allows much better frequency estimation accuracy. Index Terms: Emitter Location, Frequency Estimation, Doppler Shift, Cramer-Rao Bound Corresponding Author: Mark Fowler Department of Electrical and Computer Engineering State University of New York at Binghamton P. O. Box 6000 Binghamton, NY 13902-6000 Phone: 607-777-6973 Fax: 607-777-4464 E-mail: [email protected]
mfowler
Text Box
Submitted to IEEE Transactions on Aerospace and Electronic Systems
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I. Introduction
Passive location of a stationary emitter with unknown frequency from a single moving platform is an im-
portant problem that has been investigated recently [1] – [10]. The importance of single-platform methods
stems from the desire to provide accurate location of emitters from a single aircraft for tactical and/or strategic
uses: single aircraft ability provides flexibility over multi-platform methods [9] and therefore there has been
much interest in the development and testing of single-platform methods [4], [5]. All the single-platform ap-
proaches that have been proposed consist of measuring, at instants within some interval of time, one or more
signal features that depend on the emitter’s location according to some signal model (i.e., measurement model),
and then processing them to estimate the emitter location. For example, the measured signal feature could be
the bearing to the emitter [3], [9], the Doppler-shifted frequency of the signal [2], [6], [8], the interferometer
phase between two spaced antennas on a single platform [2], the times-of-arrival of the pulses, etc., or combi-
nations of these [1], [7].
Regardless of the feature(s) exploited, the general approach is the same: first estimate the signal features
(e.g., frequency, angle, etc.) at a set of time instants and then use a signal model that describes the relationship
between the measured signal feature(s) and the emitter location, together with some statistical inference tech-
nique (e.g., least-squares, maximum likelihood, etc.) to estimate the emitter’s location. Thus, to evaluate the
location accuracy of an emitter location method it is first necessary to evaluate the accuracy of the feature(s)
estimation.
This paper focuses on the specific case of measuring the Doppler-shifted frequency of a coherent pulse-
train signal intercepted at a single moving antenna for use in emitter location. Algorithms for estimating the
Doppler-shifted frequency of a coherent pulse train have been proposed [6], [11], where coherency from pulse
to pulse allows much better frequency accuracy and is considered to be necessary to support the subsequent
emitter location processing. Despite the publication of these algorithms for coherent frequency estimation, no
results have been available for the Cramer-Rao lower bound (CRLB) for this case. The only accuracy results
3
previously available were: simulation results from Becker’s algorithm [11], experimentally obtained accuracy
levels reported by corporations developing actual systems [4], [5], and an approximate accuracy analysis [10].
This paper derives the CRLB for estimating the Doppler-shifted frequency of a coherent pulse train and com-
pares the bound to previously published accuracy results; for comparison, the CRLB is also derived for a non-
coherent pulse train and a simple formula is derived that characterizes the improvement achievable due to ex-
ploiting coherency.
II. CRLB for Frequency Estimation of Coherent Pulse Train
A. Signal Model
A complex-valued coherent pulse train from an emitter, consisting of pulses, can be modeled by K
, (1) 1
( )
0
( ) ( )K
j tk
k
s t e p t Tω φ−
+
=
= ∑ −
p( )twhere is a single pulse (possibly complex), ω is the carrier frequency,φ is a phase offset, and the are
called the transmitted pulse times. Let be the total pulse “on-time” and assume that p(t) is time-limited on
the interval [0 with . The signal is received at a moving platform, with initial range ,
and radial velocity . The (noise-free) received signal is model as
kT
OTT
, ]OTT 1OT k kT T T −< − 0d
0v
0( ) ( )dr t s tc
α= − , (2)
01 /v cα = −where is the signal dilation coefficient from the Doppler effect, and is the signal propagation
rate. Substituting
c
(1) into (2) yields that at the receiver the signal appears as
(3) 1
0
( ) ( ),K
j t jk
k
r t e e p t Tω ϕ−
=
= ∑ −
4
where αωω =~ , cd /~
0ωφφ −= , )()(~ tptp α= and ( )0 /k kT d c T / α= + ; the will be called the pulse
times. Furthermore, under the narrowband approximation
kT
[13] the time scale factor α has a negligible effect on
the pulse so that p(αt) ≈ p(t) so the model becomes
(4) 1
0
( ) ( ).K
j t jk
k
r t e e p t Tω ϕ−
=
= ∑ −
ΔThe received signal is sampled at the Nyquist interval and the samples are assumed to be perturbed by
complex zero-mean white Gaussian noise, , with variance 2[ ]w n σ , resulting in
(5) 1
0
[ ] [ ] [ ]
( ) [K
j n jk
k
x n r n w n
e e p n T w nω φ−
Δ
=
= +
= Δ − +∑ ].
From this received x[n] we wish to estimate the parameter vector
0 1 1KT T Tω φ − . (6) ⎡ ⎤= ⎣ ⎦θ
The key parameter to be estimated is the Doppler frequencyω , and the remaining unknowns are nuisance pa-
rameters that must be factored into the analysis (at least until shown irrelevant). It is recognized that the pulse,
( )p t , is generally not known, either; however, the CRLB will depend on the form of ( )p t so numerical results
for the CRLB will depend on the specific pulse shape assumed to have been intercepted.
B. Derivation of the General Case FIM and CRLB for Doppler Frequency
In preparation for computation of the CRLB [12], the derivatives of the signal with respect to each un-
known are found to be
][][~ nnrjnr Δ=∂∂ω (7)
][][~ njrnr =∂∂φ (8)
5
1
0
[ ] [ ] ( )
( ),
Kj n j
kkl
j n jl
r n e e k l p n TT
e e p n T
ω ϕ
ω ϕ
δ−
Δ
=
Δ
∂ ′= − − Δ −∂
′= − Δ −
∑ (9)
where
( )( )l
lt n T
dp tp n Tdt =Δ −
′ Δ − = . (10)
In order to simplify the expressions in the following derivation, define:
1 1 1
2 2 20 1 2
0 0 0
1 1 1( ) ( ) ( )N N N
n n n
S p n S n p n S n p nN N N
− − −
= = =
= Δ = Δ =∑ ∑ ∑ 2Δ (11)
1 1
* *0 1
0 0
1 1Im ( ) ( ) Im ( ) ( )N N
n n
C p n p n C n p n pN N
− −
= =
⎧ ⎫ ⎧′ ′= Δ Δ = Δ⎨ ⎬ ⎨⎩ ⎭ ⎩
∑ ∑ n ⎫Δ ⎬
⎭ (12)
1
2
00
1 ( )N
n
B p nS N
−
=
′= Δ∑ (13)
( )1 1 2
1 20 0
1 K K
kk k
1kR T R
K K
− −
= =
= =∑ T∑ , (14)
where S0 is a measure of the pulse power, S1 is a measure of the pulse temporal centroid, S2 is a measure of the
pulse temporal spread, C0 and C1 are time-frequency cross-coupling (or skew [13]) measures, B is a measure of
the pulse bandwidth, R1 is the average of the pulse times, and R2 is a measure of the temporal spread of the
pulse arrival times. We also define Ik to be the index set where the kth received pulse samples are non-zero.
The elements of the Fisher Information Matrix (FIM), , under the signal plus complex WGN assumption
are given by
J
6
( )
( )
*
2
1 22
20
1 1 2 22
0 0
22 1 1 2 02
2 Re [ ] [ ]
2 ( )
2 ( )
2 2 ,
k
n
K
kk n I
K N
kk n
J r n
n p n T
n T p n
N KS KR S KR S
ωω σ ω ω
σ
σ
σ
−
= ∈
− −
= =
r n⎧ ⎫∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞= ⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭
= Δ Δ −
= Δ + Δ
⎡ ⎤= Δ + Δ +⎣ ⎦
∑
∑ ∑
∑∑
(15)
where we have assumed that each pulse has N non-zero samples. Continuing in the same fashion gives
*
2
1 12
02 20 0
2 Re [ ] [ ]
2 2( ) ,
n
K N
k n
J r n
p n KNS
φφ σ φ φ
σ σ
− −
= =
r n⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪= ⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩
= Δ =
∑
∑∑
⎭ (16)
and
[ ]
*
2
1 2
20
1 1 02
2 Re [ ] [ ]
2 ( )
2 ,
k
n
K
kk n I
J r n
n p n T
KNS NKR S
ωφ σ ω φ
σ
σ
−
= ∈
r n⎧ ⎫⎛ ⎞∂ ∂⎪ ⎪⎛ ⎞= ⎨ ⎬⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭
= Δ Δ −
= Δ +
∑
∑ ∑ (17)
and
*
2
*2
1*
20
1 02
2 Re [ ] [ ]
2 Re ( ) ( )
2 Im ( ) ( ) ( )
2 ,
k
k
Tn k
k kn I
N
kn
k
J r n r nT
j n p n T p n T
n T p n p n
N C T C
ω σ ω
σ
σ
σ
∈
−
=
⎧ ⎫⎛ ⎞∂ ∂⎪ ⎪⎛ ⎞= ⎨ ⎬⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭
⎧ ⎫⎪ ⎪′= − Δ Δ − Δ −⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫′= Δ + Δ Δ⎨ ⎬⎩ ⎭
⎡ ⎤= Δ +⎣ ⎦
∑
∑
∑
(18)
and
7
*
2
*2
02
2 Re [ ] [ ]
2 Re ( ) (
2 ,
k
k
Tn k
kn I
J r n rT
jp n T p n T
N C
φ σ φ
σ
σ
∈
⎧ ⎫⎛ ⎞⎛ ⎞∂ ∂⎪ ⎪= ⎨ ⎬⎜ ⎟⎜ ⎟ ∂∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭
)k
n
⎧ ⎫⎪ ⎪′= − Δ − Δ −⎨ ⎬⎪ ⎪⎩ ⎭
=
∑
∑ (19)
and
*
2
02
2 Re [ ] [ ]
2 ,
k kT Tn k k
J r nT T
NS B
σ
σ
r n⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪= ⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩
=
∑⎭ (20)
and
*
2
2 Re [ ] [ ]
0,
kT T kn k
J r nT Tσ
∞
≠=−∞
r n⎧ ⎫⎛ ⎞⎛ ⎞∂ ∂⎪ ⎪= ⎨ ⎬⎜ ⎟⎜∂ ∂⎝⎝ ⎠
⎟⎠⎪ ⎪⎩
=
∑⎭
(21)
where the last result is zero because the pulses are non-overlapping. These matrix elements form the FIM given
by
22 1 1 2 0 1 1 0 1 0 0 1 1 0 1 1
1 1 0 0 0 0 0
1 0 0 0 0
2
1 1 0 0 0
1 1 0 0 0
2
0 02
0
0
0 0
K
K
KS KR S KR S KS KR S C T C C T C C T C
KS KR S KS C C C
C T C C S BN
C T C C S B
C T C C S B
σ
−
−
⎡ ⎤Δ + Δ + Δ + Δ + Δ + Δ +⎢ ⎥⎢ ⎥
Δ +⎢ ⎥⎢ ⎥⎢ ⎥Δ +⎢ ⎥= ⎢ ⎥
Δ +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
Δ +⎢ ⎥⎣ ⎦
J
0
. (22)
This structure shows that in general there is correlation between the estimates of most of the parameters, except
between the pulse times. The algorithm proposed in [6] relies on cross-correlation to estimate the pulse times;
8
thus, it is clear from the above FIM that the accuracy of those pulse time estimates will impact the overall accu-
racy of the Doppler estimate.
To proceed further, defining the partitions shown in (22) as
2
2T
Nσ
⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦
A CJ
C B, (23)
allows use of standard results for inverting symmetric partitioned matrices to yield the 2×2 CRLB matrix for
Doppler frequency and phase:
( )2 11
,
12
0
2
,2
T
T
CRLBN
N S B
ω φ
σ
σ
−−
−
= −
⎛ ⎞= −⎜ ⎟
⎝ ⎠
A CB C
CCA
(24)
where the fact that matrix B is a scalar (S0B) times the identity has been used. Evaluating the form for CCT and
substituting into (24) gives
122 1 1 2 0 1 1 02
,1 1 0 0
2
2
KS KR S KR S KS KR SCRLB
N KS KR S KSω φ
σ
−⎡ ⎤Δ + Δ + Δ +⎢ ⎥
= ⎢ ⎥Δ +⎢ ⎥
⎣ ⎦
)
, (25)
with the following definitions
. (26) 2 20 0 0 0 1 1 1 0 2 2 1 0( / ) ( / ) ( /oS S C S B S S C C S B S S C S B− − −
Finding the 1,1 element of (25) gives the CRLB for the Doppler frequency estimate. Using Cramer’s rule gives
( ) ( )
( )
( ) ( )
20
220 2 1 1 2 0 1 1 0
2
2 2 20 2 0 1 0 2 1
2 2 20 0 2
2 2
12 [( / ) ( / )]
1 ,2 1 ( / )
SCRLBNK S S R S R S S R S
NKS S S S S R R
NK SNR C S B D R
ωσ
σ
=Δ + Δ + − Δ +
=Δ − + −
=− Δ +
2 (27)
where 20 /SNR S σ and
9
( ) ( ) (12 22
2 0 1 0 2 2 1 10
1/ /K
kk
)D S S S S R R R T RK
−
=
− − = ∑ − . (28)
Note that D is a “mean-normalized” measure of the pulse duration and 2R is a “mean-normalized” version of
the time spread of the pulse times. These mean-normalized measures ensure that the choice of time origin has
no effect on the CRLB for the Doppler estimate.
C. Special Cases for the FIM and CRLB
To allow some further insight into this result some special cases are now considered. Note that the values
in the upper-right and lower-left corners of the FIM in (22) depend on the quantities C0 and C1, which are
shown in (12) to be the imaginary part of an expression. Thus, if the pulse p(t) is such that the quantity inside
the Im{} in (12) is purely real then many terms in (22) become zero and yields a block diagonal form given by
Table 3: Comparison to Simulation Results for PRI of approximately 1 ms
Fs(MHz)
K
SNR (dB) / 2CRLBω π
(Hz)
Ratiofσ in
2 /f CRLB [11] ωπσ(Hz)
@ 23
@ 37f
f
dB
dB
σ
σ
23 0.36 2.3 6.5 4 37 0.07 1.5 21.1
1.5
23 0.25 2.1 8.3 5 37 0.05 1.1 21.9
1.9 100
23 0.19 1.7 8.9 6 37 0.04 1.0 26.3
1.7
23 0.50 2.3 4.6 4 37 0.10 1.3 12.9
1.8
23 0.36 1.5 4.2 5 37 0.07 1.1 15.5
1.4 50
23 0.27 1.6 5.9 6 37 0.05 0.9 16.8
1.8
23 0.71 3.6 5.1 4 37 0.14 1.3 9.1
2.8
23 0.50 1.9 3.8 5 37 0.10 1.1 10.9
1.7 25
23 0.38 1.7 4.5 6 37 0.08 0.9 11.8
1.9
22
Appendix: Evaluation of C0
The CRLB for the most general coherent pulse train case was shown to depend on the value of C0 given in
(12), which is evaluated here using an integral approximation of the summation given by
*0
0
1 1 1Im ( ) ( ) Im ( ) ( )OTT
n
C p n p n p t pN N
* t dt⎧ ⎫⎧ ⎫ ⎪ ⎪′= Δ Δ ≈⎨ ⎬ ⎨ Δ⎩ ⎭
′ ⎬⎪ ⎪⎩ ⎭
∑ ∫ . (49)
As discussed in the main body of the paper, C0 is non-zero only when p(t) is complex, where it can be ex-
pressed in polar form or rectangular form, both of which are shown here as
[ ] [ ]( )( ) ( ) ( )cos ( ) ( )sin ( )
( ) ( )
j tp t A t e A t t j A t t
a t b t
φ φ= = + φ , (50)
where A(t) is the real-valued envelope function and φ(t) is the real-valued phase function. Then
[ ][ ]
[ ]
00
0
1 1Im ( ) ( ) ( ) ( )
1 1 ( ) ( ) ( ) ( ) .
OT
OT
T
T
C a t jb t a t jbN
a t b t a t b t dtN
⎧ ⎫⎪ ⎪′ ′≈ + −⎨ ⎬Δ⎪ ⎪⎩ ⎭
′ ′= −Δ
∫
∫
t dt
. (51)
But, using (50) gives
[ ] [ ]
[ ] [ ]
( ) ( )cos ( ) ( ) ( )sin ( )
( ) ( )sin ( ) ( ) ( )cos ( )
a t A t t A t t t
b t A t t A t t t
φ φ φ
φ φ φ
′ ′ ′= −
′ ′ ′= + (52)
and
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]{ }[ ] [ ] [ ]
2 2
2 2
( ) ( ) ( )sin ( ) ( )cos ( ) ( ) ( )sin ( )
( ) ( )sin ( ) cos ( ) ( ) ( )sin ( )
( ) ( ) ( )cos ( ) ( )sin ( ) ( ) ( )cos ( )
( ) ( )sin ( ) cos ( ) ( ) ( )cos ( )
a t b t A t t A t t A t t t
A t A t t t A t t t
a t b t A t t A t t A t t t
A t A t t t A t t t
φ φ φ
φ φ φ φ
φ φ φ
φ φ φ φ
′ ′ ′= ⎡ − φ
φ
⎤⎣ ⎦
′ ′= −
′ ′ ′= +
′ ′= +
(53)
which when substituted into (51) gives
23
2
00
1 1 ( ) ( )OTT
C A tN
φ′≈ −Δ ∫ t dt
t
. (54)
Recognizing that ( ) ( )itφ ω′ = , the instantaneous frequency, we can write
2
00
1 1 ( ) ( )OTT
iC A tN
ω≈ −Δ ∫ t dt . (55)
For any signal for whom the instantaneous frequency varies symmetrically around the carrier frequency (i.e.,
the frequency to be estimated) then (55) implies that C0 will be zero; this includes linear FM pulses. It is also
approximately true for pulses that are subjected to BPSK with a large enough number of phase transitions
within each pulse. Although it is difficult to make a wide-sweeping general conclusion here, it is also likely
that most if not all typical phase modulations will give a small value of C0 because the instantaneous frequency
typically varies uniformly (at least approximately) above and below zero. Also, it should be noted that the ef-
fect of any non-zero value of C0 will be deemphasized through the division in 2 20 01 ( / )C S B⎡ ⎤−⎣ ⎦ . Thus, we con-
jecture that the parameter C0 has little effect.
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