An electronic warfare perspective on time difference of arrival estimation subject to

An electronic warfare perspective on time difference of arrival estimation subject to
An electronic warfare
perspective on time difference
of arrival estimation subject to
radio receiver imperfections
J O H A N FA L K
Licentiate Thesis
Stockholm, Sweden 2004
An electronic warfare perspective on time
difference of arrival estimation subject to
radio receiver imperfections
Johan Falk
TRITA-S3-SB-0446
ISSN 1103-8039
ISRN KTH/SB/R - - 0446 - - SE
December 2004
SIGNAL PROCESSING LABORATORY
KTH SIGNALS, SENSORS AND SYSTEMS
ROYAL INSTITUTE OF TECHNOLOGY
STOCKHOLM, SWEDEN
Submitted to the School of Electrical Engineering, Royal Institute of
Technology, in partial fulfillment of the requirements for the degree of
Technical Licentiate.
c Johan Falk, 2004
Copyright °
An electronic warfare perspective on time difference of arrival
estimation subject to radio receiver imperfections
Signal Processing Laboratory
KTH Signals, Sensors and Systems
Royal Institute of Technology (KTH)
SE-100 44 Stockholm, Sweden
http://www.s3.kth.se
The work presented in this thesis is funded by
Swedish Defence Research Agency (FOI)
Department of Electronic Warfare Systems
Division of Command and Control Systems
SE-581 11 Linköping, Sweden
http://www.foi.se
Abstract
In order to ensure secure communication in digital military radio systems,
multiple methods are used to protect the transmission from being intercepted by enemy electronic warfare systems. An intercepted transmission
can be used to estimate several parameters of the transmitted signal such
as its origin (position or direction) and of course the transmitted message
itself. The methods used in traditional electronic warfare direction-finding
systems have in general poor performance against wideband low power
signals while the considered correlation-based time-difference of arrival
(TDOA) methods show promising results.
The output from a TDOA-based direction-finding system using two spatially separated receivers is the TDOA for the signal between the receiving
sensors which uniquely describes a hyperbolic curve and the emitter is located somewhere along this curve. In order to measure a TDOA between
two digital radio receivers both receiver systems must have the same time
and frequency references to avoid degradation due to reference imperfections. However, in some cases, the receivers are separated up to 1000 km
and can not share a common reference. This is solved by using a reference
module at each of the receiver sites and high accuracy is achieved using the
NAVSTAR-GPS system but, still, small differences between the outputs
of the different reference modules occurs which degrades the performance
of the system.
In a practical electronic warfare system there is a number of factors that
degrade the performance of the system, such as non-ideal antennas, analog
receiver filter differences, and the analog to digital converter errors. In
this thesis we concentrate on the problems which arises from imperfections
in the reference modules, such as time and frequency errors.
i
ii
The performance of both time- and frequency-domain based TDOA estimators are studied and compared to the Cramér-Rao lower bound. Also,
the effects from the reference frequency errors are studied in terms of
performance. Any difference between the time reference signals between
the two receiver systems produces a biased estimate. This bias is directly
linked to the time-difference, or error, between the time reference outputs
of the two reference modules. In order to digitize, or sample, the signal it
needs to be transposed, or mixed, to near baseband using a superheterodyne receiver controlled by the reference module. A difference in phase
between the mixer frequency reference outputs using two reference modules will not affect the performance of the TDOA estimation. However,
a receiver oscillator frequency error due to the frequency difference between the frequency reference outputs of two different reference modules
will result in a noise-like degradation of the TDOA estimation process.
A measure of this degradation is presented as the estimators robustness
against a frequency error
Acknowledgements
Inspiration and encouragement comes in many forms and over the past
years I’ve been inspired and encouraged by a lot of people around me. Especially my supervisors, Peter Händel and Magnus Jansson - supervisors
like you are an inspiration to any student. It has been a pleasure working
with you.
In pursuing my degree, I have travelled the equivalent of over 3 times
around the world or 1/3 of the distance to the moon. However, I feel that
the many hours spent on trains to visit the Signal Processing group at
KTH are well spent. Thanks to you all at KTH for inviting me into a
truly inspiring and relaxed environment.
I would also like to send a word of gratitude to the Swedish Defence
Research Agency (FOI) for funding my research. The possibility to see
your theoretical models perform in the real world is truly fantastic. Also,
thanks to my friends and colleagues at FOI — I’ve spent many weekends
and evenings alone in the corridors, missing your footsteps and encouraging words.
A special thanks to my greatest source of inspiration and encouragement
— Mom and Dad. Thank you for everything!
iii
iv
Contents
Abstract
i
Acknowledgements
iii
Contents
v
1 Introduction
1.1 Direction-finding and positioning . . . . .
1.2 A basic system model and two estimators
1.3 Contributions and outline . . . . . . . . .
1.4 Notation . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
7
8
11
14
17
2 System models
2.1 Analog to digital conversion . . . . . . . . . . . . . . .
2.2 Fractional delays of sampled data . . . . . . . . . . . .
2.3 Limited acquisition intervals . . . . . . . . . . . . . . .
2.4 Channel model . . . . . . . . . . . . . . . . . . . . . .
2.5 A baseband model . . . . . . . . . . . . . . . . . . . .
2.6 Reference imperfections . . . . . . . . . . . . . . . . .
2.6.1 Time reference error . . . . . . . . . . . . . . .
2.6.2 Receiver oscillator reference errors . . . . . . .
2.6.3 Sampling frequency reference error . . . . . . .
2.7 Receiver system models . . . . . . . . . . . . . . . . .
2.7.1 Ideal receiver system model . . . . . . . . . . .
2.7.2 Receiver model for multiple sources . . . . . .
2.7.3 Receiver model with time reference error . . . .
2.7.4 Receiver model with oscillator phase error . . .
2.7.5 Receiver model with oscillator frequency error .
2.8 Summary of models . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
19
20
21
22
22
25
26
27
27
27
31
32
34
34
35
37
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
vi
Contents
3 Time- and frequency-domain TDOA estimation
3.1 A lower bound on TDOA estimation accuracy . . . . .
3.2 Time-domain TDOA estimator . . . . . . . . . . . . .
3.3 Frequency-domain TDOA estimator . . . . . . . . . .
3.3.1 Practical aspects . . . . . . . . . . . . . . . . .
3.3.2 Performance analysis . . . . . . . . . . . . . . .
3.3.3 Variance reduction using block-averaging . . .
3.4 Effects of time reference errors . . . . . . . . . . . . .
3.5 Effects of receiver frequency errors . . . . . . . . . . .
3.5.1 Effects of frequency errors with block-averaging
3.6 Robustness against frequency errors . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
39
39
42
46
47
50
53
55
58
58
64
4 Conclusions and topics for future work
71
A Useful sums
A.1 Time averaging of frequency error . . . . . . . . . . . . .
A.2 Block-averaging of frequency error . . . . . . . . . . . . .
A.3 Sum of powers . . . . . . . . . . . . . . . . . . . . . . . .
73
73
77
78
B Complex-valued random variables
B.1 The expected value of the magnitude of a complex-valued
Gaussian variable . . . . . . . . . . . . . . . . . . . . . . .
79
Bibliography
83
81
Chapter 1
Introduction
In order to ensure secure communication in digital military radio systems,
multiple methods are used to protect the transmission from being intercepted by enemy electronic warfare systems. An intercepted transmission
can be used to estimate several parameters of the transmitted signal such
as origin (position or direction), modulation, coding, encryption and of
course the transmitted message itself. During World War II, the main
method to ensure unintercepted transmissions was to use burst transmissions, such as the 454 ms transmissions used by German Wolfpack submarine groups [1]. Electronic warfare systems with intercept and directionfinding capabilities of these signals followed shortly. Modern methods to
avoid interception include both frequency-hopping spread-spectrum and
direct-sequence spread-spectrum communication systems operating with
low radiated power and directional antennas. Emerging technologies also
include filtered spreading codes to further reduce the probability of interception [15]. A typical modern tactical military communication system
uses some 100.000 hops per second and more than 10 MHz of instantaneous bandwidth. Future communication systems are expected to operate
at higher hop-rates and larger bandwidths.
The methods used in traditional electronic warfare direction-finding
systems to intercept modern frequency-hopping systems have in general
poor performance against wideband low power signals [11] while correlationbased time-difference of arrival (TDOA) methods show promising results
[3]. The considered method is applicable to other wideband signals, such
as radar, sonar or acoustic signals, as well.
7
8
Chapter 1. Introduction
Figure 1.1: Two phase-measuring directon-finding systems produces two
directions of arrival (DOAs). The intersection between these DOAs represents the estimated position of the transmitter.
1.1
Direction-finding and positioning
Traditional direction-finding systems employing the triangulation principle use an antenna array to estimate the direction of arrival by measuring phase-differences between the individual antenna elements. Each
direction-finding system then produces a direction-of-arrival estimate for
the intercepted signal. An estimated emitter position can be calculated
from the intersection of the individual directions of arrival as illustrated
in Figure 1.1. In this thesis, two dimensional direction-finding and positioning is considered.
Existing phase-measuring direction-finding systems are often designed
to perform well against frequency-hopping systems using narrowband direction-finding methods such as MUSIC or Watson-Watt which gives poor
direction-finding performance when applied to wideband low power signals
[11]. To meet the need for direction-finding and positioning of wideband
low power signals, a correlation-based TDOA method using two or more,
spatially separated receivers, is considered. The output from a TDOAbased direction-finding system is the TDOA for the signal between a pair
of receiving sensors which uniquely describes a hyperbolic curve and the
1.1. Direction-finding and positioning
9
1500
Sensor 2
Sensor 3
1000
North-South position [m]
500
0
Sensor 1
-500
-1000
-1500
-2000
-2500
-1000
Emitter
-500
0
500
1000
1500
East-West position [m]
2000
2500
3000
Figure 1.2: Two TDOAs between three sensors gives three hyperbolic
curves which, here, uniquely intersects at the emitter position.
emitter is located somewhere along this curve. More than two receiving
sensors gives three, or more, hyperbolic curves which intersect at a position describing the emitter position as illustrated in Figure 1.2. However,
in a practical scenario both bias and variance of the estimated TDOAs
will affect the hyperbolic curves which will not intersect in a point but
rather describe an area from which the emitter position is calculated.
How to calculate the emitter position from this area is a problem with
several proposed solutions [7]. If the sensor positions are selected without
consideration to the emitter position the hyperbolic curves may intersect
at more than one position. The problem of selecting suitable sensor positions to avoid more than one intersection between the hyperbolic curves is
not considered here. In this thesis the focus is on estimating the TDOAs
which are used to calculate the hyperbolic curves.
10
Chapter 1. Introduction
1500
Sensor 2
1000
North-South position [m]
500
Asymptotic direction of arrival
0
Sensor 1
-500
-1000
-1500
-2000
-2500
-1000
Emitter
-500
0
500
1000
1500
East-West position [m]
2000
2500
3000
Figure 1.3: The TDOA between two sensors separated by a distance d
gives a hyperbolic curve. For large distances this hyperbolic curve can
be approximated by a straight line giving an approximate (asymptotic)
direction of arrival.
When the distance d between the sensors is short relative the distance
between the emitter and the sensors, an approximate direction of arrival
α can be calculated as
µ
¶
v∆t
α = sin−1
.
(1.1)
d
The propagation speed of the signal is denoted v and ∆t is the TDOA
between the receiving sensors. An illustration of this approximation is
given in Figure 1.3 where the fundamental ambiguity is omitted.
To summarize, the TDOA for a signal between two receiving sensors
gives a hyperbolic curve which describes the possible locations of the emitter. Three, or more, sensors will produce three, or more hyperbolic curves
and the position of the emitter is given by the intersection between these
curves.
1.2. A basic system model and two estimators
1.2
11
A basic system model and two
estimators
In the previous chapter it is shown that the TDOA of the signal of interest
between two sensors gives a hyperbolic curve and an approximate direction
to the emitter. In the following, a basic correlation-based model is derived
to show how the TDOA can be calculated using the outputs of two generic
sensors — r1 (t) and r2 (t). Throughout this thesis a complex-valued signal
model is used and here the signal of interest is modelled as a zero-mean
complex-valued wide sense stationary baseband process denoted s (t). The
signal is thus characterized by its auto-correlation function
φs (τ ) = E {s (t + τ ) s∗ (t)} .
(1.2)
Under the assumption that s (t) is wide sense stationary the autocorrelation function is independent of time t and a function of τ only.
Furthermore, it is assumed that s(t) is strictly bandlimited into the frequency range (−W, W ) Hz and that its power spectral density is continuous in frequency. That is, the transmitted signal is assumed to be
strictly band-limited, but broadband. In the basic model, the signal s (t)
is received at two spatially separated sensors with additive noise. The
time of arrival of the signal will differ between the two sensors resulting in
a TDOA denoted ∆t . The TDOA is independent of time for non-moving
emitter and receiving sensors. Now, the outputs of the two receiving
sensors with ideal non-dispersive complex-valued additive white Gaussian
noise (AWGN) are given by
r1 (t) = s (t) + z1 (t)
(1.3)
r2 (t) = s (t − ∆t ) + z2 (t) .
(1.4)
and
The noises z1 (t) and z2 (t) are assumed zero-mean, mutually uncorrelated
and uncorrelated to s (t). The cross-correlation function (CCF) between
the two received signals r1 (t) and r2 (t) is defined as
E {r1 (t + τ ) r2∗ (t)}
(1.5)
which is evaluated using (1.3)-(1.4)
E {r1 (t + τ ) r2∗ (t)} = E {s (t + τ ) s∗ (t − ∆t )} + E {s (t + τ ) z2∗ (t)}
+E {z1 (t + τ ) s∗ (t − ∆t )} + E {z1 (t + τ ) z2∗ (t)}
(1.6)
12
Chapter 1. Introduction
where the only non-zero term is the first signal-signal component since
the noises are assumed both mutually uncorrelated and uncorrelated to
s (t). The CCF is then a function of τ only since s (t) is assumed wide
sense stationary
φ (τ ) , E {r1 (t + τ ) r2∗ (t)}
= E {s (t + τ ) s∗ (t − ∆t )} = φs (τ + ∆t )
(1.7)
where the second equality follows from (1.6) and the third from (1.2).
Considering an electronic warfare scenario there are typically several
time and/or frequency overlapping signals in the received sequences. If
the signals only are time overlapping, simple frequency filtering is achieved
by employing a frequency-domain correlation-based method. Separation
of time and frequency overlapping signals can be done using spatial filtering [9]. Based on the idealized basic model described above, two ways of
estimating the TDOA are studied — time-domain and frequency-domain
based estimators. The time-domain TDOA estimator is based on the relation in (1.7), that is finding the argument that maximizes the amplitude
of the CCF
∆t = − arg max |φ (τ )| .
(1.8)
τ
The time-domain TDOA estimator is then given by the lag τ that maximizes the amplitude of the estimated CCF
¯
¯
ˆ t = − arg max ¯¯φ̂ (τ )¯¯ .
∆
(1.9)
τ
In Figure 1.4, the CCF between the received signals from two sensors is
calculated for a white Gaussian noise signal. The peak at −1 µs corresponds to a TDOA of ∆t = 1 µs.
The considered frequency-domain TDOA estimator is based on the
cross spectral density Φ (f ), and in particular the phase slope of the cross
spectral density which is used to calculate the TDOA. The cross spectral
density is defined as the Fourier transform of the CCF [12]
Φ (f ) , F {φ (τ )} = Φs (f ) ej2πf ∆t
(1.10)
where Φs (f ) = F {φs (τ )} is the power spectral density of s (t). In (1.10),
the second equality follows from (1.7) and the time-shift property of the
Fourier transform. The phase of the cross spectral density Γ (f ) , ∠Φ (f )
is using (1.10) given by
Γ (f ) = 2πf ∆t
(1.11)
1.2. A basic system model and two estimators
13
1
0.9
0.8
Correlation | φ (τ)|
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
-1
0
1
Time difference [µs]
2
3
4
5
Figure 1.4: The CCF is calculated for an example signal in noise. Here,
the true TDOA ∆t = 1 µs corresponds to the position of the peak at −1
µs in accordance with (1.9).
since the phase of the real-valued power spectral density is zero. The
phase of the cross spectral density in (1.11) is linear in f with a slope
determined by the TDOA ∆t as illustrated in Figure 1.5 for a typical
bandlimited signal. Within the central high SNR region the phase is
linear while the low SNR region is dominated by the noise resulting in a
random, non-linear phase.
The electronic warfare scenario assumes no a priori information of
the signal and an estimator needs to perform well under such circumstances. The considered frequency-domain based TDOA estimator is a
least-squares estimator which requires no prior knowledge of the signal or
the noise. The main idea is to fit a straight line to the phase curve of the
estimated cross spectral density [18],[26]. In Chapter 3 the estimators are
analyzed in detail.
14
Chapter 1. Introduction
10
5
|Φ(f)|
0
-5
-10
-15
-20
-25
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Frequency [MHz]
0.2
0.3
0.4
0.5
3
2
∠Φ(f)
1
0
-1
-2
-3
-0.5
Figure 1.5: Typical CSD of a wideband signal. The upper graph shows
the amplitude with the central high SNR region. In the bottom graph the
linear phase within the signal bandwidth is visible.
1.3
Contributions and outline
In this thesis the focus is on modelling a practical TDOA direction-finding
system and the performance degradation of a correlation-based TDOA
estimator using imperfect receiver systems. In the considered electronic
warfare scenario the SNR is assumed low which results in TDOA estimates
with high variance. In order to reduce this variance block-averaging is
used. That is, several spectral estimates calculated from independent
data records are averaged and a TDOA estimate with reduced variance
is obtained. In the following, the effects of the imperfect receiver systems
are analyzed in terms of bias and variance of the TDOA estimates.
In previous work, especially [14] and [18], the effects of a limited acquisition interval are analyzed briefly for a perfect receiver system. However,
the effects of imperfect receiver systems in combination with a limited
acquisition interval and block-averaging have not been published in the
literature. In this thesis, the main contributions are the models of the
1.3. Contributions and outline
15
imperfect receiver systems and how these imperfections affect the TDOA
estimation process. In particular the effects of block-averaging in combination with a receiver oscillator frequency error and a limited acquisition
interval are considered from a practical viewpoint.
The following Chapters are organized as
• Chapter 2 - System models,
describing the considered system models including the effects of limited acquisition intervals with receiver and reference imperfections.
In particular the case of receiver frequency error in combination with
a limited acquisition interval is considered.
• Chapter 3 - Time- and frequency-domain TDOA estimation,
the system models derived in Chapter 2 are used in estimating the
TDOA. The effects of different errors on the proposed TDOA estimators are studied and in particular the effects of timing errors and
frequency errors are studied. The estimator performance is compared to the Cramér-Rao lower bound.
• Chapter 4 - Conclusions and topics for future research,
a summary of the contributions and conclusions to be drawn from
the results. Also, some topics for further research are presented.
The work in this thesis is in part based on the following publications
• J. Falk, P. Händel, and M. Jansson, "Estimation of a receiver frequency error in a TDOA-based direction-finding systems," Proc.
Asilomar Conference on Signals, Systems, and Computers, Nov.,
Monterey, CA, USA, 2004.
• J. Falk, P. Händel, and M. Jansson, "Multisource time delay estimation subject to receiver frequency errors," Proc. Asilomar Conference on Signals, Systems, and Computers, Nov., Monterey, CA,
USA, vol. 1, pp. 1156 - 1160, 2003.
• U. Ahnström, J. Falk, P. Händel, and M. Wikström, "Detection
and direction-finding of spread spectrum signals using correlation
and narrowband interference rejection," Proc. Nordic Matlab Conference, Oct., Copenhagen, Denmark, 2003.
16
Chapter 1. Introduction
• M. Wikström, U. Ahnström, J. Falk, and P. Händel, "Implementation of an acoustic location-finding system using TDOA measurements," Proc. Nordic Matlab Conference, Oct., Copenhagen, Denmark, 2003.
• J. Falk, P. Händel, and M. Jansson, "Effects of frequency and phase
errors in electronic warfare TDOA direction-finding systems," Proc.
Military Communications Conference, Milcom 2003, Oct., Boston,
USA, 2003.
• J. Falk, P. Händel, and M. Jansson, "Direction finding for electronic
warfare systems using the phase of the cross spectral density," Proc.
Radiovetenskap och Kommunikation 2002, June, Stockholm, Sweden, 2002.
1.4. Notation
1.4
17
Notation
x (t) , y (t)
x [n] , y [n]
x∗
pN [n] , p0M [m]
fs
∆t
∆
γ
ε
µx
ϕ
θ̂
φx (τ ) , φx [m]
φ (τ ) , φ [m]
Φx (f ) , Φx [k]
Φ (f ) , Φ [k]
Γ (f ) , Γ [k]
T
N
L
B
M
2W
U
FM {x [n]}
E{x [n]}
∠x [n]
sinc(x)
Time continuous signal
Time discrete signal
Complex conjugate of x
Rectangular windows of length N and M , respectively
Sampling frequency (Hz)
TDOA in continuous time (seconds)
TDOA in discrete time ∆ = ∆t fs (samples)
Receiver oscillator phase difference (radians)
Receiver frequency difference relative fs
Normalized frequency for signal x, µx = fx /fs
Receiver oscillator phase (radians)
Estimate of the parameter θ
Auto-correlation function of the signal x
Cross-correlation function (CCF) between two signals
Power spectral density of the signal x
Cross spectral density between two signals
Phase of the cross spectral density
Acquisition interval in seconds
Acquisition interval in samples N = T fs
Length of block used in block-averaging (samples)
Number of blocks used for non-overlapping
block-averaging N = LB
Length of CCF, M = 2N − 1 (samples)
Bandwidth of the complex-valued signal (Hz)
Number of sources in a multisource scenario
The discrete Fourier transform (DFT) of x [n]
of length M for k = 1 − N, ..., N − 1
Expected value of the random variable x [n]
Phase of the complex-valued variable x [n]
sin (πx) /πx
18
Chapter 1. Introduction
Chapter 2
System models
In the considered electronic warfare scenario the signal of interest is unknown and received at low SNR. Most modern electronic warfare systems
consist of digital radio receivers with digital signal processing to analyze
the acquired signals [22]. To model a practical electronic warfare system with digital receivers, a simple but yet detailed model using superheterodyne receivers is derived. In order to keep the digital receivers near
time- and frequency-synchronized, reference modules are used at each of
the receiver sites. In the following it is shown how imperfections in, or
more accurately, differences between the reference modules will affect the
models of the considered correlation-based TDOA direction-finding system. In previous work baseband models are often considered for simplicity
[14],[18]. However, to properly model the receiver system imperfections
such as the receiver frequency error, the limited acquisition interval and
the receiver mixing in a practical system need to be considered.
2.1
Analog to digital conversion of the
received signals
A typical modern electronic warfare system uses digital receivers and the
signal processing is performed using sampled data. The received data
are digitized using the sampling frequency fs Hz and K bits. In the
considered electronic warfare scenario the signal of interest is assumed
to have low SNR, that is the channel noise power will be in the same
order as the signal power. In a practical receiver system the analog-todigital converter (ADC) input levels are adjusted so that the full scale
19
20
Chapter 2. System models
amplitude never is reached. This leads to a fewer number of effective bits
than the maximum K bits. However, in [6] it is shown that the effects
of a reduced number of effective bits can be suppressed by using a larger
acquisition interval. Thus, in this thesis ideal sampling is assumed, that
is no amplitude quantization effects are included in the analysis since
the effects of non-ideal sampling can be reduced by a larger acquisition
interval.
In modern digital receivers the analog filters have high out-of-band attenuation, that is the effects of aliasing are negliable and is not considered
in this thesis.
2.2
Fractional delays of sampled data
In this thesis, the notation s [n − ∆] and φs [m + ∆] are commonly used
to describe delayed versions of discrete time sequences. The TDOA ∆
between the two sensors is not an integer in a practical system and the
delayed sequences are thus formally undefined. However, the fractionally
delayed sequences are only used to analyze the received sequences and are
defined as follows. Considering a noise free scenario, the digitized versions
of the received signals in (1.3)-(1.4) are given by
and
r1 [n] = s (t) | t=nfs−1 , s [n]
(2.1)
r2 [n] = s (t − ∆t ) | t=nfs−1 , s [n − ∆]
(2.2)
with ∆ = ∆t fs . The CCF between (2.1)-(2.2) is then given by
φ [m] = E {r1 [n + m] r2∗ [n]}
= E {s [n + m] s∗ [n − ∆]}
which using (2.1)-(2.2) is evaluated to
¢ ¡
¢ª
© ¡
φ [m] = E s [n + m] fs−1 s∗ [n − ∆] fs−1
¡
¢
= φs [m + ∆] fs−1 .
By defining the delayed auto-correlation function of s [n] as
¡
¢
φs [m + ∆] fs−1 , φs [m + ∆] = E {s [n + m] s∗ [n − ∆]}
(2.3)
(2.4)
(2.5)
the time-discrete CCF in (2.4) can be written on the form
φ [m] = φs [m + ∆] .
(2.6)
2.3. Limited acquisition intervals
21
Now, both s [n − ∆] and φs [m + ∆] are defined for non-integer ∆s and
the analysis is valid for all bandlimited signals s [n]. Note that it is only
in the analysis of the considered methods that the fractionally delayed
signals are used, not in the method itself.
2.3
Limited acquisition intervals
In a practical data acquisition system the acquisition interval is limited to
T seconds. This is not always considered, in theoretical models used to analyze the behavior and performance of TDOA direction-finding systems.
However, in presence of receiver imperfections the limited acquisition interval needs to be considered. In this thesis, the analysis is made using
a data acquisition interval of T seconds or N = T fs samples. That is,
the considered stochastic processes are defined for an infinite time interval
but only T seconds, or N samples, are used in the analysis. The selection
of the N samples is made using a deterministic asymmetric rectangular
window pN [n] described by
½
1 − N2 + 1 · n · N2
(2.7)
pN [n] =
0 otherwise
where for simplicity, N is always assumed to be an even integer. Now,
for any signal x̃ [n], defined for all n, pN [n] is used to choose which N
samples to include in the analysis
½
x̃ [n] − N2 + 1 · n · N2
x [n] = x̃ [n] pN [n] =
.
(2.8)
0
otherwise
The asymmetric window in (2.7) is used to select which N samples of
the received sequences to be used in calculating the CCF. The resulting
CCF will consist of M · 2N − 1 non-zero samples which is selected by a
symmetric rectangular window denoted p0M [m] defined as
½
1 −N + 1 · m · N − 1
0
.
(2.9)
pM [m] =
0 otherwise
That is, the subindex denotes the length of the window and the apostrophe shows whether or not the window is symmetric. Note the difference
between pA [n] and p0A/2 [n], that is pA [A/2] 6= p0A/2 [A/2].
22
2.4
Chapter 2. System models
Channel model
In a practical multi-channel receiver system the individual noises are, by
a practical rule-of-thumb, assumed uncorrelated if the distance between
the sensors is larger than ten wavelengths. The considered electronic warfare scenario with receiving sensors separated up to some 1000 km fulfills
this for all considered wavelengths and thus the noises are assumed uncorrelated. For simplicity, the noises are also assumed to be equal power
and additive white Gaussian. In a practical electronic warfare scenario
the acquisition interval is short and thus the effects of fading are ignored.
Considering audio or radio channels the propagation speed is independent of the frequency, that is non-dispersive propagation is assumed. The
considered models can be extended to include the effects of multipath
propagation but analysis of such models are beyond the scope of this thesis. To summarize, the channels are assumed to be time-invariant channels
with uncorrelated additive white complex-valued Gaussian noises.
2.5
A baseband model
The baseband model is used when baseband signals are acquired, such
as low frequency radio, audio and sonar applications. In this model the
outputs from two ideal digital receiving sensors are described by
r̃1 [n] = s [n] + z1 [n]
(2.10)
r̃2 [n] = s [n − ∆] + z2 [n]
(2.11)
and
where ∆ = ∆t fs samples defines the TDOA for the signal of interest between the receiving sensors. The fractional delay of the signal in (2.11) is
only used for analysis of the received sequences as discussed in Chapter
2.2. The signal s [n] is assumed to be a wideband wide sense stationary
signal which is uncorrelated to the mutually uncorrelated complex-valued
noise sequences z1 [n] and z2 [n]. Also, the noises are assumed to have
equal power σ 2z . To model the limited acquisition interval the sensor outputs are windowed using the rectangular window pN [n] in (2.7) resulting
in sequences that are non-zero for N samples each. The windowed received
sequences are then given by
r1 [n] = (s [n] + z1 [n]) pN [n]
(2.12)
r2 [n] = (s [n − ∆] + z2 [n]) pN [n] .
(2.13)
and
2.5. A baseband model
23
Note that in (2.12)-(2.13) both the signal, s [n], and the noises, z1 [n] and
z1 [n], are wide sense stationary (WSS) random processes. However, the
received sequences, r1 [n] and r2 [n], are not wide sense stationary resulting
in a cross-correlation function (CCF) that depends not only on the lag m
but also on the time n. Now, the CCF between the two receiver outputs
(2.12)-(2.13) is given by
E {r1 [n + m] r2∗ [n]} = E {s [n + m] s∗ [n − ∆]} pN [n + m] pN [n]
= φs [m + ∆] pN [n + m] pN [n]
(2.14)
where the first equality follows from the assumption of mutually uncorrelated noises and that the signal is uncorrelated to the noises as described
in Chapter 2.4. The second equality in (2.14) follows from the definition of the auto-correlation function. The function in (2.14) is dependent
on time n due to the rectangular windows pN [n + m] pN [n] from the limited acquisition interval. To eliminate the dependence of time, the CCF is
time-averaged following the strategy in [19]. That is, the time dependence
is eliminated by averaging the CCF in (2.14) over time
φ [m] =
∞
1 X
E {r1 [n + m] r2∗ [n]}
N n=−∞
= φs [m + ∆]
∞
1 X
pN [n] pN [n + m]
N n=−∞
(2.15)
for all m < ∞. The CCF in (2.15) is only non-zero for combinations of
n, m such that the windowing functions within the summation is non-zero.
Let
∞
1 X
P [m] =
pN [n] pN [n + m]
(2.16)
N n=−∞
then, for M = 2N − 1 and p0M [m] as defined in (2.9)
P [m] =
1
N
N/2
X
pN [n + m] =
n=−N/2+1
N − |m| 0
pM [m] .
N
(2.17)
The time-averaged CCF between two time discrete wide sense stationary
sequences, received using ideal baseband receivers with a limited acquisition interval, is then given by inserting (2.17) into (2.15)
φ [m] =
N − |m|
φs [m + ∆] p0M [m] .
N
(2.18)
24
Chapter 2. System models
The limited acquisition interval imposes a constraint on the TDOA range
of ∆ that is covered by a non-zero φ [m], which follows from the limited
length of the received sequences r1 [n] and r2 [n]. To obtain a non-zero
CCF, the received sequences must contain a common part of the signal
which is true only if |∆| · N − 1. Note that the amplitude of the CCF
in (2.18) is reduced due to the factor (N − |m|) /N which for large |m|
is close to zero. When estimating the CCF, |m| needs to be significantly
smaller than N for the CCF in (2.18) to give reliable results.
To obtain the frequency-domain TDOA model, the cross spectral density is calculated using the discrete Fourier transform (DFT) of the CCF.
The DFT of length M = 2N −1 samples is denoted FM {·} and is given by
the Fourier transform evaluated on a frequency grid ω k = 2πk/M for the
discrete frequency bins k = 1 − N, ..., N − 1 [17]. The frequency discrete
cross spectral density Φ [k] of the time-discrete CCF is then given by
Φ [k] = FM {φ [m]} =
N
−1
X
φ [m] e−j2πkm/M
k = 1 − N, ..., N − 1.
m=1−N
(2.19)
The time and frequency discrete model is now obtained by inserting (2.18)
into (2.19)
Φ [k] =
N−1
X
m=1−N
N − |m|
φs [m + ∆] e−j2πkm/M
N
k = 1 − N, ..., N − 1.
(2.20)
The cross spectral density in (2.20) is not easily evaluated for a general
source signal with an auto-correlation function given by φs [m]. To get
some insight into the behavior of the cross spectral density, an example
signal is considered.
White signal of interest This example signal is a complex-valued zeromean white Gaussian noise sequence with
φs [m] = σ 2s δ [m]
where δ [m] is the discrete Kronecker delta defined as
½
1 m=0
δ [m] =
0 m 6= 0
(2.21)
(2.22)
A military tactical communication system which uses direct-sequence
spread-spectrum with long and orthogonal codes have similar characteristics as this example signal.
2.6. Reference imperfections
25
Now, inserting (2.21) into (2.20) with a TDOA of ∆ samples gives
Φ [k] =
N − |∆| 2 j2πk∆/M
σs e
N
k = 1 − N, ..., N − 1.
(2.23)
The phase of Φ [k] is linear in frequency
Γ [k] = ∠Φ [k] =
2πk∆
M
k = 1 − N, ..., N − 1
(2.24)
and the TDOA ∆ can be estimated as previously described by fitting a
straight line to the phase curve. In order to analyze and compare different
models and estimators in the following, the example white noise signal is
used. From a strict mathematical point, ∆ must be an integer to give the
result in (2.23). However, as discussed in Chapter 2.2, the non-integer ∆
is not a problem in this analysis since the considered signals are digitized
versions of continuous signals where no integer constraints are made on
the TDOA.
2.6
Reference imperfections
In order to measure a TDOA between two digital radio receivers both
receiver systems must have the same time and frequency references to
avoid degradation due to reference imperfections. However, the receivers
are separated with several kilometers and can not share a common reference. In practice, this is solved by using a reference module at each of
the receiver sites. This reference module, as illustrated in Figure 2.1, is
assumed to be controlled by an external reference source. For simplicity, assume that this external source is the NAVSTAR-GPS system. The
outputs of the considered reference modules, described in Table 2.1, are
the time reference signal which gives the absolute starting time of the
acquisition, the receiver oscillator reference (frequency and phase) used in
the superheterodyne receiver to mix the signal to near baseband, and the
sampling frequency.
Output signal
Time reference
Frequency reference
Sampling frequency
ttrig
fref , ϕref
fs
Signal type
rectangular pulse
10 MHz sine wave
10 MHz sine wave
Table 2.1: The outputs from the reference module.
26
Chapter 2. System models
Figure 2.1: A typical reference module with outputs used in the spatially
separated digital receivers to obtain a near time and frequency synchronous TDOA direction-finding system.
High accuracy is achieved using the NAVSTAR-GPS system but, still,
small differences between the outputs of the different reference modules
occurs which degrades the performance of the system as will be described
in Sections 3.4 and 3.5, respectively. From a typical high-end NAVSTARGPS receiver [8] the timing error is < 50 ns and the frequency error is
< 10−10 relative the 10 MHz output signal.
If the GPS receiver fails, the system performance will quickly degrade
since both time and frequency reference signals diverges from the external reference. Of course, another way of keeping the time and frequency
references accurate is to use an atomic clock which does not rely on an external source. The main disadvantage of the needed high-end atomic clock
is its price which is approximately 500 times higher than that for a GPS
receiver. Below the errors between the reference modules are discussed in
some detail.
2.6.1
Time reference error
In a TDOA-based electronic warfare direction-finding system the TDOA
between the outputs of two spatially separated receivers give a direction of
arrival for the received signal as shown in previous chapters. A difference
between the time reference signals between the two receiver systems produces a biased estimate. This bias is directly linked to the time-difference,
or error, between the time reference outputs of the two reference modules.
If the timing error is larger than the acquisition interval — the CCF is zero
since no common part of the signal exist in the two received sequences.
The effects on the system models suffering from a timing error are modelled in Chapter 2.7.3.
2.7. Receiver system models
2.6.2
27
Receiver oscillator reference errors
In order to digitize, or sample, the signal it needs to be transposed, or
mixed, to near baseband using a superheterodyne receiver where the signal
is mixed with a high frequency carrier. This reference frequency signal is
controlled by the reference module and differences, or errors, between the
outputs of the reference modules will affect the TDOA estimation process.
The oscillator phase error is modelled in Chapter 2.7.4 and the frequency
error in Chapter 2.7.5.
2.6.3
Sampling frequency reference error
An error between the sampling frequencies in the different receiver systems severely degrades the performance due to decorrelation. However,
this error is small compared to the receiver oscillator frequency which is
created from the reference frequency output. As an example, the error
from the reference module is practically the same for both the sampling
frequency output and the frequency reference output. However, the sampling frequency is in most cases small compared to the mixing frequency
which is in parity with the carrier frequency of the signal. Consequently
the sampling frequency error is smaller than the mixer frequency error.
Under this assumption, the sampling frequency error is not considered in
this thesis.
2.7
Receiver system models
In a practical electronic warfare system, a number of factors degrades
the performance of the system. Some of these factors are wave distortion
due to non-ideal antenna placement, analog receiver filter differences resulting in a dispersive receiver system, and the analog to digital converter
errors including time and amplitude jitter. However, in this thesis we concentrate on the problems which arises from imperfections in the receiver
systems, such as time and frequency reference errors. In the following,
models are derived considering the reference imperfections followed by an
analysis of its effects on the TDOA estimation process.
In modern electronic warfare direction-finding systems digital wideband receivers are used to acquire the signal as illustrated in Figure 2.2. In
this thesis positioning in general and direction-finding in particular is considered and by using two receivers direction-finding is possible, while three
or more receivers are needed to calculate a position of the signal. Each
of the digital receivers are assumed to consist of an ideal IQ-modulator
28
Chapter 2. System models
Figure 2.2: The R sensors in a typical TDOA electronic warfare system
consists of digital receivers which are spatially separated and controlled
by the outputs of separate reference modules.
followed by an analog superheterodyne receiver and an ideal analog to
digital converter as seen in Figure 2.3. The analog superheterodyne receivers are modelled by an input bandpass filter of bandwidth 2W Hz followed by the mixer. The mixer frequency is generated using the reference
frequency signal which is one of the outputs from the reference module
described in Chapter 2.6. Differences between the reference frequency signals used in the different receivers results in a performance degradation
which is also described in Sections 2.7.5 and 3.5. Low-pass filtering with
bandwidth ±W Hz of the mixer output gives the complex-valued near
baseband component as illustrated in Figure 2.4.
In this thesis, the analysis is concentrated on one pair of receivers.
When using more than two receivers the received data sequences are still
analyzed in pairs. The signal of interest s (t), which is transmitted at
carrier frequency f0 Hz and phase ϕ0 , is received at the input of the two
receivers. After IQ-modulation the received signals
r1 (t) = s (t) ej2πf0 t+jϕ0 + z1 (t)
(2.25)
2.7. Receiver system models
29
Figure 2.3: The digital receivers are modelled by an ideal IQ-modulator
followed by the analog superheterodyne receiver and the analog to digital
converter. The digital output describes a complex-valued near baseband
signal.
Figure 2.4: The analog superheterodyne receiver is modelled as a filtermixer-filter receiver where the mixer frequency is created using the reference frequency from the reference module.
30
Chapter 2. System models
and
r2 (t) = s (t − ∆t ) ej2πf0 (t−∆t )+jϕ0 + z2 (t)
(2.26)
where z1 (t) and z2 (t) denotes the additive noises, are fed to the superheterodyne receiver. The goal is to mix these signals to near baseband
which is achieved by setting the receiver mixing frequencies f1 and f2 with
phases ϕ1 and ϕ2 , respectively, to near f0 . This results in
©
ª
r1 (t) =
s (t) ej2πf0 t+jϕ0 + z1 (t) e−j2πf1 t−jϕ1
= s (t) ej2π(f0 −f1 )t+j(ϕ0 −ϕ1 ) + z1 (t) e−j2πf1 t−jϕ1
(2.27)
and
n
o
r2 (t) = s (t − ∆t ) ej2πf0 (t−∆t )+jϕ0 + z2 (t) e−j2πf2 t−jϕ2
= s (t − ∆t ) e−j2πf0 ∆t +j2π(f0 −f2 )t+j(ϕ0 −ϕ2 ) + z2 (t) e−j2πf2 t−jϕ2 .
(2.28)
Following the mixer is the analog-to-digital converter where the
received signals are digitized. In the following the discrete frequencies
are given relative the sampling frequency fs . That is, the transmitter
frequency is now given by µ0 = f0 fs−1 and the receiver mixing frequencies
are given by µ1 = f1 fs−1 and µ2 = f2 fs−1 , respectively. The outputs of
the digital receivers in Figure 2.3 are then for input signals (2.27)-(2.28)
given by
©
ª
r̃1 [n] =
s [n] ej2πµ0 n+jϕ0 e−j2πµ1 n−jϕ1 + z1 [n]
= s [n] ej2π(µ0 −µ1 )n+j(ϕ0 −ϕ1 ) + z1 [n]
(2.29)
and
r̃2 [n] =
n
o
s [n − ∆] ej2πµ0 (n−∆)+jϕ0 e−j2πµ2 n−jϕ2 + z2 [n]
= s [n − ∆] e−j2πµ0 ∆+j2π(µ0 −µ2 )n+j(ϕ0 −ϕ2 ) + z2 [n] (2.30)
where the digitized representation of the noises include the complex rotation due to the mixing. The complex rotation will not affect the statistical
properties of the noises and is ignored in the following. In a practical receiver system only a limited number of samples are used from the received
sequences. That is, the rectangular window pN [n] described in Chapter
2.3 is applied to (2.29) and (2.30) to limit the analysis to N samples.
2.7. Receiver system models
2.7.1
31
Ideal receiver system model
In some scenarios the receivers are located close to each other and a common reference module can be used. In such a case, the receivers can be
connected by a cable and perfect synchronization is achieved. That is,
there are no differences between the reference signals leading to µ1 = µ2
and ϕ1 = ϕ2 . Moreover, in the ideal receiver system the transmitter frequency and phase are assumed known and the receiver mixing frequency
is set to the transmitter frequency µ0 = µ1 and is phase-locked to the
transmitter ϕ0 = ϕ1 . Now, (2.29)-(2.30) are evaluated to
r1 [n] = (s [n] + z1 [n]) pN [n]
and
¡
¢
r2 [n] = s [n − ∆] e−j2πµ0 ∆ + z2 [n] pN [n]
(2.31)
(2.32)
which is recognized as the same model as in (2.12)-(2.13) but with a
complex rotation e−j2πµ0 ∆ due to the receiver mixing residual in receiver
channel 2. The CCF between the received sequences (2.31)-(2.32) then
follows from calculations similar to (2.14)-(2.18).
φ [m] =
N − |m|
φs [m + ∆] ej2πµ0 ∆ p0M [m] .
N
(2.33)
The cross spectral density is given by the time discrete Fourier transform
of (2.33). However, the cross spectral density is not easily evaluated for
a general source signal with an auto-correlation function given by φs [m].
Evaluating the cross spectral density for the white example signal (2.21),
with similarities to wideband military communication sources, gives
Φ [k] =
N − |∆| 2 j2π( Mk +µ0 )∆
σs e
N
k = 1 − N, ..., N − 1.
(2.34)
The corresponding phase of the cross spectral density Γ [k] = ∠Φ [k] is
then given by
¶
µ
k
k = 1 − N, ..., N − 1.
(2.35)
+ µ0 ∆
Γ [k] = 2π
M
The mixing residual 2π∆µ0 will only affect the bias of the phase curve
but does not affect the performance of the TDOA estimation process since
the bias and slope estimation decouples as shown in Chapter 3.3.
32
2.7.2
Chapter 2. System models
Receiver model for multiple sources
In an electronic warfare scenario, the intercept receivers collect all signals within their bandwidth. Clearly, multisource as well as multipath
propagation are common in many practical scenarios. In this chapter,
a model for multiple sources is introduced. The number of uncorrelated
signals is denoted U . As in the previous chapter, deriving a model for the
time-averaged CCF is the goal.
Only two receivers are used leading to only one mixer in each receiver
for all the signals and the mixer oscillator frequencies in the two receivers
are again given by µ1 and µ2 , respectively. In the same way, the mixer
oscillator phases are given by ϕ1 and ϕ2 , respectively. The carrier frequency and phase of source u are denoted µu and ϕu , respectively. Now,
the received sequences containing U uncorrelated sources are given by
" U
#
X £
¤
1
r [n] =
s n − ∆1 ej2πµu (n−∆u )+jϕu −j2πµ1 n−jϕ1 + z [n] p [n]
1
u
u
1
N
u=1
(2.36)
and
r2 [n] =
"
U
X
#
£
¤ j2πµ (n−∆2 )+jϕ −j2πµ n−jϕ
2
u
u
2
2 + z [n] p
u
su n − ∆u e
2
N [n]
u=1
(2.37)
where ∆iu denotes the time-delay, not difference, from source u to receiver i. In the ideal receiver case the receiver oscillators are assumed
phase and frequency synchronized, that is µ1 = µ2 and ϕ1 = ϕ2 . The
sources su [n] are assumed zero-mean and mutually uncorrelated, that is
E{su [n] s∗v [n]} = 0 for u 6= v. The multisource CCF is now given by
E{r1 [n + m]r2∗ [n]} =
U
X
φsu[m + ∆u ]ej2πµu(m+∆u )−j2πµ1m pN [n + m]pN [n]
u=1
(2.38)
where ∆u = ∆2u − ∆1u describes the TDOA for source su [n] between the
two receivers. Without loss of generality, the transmitter frequencies µu
are assumed to be equal and known, that is µu = µ0 . If there are different transmitter frequencies this can be modelled in the auto-correlation
functions φsu [m] of the individual sources. Under the above assumption
2.7. Receiver system models
33
the CCF (2.38) becomes
E {r1 [n + m] r2∗ [n]} =
U
X
φsu [m + ∆u ] ej2πµ0 ∆u pN [n + m] pN [n]
u=1
(2.39)
which is equal to a sum of single source models (2.14) with a specific
complex rotation for each of the uncorrelated sources. Applying timeaveraging to remove the dependence of n gives the time-averaged multisource CCF
φ [m] =
U
N − |m| X
φsu [m + ∆u ] ej2πµ0 ∆u p0M [m] .
N
u=1
(2.40)
Note that for U = 1 the expression in (2.40) is reduced to the single
source result in (2.33). The expression in (2.40) is valid for U uncorrelated
sources which is a sum of single source CCFs. The practical implication
of the result in (2.40) is that the multisource CCF has U peaks describing
the TDOAs for the individual sources. However, when the TDOA of the
different sources are about the same, the peaks will overlap in the resulting
multisource CCF. In this case the multiple sources can not be separated
in the spatial domain. However, both time and frequency-domain can
be used in combination with the spatial domain to separate the different
sources. That is, the source can be separated if they are non-overlapping
in either time, frequency or spatial domain.
Calculating the cross spectral density without any spatial filtering of
the multisource CCF in (2.40) gives a sum of the individual cross spectrums. Considering U white noise sources (2.21) with different TDOAs
yields
Φ [k] =
U
X
N − |∆u | 2 j2π(k/M+µ0 )∆u
σ su e
N
u=1
k = 1 − N, ..., N − 1. (2.41)
The phase of (2.41) is given by Γ [k] = ∠Φ [k] and shows that the phase is
given by an amplitude weighted sum of all the sources. However, spatial
filtering (time-domain filtering in [9]) can be used to separate the multiple
TDOAs.
34
Chapter 2. System models
2.7.3
Receiver model with time reference error
A difference, or error, between the time reference outputs of the reference
modules results in a biased estimate of the TDOA. The received sequences
in (2.29)-(2.30) are, for a timing error ∆ref , given by
³
´
r1 [n] = s [n] ej2π(µ0 −µ1 )n+j(ϕ0 −ϕ1 ) + z1 [n] pN [n]
(2.42)
and
³
´
r2 [n] = s[n−∆−∆ref ]e−j2πµ0 (∆+∆r e f)+j2π(µ0 −µ2 )n+j(ϕ0 −ϕ2 )+z2 [n] pN[n] .
(2.43)
For simplicity, let µ0 = µ1 = µ2 and ϕ0 = ϕ1 = ϕ2 , then the received
sequences are
r1 [n] = (s [n] + z1 [n]) pN [n]
(2.44)
and
³
´
r2 [n] = s [n − ∆ − ∆ref ] e−j2πµ0 (∆+∆r e f ) + z2 [n] pN [n] .
(2.45)
N − |m|
φs [m + ∆ + ∆ref ] ej2πµ0 (∆+∆r e f ) p0M [m] .
N
(2.46)
In this case, the time-averaged CCF follows directly from the calculations
presented in Chapter 2.7.1 with ∆ replaced by ∆ + ∆ref . The result reads
φ [m] =
Accordingly, the peak of the averaged CCF is translated from the actual
TDOA. Note that the timing error results in an extra amplitude reduction
as seen in (2.34) for ∆ → ∆ + ∆ref .
2.7.4
Receiver model with oscillator phase error
The difference in phase between the mixer frequency reference outputs
using two reference modules affects the model but not the performance
in terms of TDOA estimation as seen in the following. The reference
oscillator phase difference denoted γ is defined as the difference between
the phases of the individual oscillators, γ = ϕ2 − ϕ1 and the received
sequences suffering from an oscillator phase error are given by
³
´
(2.47)
r1 [n] = s [n] ej2π(µ0 −µ1 )n+j(ϕ0 −ϕ1 ) + z1 [n] pN [n]
and
³
´
r2 [n] = s [n − ∆] e−j2πµ0 ∆+j2π(µ0 −µ2 )n+j(ϕ0 −ϕ2 ) + z2 [n] pN [n] .
(2.48)
2.7. Receiver system models
35
To simplify the analysis, let µ0 = µ1 = µ2 . Once again, similar calculations as (2.14)-(2.18) results in
φ [m] =
N − |m|
φs [m + ∆] ej2πµ0 ∆+jγ p0M [m] .
N
(2.49)
The complex rotation caused by the receiver oscillator phase difference γ
does not affect the magnitude of φ [m]. Thus, it will not affect the TDOA
estimators based on the magnitude of (2.49).
In frequency domain, this is illustrated by the white example signal in
(2.21). The spectral properties are now
Φ [k] =
N − |∆| 2 j2π( Mk +µ0 )∆+jγ
σs e
N
and
Γ [k] = ∠Φ [k] = 2π
µ
¶
k
+ µ0 ∆ + γ
M
k = 1 − N, ..., N − 1
(2.50)
k = 1 − N, ..., N − 1. (2.51)
The phase error γ results in a bias of the phase curve but will not affect
the slope of the phase curve. Thus, the receiver oscillator phases do not
affect the result in terms of TDOA estimation. A result which follows
from this, is that the absolute phases of the transmitter or the receivers
will not affect the performance of the considered TDOA estimators.
2.7.5
Receiver model with oscillator frequency error
The receiver oscillator frequency error is due to the frequency difference
between the frequency reference outputs of two different reference modules. Accordingly, the receiver frequency imperfections result in a frequency shift which must not be confused with doppler (time scaling)
introduced by moving transmitters or receivers [5]. For simplicity, assume the transmitter frequency and phase to be known, µ0 = µ1 and
ϕ0 = ϕ1 = ϕ2 , respectively. A frequency difference, or error, between the
two receiver oscillators is defined as ε = µ2 − µ1 which here is assumed
small, that is ε ¿ 1/N . The effects of large ε are considered in Chapter
3.6. Now, the received sequences are
r1 [n] = (s [n] + z1 [n]) pN [n]
and
(2.52)
³
´
s [n − ∆] e−j2πµ0 ∆+j2π(µ0 −µ2 )n + z2 [n] pN [n]
¡
¢
= s [n − ∆] e−j2πµ0 ∆−j2πεn + z2 [n] pN [n] .
(2.53)
r2 [n] =
36
Chapter 2. System models
The CCF between (2.52) and (2.53) is
E{r1 [n + m]r2∗ [n]} = φs [m + ∆] ej2πµ0 ∆
N −1
1 X j2πεn
e
pN[n + m]pN[n]
N
n=1−N
(2.54)
which is dependent on time n. Time-averaging is performed in accordance
with (2.18)
φ [m] = φs [m + ∆] ej2πµ0 ∆
1
N
N/2
X
ej2πεn pN [n + m] .
(2.55)
n=−N/2+1
An explicit formula for the sum in (2.55) is given by Appendix A.1. The
resulting CCF is
N − |m|
sinc (ε (N − |m|)) 0
φs [m + ∆] ej2πµ0 ∆+jπε(1−m)
pM [m] .
N
sinc (ε)
(2.56)
Again, considering the white example signal in (2.21), the cross spectral
density of (2.56) is given by
φ [m] =
Φ [k] =
N − |∆| 2 j2πk/M+j2πµ0 ∆+jπε(1+∆) sinc (ε (N − |∆|))
σs e
N
sinc (ε)
(2.57)
for k = 1 − N, ..., N − 1. The phase of (2.57) is then
Γ [k] =
2πk∆
+ 2πµ0 ∆ + πε (1 + ∆)
M
k = 1 − N, ..., N − 1
(2.58)
where it is seen that the receiver oscillator frequency error gives a biased
phase curve which not does affect the slope. However, as seen in (2.56)
and (2.57) the sinc-factor results in an amplitude reduction given ε, ∆
and N . This amplitude reduction leads to a noise-like degradation in
the estimation process, as shown in Chapter 3. The robustness against
frequency errors is further analyzed in Chapter 3.6 where the effects of
large frequency errors also are studied.
2.8. Summary of models
2.8
37
Summary of models
Several models of a two channel TDOA direction-finding system are derived and studied. In Table 2.2 the key properties of the considered models are presented. The baseband model is to be used with non-mixing
receivers, such as low frequency radio, audio and sonar applications. The
ideal receiver system model is used when the considered mixing receivers
are controlled by the same reference module. When the mixing receivers
use separate reference modules the effects of errors between the outputs of
the reference modules are considered in the receiver system models subject
to timing error, oscillator phase error and oscillator frequency error.
Model
Baseband
Ideal receivers
Timing error
Phase error
Frequency error
Variable
—
µ0
∆ref
γ
ε
φ [m]
(2.18)
(2.33)
(2.46)
(2.49)
(2.56)
Γ [k] for example signal
k
M∆ ¢
¡ 2π
k
M +
¡ 2π
¢ µ0 ∆
k
2π M
+
µ
(∆
0
¡k
¢ + ∆ref )
2π
+
µ
¡ k M ¢ 0 ∆+γ
2π M
+ µ0 ∆ + πε (1 + ∆)
Table 2.2: Summary of system models
38
Chapter 2. System models
Chapter 3
Time- and frequencydomain TDOA estimation
Correlation based TDOA estimation can be made using time- or
frequency-domain estimators. In previous work it is shown that these
estimators have similar characteristics and that both the time-domain
estimator [14] and the frequency-domain estimator [18] are unbiased and
attain the Cramér-Rao lower bound (CRLB) for large enough data records
(asymptotically efficient). In an electronic warfare scenario, suppression
of unwanted signals is an important feature. Using a frequency-domain
estimator yields simple frequency filtering while a time-domain estimator
allows simple spatial filtering. Promising results from measurements of
actual radio transmitters using the considered estimators are described
in [25]. In a practical TDOA-based direction-finding system, a combined
time- and frequency-domain estimator can be used. That is, the CCF is
estimated and spatial filtering is applied [9]. The filtered CCF is then used
to calculate the estimated cross spectral density where frequency filtering
is performed. In terms of bias and variance the performance is similar
between the time- and frequency-domain estimators. In this thesis, the
focus is on frequency-domain estimators due the simple frequency filtering
which is needed in electronic warfare scenarios.
3.1
A lower bound on TDOA estimation
accuracy
A well known and commonly used lower bound on the variance in estimation problems is the CRLB [13]. The extensive derivation of the CRLB
for correlation based TDOA estimation is presented in [14] with examples
39
40 Chapter 3. Time- and frequency-domain TDOA estimation
of its application given in [18] and [26]. In this thesis the main focus is
on time discrete models but for simplicity the following calculation of the
CRLB is made for a time continuous model. Basically the CRLB is given
by [14]
¸−1
Z ∞ 2
1
f C (f )
CRLB (∆t ) = 2 2
(3.1)
df
8π T
−∞ 1 − C (f )
where C (f ) is the squared coherence function defined by the cross spectral
density and the power spectral density of the sensor outputs, Φ1 (f ) and
Φ2 (f ), as
|Φ (f )| 2
.
(3.2)
C (f ) =
Φ1 (f ) Φ2 (f )
Under an assumption of equal power in the two sensor outputs in bandlimited additive white complex-valued Gaussian noise with variance σ 2z ,
the squared coherence function is for any model described in Chapter 2
given by
Φ2s (f )
.
(3.3)
C (f ) =
[Φs (f ) + σ 2z ]2
The SNR between the signal and the channel noise is defined as
SNR (f ) =
Φs (f )
σ 2z
(3.4)
where Φs (f ) denotes the power spectral density of the signal of interest
s (t). For example, a flat spectrum signal with power σ2s results in a
constant SNR within the bandwidth of the noise
SNR =
σ 2s
σ 2z
(3.5)
and (3.3) is evaluated to
C (f ) =
SNR2
[SNR + 1]2
.
(3.6)
Inserting (3.6) into (3.1) gives the CRLB for a flat spectrum signal in
additive white complex-valued Gaussian noise
#−1
"Z
W
1 1 + 2SNR
2
CRLB (∆t ) =
f df
8π 2 T SNR2
−W
=
3
16π 2 T W 3
1 + 2SNR
.
SNR2
(3.7)
3.1. A lower bound on TDOA estimation accuracy
10
10
2
Variance [s ]
10
10
10
10
10
10
41
-10
T = 1 ms, W = 100 kHz
T = 1 ms, W = 10 MHz
T = 1 s, W = 100 kHz
T = 1 s, W = 10 MHz
-12
-14
-16
-18
-20
-22
-24
-20
-15
-10
-5
0
SNR [dB]
5
10
15
20
Figure 3.1: The CRLB (3.7) for the example signal in AWGN is plotted
for different acquisition times T and bandwidths W .
The CRLB in (3.7) is evaluated over the signal bandwidth - 2W Hz. Note
that the CRLB decreases with W 3 while it is linear in T and SNR as illustrated in Figure 3.1 for the white example signal. In [11], it is shown that
the variance of a traditional phase-measuring direction-finding system decreases with W . This implies that a correlation based direction-finding
system potentially outperforms a traditional phase-measuring system for
wideband signals.
For sampled data, the CRLB for ∆ = ∆t fs is sought. Straightforward
calculations gives
CRLB (∆) = fs2 CRLB (∆t ) .
(3.8)
Considering Nyquist sampling with fs = 2W yields by inserting fs = 2W
into (3.7)
2SNR + 1
3
CRLB (∆) = 2
.
(3.9)
2π T fs SNR2
Finally, observing that the number of samples N equals N = T fs results
42 Chapter 3. Time- and frequency-domain TDOA estimation
in
3 2SNR + 1
.
(3.10)
2π 2 N SNR2
That is, the CRLB for a flat spectrum signal, using Nyquist sampling,
is for N samples given by (3.10). The CRLB can be calculated for any
signal with known spectral characteristics (3.2). In a practical system,
targeting military communication systems, the acquisition interval and
signal bandwidth are in the range T > 1 ms and W > 100 kHz. The
range of the CRLB is then larger than 1010 considering typical military
communication systems as illustrated in Figure 3.1.
CRLB (∆) =
3.2
Time-domain TDOA estimator
In 1976, the generalized method for estimation of time delay using the
generalized CCF was introduced [14]. This method is considered a standard reference by most researchers in this field. The basic idea of the
generalized CCF is to compare, or correlate, the outputs of two sensors
and determine the time delay between the two channels. Alternatives
to the generalized CCF method are the average-square-difference function and the average-magnitude-difference function [10],[16] which both
are less complex to calculate than the generalized CCF. For medium and
high SNR the two alternative methods show results in parity with the
considered methods based on the generalized CCF. However, for low SNR
the alternative methods fail while the generalized CCF produces usable
results. Accordingly, in the considered electronic warfare scenario, under
the low SNR assumption, the generalized CCF method is considered.
The generalized TDOA method presented in [14] is reviewed in Chapter 1.2. However, in this thesis, the effects of a limited acquisition interval
for time-discrete data are considered. The method in [14] is adjusted for
these effects in the following. Also, with the electronic warfare scenario
no signal characteristics are known and the estimator needs to function
without any a priori information of the signal. The considered estimators
of ∆ rely on an estimate of the cross-correlation function, that is
φ̂ [m] =
1
N
N−1
X
r1 [n + m] r2∗ [n] .
(3.11)
n=1−N
Below it is shown that (3.11) is an unbiased estimate of the timeaveraged CCF in (2.18) for the baseband model (2.12)-(2.13). The ex-
3.2. Time-domain TDOA estimator
Step
1.
2.
3.
4.
5.
43
Procedure
Acquire the outputs from two spatially separated sensors,
r1 [n] and r2 [n] of length N samples each.
Calculate φ̂ [m] given by (3.11).
ˆ m.
Solve (3.14) and denote the result ∆
ˆ m and its two
Fit a second degree polynomial Π (∆) to ∆
ˆ m−1 and ∆
ˆ m+1 .
nearest neighbors, ∆
ˆ = arg max Π (∆)
Calculate the refined estimate using ∆
ˆ)
∂Π(∆
=0
or ∂ ∆
ˆ
Table 3.1: An outline for a time domain TDOA estimator.
pected value of (3.11) is calculated by inserting (2.12)-(2.13) into (3.11)
n
o
E φ̂ [m]
=
1
N
N
−1
X
E {s [n + m] s∗ [n − ∆]} pN [n + m] pN [n]
n=1−N
= φs [m + ∆]
1
N
N−1
X
pN [n + m] pN [n]
(3.12)
n=1−N
where φs [m] = E{s [n + m] s∗ [n]}. The definition and closed form expression of P [m] in (2.16)-(2.17) is used to evaluate (3.12) and it is seen that
this estimator of the CCF is an unbiased estimator of the time-averaged
CCF in (2.18)
n
o
E φ̂ [m]
=
N − |m|
φs [m + ∆] p0M [m]
N
= φ [m] .
(3.13)
That is, the considered time-domain CCF estimator in (3.11) is unbiased.
The time-domain TDOA estimator is given by the maximizing argument of the estimated CCF
¯
¯
ˆ = − arg max ¯¯φ̂ [m]¯¯ .
∆
(3.14)
m
An outline for estimating the TDOA using a time-domain estimator is
given in Table 3.1. The estimator in (3.14) provides a rough estimate of
ˆ is in its
∆. Note that φ̂ [m] is an unbiased estimate of φ [m], while ∆
simplest form biased due to the time-discrete CCF. The performance in
terms of mean square error (MSE) of the time-domain TDOA estimator
44 Chapter 3. Time- and frequency-domain TDOA estimation
60
40
∆ = 500 samples
20
MSE [dB]
∆ = 0 samples
0
-20
CRLB
-40
-60
-80
-20
-15
-10
-5
0
SNR [dB]
5
10
15
20
Figure 3.2: The MSE for the example signal is plotted using the timedomain TDOA estimator for ∆ = 0 and ∆ = 500 samples using N =
1024 samples and 500 Monte-Carlo simulations. The CRLB is given for
comparison.
in (3.14) is for the white example signal presented in Figure 3.2 as a
function of the SNR. For low SNR, the estimator performance deviates
from the CRLB, and a sharp delay-dependent threshold effect is observed.
For SNRs below the threshold, typically the estimator identifies a peak
in the CCF originating from the noise as the maximum peak. When this
happens the variance of the time-domain estimator will be much larger
than the CRLB due to the large errors [24] which can be seen in Figure
3.2 for low SNRs where the MSE differs from the CRLB. In the considered
electronic warfare scenario the acquisition interval is large but the SNR
is low causing large estimation errors. The method of block-averaging
described in Chapter 3.3.3 can be used to suppress the effects of the noise
in the estimated CCF, that is lower the variance of φ̂ [m] to reduce the
occurrences of large errors.
Note that in (3.13) the magnitude of the expected CCF is a function
of ∆ and an increased delay (positive or negative) reduces the magnitude
|φ[m]|
|φ[m]|
3.2. Time-domain TDOA estimator
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1000
45
∆ = 0 samples
-800
-600
-400
-200
0
200
400
600
800
1000
400
600
800
1000
∆ = 500 samples
-800
-600
-400
-200
0
200
Time difference [samples]
Figure 3.3: The CCF is estimated using the example signal with ∆ =
{0, 500} and N = 1024 samples. Note the reduction in amplitude due to
the factor (N − |∆|) /N (dotted line).
due to the factor (N − |m|) /N . Thus, the performance of (3.14) is a
function of ∆. The CCFs for two realizations of the white example signal
with different ∆ = {0, 500} is shown in Figure 3.3 from which it is evident
that the amplitude reduction for ∆ = 500 with N = 1024 is drastic and
affects the performance of the estimator. According to Figure 3.2, the
estimator is statistically efficient for ∆ = 0 while an increased ∆ results
in an increased estimation error using the white example signal (2.21). In
this example the amplitude loss is approximately |∆| /N = 500/1024, or
≈ 3 dB. However, for N → ∞ the estimator in (3.14) is efficient for all
limited ∆ since (N − |m|) /N → 1.
46 Chapter 3. Time- and frequency-domain TDOA estimation
3.3
Frequency-domain TDOA estimator
The frequency-domain TDOA estimator is based on the estimated cross
spectral density Φ̂ [k] which is calculated using the estimated CCF
n
o
Φ̂ [k] = FM φ̂ [m]
(3.15)
where FM {·} denotes the discrete Fourier transform (DFT) for
k = 1 − N, ..., N − 1. For the models in Chapter 2 the phase of the cross
spectral density is linear. This can be used in a frequency-domain estimator which estimates the linear slope using a linear least squares estimator
(LLSE). Generally the estimated phase curve is modelled as
Γ̂ [k] = Γ [k] + v [k]
k = 1 − N, ..., N − 1
(3.16)
where Γ [k] is the modelled phase and v [k] is the disturbance caused by
noise, receiver imperfections and the limited acquisition interval. The
LLSE is found by minimizing the least squares error cost function J (∆)
[4], that is minimizing the quadratic difference between the measured and
modelled phase curves. Here, let γ̂ denote the bias of the phase from
the reference imperfections, such as the time, phase and frequency errors
described in Table 2.2. The expected phase curve is then given by
Γ [k] =
2πk∆
+ γ̂
M
k = 1 − N, ..., N − 1.
(3.17)
Now, the least squares cost function is given by
J (∆) =
N
−1
X
k=1−N
=
N
−1
X
k=1−N
³
´2
Γ̂ [k] − Γ [k]
µ
¶2
2πk∆
Γ̂ [k] −
− γ̂ .
M
(3.18)
Note that this cost function is valid for all of the models discussed in this
thesis. The minimum of the cost function in (3.18) is found by setting its
first derivative to zero. That is,
³ ´
ˆ
N
−1
∂J ∆
X
ˆ
2πk2 ∆
−k Γ̂ [k] +
+ kγ̂ = 0
(3.19)
=
ˆ
M
∂∆
k=1−N
3.3. Frequency-domain TDOA estimator
47
ˆ gives
which solved for ∆
ˆ = M
∆
2π
PN −1
k=1−N
PN−1
k Γ̂ [k] − γ̂ k=1−N k
.
PN−1
2
k=1−N k
(3.20)
This estimator is independent of any phase bias since
N
−1
X
k≡0
(3.21)
k=1−N
which gives the frequency-domain TDOA estimator using (A.22)
ˆ =
∆
N
−1
X
3
k Γ̂ [k] .
2πN (N − 1)
(3.22)
k=1−N
A simple outline for estimating the TDOA using this frequency-domain
estimator is given in Table 3.2 and is analyzed in the following chapters.
In particular, the effects of a limited acquisition interval and the effects
of a receiver error are studied.
Step
1.
2.
3.
4.
5.
Procedure
Acquire the outputs from two spatially separated sensors,
r1 [n] and r2 [n] of length N samples each.
Calculate φ̂ [m] given by (3.11).
Calculate Φ̂ [k], which is the DFT of φ̂ [m].
Calculate the phase curve Γ̂ [k] = ∠Φ̂ [k]
Estimate ∆ using (3.22).
Table 3.2: An outline for a frequency domain TDOA estimator.
3.3.1
Practical aspects
In a practical TDOA direction-finding system the receivers are normally
spatially separated several kilometers and the received sequences need to
be transmitted to a remote processing station where the digital signal
processing takes place. By using the frequency-domain TDOA estimator
in Table 3.2, simple frequency filtering is possible and the amount of data
to be transmitted is reduced, as shown in the following. The data reduction follows from the method of calculating the estimated cross spectral
48 Chapter 3. Time- and frequency-domain TDOA estimation
density from the DFT of the two received sequences, that is the cross spectral density is estimated using the M sample DFT of the two received N
sample sequences
1
(3.23)
FM {r1 [n]} FM {r2∗ [n]} .
N
The result in (3.23) is well known in spectral estimation and is based
on the Wiener-Kinchin theorem [12] stating that the Periodogram can
be estimated from either the signal itself or its auto-correlation function.
The generalization to the cross spectral density is straightforward, but
included for reference. Normally the estimated CSD is calculated for
k = 1 − N...N − 1 as
n
o
Φ̂ [k] = FM φ̂ [m]
Φ̂ [k] =
1
N
=
N
−1
X
N
−1
X
r1 [n + m] r2∗ [n] e−j2πkm/M .
(3.24)
m=1−N n=1−N
However, substituting n + m with q in (3.24) gives
Φ̂ [k] =
1
N
N
−1
X
NX
−1+n
r1 [q] r2∗ [n] e−j2πk(q−n)/M .
(3.25)
n=1−N q=1−N+n
The received sequences, r1 [n] and r2 [n], are zero outside the range
−N/2 + 1 · n · N/2 due to the limited acquisition interval. Accordingly, the summation
NX
−1+n
r1 [q] e−j2πkq/M
(3.26)
q=1−N +n
is non-zero only for −N/2 + 1 · q · N/2. That is, the limits of the
summation is changed since 1 − N + n · −N/2 + 1 and N − 1 + n ≥ N/2
must be fulfilled to achieve a non-zero result. The sum in (3.26) is then
written as
N
−1
X
r1 [q] e−j2πkq/M .
(3.27)
q=1−N
Now, the CSD in (3.25) is simplified to
Φ̂ [k] =
=
1
N
N
−1
X
N
−1
X
r1 [q] r2∗ [n] e−j2πk(q−n)/M
n=1−N q=1−N
1
FM {r1 [n]} FM {r2∗ [n]} .
N
(3.28)
3.3. Frequency-domain TDOA estimator
49
20
15
10
∠Φ[k] [rad]
5
0
-5
-10
-15
-20
-0.5
Expected phase - Before unwrapping
High SNR - Successful unwrapping
Low SNR - Unsuccessful unwrapping
-0.4
-0.3
-0.2
-0.1
0
0.1
Frequency [f ]
0.2
0.3
0.4
0.5
s
Figure 3.4: The phase of the CSD for the example signal is shown before
and after unwrapping. The unwrapping is needed to resolve the steps but
fails for low SNR.
The estimator in (3.22) uses Γ̂ [k] = ∠Φ̂ [k] which now is given by
Γ̂ [k] = ∠FM {r1 [n]} − ∠FM {r2 [n]} .
(3.29)
That is, after applying the DFT locally at the individual receivers to
the received sequences, only the phase within the signal bandwidth is
needed to calculate the TDOA. Accordingly, the amplitude information
is not needed and is not transmitted thus reducing the total amount of
transmitted data by 50%.
For low SNR, the estimated cross spectral density suffers from large
estimation errors caused by the angle-operator (∠) whose output is limited
to [−π, π]. An unwrap-operator in combination with the angle-operator
is used to produce an output in the range of [−∞, ∞] where the unwrap
operator unwraps the radian phases changing absolute jumps greater than
π to their 2π-complement. In Figure 3.4, it is seen that for high SNR
the unwrap-operator is successful in producing a correct result, that is
50 Chapter 3. Time- and frequency-domain TDOA estimation
Step
1.
2.
3.
Procedure
Follow steps 1-3 in the procedureh outlinedi in Table 3.1
ˆ m to move the
Form the shifted CCF φ̄ [k] → φ̂ k + ∆
peak to lag zero.
Follow steps 3-5 in the procedure outlined in Table 3.2
Table 3.3: An outline for a frequency domain TDOA estimator with reduced uwrapping problems.
a straight line within the signal bandwidth. For low SNR the unwrap
operator fails to resolve the limited output of the angle-operator. To
reduce the occurrences of unwrapping failure the variance of the estimated
slope needs to be reduced, which in Chapter 3.3.3 is achieved using blockaveraging. In practice, an additional method can be used to reduce, not
eliminate, the unwrapping problem and is briefly presented in Table 3.3.
That is, by moving the peak of the CCF to lag zero, step 2 in Table 3.3,
the slope of the cross spectral density is limited so that the occurrences
of unwrapping failures are reduced. However, large errors still occur for
low SNR.
3.3.2
Performance analysis
The performance of the considered frequency-domain TDOA estimator
presented in Table 3.2 is analyzed below. It is shown to be unbiased and
with variance close to the CRLB for high SNR.
The expected value of the considered frequency-domain estimator follows from (3.22)
n o
ˆ =
E ∆
N
−1
n
o
X
3
kE Γ̂ [k] .
2πN (N − 1)
(3.30)
k=1−N
Using (3.16) the expected value of the phase curve is
n
o
E Γ̂ [k] = ∠Φ [k]
k = 1 − N, ..., N − 1
(3.31)
for any of the models given in Table 2.2. Note that the phase curves of
the models presented in Table 2.2 are all on the form
2πk∆
+ γ̂
M
(3.32)
3.3. Frequency-domain TDOA estimator
51
where γ̂ is a bias that includes the effects of the imperfections as discussed
around (3.16). Now, using (A.22) and (3.21) the expression in (3.30) is
n o
ˆ
E ∆
=
µ
¶
N−1
X
3
2πk∆
k
+ γ̂
2πN (N − 1)
M
k=1−N
= ∆
3
N (N − 1) M
N
−1
X
k 2 + γ̂
k=1−N
N
−1
X
k.
(3.33)
k=1−N
The evaluation of the first sum is found in Appendix A.3 and the last sum
is strictly zero leading to
n o
ˆ = ∆.
E ∆
(3.34)
That is, the frequency-domain TDOA estimator is unbiased for all the
models presented in this thesis.
In previous work the variance of the frequency-domain TDOA estimator is shown to (asymptotically) attain the CRLB assuming small estimation errors [18]. The following analysis follows [18] but is adjusted for
a digitized complex-valued model. Now, the variance of the estimator in
(3.22) is calculated for a complex-valued system model
à N−1
!
³ ´
X
9
ˆ
var ∆ =
k Γ̂ [k] .
(3.35)
2 var
4π2 N 2 (N − 1)
k=1−N
In [4] it is shown that for Gaussian noise and large enough time-bandwidth
products (W T > 8), the bins of the estimated phase curve Γ̂ [k] are uncorrelated. Here, the noises are assumed Gaussian and the time-bandwidth
condition is easily met in practice when considering wideband signals.
Accordingly, the bins of the phase curve are assumed uncorrelated which
leads to
à N −1
!
N−1
³
´
X
X
var
k Γ̂ [k] =
k2 var Γ̂ [k] .
(3.36)
k=1−N
k=1−N
ˆ is given by
Now, the variance of ∆
³ ´
ˆ =
var ∆
9
2
4π2 N 2 (N − 1)
N
−1
X
k=1−N
³
´
k 2 var Γ̂ [k] .
In (3.37) the variance is approximately given by [2]
³
´ 1 − C [k]
12
var Γ̂ [k] ≈
C12 [k]
(3.37)
(3.38)
52 Chapter 3. Time- and frequency-domain TDOA estimation
where C12 [k] is the (magnitude squared) coherence function given by the
cross spectral density and the power spectral densities of the received
sequences
2
|Φ [k]|
C12 [k] =
.
(3.39)
Φ1 [k] Φ2 [k]
In (3.39), Φ1 [k] and Φ2 [k] denotes the power spectral densities
received sequences r1 [n] and r2 [n], respectively. For the white
ple signal (2.21) used with any of the models derived in Chapter
coherence function in (3.39) is given by
µ
¶2
N − |∆|
σ 4s
C12 [k] =
2.
N
(σ 2s + σ 2z )
of the
exam2, the
(3.40)
where σ 2z denotes the power of the noises. The SNR is then defined as
SNR =
σ 2s
φs [0]
=
σ 2z
σ 2z
(3.41)
which inserted into (3.40) gives the coherence function for a flat spectrum
signal with equal SNR in the two receiver channels
µ
¶2
SNR2
N − |∆|
(3.42)
C12 [k] =
2.
N
(SNR + 1)
The variance of the phase curve in (3.38) is then given by
"
µ
¶2 #
³
´ µ N − |∆| ¶−2 2SNR + 1
N − |∆|
var Γ̂ [k] ≈
+1−
N
N
SNR2
(3.43)
which is independent of frequency k since both the signal and noises are
flat within the receiver bandwidth. Now, the variance of the considered
TDOA estimator is found by inserting (3.43) into (3.37)
µ
µ
¶−2 "
¶2 #
³ ´
3
(2N
−1)
2SNR+1
N
−|∆|
N
−
|∆|
ˆ ≈
var ∆
. (3.44)
+1−
4π2 N (N −1)
N
N
SNR2
The approximation in (3.38), and consequently (3.44), is not valid for
large estimation errors (low SNR or small N in combination with large
∆) when the phase curve suffers from unwrapping problems as discussed
in Chapter 3.3.1. Note that the amplitude reduction from the limited
acquisition interval increases the variance. However, for large acquisition
intervals and small TDOAs, (3.44) is close to the CRLB in (3.10).
3.3. Frequency-domain TDOA estimator
3.3.3
53
Variance reduction using block-averaging
In order to reduce the variance of the estimated cross spectral density,
block-averaging is used. Block-averaging is used in a variety of applications, for example in spectral estimation both the averaged periodogram
and the Welch periodogram use block-averaging [12]. The block-averaging
is performed by dividing the acquired sequences of length N samples into
B blocks of length L samples. For simplicity the blocks are assumed nonoverlapping but in a practical scenario an increase in performance can be
achieved using overlapping blocks. An estimated block-cross spectral density Φ̂b [k] is calculated for the individual blocks, with individual length
A = 2L − 1 samples. The B blocks are averaged to form an averaged
estimate of the cross spectral density
Φ̂avg [k] =
B
1 X
Φ̂b [k]
B
k = 1 − L, ..., L − 1
(3.45)
b=1
where Φ̂b [k] is given by (3.15), substituting M by A. Note that blockaveraging reduces the span of delays that can be observed from the cross
spectral density. Due to the averaging, the span of observable TDOAs is
reduced from |∆| · N − 1 to |∆| · L − 1.
Now, using the phase curve of the block-averaged cross spectral density
in (3.22) with N replaced by the block length L gives
ˆ =
∆
L−1
X
3
kΓ̂avg [k]
2πL (L − 1)
(3.46)
k=1−L
where the phase of the averaged cross spectral density is given by
Γ̂avg [k] = ∠Φ̂avg [k]
k = 1 − L, ..., L − 1.
(3.47)
In [18] the variance of the estimated TDOA is presented in terms of the
estimated cross spectral density variance. The following analysis follows
[18] closely but is adjusted for the considered discrete complex-valued
model and the block-averaging. Following (3.35)-(3.43) with N replaced
by L, the variance of the estimator in (3.46) is
à L−1
!
³ ´
X
9
ˆ
var ∆ =
k Γ̂avg [k] .
(3.48)
2 var
4π 2 L2 (L − 1)
k=1−L
The small error variance reduction in the uncorrelated bins of the blockaveraged cross spectral density are reduced by a factor B and hence the
54 Chapter 3. Time- and frequency-domain TDOA estimation
variance of the slope of the block-cross spectral density is also reduced by
a factor B
´
³
´ var Γ̂b [k]
³
var Γ̂avg [k] =
k = 1 − L, ..., L − 1.
(3.49)
B
The variance of (3.46) is then given by
³ ´
ˆ =
var ∆
9
4π 2 N L (L − 1)2
L−1
X
k=1−L
³
´
k 2 var Γ̂b [k] .
(3.50)
In order to analyze the effects of the block-averaging in terms of variance,
the white example signal (2.21) is used. The variance for each of the phase
curves is given by (3.43) with N replaced by the block length L
"
µ
¶2 #
³
´ µ L − |∆| ¶−2 2SNR + 1
L − |∆|
var Γ̂b [k] ≈
+1−
(3.51)
L
L
SNR2
which inserted into (3.50) gives the variance, for the white example signal
(2.21), of the frequency-domain TDOA estimator with block-averaging
µ
µ
¶−2"
¶2 #
³ ´
3 (2L − 1) L−|∆|
L−|∆|
2SNR + 1
ˆ
var ∆ ≈ 2
+1−
.
4π N (L − 1)
L
L
SNR2
(3.52)
The performance in terms of variance of the considered block-averaged
frequency-domain TDOA estimator is for large N and L with small ∆ close
to the CRLB (3.10). However, for a not so large acquisition interval one
may be led to believe that the smallest variance is obtained by choosing
L = N (B = 1) which is not true since (3.52) only describes the small
error variance. By choosing a small B the estimation of Γ̂avg [k] suffers
from unwrapping problems causing large errors in the TDOA estimation.
These problems are suppressed by using a higher degree of averaging, that
is a larger B which leads to a lower variance of Γ̂avg [k]. In Figure 3.5 the
effects of averaging are seen for a signal with similar characteristics as the
white example signal (2.21). The figure shows the phase of the estimated
cross spectral density using N = 1024 samples with no averaging (B = 1)
and with averaging (B = 16). The effects of failed unwrapping are seen as
steps of 2π while unwrapping of the averaged curve shows a linear phase.
Failed unwrapping will result in large TDOA estimation errors since the
straight line will be fitted to a non-linear phase curve.
3.4. Effects of time reference errors
55
5
B=1
B=16
0
∠Φ[k] [rad]
-5
-10
-15
-20
-25
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Frequency [f ]
0.2
0.3
0.4
0.5
s
Figure 3.5: The phase of an estimated cross spectral density for a signal
received at 10 dB SNR using N = 1024 samples with no averaging B = 1
and averaging using B = 16 are presented. The failed unwrapping is seen
as 2π-steps.
In Figure 3.6, the effects on the mean square error (MSE) using blockaveraging for the white example signal are seen. The large errors for low
degrees of averaging are due to the problems with the unwrap operator.
These problems are suppressed when using higher degrees of averaging
since the variance of the phase is reduced. However, for really high degrees
of averaging the variance will increase according to (3.52).
3.4
Effects of time reference errors
In a TDOA-based electronic warfare direction-finding system the TDOA
between the outputs of two spatially separated receivers is calculated.
When there is an error in the time reference signal between the two receiver systems, a biased estimate is produced. This bias is directly linked
to the time-difference, or error, between the time reference outputs of the
56 Chapter 3. Time- and frequency-domain TDOA estimation
30
20
10
MSE [dB]
0
-10
-20
-30
-40
-50
-60
-20
B=1
B=4
B=16
B=128
CRLB
-15
-10
-5
0
SNR [dB]
5
10
15
20
Figure 3.6: The MSE for the example signal is plotted using the frequencydomain TDOA estimator with different degrees of averaging and compared
to the CRLB using N = 1024 samples and ∆ = 1 sample.
two reference modules presented in Chapter 2.7.3.
The effects of a timing error are primarily dependent on three factors
— the distance between the receivers, the direction of arrival and the propagating speed of the signal. For any given timing error and propagating
speed, the directional error is minimized by placing the receivers as far
apart as possible. That is, the distance between the receivers gives the
maximum TDOA. For a short receiver distance the timing error causes a
larger directional error since the ratio between the timing error and the
maximum TDOA is large.
In a practical electronic warfare scenario, where the receivers are positioned several kilometers apart, a timing error smaller than 100 ns is sufficient in most cases. This time accuracy is achieved using the NAVSTARGPS system. However, the effect of the timing error in terms of directional
error is also dependent on the absolute direction of arrival of the signal.
This is seen using the approximate formula (1.1) in the following example.
3.4. Effects of time reference errors
57
1500
Sensor 2
1000
∆ = -0.5 ± 0.1 µs
t
North-South position [m]
500
Sensor 1
0
-500
-1000
-1500
∆ = 3.2 ± 0.1 µs
t
-2000
-2500
-1000
-500
0
500
1000
1500
East-West position [m]
2000
2500
3000
Figure 3.7: A time reference error gives rise to different directional errors
depending on the direction of arrival of the signal. Here, the time reference error is ±0.1 µs while the distance between the receivers is 1000
meters. The two transmitters are located at ∆ = −0.5 µs and ∆ = 3.2
µs, respectively.
A practical example of a time reference error Consider a scenario
where two intercept receivers are placed 1000 meters, or 3.33 µs apart, as
illustrated in Figure 3.7. Two signals are received one at a time by the two
receivers and the only difference between the two signals are the TDOAs
which are ∆ = −0.5 µs and ∆ = 3.2 µs, respectively. Assuming that the
TDOA direction-finding system suffers from a time reference error of ±0.1
µs, then the actual directional error for the first signal is approximately
±2◦ while the error for the second signal is approximately ±8◦ using the
approximate formula in (1.1) with propagating speed v = 3 · 108 m/s.
58 Chapter 3. Time- and frequency-domain TDOA estimation
3.5
Effects of receiver frequency errors
In order to digitize the received signal it needs to be transposed, or mixed,
to near baseband using a superheterodyne receiver, as outlined in Chapter
2.7. The oscillators in the receivers are controlled by the reference modules
and in particular their frequency reference outputs. If the frequencies of
these reference signals differ, the performance of the TDOA estimator
is degraded as shown in the following. The receiver oscillator frequency
error ε is due to the frequency difference between the frequency reference
outputs of the two different reference modules. In Chapter 2.7.5 it is
shown that an error, or difference, in frequency between the two receiver
oscillators leads to a reduced amplitude of the CCF (2.56).
Using the model presented in (2.52)-(2.53) the estimated CCF is given
by (3.11). To get an understanding of how receiver oscillator frequency
errors affect this estimator its expected value is calculated
n
o N − |m|
sinc (ε (N − |m|)) 0
E φ̂ [m] =
φs [m + ∆]ej2πµ0 ∆+jπε(1−m)
pM [m]
N
sinc (ε)
(3.53)
which is recognized as (2.56). In a practical system the sinc-factor reduces the contribution from the signal while the power of the noises are
unaffected by the frequency error. This leads to a reduced SNR due to
the frequency error which is discussed in the following.
3.5.1
Effects of frequency errors with block-averaging
In Chapter 3.3.3, block-averaging is used to reduce the variance of the estimated quantities. When block-averaging is used in presence of a receiver
oscillator frequency error the block-averaging results in an amplitude reduction of the estimated CCF, or cross spectral density, and reduces the
performance of the TDOA-estimator. However, some block-averaging still
is needed to reduce the effects of large estimation errors. How to choose
the amount of block-averaging in presence of non-zero ε is discussed in
Chapter 3.6. Now, in order to reduce the variance of the estimated cross
spectral density, block-averaging is used. That is, the received sequences
of length N samples are divided into B blocks each consisting of L samples. An estimate of the CCF is formed for each block, φ̂b [k], and then a
block-averaged estimate of the CCF is calculated as
φ̂avg [m] =
B
1 X
φ̂b [m] .
B
b=1
(3.54)
3.5. Effects of receiver frequency errors
59
Figure 3.8: The received sequences are divided into blocks of length L
samples each.
The individual block-CCFs are estimated as
φ̂b [m] =
N −1
1 X
∗
r1b [n + m] r2b
[n]
L
(3.55)
n=1−N
where
r1b [n] = (s [n] + z1 [n]) pL n +
and
¡
−j2πεn
r2b [n] = s [n − ∆] e
¸
L
(B + 1) − bL
2
¸
¢
L
+ z2 [n] pL n + (B + 1) − bL .
2
(3.56)
(3.57)
The limits on n in the received blocks (3.56)-(3.57) are directly given by
how the received sequences are divided into blocks. In Figure 3.8 the
limits on n for B = 4 is presented.
Now, the block-averaging will not always increase the performance due
to the frequency error which is seen by calculating the expected value of
the block-averaged CCF
B
n
o
o
1 X n
E φ̂avg [m] =
E φ̂b [m]
B
(3.58)
b=1
which is, using (3.55)-(3.57)
L
N −1+ 2 (B+1)−bL
n
o 1
X
L
E φ̂b [m] = φs [m + ∆]
ej2πε(n− 2 (B+1)−bL)pL [n + m]pL [n] .
L
L
n=1−N + 2 (B+1)−bL
(3.59)
60 Chapter 3. Time- and frequency-domain TDOA estimation
The expected value of the block-averaged CCF estimate is then given by
inserting (3.59) into (3.58)
n
o
E φ̂avg [m] =
B
3N/2−1+ L
X 2 −bL
j2πεn
X
L
1
= φs[m+∆]e−j2πε 2 (B+1)
e−j2πεbL
N
b=1
e
pL [n+m]pL [n] .
n=1−N/2+ L
2 −bL
(3.60)
The last summation is rewritten as
3N/2−1+ L
X 2 −bL
j2πεn
e
pL [n + m]pL [n] =
n=1−N/2+ L
2 −bL
= (L − |m|)ejπε(1−m)
L/2
X
ej2πεn pL [n + m]
n=−L/2+1
sinc (ε (L − |m|)) 0
pA [m]
sinc (ε)
(3.61)
where the second equality follows from Appendix A.1. The summation
over the block-averaging is, using Appendix A.2
B
sinc (εN )
1 X −j2πεbL
e
= e−jπεL(B+1)
.
B
sinc (εL)
(3.62)
b=1
Now, the expected value of the block-averaged CCF in (3.60) is using
(3.61)-(3.62) given by
n
o
E φ̂avg [m] =
=
L−|m|
sinc (εN ) sinc (ε (L − |m|)) 0
φs [m+∆]ejπε(1−m−2L(B+1))
pA [m]
L
sinc (ε)
sinc (εL)
(3.63)
where the sinc-factors gives an additional amplitude reduction due to
the block-averaging in presence of a receiver frequency error. The blockaveraged CCF is used to calculate the block-averaged estimate of the cross
spectral density
o
n
Φ̂avg [k] = FA φ̂avg [m] .
(3.64)
The expected value of the estimated averaged cross spectral density is
given by the expected value of the estimated averaged CCF as
o
oo
n n
n
(3.65)
E Φ̂avg [k] = FA E φ̂avg [m]
3.5. Effects of receiver frequency errors
61
with A = 2L − 1. For the white example signal in (2.21), the blockaveraged cross spectral density is
n
o
E Φ̂avg [k]
=
=
L − |∆| 2 jπε(1+∆−2L(B+1))+j2πk∆/A sinc (εN ) sinc (ε (L − |∆|))
σs e
L
sinc (ε)
sinc (εL)
(3.66)
for k = 1 − L, ..., L − 1. The (expected) slope of the cross spectral density
in (3.66) used in the TDOA estimator (3.22) is then for the white example
signal given by
n
o
∠E Φ̂avg [k] = 2πk∆/A + πε (1 + ∆ − 2L (B + 1)) k = 1 − L, ..., L − 1
(3.67)
where it is seen that a frequency error in combination with block-averaging
only affects the bias of the slope. However, the frequency error in combination with block-averaging yield an additional amplitude degradation
(3.63) which results in a performance degradation of the TDOA estimator.
Numerical simulations based on sampled data are run to evaluate the
performance in presence of a frequency error. The presented results are
based on 1000 independent Monte-Carlo simulations. Considering crystal
oscillators or NAVSTAR-GPS disciplined Rubidium oscillators to be used
with the intercept receivers, then the typical relative frequency errors
are 10−5 and 10−10 , respectively. In this example the assumed maximum receiver mixer frequency is 1010 Hz which for sampling frequency
fs = 107 Hz yields the interval of interest for ε. That is, ε < 10−2 and
accordingly the numerical simulations are run with 10−7 < ε < 10−2 for a
high SNR case (20 dB) and a low SNR case (0 dB). The results, using the
white example signal, are presented in Figures 3.9 and 3.10 using different degrees of averaging. From the figures it is evident that an ordinary
crystal oscillator with ε ≈ 10−2 is not stable enough while a NAVSTARGPS disciplined oscillator with ε ≈ 10−7 clearly is stable enough since the
mean square error attains the CRLB (ε = 0) for small enough ε. In the
high SNR case it is seen that the CRLB is attained for small frequency
errors and a suitable degree of averaging. The low SNR case is shown in
Figure 3.10 where the CRLB is attained for a suitable choice of B but
since the SNR is lower, a higher degree of averaging is needed to attain
the CRLB.
62 Chapter 3. Time- and frequency-domain TDOA estimation
20
10
0
MSE [dB]
-10
-20
-30
-40
B=1
B=2
B=32
B=256
CRLB
-50
-60 -7
10
10
-6
-5
-4
10
10
Frequency error ε [f ]
10
-3
10
-2
s
Figure 3.9: The MSE is determined by numerical simulations using the
example signal. Here, the SNR is 20 dB and B = 32 (N = 1024) gives
the lowest MSE for all small frequency errors (ε < 1/N ).
3.5. Effects of receiver frequency errors
63
30
20
MSE [dB]
10
0
-10
-20
B=1
B=16
B=128
B=256
CRLB
-30
-40 -7
10
10
-6
-5
-4
10
10
Frequency error ε [f ]
10
-3
10
-2
s
Figure 3.10: The MSE is determined by numerical simulations using the
example signal. Here, the SNR is 0 dB and B = 128 (N = 1024) gives
the lowest MSE for all small frequency errors (ε < 1/N ).
64 Chapter 3. Time- and frequency-domain TDOA estimation
3.6
Robustness against frequency errors
In previous chapters it is shown that the performance of the TDOA estimator is degraded in presence of a frequency error between the receiver
oscillators. In this chapter a power ratio Q is defined which describes the
power-loss due to the frequency error resulting in the TDOA-estimator
performance degradation. Using the Q-value it is possible to calculate
the power-loss in the system due to the frequency error for the given
signal. The Q-value is defined as
Q=
Pout
Pin
(3.68)
where Pout is the total power of the calculated cross spectral density
between the sensor outputs
Pout =
1
M
N
−1
X
|Φ [k]|
(3.69)
k=−(N−1)
and Pin is the power of the signal, that is Pin = φs [0]. The power
ratio Q describes the relative loss of power due to the frequency error ε
described by (3.63) and (3.66). Ideally, the Q-value should be close to
unity, indicating no loss in power. For example, consider the basic model
in Chapter 1.2 where ∆t = ∆fs−1
Z ∞
¯
¯
¯Φs (f ) ej2πf ∆t ¯ df = φs (0)
Pout =
(3.70)
−∞
and thus Q ≡ 1. One may note that the Q-value defined in (3.68) depends
on the auto-correlation function of the source signal. The Q-value can also
be numerically evaluated for different values of the influencing parameters,
that is physical conditions (∆, SNR, φs [m]), system configuration (N, fs )
and system imperfections (ε).
In order to get some insight into the behavior of the Q-value, an example is considered. Given the white example signal (2.21) with a TDOA
described by ∆, the output power Pout is given by inserting the cross
spectral density in (2.57) into (3.69). A straightforward calculation gives
¯
¯
N − |∆| ¯¯ sinc (ε (N − |∆|)) ¯¯ 2
Pout =
(3.71)
¯
¯ σs
N
sinc (ε)
while the input power is
Pin = σ 2s .
(3.72)
3.6. Robustness against frequency errors
65
1
Q-value
0.8
0.6
0.4
0.2
Estimate
Model
0 -7
10
10
-6
-5
-4
10
10
Frequency error ε [f ]
10
-3
10
-2
s
Figure 3.11: The Q-value for the example signal with N = 1024 and
∆ = 10 samples. Note the difference between the model and the estimated
curves for large frequency errors (ε > 1/N ).
The resulting Q-value becomes
¯
¯
N − |∆| ¯¯ sinc (ε (N − |∆|)) ¯¯
Q=
¯
¯
N
sinc (ε)
(3.73)
and, clearly, it depends depends on ε, ∆ and N . In Figure 3.11, the Qvalue (3.73) is shown for the white example signal with N = 1024 and
∆ = 10 samples. From (3.73) one may note that Q → (N − |∆|) /N ≈ 1
when ε → 0 as expected. Also note that for a fixed ε, Q → 0 as N → ∞
due to the decorrelation between the channels from the frequency error.
However, to better describe the effects of the frequency error in a
practical scenario, the Q-value needs to be estimated from the received
sequences. This estimated Q-value is now defined as
Q̂ =
P̂out
P̂in
(3.74)
where the estimated output power is calculated using the estimated cross
66 Chapter 3. Time- and frequency-domain TDOA estimation
spectral density
P̂out =
1
M
N−1
X
k=1−N
¯
¯
¯
¯
¯Φ̂ [k]¯
(3.75)
and P̂in is an estimate of the signal power Pin . As illustrated in Figure
3.11 the estimated Q̂ attains the modelled Q for small frequency errors.
However, for large frequency errors (ε > 1/N ) the modelled and the estimated Q-values differ. To illustrate these effects the white example signal
(2.21) with ∆ = 0 in a noise-free scenario is used. The estimated cross
spectral density in (3.75) is given by (3.23)
Φ̂ [k] =
1
R1 [k] R2∗ [k]
N
k = 1 − N, ..., N − 1
(3.76)
where R1 [k] = FM {r1 (n)} and R2 [k] = FM {r2 (n)}. The expected
value of the estimated output power is calculated to get some insight into
the behavior of the estimated Q-value
n
o
E P̂out
=
1
M
k=1−N
1
MN
=
N
−1
X
¯o
n¯
¯
¯
E ¯Φ̂ [k]¯
N−1
X
E {|R1 [k] R2∗ [k]|} .
(3.77)
k=1−N
The large frequency error decorrelates the DFTs of the received sequences
which leads to
o
n
E P̂out =
1
MN
N
−1
X
E {|R1 [k]|} E {|R2∗ [k]|} .
(3.78)
k=1−N
Calculating the DFT of the white example signal gives a circular, complexvalued Gaussian sequence with variance N σ 2s . The expectations in (3.78)
is then, for a noise free analysis, calculated using Appendix B.1
r
N πσ 2s
E {|R1 [k]|} = E {|R2 [k]|} =
.
(3.79)
4
By inserting (3.79) into (3.78), a straightforward calculation gives
n
o πσ 2
s
E P̂out =
4
(3.80)
3.6. Robustness against frequency errors
67
with the input power given by σ2s . The (expected) estimated Q-value
for the white example signal suffering from large frequency errors is then
given by
o
n
n o E P̂out
π
= ≈ 0.785.
E Q̂ =
(3.81)
σ 2s
4
This means that the estimated Q-value for a white noise signal always
is between unity and π/4. When Q̂ attains the threshold at π/4 the frequency error is large, ε > 1/N , and the TDOA-estimator fails to produce a
reliable estimate due to the decorrelation between the received sequences.
Block-averaging of the cross spectral density will change both the range
and the level of the threshold since the blocklength determines at which ε
the TDOA estimator fails to produce a reliable result, that is the estimated
Q-value reaches the threshold level for large frequency errors given by
ε > 1/L. In the following, the effects of block-averaging on the estimated
Q-value are discussed. The estimated output power is now calculated
using the block-averaged cross spectral density
B
1 X
Φ̂avg [k] =
Φ̂b [k]
B
k = 1 − L, ..., L − 1
(3.82)
b=1
where the block-cross spectral densities Φ̂b [k] are calculated using the
∗
block DFTs R1b [k] and R2b
[k]
1
∗
[k]
k = 1 − L, ..., L − 1.
R1b [k] R2b
L
The estimate of the output power is then
Φ̂b [k] =
P̂out =
L−1
¯
1 X ¯¯
¯
¯Φ̂avg [k]¯
A
(3.83)
(3.84)
k=1−L
where A = 2L − 1. To get some understanding of the behavior of the
output power, the expected value is calculated using the white example
signal (2.21). The expected value of (3.84) is then given by
L−1
¯o
n
n¯
o
1 X
¯
¯
E P̂out =
E ¯Φ̂avg [k]¯
A
(3.85)
k=1−L
which, using (3.66) with P̂in = σ 2s gives the expected value of Q̂ using
block-averaging
n o L − |∆| ¯¯ sinc (εN ) ¯¯ ¯¯ sinc (ε (L − |∆|)) ¯¯
¯
¯¯
¯.
(3.86)
E Q̂ =
¯ sinc (ε) ¯ ¯
¯
L
sinc (εL)
68 Chapter 3. Time- and frequency-domain TDOA estimation
Note, that for no averaging with B = 1 (L = N ) the expression in (3.86)
becomes (3.73).
For large frequency errors (ε > 1/L) the analysis above will not hold
∗
since R1b [k] and R2b
[k] inn¯(3.83) becomes
uncorrelated due to the large
¯o
¯
¯
frequency error. Now, E ¯Φ̂avg [k]¯ in (3.85) is calculated using the
definition of the variance as
s ½
¯o
¯
¯2 ¾
¯o
n¯
n¯
¯
¯
¯
¯
¯
¯
E ¯Φ̂avg [k]¯ = E ¯Φ̂avg [k]¯ − var ¯Φ̂avg [k]¯ .
(3.87)
Note that
and
¯o
¯o
n¯
n¯
1
¯
¯
¯
¯
var ¯Φ̂avg [k]¯ = var ¯Φ̂b [k]¯
B
½¯
¯2 ¾
o
o
n
n
1
¯
¯
E ¯Φ̂avg [k]¯ = var Φ̂avg [k] = var Φ̂b [k]
B
(3.88)
(3.89)
since the blocks are uncorrelated and the white example signal is zeromean. Inserting (3.88) and (3.89) into (3.87) gives
¯o r 1 ³
¯o´
n¯
o
n¯
n
¯
¯
¯
¯
E ¯Φ̂avg [k]¯ =
(3.90)
var Φ̂b [k] − var ¯Φ̂b [k]¯ .
B
Again, use the definition of the variance to obtain
½¯
¯2 ¾
o
o
n
n
¯
¯
var Φ̂b [k] = E ¯Φ̂b [k]¯ − E2 Φ̂b [k]
and
½¯
¯o
¯2 ¾
¯o
n¯
n¯
¯
¯
¯
¯
¯
¯
var ¯Φ̂b [k]¯ = E ¯Φ̂b [k]¯ − E2 ¯Φ̂b [k]¯
which inserted into (3.90) gives
¯o r 1 ³ n¯
¯o
n¯
o´
n
¯
¯
¯
¯
E ¯Φ̂avg [k]¯ =
E2 ¯Φ̂b [k]¯ − E2 Φ̂b [k] .
B
(3.91)
(3.92)
(3.93)
The block DFTs are uncorrelated for large frequency errors (ε > 1/L) and
the white example signal is zero-mean, which gives
n
o
1
1
∗
∗
E Φ̂b [k] = E {R1b [k] R2b
[k]} = E {R1b [k]} E {R2b
[k]} = 0.
A
A
(3.94)
3.6. Robustness against frequency errors
69
Now, (3.93) is evaluated using (3.83), (3.94) and Appendix B.1
¯o
¯o
n¯
n¯
1
πσ 2
¯
¯
¯
¯
E ¯Φ̂avg [k]¯ = q E ¯Φ̂b [k]¯ = q s
B
4 B
(3.95)
and the expected value of the output power in (3.85) is
n
o
πσ 2
E P̂out = q s .
4 B
(3.96)
The estimated Q-value is for the white example signal with blockaveraging then given by
n o
π
E Q̂ ≈ q .
(3.97)
4 B
In Figure 3.12 it is seen that block-averaging of the cross spectral density
helps reduce the threshold effect following the large frequency errors for
a noise-free scenario. However, the degree of block-averaging B is to be
chosen with respect to both the frequency error and the SNR as shown
in Chapter 3.3.3 to obtain an unbiased TDOA estimate with low variance. Note that the ripple for large ε in Figure 3.12 also is visible in
Figures 3.9 and 3.10. Also note that for a very high degree of averaging
(here B = 256) the amplitude is reduced since the ratio
L − |∆|
L
becomes small in (3.63).
(3.98)
70 Chapter 3. Time- and frequency-domain TDOA estimation
B=1
1
B = 16
0.8
Q-value
B = 256
0.6
0.4
0.2
Model
Estimate
0 -7
10
10
-6
-5
-4
10
10
Frequency error ε [f ]
10
-3
10
-2
s
Figure 3.12: The Q-value for the example signal with N = 210 and ∆ = 1
with different degrees of block-averaging.
Chapter 4
Conclusions and topics
for future work
In this thesis both time- and frequency-domain based TDOA estimators
are derived and the performance in terms of bias and variance is calculated
using the white example signal in (2.21). Numerical simulations based
on sampled data are used to evaluate the estimator performance and is
compared to the CRLB in Chapter 3.1. It is shown that the considered
estimators have similar performance and that the CRLB is attained for
large data records. In the considered electronic warfare scenario the SNR
is assumed low and block-averaging of the spectral estimators are used to
reduce the variance of the TDOA estimators.
In a practical system, reference imperfections will degrade the performance of a TDOA-based direction-finding system. In particular the effects of time and frequency reference errors are studied. It is shown that
a time error gives a biased TDOA estimate while a frequency error results
in a noise like degradation due to an amplitude reduction in the CCF
(2.56). Moreover, block-averaging will increase the amplitude reduction
when a frequency error exist (3.63), that is the effects of a frequency error
will be more serious if block-averaging is used. In Chapter 3.6 the robustness against a frequency error for a specific scenario is calculated using
the Q-value which describes the amplitude reduction due to a frequency
error.
71
72
Chapter 4. Conclusions and topics for future work
The next natural step is to estimate the frequency error from the received
sequences. A method of joint estimation of the TDOA and the frequency
error is presented in [21] and [23]. The performance degradation due to the
frequency error is then reduced and the variance of the TDOA estimator
is reduced so that the CRLB is (asymptotically) attained.
In the considered electronic warfare scenario with low SNR, blockaveraging is used to reduce the variance of the TDOA estimates to obtain
useful results. However, for any given scenario, regardless of any reference
errors, the parameters used in the block-averaging must be chosen with
respect to the actual scenario. If chosen correct, the TDOA estimator
variance is minimized for that particular scenario. However, the optimum
choice of the block-averaging parameters is not discussed in this thesis
and is left for future work.
Several system models considering TDOA-estimation with receiver reference imperfections are derived and analyzed in this thesis. However, no
multipath scenarios are considered and hence no system models containing the effects of multipath are derived. An interesting path for future
work would include system models for multipath scenarios where a combination of the auto-correlation functions and the cross-correlation function
can be used to identify direct and indirect paths for increased accuracy in
the TDOA estimation process.
Appendix A
Useful sums
A.1
Time averaging of frequency error
When analyzing the effects of limited acquisition intervals in combination with a frequency error the CCF needs to be time-averaged. This
time-averaging gives rise to a summation over a complex rotation and a
rectangular pulse as shown below. The result is a complex rotation and
an amplitude reduction due to the frequency error.
Claim:
N/2
X
ej2πεn pN [n + m] = (N −|m|) ejπε(1−m)
n=−N/2+1
sinc (ε (N −|m|)) 0
pM[m]
sinc (ε)
(A.1)
Proof:
First, identify the two separate cases for positive or negative m given by
the properties of the rectangular window
N/2
X
n=−N/2+1
j2πεn
e
pN [n + m] =
( PN/2−m
Pn=−N/2+1
N/2
ej2πεn p0M [m]
j2πεn 0
pM
n=−N/2+1−m e
m≥0
[m] m < 0
.
(A.2)
For negative m, rewrite the summation limits to comply with standard
73
74
Appendix A. Useful sums
summation formulas
N/2
X
ej2πεn = ej2πε(1−N/2−m)
N −1+m
X
ej2πεn
(A.3)
n=0
n=−N/2+1−m
where the last sum is evaluated using Eulers formula
sin (x) =
ejx − e−jx
.
2j
(A.4)
That is,
N −1+m
X
ej2πεn =
n=0
sin (πε (N + m)) ejπε(N+m)
.
sin (πε)
ejπε
(A.5)
Inserting (A.5) into (A.3) gives
N/2−m
X
ej2πεn
= ej2πε(1−N/2−m)
n=−N/2+1
= ejπε(1−m)
sin (πε (N + m)) ejπε(N +m)
sin (πε)
ejπε
sin (πε (N + m))
.
sin (πε)
(A.6)
For m ≥ 0
N/2−m
X
ej2πεn = ej2πε(1−N/2)
N−1−m
X
ej2πεn
(A.7)
sin (πε (N − m)) ejπε(N−m)
.
sin (πε)
ejπε
(A.8)
n=0
n=−N/2+1
where the last summation is evaluated to
N−1−m
X
n=0
ej2πεn =
Inserting (A.8) into (A.7) gives
N/2−m
X
ej2πεn
= ej2πε(1−N/2)
n=−N/2+1
= ejπε(1−m)
sin (πε (N − m)) ejπε(N −m)
sin (πε)
ejπε
sin (πε (N − m))
.
sin (πε)
(A.9)
A.1. Time averaging of frequency error
75
Now, it is seen that (A.6) and (A.9) are equal in most parts and a formula
for all m is achieved using
N/2
X
ej2πεn pN [n + m] = ejπε(1−m)
n=−N/2+1
sin (πε (N − |m|)) 0
pM [m]
sin (πε)
= (N − |m|) ejπε(1−m)
where
sinc (x) =
Note that
sinc (ε (N − |m|)) 0
pM [m]
sinc (ε)
(A.10)
sin (πx)
.
πx
sinc (ε (N − |m|))
= 1 for |ε| = 0, 1, 2, ...
sinc (ε)
(A.11)
(A.12)
and
sinc (ε (N − |m|))
= 0 for |ε (N − |m|)| = 1, 2, ... and 0 < |ε| < 1.
sinc (ε)
(A.13)
The effects of the amplitude loss due to limited acquisition interval in
combination with a frequency error are shown in Figure A.1 where it is
seen that the amplitude loss depends on the acquisition interval N as well
as the frequency error ε.
76
Appendix A. Useful sums
1
0.9
0.8
0.7
Q-value
0.6
0.5
0.4
0.3
0.2
0.1
0 -7
10
10
N=2
16
N=2
10
-6
-5
-4
10
10
Frequency error ε [f ]
10
-3
10
-2
s
Figure A.1: The amplitude loss due to ε varies with N . Large N gives a
higher amplitude loss since 1/N is smaller.
A.2. Block-averaging of frequency error
A.2
77
Block-averaging of frequency error
When analyzing block-averaging in combination with a frequency error a
summation over a complex rotation arises. The summation is calculated
over B blocks each of length L samples for a total of N samples. The
result is a complex rotation and an amplitude reduction due to the blockaveraging.
Claim:
B
1 X −j2πεbL
sinc (εN )
e
= e−jπεL(B+1)
B
sinc (εL)
(A.14)
b=1
Proof:
First, let α = e−j2πεL for notational simplicity, then
B
B
B−1
1 X b
α X b
1 X −j2πεbL
e
=
α =
α
B
B
B
b=1
b=1
(A.15)
b=0
where the last sum is a standard sum
B−1
α X b
α 1 − αB
α(B−1)/2 α−B/2 − αB/2
α =
.
=
B
B 1−α
B
α−1/2 − α1/2
(A.16)
b=0
Use Eulers formula on (A.16) with α = e−j2πεL and N = BL
B
X
e−j2πεbL
= e−jπεL(B+1)
ejπεN − e−jπεN
ejπεL − e−jπεL
= e−jπεL(B+1)
sin (πεN )
sin (πεL)
b=1
(A.17)
and (A.15) is for sinc(x) = sin (πx) /πx evaluated to
B
sinc (εN )
1 X −j2πεbL
e
= e−jπεL(B+1)
.
B
sinc (εL)
(A.18)
b=1
Note the amplitude variations for the sinc-factors with extremes at
and
since N = BL.
sinc (εN )
= 1 for |εL| = 0, 1, 2, ...
sinc (εL)
(A.19)
sinc (εN )
= 0 for |εN | = 1, 2, ...
sinc (εL)
(A.20)
78
A.3
Appendix A. Useful sums
Sum of powers
Three standard sums of powers are used:
Claim [20]:
N
−1
X
N (N − 1)
2
(A.21)
N (N − 1) (2N − 1)
6
(A.22)
k=
k=0
Claim [20]:
N
−1
X
k=0
k2 =
Claim [20]:
N
−1
X
k=0
αk =
1 − αk
1−α
(A.23)
Appendix B
Complex-valued random
variables
In this thesis complex-valued random variables are used and are defined
below. This definition follows closely the definition in [13]. Let z (t) be a
complex-valued random variable defined as
z (t) = u (t) + jv (t)
(B.1)
where u (t) and v (t) are real-valued, mutually uncorrelated and identically
distributed random variables. Let
E {u (t)} = E {v (t)} = ū
and
with
©
ª
©
ª
E u2 (t) = E v 2 (t) = σ 2z /2 + ū2
var {u (t)} = var {v (t)} = σ 2z /2.
(B.2)
(B.3)
(B.4)
Then the complex-valued random variable z (t) have the following properties
z̄ = E {z (t)} = E {u (t)} + jE {v (t)} = ū (1 + j)
(B.5)
and
n
o
ª
©
ª
¡
©
¢
2
E |z (t)| = E u2 (t) + E v 2 (t) = 2 σ 2z /2 + ū2 .
79
(B.6)
80
Appendix B. Complex-valued random variables
The variance of the complex-valued random variable follows from
n
o
2
var {z (t)} = E |z (t) − E {z (t)}|
n
o
= E |z (t)| 2 − |E {z (t)}| 2
(B.7)
which is evaluated to
©
ª
var {z (t)} = 2E u2 (t) − 2E2 {u (t)}
= 2var {u (t)}
= σ 2z .
(B.8)
For a complex valued Gaussian variable z (t) = u (t) + jv (t) where
u¢
¡
and v are independent and identically distributed denoted u, v ∈ N ū, σ 2
the probability density function (PDF) is given by the joint PDF of u, v
¸
¸
1
1
1
1
p (u, v) = p
exp − 2 (u − ū)2 · p
exp − 2 (v − ū)2
σz
σz
πσ2z
πσ 2z
¸
³
´
1
1
2
2
exp − 2 (u − ū) + (v − ū)
.
(B.9)
=
2
πσz
σz
The PDF of the complex-valued Gaussian variable z (t) is then
¸
1
1
2
p (z) =
exp − 2 |z − z̄|
πσ 2z
σz
which is denoted
¡
¢
z ∼ CN z̄, σ 2z .
(B.10)
(B.11)
B.1. The expected value of the magnitude of a complex-valued
Gaussian variable
81
B.1
The expected value of the magnitude of
a complex-valued Gaussian variable
Claim:
The expected value of the magnitude of a complex-valued zero mean
Gaussian variable z with variance σ 2z is given by
r
πσ 2z
E {|z|} =
(B.12)
4
Proof:
¢
¡
A complex-valued Gaussian random variable z ∼ CN z̄, σ 2z with a PDF
according to (B.10) is used to calculate the expected value from the definition
E {|z|} = E {|u + jv|}
¸
Z ∞Z ∞
¢
1
1 ¡ 2
2
=
|u + jv| exp − 2 u + v dudv. (B.13)
πσ2z −∞ −∞
σz
The integral is evaluated with a transformation of z into polar coordinates
p
(B.14)
r = |z| = u2 + v 2
³ ´
−1 v
θ = tan
(B.15)
u
dudv = rdrdθ
(B.16)
which inserted into (B.13) gives
E {|z|} =
=
¸
Z ∞ Z 2π
1
1 2
2
r exp − 2 r drdθ
πσ 2z 0
σ
0
¸ z
Z ∞
2
1
r2 exp − 2 r2 dr.
σ 2z 0
σz
(B.17)
The solution of the remaining integral is found in a table of integrals [20]
r
2 σ 2z p 2
σ 2z π
E {|z|} = 2
σz π =
.
(B.18)
σz 4
4
82
Appendix B. Complex-valued random variables
Bibliography
[1] A. O. Bauer, R. Erskine, K. Herold, Funkpeilung als alliierte Waffe
gegen deutsche U-Boote 1939-1945, Rheinberg, Germany, 1997.
[2] J. S. Bendat, "Statistical errors in measurement of coherence functions and input/output quantities," Journal of Sound and Vibration,
vol. 59(3), pp. 405 - 421, 1978.
[3] G. C. Carter, Ed., Coherence and time delay estimation: an applied
tutorial for research, development, test, and evaluation engineers,
New York, IEEE Press, 1993.
[4] Y. T. Chan, R. V. Hattin, and J. B. Plant, "The least squares estimation of time delays and its use in signal processing," IEEE trans.
Acoustics, Speech, and Signal Processing, vol. ASSP-26, no. 3, pp.
217—222, 1978.
[5] C. Couvreur, and Y. Bresler, "Doppler-based motion estimation for
wide-band sources from single passive sensor measurements," Proc.
IEEE International Conference on Acoustics, Speech, and Signal
Processing, ICASSP-97, vol. 5, pp. 3537-3540, 1997.
[6] A. Fertner, and A. Sjölund, "Comparison of various time delay estimation methods by computer simulation," IEEE trans. Acoustics,
Speech, and Signal Processing, Oct., vol. 34, iss. 5, pp. 1329 - 1330,
1986.
[7] F. Gustafsson, and F. Gunnarsson, "Positioning using time-difference
of arrival measurements," Proc. IEEE International Conference on
Acoustics, Speech, and Signal Processing, ICASSP ’03, April, vol. 6,
pp. VI-553-6, 2003.
83
84
Bibliography
[8] R. M. Hambly, and T. A. Clark, "Critical evaluation of the Motorola M12+ GPS timing receiver vs. the master clock at the United
states naval observatory, Washington, D.C," Proc. 34th Annual Precise Time and Time Interval (PTTI) Meeting, Dec., pp. 109 - 116,
Reston, VA, USA, 2002.
[9] A. W. Houghton, and C. D Reeve, "Direction finding on spreadspectrum signals using the time-domain filtered cross spectral density," Proc. IEE Radar, Sonar and Navigation, Dec., vol. 144, no. 6,
pp. 315-320, 1997.
[10] G. Jacovitti, and G. Scarano, "Discrete time techniques for time delay
estimation," IEEE Trans. Signal Processing, Feb., vol. 41, no. 2, pp.
525-533, 1993.
[11] H. J. Jenkins, Small-Aperture Radio Direction-Finding, London,
Artech House, 1991.
[12] S. M. Kay, Modern spectral estimation, London, Prentice Hall, 1988.
[13] S.M Kay, Fundamentals of Statistical Signal Processing: Estimation
Theory, London, Prentice Hall, 1993.
[14] C. H. Knapp, and G. C. Carter, "The generalized method for estimation of time delay," IEEE Transactions on Acoustics, Speech, and
Signal Processing, Aug., vol. 24, no. 4, pp. 320-327, 1976.
[15] A. L. Lindblad and A. M. Wik, "Novel LPI concept using filtered
spreading codes," Proc. IEEE Military Communications Conference,
MILCOM 96, Oct., McLean, VA, USA, 1996.
[16] D. L. Maskell, and G. S. Woods, "The estimation of subsample time
delay of arrival in the discrete-time measurement of phase delay,"
IEEE trans. Instrumentation and Measurement, Dec., vol. 48, iss. 6,
pp. 1227 - 1231, 1999.
[17] A. V. Oppenheim, R. W. Schafer, Discrete-time signal processing,
London, Prentice Hall, 1989.
[18] A. G. Piersol, "Time delay estimation using phase data," IEEE
Trans. Acoustics, Speech, Signal Processing, June, vol. 29, no. 3, pt.
2, pp. 471-477, 1981.
[19] J. G. Proakis, M. Salehi, Communication systems engineering, Prentice Hall, 1994.
Bibliography
85
[20] L. Råde, and B. Westergren, BETA Mathematics Handbook, 2nd ed.,
Lund, Sweden, Studentlitteratur, 1993.
[21] S. Stein, "Algorithms for ambiguity function processing," IEEE
Trans. Acoustics, Speech, and Signal Processing, June, vol. ASSP29, no. 3, 1981.
[22] J. Tsui, Digital techniques for wideband receivers, Norwood, Artech
House, 2001.
[23] R. J. Ulman , and E. Geraniotis, "Wideband TDOA/FDOA processing using summation of short-time CAF’s," IEEE Trans. Signal
Processing, Dec., vol. 47, no. 12, 1999.
[24] E. Weinstein, and A. J. Weiss, "Fundamental Limits in passive
time delay estimation - part II: Wide-band systems," IEEE trans.
Acoustics, Speech, and Signal Processing, Oct., vol. 32, no. 5,
pp.1064-1077, 1984.
[25] D. P. Young, C. M. Keller, D. W. Bliss, and K. W. Forsythe, "Ultrawideband (UWB) transmitter location using time difference of arrival
(TDOA) techniques," Proc. Asilomar Conference on Signals, Systems, and Computers, Nov., Monterey, CA, USA, vol. 2 , pp. 1225 1229, 2003.
[26] Zhao Zhen, and Hou Zi-qiang, "The generalized phase spectrum
method for time delay estimation," Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP’84, Mar.,
vol. 3, pp. 46.2/1-4, 1984.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Related manuals

Download PDF

advertisement