PERFORMANCE ANALYSIS OF SPACE-TIME
ADAPTIVE ANTENNA ELECTRONICS FOR
TIME-OF-ARRIVAL ESTIMATION
A Thesis
Presented in Partial Fulfillment of the Requirements for
the Degree Master of Science in the
Graduate School of The Ohio State University
By
Andrew O’Brien, B.S.
* * * * *
The Ohio State University
2007
Master’s Examination Committee:
Prof. Prabhakar H. Pathak, Adviser
Dr. Inder J. Gupta
Prof. Lee Potter
Approved by
Adviser
Graduate Program inElectrical and Computer
Engineering
c© Copyright by
Andrew O’Brien
2007
ABSTRACT
Antenna arrays with adaptive signal processing are commonly utilized to allow the
reception of signals in harsh interference environments. Modern interference suppres-
sion typically involves the use of space-time adaptive processing (STAP). The primary
applications of STAP have been radar and communications systems, where interfer-
ence suppression performance is commonly optimized for output signal-to-noise ratio
(SINR); however, there is increasing demand for STAP in time-of-arrival estimation
applications, such as geolocation. Since output SINR is an inadequate indication
of TOA estimation performance, reconsideration of STAP for these applications is
required. It is understood that TOA estimation variance is primarily dependent on
post-correlation carrier-to-noise ratio (C/N). Consequently, the present work develops
a novel STAP algorithm which maximizes C/N. The distinction between SINR and
C/N will be established, and a performance analysis of the common STAP methods
will be performed in the context of this new algorithm. Additionally, the analysis
is extended to the TOA estimation variance. It will be demonstrated that optimal
STAP systems for TOA estimation follow from the optimal C/N algorithm. In the
end, this contributes not only useful performance bounds, but insight into the proper
design of STAP systems specifically for TOA estimation applications.
ii
VITA
January 24, 1981 . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Akron, OH
June 9, 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B.S. Electrical Engineering, The OhioState University, Columbus, Ohio,USA
June , 2004-present . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, Electro-Science Lab, Ohio State University
PUBLICATIONS
Research Publications
M.L. Rankin, A. O’Brien and I.J. Gupta, “Effects of Sampling Rate of STAP-BasedRFI Suppression Systems“. 2006 IEEE International Symposium on Antennas and
Propagation, Albuquerque,NM, Jul. 2006.
I.J. Gupta, C. Church, A. O’Brien, and C.Slick “Prediction of Antenna and AntennaElectronics Induced Biases in GNSS Receivers“. Institute of Navigation National
Technical Meeting, San Diego, CA, Jan. 2007.
A. O’Brien and I.J. Gupta, “Relation between Output SINR of GNSS Adaptive An-tennas and Receiver C/N Performance“. Joint Navigation Conference, Orlando, FL,Apr. 2007.
C. Church, A. O’Brien and I.J. Gupta, “Adaptive Antenna Induced Biases in GNSSReceivers“. Institute of Navigation Annual Meeting, Cambridge, MA, Apr. 2007.
iii
FIELDS OF STUDY
Major Field: Electrical and Computer Engineering
Studies in:
Space-Time Adaptive ProcessingGlobal Navigation Satellite SystemsElectromagnetics
iv
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Chapters:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. STAP System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Signal Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Antenna Array and Front-End Electronics . . . . . . . . . . . . . . 72.3 Space-Time Adaptive Processing . . . . . . . . . . . . . . . . . . . 92.4 STAP Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3. SINR and Carrier-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . 18
3.1 SINR and C/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Vector Form of Post-Correlation SINR . . . . . . . . . . . . . . . . 223.3 Post-Correlation Performance Results . . . . . . . . . . . . . . . . 253.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4. Max C/N Adaptive Beamforming . . . . . . . . . . . . . . . . . . . . . . 37
4.1 Max C/N Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Max C/N Performance . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Alternative Form of the Max. Post-Corr. SINR . . . . . . . . . . . 47
v
4.4 Convergence with Increasing Taps . . . . . . . . . . . . . . . . . . 484.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5. Optimum STAP Weights for TOA Estimation . . . . . . . . . . . . . . . 53
5.1 Optimum TOA Estimation . . . . . . . . . . . . . . . . . . . . . . 535.2 Derivation of the CRLB for TOA and Phase . . . . . . . . . . . . . 565.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Non-Ideal TOA Estimator . . . . . . . . . . . . . . . . . . . . . . . 655.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
vi
LIST OF FIGURES
Figure Page
2.1 Array of isotropic antenna elements used for simulations. . . . . . . 7
2.2 The space-time adaptive filter model. . . . . . . . . . . . . . . . . . 9
3.1 Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 24MHz system bandwidth in the absence ofinterference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 A closer view of one part of Figure 3.1. . . . . . . . . . . . . . . . . 29
3.3 Processing gain corresponding the results shown in Figure 3.1. . . . 30
3.4 The total system response (Hsys) of the antenna and STAP processorin the desired signal direction (θ = 0). . . . . . . . . . . . . . . . . . 30
3.5 The noise power spectrum (normalized) at the output of the STAPprocessor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 24MHz system bandwidth in the presence ofan interferer at θ = 80◦. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 A closer view of one part of Figure 3.6. . . . . . . . . . . . . . . . . 33
3.8 Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 34MHz system bandwidth in the absence ofinterference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.9 A closer view of one part of Figure 3.8. . . . . . . . . . . . . . . . . 35
vii
4.1 Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms (including the optimum C/N algorithm) with a 24MHzsystem bandwidth in the absence of interference. . . . . . . . . . . . 42
4.2 A closer view of one part of Figure 4.1. . . . . . . . . . . . . . . . . 43
4.3 The total system response (Hsys) of the antenna and STAP processorin the desired signal direction (θ = 0). . . . . . . . . . . . . . . . . . . 44
4.4 The normalized undesired component power spectral density at theoutput of the STAP processor. . . . . . . . . . . . . . . . . . . . . . 44
4.5 Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms (including the optimum C/N algorithm) with a 24MHzsystem bandwidth in the presence of an interferer at θ = 80◦. . . . . 45
4.6 A closer view of one part of Figure 4.5. . . . . . . . . . . . . . . . . 46
5.1 STAP/GNSS system model with two Cramer-Rao lower bounds forTOA estimation in non-white Gaussian noise. . . . . . . . . . . . . . 55
5.2 Minimum achievable TOA standard deviation for different STAP al-gorithms corresponding to the scenario in Figure 4.1. . . . . . . . . . 61
5.3 The results of Figure 5.2 but represented as variance and inverted. Thisform is directly comparable to the post-correlation SINR in Figure 4.1(no interference). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Minimum achievable TOA standard deviation for different STAP al-gorithms corresponding to the scenario in Figure 4.3. . . . . . . . . . 63
5.5 The results of Figure 5.4 but represented as variance and inverted. Thisform is directly comparable to the post-correlation SINR in Figure 4.3(one interferer). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.6 Cross-correlation functions for different STAP algorithms, correspond-ing to the system responses in Figure 3.5. . . . . . . . . . . . . . . . 66
5.7 TOA standard deviation of an NELP for different STAP algorithmscorresponding to the scenario in Figure 4.1. . . . . . . . . . . . . . . 68
5.8 TOA inverse variance of an NELP for different STAP algorithms cor-responding to the scenario in Figure 4.1. . . . . . . . . . . . . . . . . 69
viii
5.9 TOA standard deviation of an NELP for different STAP algorithmscorresponding to the scenario in Figure 4.3. . . . . . . . . . . . . . . 70
5.10 TOA inverse variance of an NELP for different STAP algorithms cor-responding to the scenario in Figure 4.3. . . . . . . . . . . . . . . . . 71
ix
CHAPTER 1
Introduction
Time-of-arrival (TOA) estimation forms the basis for a number of applications,
including ranging and navigation. As dependency on these systems increases, so does
demand for robust receivers that are able to operate in the presence of interference.
Particularly in the case of weak signals, these systems are vulnerable to interference
unless the receiver system has been designed to mitigate such interference. Without
mitigation, the interference power needs only to overcome the processing gain of the
spread spectrum ranging signals to prevent acquisition or introduce positioning errors.
The interference itself can come from a variety of sources. A receiver may be placed
in an environment with unintentional or natural radio frequency interference (RFI).
On the other hand, the interference could also be intentional jamming. While limited
interference suppression techniques have been developed for single antenna receiver
systems, significant suppression requires the use of a multi-antenna array with an
adaptable reception pattern. This adaptive array allows greater signal-to-interference-
plus-noise ratio (SINR) performance over a single element antenna by virtue of its
ability to provide beamforming/null steering in specific directions, thus giving the
system spatial degrees of freedom with which to separate the desired and undesired
signals. The potential application of adaptive antennas to interference suppression
in TOA applications, such as satellite navigation, has garnered increasing attention,
and recent research has been devoted to studying the performance of such systems.
The pattern of a modern adaptive antenna is adapted via digital filters whose
complex weights combine the signal from each antenna element. Each antenna ele-
ment is proceeded with its own finite impulse response (FIR) filter, the outputs of
which are combined together to produce a single output signal. The weights are
1
chosen to destructively cancel interference while constructively preserving the signal
of interest (SOI). Weighting algorithms exist which provide various capabilities such
as maximizing signal-to-noise ratio (SINR) or minimizing the mean-squared error
(MSE) between the incident and reference signals. If the FIR filters have a single tap,
then the resulting process is known as space-only processing. Space-only adaptive
processing permits sufficient rejection of narrow-band interference; however, greater
interference suppression can be accomplished by having a multi-tap filter behind each
antenna element. This adds a temporal filtering aspect to the spatial filtering, and the
resulting filter is commonly referred to as a space-time adaptive processor (STAP).
STAP filters allow better suppression of wideband interference and compensation of
array dispersion effects. These capabilities of STAP make it an attractive solution
to enhancing the robustness of receivers in harsh RFI and multipath environments.
Unfortunately, although STAP techniques have proven interference suppression abil-
ity, they were not designed specifically for receivers which perform TOA estimation.
Thus far, the effect of STAP on positioning systems has only been expressed by either
running numerical trials or limited analytic treatments. There exist no performance
bounds for the complete STAP receiver system, making the results of such trials only
relevant in measuring relative performance.
This thesis models and analyzes the combined STAP and receiver system in or-
der to design an optimal STAP algorithm specifically tailored for TOA estimation
applications. The particular application studied in the present work is global naviga-
tion satellite systems (GNSS). Toward this end, the present work contributes three
primary accomplishments:
1. The identification of performance bounds. Popular study of STAP for GNSS
systems currently lack a useful set of performance bounds; rather bounds are
used for either STAP or GNSS receivers independently. This paper will provide
bounds for optimal STAP performance in terms of post-correlation SINR as
well as delay and phase estimation variance. Simulations demonstrate that
existing techniques perform satisfactorily close to these bounds. In this way,
these bounds provide a useful perspective on the performance of current systems.
2
2. The identification of an optimal, theoretical system which achieves those bounds.
Thus far, the identification of an optimal system has been unavailable. Insight
can be gained by studying the behavior of an optimal STAP system. This study
identifies the optimal system with respect to post-correlation carrier-to-noise ra-
tio (C/N) which it calls maximum C/N beamforming. This work exhibits the
behavior of this ideal system and compares that behavior to existing algorithms.
3. The collection of a complete set of performance metrics. Traditional STAP
performance analysis observed general metrics such as output SINR and im-
plied that GNSS receiver performance would follow. However, this paper will
demonstrate how metrics such as pre-correlation SINR can be deceiving when
comparing relative performance of different STAP algorithms. By identifying
better metrics, STAP performance for TOA estimation applications can be more
accurately and confidently analyzed.
The document is organized as follows: Chapter 2 provides an overview of the
STAP receiver system model that we will use as the basis for this paper. It defines
a collection of equations general enough to encompass the major aspects of a mod-
ern adaptive antenna and GNSS receiver. Chapter 3 examines the difference between
pre-correlation SINR and post-correlation C/N performance measures. Although pre-
correlation SINR is a good performance measure, C/N is a more appropriate measure
to GNSS receivers. In fact, it will be shown that different STAP algorithms will have
different relative performance if one looks at their SINR as compared to their C/N.
Chapter 4 describes the maximum performance bound on C/N. Current STAP de-
signs are simulated and compared to this bound. Furthermore, a STAP beamforming
algorithm that achieves this bound is derived and its behavior studied. Chapter 5
brings STAP performance close to navigation performance by studying the variance of
time-of-arrival (TOA) estimation of a signal. The Cramer-Rao lower bound (CRLB)
for TOA estimation is found, and the optimum beamformer in relation to TOA esti-
mation variance is found. It will be shown that the maximum C/N beamformer with
a large number of taps converges to this bound. Additionally, by pairing different
STAP methods with an optimal TOA estimator, the performance bound for each
STAP method can be defined. The second half of Chapter 5 deals with a non-ideal,
3
implementable GNSS tracking loop. The performance of common STAP methods as
well as the maximum C/N algorithm will be studied in this context. Finally, Chapter
6 provides a summary and some conclusions.
4
CHAPTER 2
STAP System Model
We begin by developing a comprehensive analytic model of an adaptive antenna
and receiver. The application under consideration is global navigation satellite sys-
tems (GNSS), which is fundamentally a TOA estimation problem. Each of the fol-
lowing sections describes a particular component in detail. Section 2.1 provides an
overview of signal scenario which includes GNSS signals and interference incident on
the receiver. The structure of these signals is described in detail. Section 2.2 defines
the antenna and front-end hardware. The antenna is assumed to be an antenna array
whose individual antenna elements are treated generically as a direction-dependent
transfer function. The front-end electronics, which perform downconversion to base-
band and analog-to-digital conversion, will be similarly represented as transfer func-
tions. Section 2.3 goes into detail regarding the STAP processor. The STAP processor
performs interference suppression and digital beamforming by using an adaptive al-
gorithm which adjusts the filter weights based on the incident signals. The different
STAP algorithms discussed in this work are reviewed in Section 2.4. The antenna
array, front-end electronics, and STAP processor compose what is collectively known
as the adaptive antenna. The signal at the output of the adaptive antenna enters the
GNSS receiver, a simplified model of which is discussed in Section 2.5.
2.1 Signal Scenario
The work done in this study applies generally to any GNSS system, and specific
simulation parameters were chosen to correspond to common aspects thereof [1].
Each satellite transmits a spread-spectrum ranging signal unique to that satellite. A
receiver tracks four or more satellites simultaneously in order to determine navigation
5
information. However, for the purposes of this paper, we will only be focused on a
single signal, which will be referred to generally as the signal-of-interest (SOI). The
incident signal directions are represented by elevation angle, θ, and azimuth angle,
φ, where θ = 0◦ corresponds to zenith. The direct-path of the SOI is defined as
originating from direction (θd, φd), and no multipath will be considered in this study.
Each satellite transmits on multiple frequency ranges which contain multiple signals.
Each signal is composed of a ranging code modulated onto a carrier frequency. The
codes are pseudorandom noise (PRN) sequences of fixed length that are known by
the receiver, which correlates the locally generated reference signal, d(t), with the
received signal, d(t− τ0)ejψ, in order to determine the signal’s relative delay (τ0) and
phase (ψ) information. Each code is designed to be minimally correlated with the
other codes so that multiple signals can be transmitted on the same frequency by all
satellites. On top of the ranging code is low bitrate navigation data; however, this
work will not consider issues dealing with acquisition of data bits and will consider
the signal as only containing the PRN spreading code.
The primary modulation scheme used in GNSS today is binary phase shift keying
(BPSK) modulation. The ranging code creates a spread spectrum signal which occu-
pies a particular bandwidth based on its chipping rate. It is typically only necessary
to consider the SOI power spectral density,
Sd(f) = CdGd(f), (2.1)
where Cd is the signal power and Gd(f) is the normalized power spectral density,∫Gd(f)df = 1. (2.2)
Simulations in this study will work with a BPSK modulation with a 10 MHz chipping
rate which gives the signal a sinc-squared spectra with a 20MHz wide main lobe. The
power level is assumed to be 30dB below the noise floor before any antenna gain.
The interference is composed of M independent sources. The mth interference
signal will be denoted si,m(t), and its angle of arrival is (θi,m, φi,m). The interference
will be modeled as wide-sense stationary, zero-mean Gaussian noise, where the power
spectral density of the mth interferer is
Si,m(f) = Ci,mGi,m(f), (2.3)
6
where Ci,m is the power and Gi,m(f) is the normalized power spectral density,
∫Gi,m(f)df = 1. (2.4)
For simulations, the interference will have a flat power spectrum with a 20MHz band-
width and 40dB interference-to-noise ratio (INR). The entire signal scenario will be
considered constant over the simulation.
2.2 Antenna Array and Front-End Electronics
The antenna model used in this study represents a K-element antenna array. The
in situ volumetric pattern of the kth antenna element is represented by the direction
dependent transfer function Hka (θ, φ, f). For GNSS applications, the antenna should
be circular-polarized and operate over multiple frequency bands as required by the
GNSS receiver. The use of the antenna element transfer functions allow this study to
apply generally to any antenna; however, simulations performed will utilize a specific
7-element planar array of isotropic elements. A diagram of the layout is shown in
Figure 2.1. The array is a uniformly spaced circular arrangement of 6 elements at a
diameter of a half wavelength at the carrier frequency. The seventh element is placed
at the center. Though an antenna of this type ignores the realities of complex antenna
patterns and mutual coupling, the purpose of the simulations are merely to provide
added perspective to the more general equations.
Figure 2.1: Array of isotropic antenna elements used for simulations.
7
Directly attached to each antenna element is the front-end hardware, which down
converts the signal to baseband and performs analog-to-digital conversion (ADC).
Each antenna element in the array has its own RF channel with independent com-
ponents which typically begin with a low-noise amplifier (LNA). Subsequently, the
signal passes through mixers and band-pass filters that down convert the signal to
various intermediate frequencies (IF). The front-end hardware determines the system
bandwidth, which is typically slightly wider than the bandwidth of the SOI. This
work will assume the rate of sampling (both I and Q) of the ADC is equal to the
inverse of the system bandwidth. The complete system may resample the signal at
different points, such as before STAP processing or before the receiver. This paper
will assume a constant sampling rate, and simulations will use a system bandwidth
of 24MHz. If the antenna and front-end are designed to handle signals in different
frequency bands, then the front-end will often have separately designed channels for
each band. In this study, it is assumed that the front-end channel is a linear pro-
cess, independent of incident signal direction and is identical and equalized for all
channels. The front-end channel is represented by the transfer function Hfe(f) . For
simulations, it will be assumed that Hfe(f) is unity within the system bandwidth and
is zero outside of it so that it is essentially ignored. In equations, use of frequency, f ,
will refer to baseband, including antenna transfer functions and signal power spectra.
Aside from interference, the primary noise contribution is the thermal noise as
determined by the first amplifier in each front-end channel. The noise on the kth
channel will be denoted sn,k(t) and will be modeled as zero-mean Gaussian noise
which is independent between channels. Its power spectral density is assumed to be
constant over all frequencies as given by
Sn(f) = N0. (2.5)
Therefore, the total thermal noise power is
Cn =
∫Sn(f)df. (2.6)
Although noise is assumed to be white at the first stages of the front-end, as it passes
through the down-conversion process, it may become correlated between samples.
8
The digital signal at the output of the kth front-end channel will be denoted xk[n]
and is composed of three primary components,
xk[n] = xkd[n] + xkn[n] + xki [n] (2.7)
where xkd is the SOI component, xkn is the thermal noise component, and xki [n] contains
all of the interference.
Figure 2.2: The space-time adaptive filter model.
2.3 Space-Time Adaptive Processing
K digital baseband signals enter the STAP processor, one from each antenna front-
end channel. The STAP processor filters and combines these signals to destructively
remove interference while constructively preserving the SOI. The model is depicted in
Figure 2.2. Optimally, a STAP processor will be focused on preserving a single SOI,
and the digital antenna electronics will contain separate beamforming channels for
each SOI. This study will assume that this is the case and that each STAP processor
is designed to focus on a single SOI in a single frequency band. The STAP processor
9
is modeled as an N-tap digital FIR filter attached to each front-end channel output.
Each complex weight in the STAP filter is designated wkn, which corresponds to the
nth weight of the kth filter. The total kth filter is represented by the Nx1 vector
hkw = [ wk1 wk2 ... wkn ]T (2.8)
and the equivalent frequency domain transfer function of the filter is
Hkw(f) =
N∑
n=1
wknej2πf(τr−(n−1)T0). (2.9)
Here, τr represents the delay of the reference tap
τr = (nref − 1)T0, (2.10)
where T0 is the period between samples, and nref is an integer denoting the reference
tap. The STAP filters under consideration will have an odd number of taps, and nref
will be chosen as the center tap. Most commonly, the entire filter is represented in a
single KNx1 stacked vector
w =
h1w...
hKw
. (2.11)
The digital output of each front-end channel is denoted by xk[n], and the instan-
taneous signal snapshot on the taps of the kth filter is denoted
xk[n] =[xk[n+N − 1] . . . xk[n]
]T. (2.12)
The output of the kth filter, yk[n], is then
yk[n] = (xk[n])Thkw(2.13)
and all of the individual output signals are summed to produce the single output
signal of the entire STAP filter,
y[n] =K∑
k=1
yk[n]. (2.14)
The output STAP signal can be represented as the sum of the received signal snapshot
vector multiplied by the weight vector
y[n] = xT [n]w (2.15)
10
where x is the signal snapshot vector
x[n] =
x1[n]...
xK [n]
. (2.16)
In a manner similar to Equation 2.7, the signal snapshot vector can be decomposed
into independent components
x[n] = xd[n] + xu[n] , (2.17)
where xd and xu are the desired and undesired components, respectively. Sometimes
it will be useful to refer to the thermal noise and interference components of the
undesired part separately
xu[n] = xi[n] + xn[n]. (2.18)
The output signal can also be decomposed into components
y[n] = xT [n]w
= (xd[n] + xn[n] + xi[n])T w
= yd[n] + yn[n] + yi[n] (2.19)
where yd, yn and yi correspond to the respective signal components. This applies
similarly for each of the k components in Equation 2.13.
At the output of STAP processor, a signal incident on the antenna has traveled
through the entire adaptive antenna system. For each antenna element, it has been
processed by the antenna element response, the front-end response, and the STAP FIR
filter for that element. These components can be combined into a single, composite
transfer function which will be referred to as the adaptive antenna response or system
response. We define the direction dependent system response of the complete adaptive
antenna as
Hsys(θ, φ, f) =K∑
k=1
Hfe(f)Hka (θ, φ, f)Hk
w(f) . (2.20)
So, for instance, the desired component at the output of the adaptive antenna can be
represented as the convolution of the SOI with the time-domain system response
yd[n] = hsys(θd, φd) ∗ d(nT0 − τ0)ejψ. (2.21)
11
Similarly, the interference component is composed of M separate interference signals,
yi[n] =M∑
m=1
yi,m[n], (2.22)
where it follows that
yi,m = hsys(θm, φm) ∗ si[n] (2.23)
The power spectra at the output of the STAP filter follow simply from the above
definitions. The power spectral density of the desired component is
Sd(f) = F−1 {E {y∗dyd}}
= |Hsys(θd, φd, f)|2 Sd(f)
= Cd |Hsys(θd, φd, f)|2Gd(f). (2.24)
Since the interference sources are assumed be WSS and independent, the power
spectral density of the interference component is
Si(f) = F−1 {E {y∗i yi}}
= F−1
{M∑
m=1
E{yHi,myi,m
}}
=M∑
m=1
|Hsys(θm, φm, f)|2 Si,m(f)
=M∑
m=1
Ci,m |Hsys(θm, φm, f)|2Gi,m(f), (2.25)
and the power spectral density of the thermal noise component is
Sn(f) = F−1 {E {y∗nyn}}
=K∑
k=1
Hkw(f)Hfe(f)Sn(f)
= |Hn(f)|2Cn, (2.26)
where Hn is defined as the effective STAP system response on the thermal noise,
Hn(f) =
√√√√K∑
k=1
|Hkw(f)Hfe(f)|2 . (2.27)
12
2.4 STAP Algorithms
While the previous section referred to the STAP weights generally as w, this sec-
tion will describe specific STAP algorithms used in simulations in this study. The
STAP processor determines the weights using an adaptive algorithm which processes
partial information about the SOI in combination with received signal snapshots.
Conventionally, the weights are updated periodically in a weight update loop; how-
ever, this study will only consider the weights in the steady state.
The STAP processor forms a received signal correlation matrix estimate using N
received signal snapshots as in
Φ =1
N
N∑
n=1
x∗[n]xT [n]. (2.28)
Since the signal scenario is in the steady state, this paper will use the analytic corre-
lation matrix for calculations,
Φ = E{x∗[n]xT [n]
}. (2.29)
Since the components of the received signal are defined as independent, Φ can be
decomposed as
Φ = E{x∗xT
}(2.30)
= E{
(xd + xn + xi)∗ (xd + xn + xi)
T}
(2.31)
= E{x∗
dxTd
}+ E
{x∗
nxTn
}+ E
{x∗
ixTi
}(2.32)
= Φd + Φn + Φi, (2.33)
where Φd, Φn and Φi correspond to the desired, thermal noise and interference com-
ponents, respectively. Φu will be used to refer to the total undesired component,
Φn + Φi. The NxN submatricies of Φ can be written as
Φkl = E{x∗
k[n]xTl [n]}. (2.34)
where the elements of this submatrix for each component are
[Φd,kl]pq =
∫Sd(f)|Hfe(f)|2Hk∗
a (•)H la(•)e−j2πf(q−p)T0df (2.35)
[Φn,kl]pq =
∫Sn(f)|Hfe(f)|2e−j2πf(q−p)T0df (2.36)
[Φi,kl]pq =
∫ ( M∑
m=1
Si,m(f)|Hfe(f)|2Hk∗a (•)H l
a(•))e−j2πf(q−p)T0df (2.37)
13
and the antenna responses are taken in the direction of the corresponding SOI or
interference.
Four different STAP algorithms will be used for simulations in this study. They
will only be discussed briefly here, because the purpose of this paper is not to an-
alyze any particular one in detail. They were chosen collectively to be diverse and
representative of popular STAP techniques.
The first algorithm solves for the weights that maximize the pre-correlation SINR
at the output of the STAP processor. The weights satisfy the eigenvalue equation [3]
Φ−1u Φdw = λw. (2.38)
for the maximum eigenvalue, λ. It is not typically implemented since the STAP
processor will not have access to the desired and undesired correlation matrices sep-
arately; however, these weights provide a useful performance bound when observing
the SINR of different STAP algorithms. The use of these weights will be referred to
as the max SINR method.
The second method constrains the array such that the STAP output produces the
minimum mean-squared error (MMSE) with respect to some reference signal. The
weights are given by
w = Φ−1s, (2.39)
where s is known as the reference correlation vector. It is given by the equation
s = E {r[n]x∗[n]} , (2.40)
where r is the reference signal which is chosen to be correlated with the SOI, and x
are the received signal snapshots. In this case, the reference signal is chosen to be the
locally generated reference signal, d. The MMSE method is commonly implemented
as least-mean-squared (LMS) adaptive filters as discussed by Widrow et al. [4]. The
method is advantageous since it achieves near optimum SINR performance.
The next method is commonly referred to as simple beam steering. It constrains
the array to provide gain in the SOI direction while minimizing the total output power
of the STAP filter. Its weights are given by
w = Φ−1u (2.41)
14
where u corresponds to the steering vector,
u =
u1...
uk
. (2.42)
The kth subvector is given by
uk = [ 0 0 . . . u∗k . . . 0 0 ]T (2.43)
and u∗k (located at the reference tap) represents the conjugate of the voltage that
would be induced on the kth element due to the SOI. As a result, this method
requires knowledge of the SOI direction and the antenna array pattern. This method
is also known as directionally constrained minimum power and is dealt with by Van
Trees [5, pp. 513-516].
The last STAP algorithm, commonly implemented for its simplicity, constrains
the array such that the output power is minimized while a specified reference tap
remains on. This method is called simple power minimization and only performs
interference suppression. It does not provide gain by forming a beam in the SOI
direction; however, it also does not require knowledge of the antenna or SOI. This
method suppresses all strong signals indiscriminantly, and it should only be used
when the SOI is sufficiently weak, as is the case in GNSS systems. The weights are
given in an identical form as Equation 2.41, except that
uk =
{[ 0 0 . . . 1 . . . 0 0 ]T , if k = kr
0, if k 6= kr(2.44)
where kr is the index of the reference element. For our simulations, the center element
of the circular array will be chosen to be the reference element, as is typically the
case. A more detailed explanation is given by Compton [6]. It is also called power
inversion, and simulations will denote it as ’PI’.
All simulations are performed using STAP filters with a length of 7 taps.
2.5 Receiver
The output of the adaptive antenna enters the GNSS receiver. Modern receivers
have multiple receiver channels for simultaneous processing of multiple navigation
15
signals, and this study will focus on a single channel. Receivers have many modes of
operation, such as acquisition and tracking. This study will assume that the receiver
has acquired the signal and is operating in tracking mode. This implies that a good
estimate of the signal’s delay and phase has already occurred and that the receiver
is making incremental updates to that estimate. Tracking is typically implemented
as a tracking loop where a TOA (time-of-arrival) and phase estimator makes and
initial estimate of the relative delay and phase of the SOI. This estimate is then
combined with previous estimates and other sources of information to ultimately
create navigation estimates which are fed back into the system and tracking loop.
This work will focus only on the unprocessed estimate and not the dynamics of the
loop itself. As described elsewhere [9], to analyze the overall tracking performance it
is sufficient to measure noise and bias performance of the unprocessed TOA estimate.
Implementations of the TOA and phase estimators are considered to have two primary
stages. First, the receiver channel performs a finite correlation of the input signal with
a locally generated reference signal. This essentially estimates the value of the cross-
correlation function at different delays. Secondly, the estimates are processed by a
discriminator which produces delay and phase estimates.
The receiver generates a reference GNSS PRN code signal, d(t) and correlates it
with y[n] to produce the cross-correlation function estimate, Ryd(τ). For convenience,
this work will assume the reference signal is pure real and has unit magnitude. If we
assume the correlation is performed over N samples, Ryd has the form
Ryd(τ) =1
N
N∑
n=1
y[n]d(nT0 + τ) (2.45)
It can be decomposed into components
Ryd(τ) =1
N
N∑
n=1
(yd[n] + yu[n])d(nT0 + τ) (2.46)
= Rdyd(τ) + Ru
yd(τ) (2.47)
where Rdyd(τ) contains the desired signal component and Ru
yd(τ) contains the undesired
noise and interference. Since the desired portion is correlated with the reference signal
16
and N is sufficiently large, Rdyd(τ) can be approximated as its mean,
Rdyd(τ) ≈ Rd
yd(τ) (2.48)
=
∫Hsys(θd, φd, f)
√CdGd(f)ej2πfτdf. (2.49)
It is clear from Equation 2.49 that the total adaptive antenna response, Hsys, has the
potential to introduce distortion into the ideal cross-correlation function,
Rdd(τ) =
∫Gd(f)ej2πfτdf (2.50)
This distortion takes the form of phase bias, group delay bias, and general distortion of
the mainlobe of the cross-correlation function. These bias errors will not be discussed
in great length in this work. Rather, this work will focus on noise errors only.
The GNSS receiver finds estimates for the cross-correlation function at different
delays simultaneously. These estimates are then fed into a discriminator, which forms
phase and TOA estimates. There are many ways to implement the discriminator
function, each with different sensitivities to aspects of the cross-correlation estimates.
That is, each may produce different bias and noise errors in the TOA and phase
estimates depending on the different bias and noise errors in the cross-correlation
function estimate. Specific discriminators will be dealt with in more details in Chapter
5. Suffice it to say, the performance of such measurements are directly related to the
quality of the cross-correlation estimates, as will be proven.
2.6 Summary
This chapter introduced the collection of equations and assumptions which define
the STAP/GNSS model. This model is a representation of a STAP/GNSS receiver
beginning from the antenna and ending at delay and phase estimation. Using this
model, an optimal STAP algorithm will be derived for GNSS applications. As a first
step, the proceeding chapter will use this model to define an appropriate performance
metric to optimize.
17
CHAPTER 3
SINR and Carrier-to-Noise Ratio
Traditionally, the interference suppression performance of a STAP system is ana-
lyzed by observing its output signal-to-interference-plus-noise ratio (SINR). Although
SINR is a good measure of performance, it is not the absolute measure for GNSS re-
ceivers. In most GNSS receiver analysis, signal quality is measured by the effective
post-correlation carrier-to-noise ratio (C/N) [7]. Unlike the pre-correlation SINR that
only deals with the signal powers, C/N also takes into account the power-spectral den-
sities of the reference GNSS signal used in the correlator and the STAP output noise.
It will be shown that the noise at the output of the STAP filter is not necessarily
white and that different STAP algorithms have different output noise power spectra.
As a consequence, the receiver processing gain varies, and the pre-correlation SINR
performance of different algorithms will not necessarily be directly proportional to
post-correlation C/N performance. Using simulations, it will be demonstrated that
if only pre-correlation SINR is observed, one can be deceived into concluding that
certain STAP algorithms produce better noise suppression for a GNSS application
when the opposite is true. In fact, algorithms that lead to maximum SINR will not
lead to maximum C/N. In general, the post-correlation performance of the different
algorithms becomes more similar and, thus, the perceived advantages of some algor-
tihms over others becomes diminished. In all, the objective of this chapter will be to
motivate the usage of C/N as a more relevant metric for the analysis of STAP/GNSS
systems.
18
3.1 SINR and C/N
In our GNSS/STAP receiver model, the STAP filter is immediately followed by
a correlator. Therefore, the SINR at the STAP output is commonly refered to as
pre-correlation (or pre-integration) SINR. SINR itself is defined simply as the ratio
of output powers
ρ0 =Pd
Pn + Pi, (3.1)
where Pd, Pn, and Pi are the desired GNSS signal power, thermal noise power, and
interference power, respectively, at the output of the STAP processor. Common
vector expressions for these component powers are
Pd = 12E{|yd[n]|2
}=
1
2wTΦdw (3.2)
Pn = 12E{|yn[n]|2
}=
1
2wTΦnw (3.3)
Pi = 12E{|yi[n]|2
}=
1
2wTΦiw. (3.4)
It follows that Equation 3.1 can be expressed as
ρ0 =wHΦdw
wH(Φn + Φi)w. (3.5)
Equivalently, since these three signal components are modeled as WSS stochastic
processes, their powers can be represented using integrals of their respective power
spectral densities,
Pd =
∫Sd(f)df (3.6)
Pn =
∫Sn(f)df (3.7)
Pi =
∫Si(f)df. (3.8)
where their power spectral densities were derived in Chapter 2 to be
Sd = Cd |Hsys(θd, φd, f)|2Gd(f) (3.9)
Sn = |Hn(f)|2Gn(f) (3.10)
Si =M∑
m=1
Ci,m |Hsys(θm, φm, f)|2Gi,m(f). (3.11)
19
Here, it is implied that they have been bandlimited to the system bandwidth by the
front-end response. This allows the pre-correlation SINR to be represented in terms
of integrals
ρ0 =Cd∫|Hsys(θd, φd, f)|2Gd(f)df∫Sn(f) + Si(f)df
. (3.12)
While pre-correlation SINR is a convenient means of understanding STAP perfor-
mance, the performance analysis of GNSS receivers typically involves observing the
post-correlation SINR. The coherent post-correlation SINR is defined as the squared
mean of the correlator output ( when the reference signal is aligned in delay, frequency,
and phase ) divided by its variance [7],
ρ0 =
∣∣∣E{Ryd(τ0)}∣∣∣2
var{Ryd(τ0)}, (3.13)
where τ0 is the peak of Ryd(τ). Thus, τ0 corresponds to the relative delay between the
incident GNSS signal and the locally generated reference signal, including any biases
introduced by the adaptive antenna. In this paper we will use the tilde to distinguish
post-correlation properties from pre-correlation ones. Since the noise was defined to
be zero mean, the numerator contains only the desired component
E{Ryd(τ)} = Rdyd(τ). (3.14)
Thus, the numerator represents the post-correlation desired component power, which
is simply the square of the desired component of the cross-correlation function,
Pd(τ) =∣∣Rd
yd(τ)∣∣2
=
∣∣∣∣∫ √
CdHsys(θd, φd, f)Gd(f)ej2πfτ df
∣∣∣∣2
= Cd
∣∣∣∣∫Hsys(θd, φd, f)Gd(f)ej2πfτ df
∣∣∣∣2
(3.15)
Since the reference signal is independent of the output noise, it is well understood that,
statistically, the correlation acts as a filter with the same spectrum as the reference
signal and is independent of τ [8]. Therefore, the power spectral density of the output
20
noise is
Su(f) =1
TGd(f)Su(f)
=1
TGd(f)(Sn(f) + Si(f))
=1
TGd(f)Sn(f) +
1
TGd(f)Si(f)
= Sn(f) + Si(f). (3.16)
Since the desired component represents the mean, the variance is caused by the
undesired components and it follows that the post-correlation noise power is
Pu = var{Ryd(τ0)}
=
∫Su(f)
=1
T
∫Gd(f)Sn(f) +Gd(f)Si(f) df, (3.17)
which is independent of τ . This yields an equation for the post-correlation SINR
ρ0 =Pd(τ0)
Pu=TCd
∣∣∫ Hsys(θd, φd, f)Gd(f)ej2πfτ df∣∣2
∫Gd(f)Sn(f) +Gd(f)Si(f) df
. (3.18)
The effective post-correlation carrier-to-noise density ratio, C/N , is related to post-
correlation SINR by the integration length, T seconds, by
C/N =ρ0
T. (3.19)
By comparing the pre-correlation integrals in Equation 3.12 to the post-correlation
case in Equation 3.18, we see that the post-correlation undesired powers take into ac-
count the ”‘coupling”’ between the reference signal spectrum and the noise spectrum.
One way to quantify this effect is to measure the ratio of post-correlation SINR to
pre-correlation SINR,
Gp =ρ0
ρ0
. (3.20)
Since this gain is related to the despreading of the SOI, it can be thought of generally
as processing gain. If the STAP ouput noise is white (Su(f) = N0) and the total
system response on the desired signal is constant (Hsys(θd, φd, f) = α,τ0 = 0), then
21
the processing gain is simply T/T0,
Gp =ρ0
ρ0
=T∣∣∣∫Sd(f)ej2πfτ0 df
∣∣∣2
∫Gd(f)Su(f) df
∫Su(f) df∫Sd(f) df
=T∣∣√Cd
∫Hsys(θd, φd, f)Gd(f)ej2πfτ0 df
∣∣2 ∫ N0 df
C2d
∫Gd(f)N0df
∫|Hsys(θd, φd, f)|2Gd(f) df
=TCdα
2∣∣∫ Gd(f)ej2πfτ0 df
∣∣2N0B
N0Cdα2∫Gd(f)df
∫Gd(f)df
= T/T0, (3.21)
which is the common interpretation of processing gain for an unperturbed DS-SS
signal in white noise. However, the conditions on the noise spectrum and desired
signal system response that allow for this result are not true in a realistic STAP
filter. It will be shown via the result of simulations in Section 3.3 that different
STAP algorithms can produce a variety of noise power-spectral densities and system
responses in the desired signal direction. While these distortions are not enough to
prevent SINR from being a useful metric, it will be shown that they are enough to
change the relative performance of different STAP algorithms.
3.2 Vector Form of Post-Correlation SINR
The SINR at the output of the STAP processor is commonly expressed in a vector
form as in Equation 3.5. The following is the derivation of an equation for post-
correlation SINR in a similar form. As introduced in Chapter 2, the receiver cross-
correlation function estimate between the STAP output, y, and the reference signal,
d, is
Ryd(τ) =1
N
N∑
n=1
y[n]d(nT0 + τ), (3.22)
where T0 is the sample spacing. If the STAP weight vector is assumed constant over
the interval n ∈ {1, ..., N}, then we can write the cross-correlation function in the
22
form
Ryd(τ) =1
N
N∑
n=1
y[n]d(nT0 + τ)
=1
N
N∑
n=1
(xT [n]w)d(nT0 + τ)
=
[1
N
N∑
n=1
xT [n]d(nT0 + τ)
]w
= sT (τ)w (3.23)
where we have defined the correlation signal vector s(τ) as
s(τ) =1
N
N∑
n=1
x[n]d(nT0 + τ). (3.24)
Equation 3.23 is notable in that it demonstrates that a STAP filter can be equiva-
lently applied post-correlation to produce the same cross-correlation values. This is
commonly known as post-correlation beamforming, and has been utilized elsewhere
[12].
Starting with Equation 3.24, if x[n] is separated into its independent components,
s can be represented as independent components similarly as
s(τ) =1
N
N∑
n=1
x[n]d(nT0 + τ) (3.25)
=1
N
N∑
n=1
(xd[n] + xn[n] + xi[n]) d(nT0 + τ) (3.26)
= sd(τ) + sn(τ) + si(τ) (3.27)
where sd, sn and si are the desired, thermal noise, and interference components,
respectively. Since xd contains the desired signal, sd(τ) will quickly converge for τ
near τ0 and can be approximated as its mean,
sd(τ) = E{sd(τ)}
= E{xd[n]d(nT0 + τ)}. (3.28)
If τ = τ0, then sd(τ) is commonly refered to as the reference correlation vector [2].
However, this work will consider it for arbitrary τ . It is a KNx1 vector whose elements
23
are
[sd(τ)]kn =
∫ √CdGd(f)Hk
a (θd, φd, f)ej2πf((nref−n)T0+τ)df. (3.29)
The equation for a single estimate of Ryd(τ) can be decomposed as
Ryd(τ) = sT (τ)w
= (sTd (τ) + sTn (τ) + sTi (τ))w
= sTd (τ)w + (sTn (τ) + sTi (τ))w
= Rdyd(τ) + Ru
yd(τ), (3.30)
where Rdyd and Ru
yd are the desired and undesired components, respectively. Since
the correlation in Equation 3.24 is finite, s is still a random variable and, as a result,
Ryd(τ) is a random variable as well. Since the noise and interference have zero mean,
then the mean of Equation 3.30 is based on the desired component and the variance
is the noise power at a particular τ . We have defined the noise and interference com-
ponents as uncorrelated with the reference signal, d(t), and it is thereby independent
of τ . Generalzing Equation 3.13 to be a function of τ , the SINR of the correlator
output is then given by
ρ0(τ) =
∣∣∣E{Ryd(τ)}∣∣∣2
var{Ryd(τ)}
=
∣∣Rdyd(τ)
∣∣2
E{∣∣∣Ru
yd(τ)∣∣∣2
}
=
∣∣sTd (τ)w∣∣2
E{|(sn(τ) + si(τ))Tw|2}
=wHs∗d(τ)s
Td (τ)w
E{wH (sn + si)∗(sn + si)Tw}
=wHs∗d(τ)s
Td (τ)w
wHE{(sn + si)∗(sn + si)T}w
and finally
ρ0(τ) =wHΦd(τ)w
wH(Φn + Φi)w(3.31)
24
where we have defined the respective correlation matricies of the post-correlation
components as
Φd(τ) = s∗d(τ)sTd (τ) (3.32)
Φn = E{s∗ns
Tn
}(3.33)
Φi = E{s∗i s
Ti
}. (3.34)
The undesired correlation matrices Φn and Φi have simple interpretations that can
be understood from Equation 3.16. The two matricies have forms similar to 2.36
and 2.37 except with an additional response, Gd(f)/T , which represents the effect of
correlation with the reference signal,
[Φn,kl
]pq
=
∫1
TSn(f)Gd(f)e−j2πf(q−p)T0df (3.35)
[Φi,kl
]pq
=
∫1
TSi(f)Gd(f)e−j2πf(q−p)T0df. (3.36)
3.3 Post-Correlation Performance Results
This section presents simulation results comparing pre-correlation and post-correlation
SINR of different STAP algorithms in interference scenarios. The objective is to
demonstrate that there can be significant discrepencies between pre-correlation and
post-correlation SINR, specifically in gauging the relative performance of different
STAP algorithms. Details of the simulation setup is as described in Chapter 2. The
antenna array of 7 isotropic elements was simulated in the presence of a single SOI.
The SOI was weak ( 30dB below the noise floor) and had a sinc-squared power spec-
trum with a 20MHz wide mainlobe. Its angle-of-incidence was φ = 0◦ and its elevation
was swept from θ = −90◦ to 90◦. The SINR performance was measured for each of
these angles using the the 4 different STAP algorithms previously discussed. All have
7 taps, and the system bandwidth is 24 MHz unless otherwise noted.
Figure 3.1 shows the pre-correlation and post-correlation SINR performance of
different STAP algorithms in the absence of interference. Despite both pre-correlation
and post-correlation SINR curves being very similar in shape, upon closer inspection
it is apparent that there are differences. Figure 3.2 shows a closer view of a portion
of Figure 3.1. Comparing the SINR performance of the different beamformers, it is
25
clear that the maximum SINR beamformer, as expected, achieves the optimum pre-
correlation SINR. MMSE beamforming is one half dB below it, followed by the simple
beam steering and power inversion methods. However, if we compare their relative
performance in post-correlation SINR, we see a different order. MMSE is now 1.5
dB better than the max SINR weights. In fact, simple beam steering is a full 2 dB
better the max SINR. Power inversion has closed the performance gap in relation to
the other algorithms significantly. The different conclusions from different metrics are
due to different processing gains for each method and is significant enough to change
the relative performance between the different methods. Furthermore, the separation
between the best and worst performing STAP methods have changed from 13 dB to
7.5 dB.
To understand why there is such a discrepency between the pre-correlation and
post-correlation SINR, we will first consider the processing gain. Going back to the
case in Figure 3.1 with 24 MHz system bandwidth, Figure 3.3 shows the processing
gain for the different algorithms. There is an additional line which shows the white
noise processing gain. It is clear that processing gain for the max SINR and MMSE
algorithms are worse than the standard processing gain for white noise. The reason for
this is evident in Figure 3.4 and 3.5. Figure 3.4 shows the normalized system response
of the complete adaptive antenna in the SOI direction for θ = 0. It is clear that the
max SINR and MMSE methods approximately form matched responses, while power
inversion and simple beam steering (whose plots overlap) have flat responses.
Figure 3.5 shows the normalized output noise power spectral density of the STAP
processor. The conclusions are similar to those made for Figure 3.4. Some algorithms
produce an approximately matched spectrum. A matched spectrum improves the
pre-correlation SINR performance but is detrimental to the processing gain. This
trade-off is what accounts for the significant discrepancy between pre-correlation and
post-correlation SINR performance.
Figure 3.6 shows the same system in the presence of a single interferer. The
interference is wideband (20 Mhz) and strong (40dB above the noise floor) with a flat
power spectral density. It was incident from φ = 0◦ and θ = 80◦. Figure 3.7 shows
a zoomed-in portion of the curves. Even in the presence of interference, the previous
conclusions hold. This is true both close to the interference as well as far away
26
from it. It should be noted that the power minimization algorithm does not perform
beamforming, just null steering. For this reason, its performance (and especially its
null width) is significantly worse that the other STAP methods. Despite this fact, its
relative performance is drastically different based upon which metric is used.
Another advantage of observing post-correlation SINR instead of pre-correlation
SINR is that it accurately handles the effects of altering the system bandwidth. Figure
3.8 shows the identical scenario as Figure 3.1 except the system bandwidth has been
increased from 24 to 34 MHz. The thermal noise power has been maintained at a fixed
total power over the entire bandwidth. Figure 3.9 shows a section of the curves more
closely. If Figure 3.9 (a) was observed in isolation, it might be erroneously concluded
that some algorithms are 6dB better than others; however, there is actually less than
1dB difference in post-correlation SINR between them (ignoring the PI case). When
comparing these results to the 24 MHz system bandwidth case in Figure 3.1, it is
clear that the pre-correlation SINR curves have changed while the post-correlation
metrics have remained approximately the same.
27
−80 −60 −40 −20 0 20 40 60 80−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation
−80 −60 −40 −20 0 20 40 60 80−10
−5
0
5
10
15
20
25
30
35
40
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation
Figure 3.1: Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 24MHz system bandwidth in the absence of interference.
28
−70 −60 −50 −40 −30 −20−40
−35
−30
−25
−20
−15
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation (zoomed)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation (zoomed)
Figure 3.2: A closer view of one part of Figure 3.1.
29
−80 −60 −40 −20 0 20 40 60 8047
48
49
50
51
52
53
54
55
Pro
cess
ing
Gai
n (d
B)
Elevation (deg)
PIBSMMSEMax SINRWhite
Figure 3.3: Processing gain corresponding the results shown in Figure 3.1.
−10 −5 0 5 10−30
−25
−20
−15
−10
−5
0
5
Hsy
s (dB
)
Freq. (MHz)
PIBSMMSEMax SINR
Figure 3.4: The total system response (Hsys) of the antenna and STAP processor inthe desired signal direction (θ = 0).
30
−10 −5 0 5 10−30
−25
−20
−15
−10
−5
0
5
Su ,p
re−
corr
(dB
)
Freq. (MHz)
PIBSMMSEMax SINR
Figure 3.5: The noise power spectrum (normalized) at the output of the STAP pro-cessor.
31
−80 −60 −40 −20 0 20 40 60 80−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation
−80 −60 −40 −20 0 20 40 60 80−10
−5
0
5
10
15
20
25
30
35
40
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation
Figure 3.6: Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 24MHz system bandwidth in the presence of an interfererat θ = 80◦.
32
−70 −60 −50 −40 −30 −20−40
−35
−30
−25
−20
−15
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation (zoomed)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation (zoomed)
Figure 3.7: A closer view of one part of Figure 3.6.
33
−80 −60 −40 −20 0 20 40 60 80−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation
−80 −60 −40 −20 0 20 40 60 80−10
−5
0
5
10
15
20
25
30
35
40
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation
Figure 3.8: Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms with a 34MHz system bandwidth in the absence of interference.
34
−70 −60 −50 −40 −30 −20−40
−35
−30
−25
−20
−15
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(a) SINR, pre-correlation (zoomed)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINR
(b) SINR, post-correlation (zoomed)
Figure 3.9: A closer view of one part of Figure 3.8.
35
3.4 Summary
It has been demonstrated that pre-correlation SINR can be a deceiving perfor-
mance metric if used to analyze the relative performance of different STAP algo-
rithms. The processing gain cannot be accurately approximated simply by the inte-
gration time since the effect of the STAP filter on the desired signal and thermal noise
is significant. Some STAP algorithms produce matched responses which improve pre-
correlation SINR but are detrimental to processing gain. Post-correlation SINR or
effective post-correlation C/N is a much more applicable metric, and this chapter has
presented a simple vector form for it. By the nature of this form’s similarity common
STAP vector equations for the pre-correlation SINR, the proceeding chapter will be
able to bring C/N into the context of STAP analysis.
36
CHAPTER 4
Max C/N Adaptive Beamforming
When observing pre-correlation SINR performance, the weights which maximize
the SINR provide a useful bound with which to objectively compare different STAP
algorithms. However, the previous chapter demonstrated that there is a noticeable
discrepancy between pre-correlation SINR and post-correlation SINR performance.
Since post-correlation performance is a more appropriate metric for STAP/GNSS
systems, it would be preferable to establish a maximum post-correlation SINR bound
instead. This chapter will derive this bound and the weights that achieve it, which
will be referred to as the max C/N weights. Simulations will compare the performance
and behavior of these optimum weights to current STAP techniques. Additionally,
the convergent behavior of the max C/N algorithm will be studied as the number of
taps in the STAP filter become large. This will provide a general post-correlation
SINR bound which is independent of the STAP filter length. While the max C/N
algorithm achieves the theoretical bound, its main utility will be in demonstrating
that many common implementations provide performance very close to this bound.
4.1 Max C/N Weights
In this section, the weights which maximize the effective post-correlation C/N
will be derived. Let the weight vector which achieves the maximum pre-correlation
SINR be denoted w0. It has been shown [2] that w0 satisfies the maximum eigenvalue
equation
Φ−1u Φdw0 = λw0. (4.1)
That is, w0 is the eigenvector of Φ−1u Φd corresponding to the maximum eigenvalue.
This solution applies generally to any equation of the same form as Equation 3.5.
37
Therefore, to find the maximum post-correlation SINR, Equation 3.31 is used, and
the resulting optimal weights, w0, are
Φ−1u Φd(τ)w0 = λw0. (4.2)
Since the post-correlation SINR and C/N are proportionate via Equation 3.19, the
weights, w0, maximize the effective post-correlation C/N. As a result, this study will
refer to them as the max C/N weights. As is evident in the above discussion, the
max C/N weights can be understood as the maximum SINR algorithm applied in a
post-correlation beamformer. This assumes that Φu is invertible, which is the case
for a full bandwidth Gd(f).
There is no need to directly solve the eigenvalue problem in Equation 4.2 to find the
weights. Rather, by simplifying the equation, the weights are shown to be equivalent
to
w0 = Φ−1u sd(τ),
where sd is the reference correlation vector defined in Equation 3.28. To show this,
first note that the matrix Φ−1u Φd is Hermitian and positive semi-definite. Since Φd
has the form Φd(τ) = s∗d(τ)sTd (τ), it follows that rank(Φ−1
u Φd) = 1. Therefore, Φ−1u Φd
has only 1 non-zero eigenvalue which is also the maximum eigenvalue, and the weights
can be put into the form
λw0 = Φ−1u s∗d(τ)s
Td (τ)w0 (4.3)
λw0 = Φ−1u s∗d(τ)(s
Td (τ)w0) (4.4)
w0 = (sTd (τ)w0
λ)Φ−1
u s∗d(τ) (4.5)
w0 = cΦ−1u s∗d(τ), (4.6)
where c is some complex scalar. The scalar does not effect the SINR performance of
the STAP filter since scaling the filter scales all components equally. The fact that
the scalar is complex means it may introduce phase bias into the SOI; however, this
work is not concerned with bias errors. Ignoring the scalar leads to the simplified
form for the max C/N weights,
w0(τ) = Φ−1u sd(τ), (4.7)
38
which have been defined as a function of τ , the relative delay between the incident
and reference signals. The filter, w0, is chosen at a particular τ , which optimizes the
C/N for that delay. Since these weights maximize the effective post-correlation C/N,
an appropriate name for their use is maximum C/N adaptive beamforming.
Correlation with the reference signal causes a wideband SOI to have a rank 1
correlation matrix. This property, used in the above conclusion, is the same used to
define the maximum pre-correlation SINR weights for CW desired signals [2]. The
fact that they are both rank 1 is a useful connection since there has been extensive
STAP research which involves the SINR of CW desired signals. As a result, much
of the mathematics used to analyze the SINR of CW desired signals will analogously
apply to the post-correlation SINR of wideband signals.
An alternative derivation of the max C/N weights follows from the minimization
of mean-squared error (MSE) on the cross-correlation function. Consider the error
between the ideal cross-correlation function and the estimated one at a particular
delay, τ ,
ε = Rdd(τ) − Ryd(τ) (4.8)
= Rdd(τ) − wH s(τ). (4.9)
The MSE is then
MSE = E{|ε|2}
(4.10)
= |Rdd(τ)|2 + wHΦw − 2Re{Rdd(τ)sTd (τ)w∗} (4.11)
Taking the gradient to solve for the weights
∇ = 0 + 2Φ(τ)w − 2Rdd(τ)sd(τ) = 0 (4.12)
w0 = Rdd(τ)(Φ(τ))−1sd(τ). (4.13)
This form can be further reduced by following the steps in Compton [2, pp. 55].
First, to total correlation matrix is given by
Φ(τ) = Φu + csd(τ)s∗
d(τ). (4.14)
Since Φu is nonsingular, the matrix inversion lemma leads to
(Φ(τ))−1 = Φ−1u + βΦ−1
u sd(τ)s∗
d(τ)Φ−1u , (4.15)
39
where
β =c
1 + cs∗d(τ)Φ−1u sd(τ).
(4.16)
Multiplying both sides by sd(τ) yields,
(Φ(τ))−1sd(τ) = Φ−1u sd(τ) + βΦ−1
u sd(τ)s∗
d(τ)Φ−1u sd(τ) (4.17)
= (1 + βs∗d(τ)Φ−1u sd(τ))Φ
−1u sd(τ). (4.18)
So Φ−1u sd(τ) is just a scalar multiple of Φ−1
u sd(τ), and Equation 4.13 is equivalent to
the max C/N weights
w0(τ) = Φ−1u sd(τ). (4.19)
Therefore, max C/N beamforming minimizes the error at a particular τ on the cross-
correlation function. Since the error corresponds to noise on the cross-correlation
estimate, it makes sense that maximizing the post-correlation SINR at that delay
would minimize that error.
4.2 Max C/N Performance
This section presents simulation results comparing pre-correlation and post-correlation
SINR of different STAP algorithms in interference scenarios. The results are similar
to those seen in Chapter 3; however, they include the results for the max C/N al-
gorithm. Details of the simulation setup is as described in Chapter 2. The antenna
array of 7 isotropic elements was simulated in the presence of a single SOI. The SOI
was weak signal ( 30dB below the noise floor) and had a sinc-squared power spectrum
with a 20MHz wide mainlobe. Its angle-of-incidence was φ = 0◦ and its elevation was
swept from θ = −90◦ to 90◦. The SINR performance was measured for each of these
angles using the the 4 different STAP algorithms previously discussed. All have 7
taps, and the system bandwidth is 24 MHz.
Figure 4.1 shows the pre-correlation and post-correlation SINR performance of
different STAP algorithms in the absence of interference. The performance of the dif-
ferent STAP methods can be considered in the context of the maximum performance
bound. Figure 4.2 shows a closer view of a portion of Figure 4.1. Comparing the
pre-correlation SINR performance of the different beamformers, it is interesting to
40
note that the max C/N weights appear to perform poorly. This conforms to the con-
clusion in Chapter 3 which showed pre-correlation SINR can be a deceiving metric.
If their relative performance is compared by observing post-correlation SINR, it is
clear the max C/N weights achieve optimum performance. However, there are other
algorithms which perform within a dB of the post-correlation SINR bound. On the
other hand, those same algorithms are very distant from the max SINR bound in the
pre-correlation SINR performance.
Figure 4.3 shows the total system response of the max C/N adaptive antenna
in the SOI direction. It is clear that, unlike some algorithms in Figure 3.5, max
C/N is not forming a matched filter. Section 4.4 will provide exact equations for the
system response as the number of taps become large. Figure 4.4 shows normalized
power spectrum of the undesired noise at the output of the STAP processor. The
response on the noise is flat and the thermal noise will remain essentially white at
the output of the STAP system, though the exact power spectral density will be
given in Section 4.4 . This behavior verifies that the optimal post-correlation SINR
performance does not come from forming a matched filter like is the case for the max
SINR and MMSE methods. Furthermore, the poor pre-correlation SINR performance
of the max C/N algorithm demonstrates that it is a deceiving performance metric
and that post-correlation SINR should be used instead.
Figure 4.5 shows the case where a single interferer is present. The interference
was wideband (20 Mhz) and strong (40dB above the noise floor) with a flat power
spectral density. It was incident from φ = 0◦ and θ = 80◦. Figure 4.6 shows a zoomed
in portion of the curves. Again, it is clear that even in the presence of interference,
the previous conclusions hold.
41
−80 −60 −40 −20 0 20 40 60 80−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) SINR, pre-correlation
−80 −60 −40 −20 0 20 40 60 80−10
−5
0
5
10
15
20
25
30
35
40
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) SINR, post-correlation
Figure 4.1: Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms (including the optimum C/N algorithm) with a 24MHz systembandwidth in the absence of interference.
42
−70 −60 −50 −40 −30 −20−40
−35
−30
−25
−20
−15
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) SINR, pre-correlation (zoomed)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) SINR, post-correlation (zoomed)
Figure 4.2: A closer view of one part of Figure 4.1.
43
−10 −5 0 5 10−30
−25
−20
−15
−10
−5
0
5
Hsy
s (dB
)
Freq. (MHz)
PIBSMMSEMax SINRMax C/N
Figure 4.3: The total system response (Hsys) of the antenna and STAP processor inthe desired signal direction (θ = 0).
−10 −5 0 5 10−30
−25
−20
−15
−10
−5
0
5
Su ,p
re−
corr
(dB
)
Freq. (MHz)
PIBSMMSEMax SINRMax C/N
Figure 4.4: The normalized undesired component power spectral density at the outputof the STAP processor.
44
−80 −60 −40 −20 0 20 40 60 80−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) SINR, pre-correlation
−80 −60 −40 −20 0 20 40 60 80−10
−5
0
5
10
15
20
25
30
35
40
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) SINR, post-correlation
Figure 4.5: Comparison of pre-correlation and post-correlation SINR of differentSTAP algorithms (including the optimum C/N algorithm) with a 24MHz systembandwidth in the presence of an interferer at θ = 80◦.
45
−70 −60 −50 −40 −30 −20−40
−35
−30
−25
−20
−15
Pre
−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) SINR, pre-correlation (zoomed)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
Pos
t−C
orre
latio
n S
INR
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) SINR, post-correlation (zoomed)
Figure 4.6: A closer view of one part of Figure 4.5.
46
4.3 Alternative Form of the Max. Post-Corr. SINR
In this section, we will discuss an alternative form of the post-correlation SINR
which will have utility throughout the rest of this paper. As discussed earlier, the
post-correlation SINR can be written in vector form as
ρ0(τ) =wHΦd(τ)w
wHΦuw. (4.20)
Equation 4.20 can be represented alternatively as
ρ0(τ) =tr{Φd(τ)W
}
tr{ΦuW
} (4.21)
where
W = wwH . (4.22)
In the case of max C/N beamforming, the weight matrix corresponds to
w0wH0 = Φ−1
u Φd(τ)Φ−1u , (4.23)
since Φ−1u is Hermetian. Plugging this into Equation 4.21,
ρ0(τ) =tr{Φd(τ) Φ−1
u Φd(τ) Φ−1u
}
tr{Φu Φ−1
u Φd(τ) Φ−1u
} (4.24)
=
tr
{(Φ−1u Φd(τ)
)2}
tr{
Φ−1u Φd(τ)
} (4.25)
= tr{
Φ−1u Φd(τ)
}(4.26)
=⟨
Φ−1u , Φ∗
d(τ)⟩
(4.27)
This form has removed the ratio of powers from Equation 4.21, leaving a single trace
or, equivalently, an inner product between NxN matrices as
〈A,B〉 =∑
i
∑
j
a∗i,jbi,j. (4.28)
The correlation matrices can be interpreted as statistical spatio-temporal represen-
tations of the desired and undesired signals, and the maximum C/N is related to
47
the degree of their independence or separability. Furthermore, the maximum post-
correlation SINR is equivalent to the single eigenvalue of Φ−1u Φd(τ),
tr{Φ−1u Φd(τ)
}= λ, (4.29)
which, is similar to the pre-correlation SINR for CW desired signals [2].
4.4 Convergence with Increasing Taps
It has been widely observed that the performance of a STAP processor converges
as the number of taps increases. This section will analyze the behavior of max C/N
beamforming as the number of taps becomes very large. Specifically, equations will
be derived for the converged maximum C/N which will provide a bound that does
not depend on the number of taps. Furthermore, the converged output noise power-
spectral density and the system response in the desired signal direction will also be
derived.
The post-correlation undesired component power can be represented as
Pu =
∫Su(f)df (4.30)
=1
T
∫Su(f)Gd(f)df (4.31)
=1
T
∫(Si(f) + Sn(f))Gd(f)df (4.32)
=1
T
∫ ( M∑
m
Si,m(f) + Sn(f)
)Gd(f)df (4.33)
=1
T
∫ ( M∑
m
Ci,m |Hsys(•)|2Gi,m(f)
+ Cn |Hn(f)|2Gn(f)
)Gd(f)df (4.34)
=1
T
∫ ( M∑
m
Ci,m
( K∑
i
K∑
j
Hwi (f)Hw
j (f) Ha∗i (•)Ha
j (•))Gi,m(f)
+ Cn
K∑
k
|Hwk (f)|2Gn(f)
)Gd(f)df (4.35)
=⟨H∗
w, Hu
⟩(4.36)
48
where the matrices Hw and Hu are KxK with the following functions as elements
Hw[i, j] = Hw∗i (f)Hw
j (f) (4.37)
Hu[i, j] = Hi[i, j] + Hn[i, j] (4.38)
Hi[i, j] =1
T
M∑
m=1
Ha∗i (θm, φm, f)Ha
j (θm, φm, f)Ci,mGi,m(f)Gd(f) (4.39)
Hn[i, j] =
{1TCnGn(f)Gd(f) i = j
0 i 6= j, (4.40)
and the inner product is defined as
〈A,B〉 =K∑
i=1
K∑
j=1
∫a∗i (x)bj(x) dx. (4.41)
The pre-correlation undesired matrices follow similarly,
Hu =1
THuHs (4.42)
where Hs is a diagonal matrix containing the SOI power spectrum
Hs[i, j] =
{Gd(f) i = j
0 i 6= j. (4.43)
To begin, the cross-correlation function can be decomposed into k components,
one for each element,
Rdyd(τ) =
∫Syd(f)e−j2πfτ df (4.44)
=
∫ √CdHsys(θd, φd, f)Gd(f)e−j2πfτ df (4.45)
=
∫ K∑
k
√CdH
ak (θd, φd, f)Hw
k (f)Gd(f)e−j2πfτ (4.46)
=K∑
k
(∫ √CdH
ak (θd, φd, f)Hw
k (f)Gd(f)e−j2πfτ df
)(4.47)
=K∑
k
(∫Syd,k(f)e−j2πfτ df
). (4.48)
49
The post-correlation desired signal power, Pd, is then
Pd(τ) =∣∣Rd
yd(τ)∣∣2 (4.49)
=K∑
i
K∑
j
(∫Syd,i(f1)e
−j2πf1τdf1
)∗(∫
Syd,j(f2)e−j2πf1τdf2
)(4.50)
=K∑
i
K∑
j
∫ ∫S∗
yd,i(f1)Syd,j(f2)e−j2π(f2−f1)τdf1df2 (4.51)
=K∑
i
K∑
j
∫ ∫Hw∗i (f1)H
wj (f2)H
a∗i (•)Ha
j (•)
CdGd(f1)Gd(f2) e−j2π(f2−f1)τ df (4.52)
where, if only the diagonal (f1 = f2) terms of the integral is taken, we define
Hd[i, j] = CdG2d(f)Ha∗
i (θd, φd, f)Haj (θd, φd, f). (4.53)
For the purposes of this paper, it will be assumed that Hs is invertible. Now the
post-correlation SINR can be written as, given the results above,
ρ0(τ) =Pd(τ)
Pu(4.54)
which is the same form as Equation 4.21. This implies that it should similarly reduce
to a form like Equation 4.27. It should be noted that Hw is the only part that
accounts for the number of taps. We will let the number of taps be large enough to
make the following equality and set
Hw = H−1u HdH
−1∗u , (4.55)
which implements a max C/N beamformer as in Equation 4.23. Substituting this
HW back into the post-correlation SINR equation ultimately yields
ρ0(τ) =⟨H−1u , Hd
⟩(4.56)
which has the same form as 4.27. This is the converged maximum post-correlation
SINR and is useful as a performance bound independent of the number of taps in the
STAP filter. The inverse of a matrix of functions can be efficiently found by treating
each element as a diagonal matrix and using a blockwise matrix inversion formula.
50
This makes it much quicker to calculate than could be achieved by simply choosing
an arbitrarily large number of taps.
To determine the converged power spectral density of the noise at the output of
the max C/N STAP filter, we begin with the post-correlation undesired power given
in Equation ??,
Pu =⟨H−1u , H∗
d
⟩(4.57)
=
⟨1
TH−1u H−1
s , H∗
d
⟩(4.58)
=1
T
K∑
i
K∑
j
∫CdG
2d(f)Ha∗
i (•)Haj (•)Hv
i,j(f)G−1d (f)df (4.59)
=
∫ ( K∑
i
K∑
j
CdHa∗i (•)Ha
j (•)Hvi,j(f)
)Gd(f)
Tdf (4.60)
=
∫Su(f)
Gd(f)
Tdf (4.61)
which implies that the undesired power spectral density is
Su(f) =K∑
i
K∑
j
CdHa∗i (θd, φd, f)Ha
j (θd, φd, f)Hvi,j(f) (4.62)
Here, Hvi,j(f) refers to the elements of H−1
u . Similarly for the system response in the
SOI direction, beginning with the desired power in Equation ??,
Pd(τ) =∣∣∣⟨H−1u , Hd
⟩∣∣∣2
(4.63)
=
∣∣∣∣∫ √
CdHsys(θd, φd, f)Gd(f) df
∣∣∣∣2
(4.64)
which implies that
⟨H−1u , Hd
⟩=
∫ √CdHsys(θd, φd, f)Gd(f) df (4.65)
=
∫ (√CdHsys(θd, φd, f)T
) 1
TGd(f) df. (4.66)
Observation of Equation 4.62 yields an Hsys of
Hsys(θd, φd, f) =1
T√CdSu(f). (4.67)
This implies that if the max C/N weights are chosen, then there would be no delay
bias introduced by the adaptive antenna on the desired signal.
51
4.5 Summary
The maximum performance bound on the post-correlation SINR has been identi-
fied for a STAP/GNSS system. The bound provided a useful perspective on existing
algorithms. There exist implementable STAP algorithms that come satisfactorily
close to this bound, such as simple beam steering. The maximum C/N performance
bound can be defined for a fixed or infinite length STAP filter, which makes its ap-
plicability even more general. The weights which achieve this theoretical bound were
identified and their behavior studied. It was clear that the max pre-correlation SINR
and max post-correlation C/N algorithms are distinct. Optimal post-correlation per-
formance does not follow from a STAP filter which forms a matched response, as is
the case with some common STAP algorithms. The next chapter will extend these
conclusions to show that the max C/N algorithm has repercussions on optimal delay
and phase estimation performance as well.
52
CHAPTER 5
Optimum STAP Weights for TOA Estimation
A GNSS receiver system determines the range to satellites by estimating the delay
and phase of the satellite’s transmitted signal relative to a local receiver clock. In
this sense, the problem of GNSS navigation is fundamentally one of time-of-arrival
(TOA) and phase estimation, and to optimize navigation performance in GNSS sys-
tems is to optimize the quality of this estimation. Since the estimator measures the
delay and phase relative to a locally generated replica of the signal, it is natural that
TOA estimators typically involve the cross-correlation of these two signals. It follows
that the work done in the previous chapters (which discussed optimal post-correlation
properties ) would then yield some application to this problem. This chapter will pro-
vide the mathematical confirmation of this. It introduces the estimator into STAP
performance analysis. It will begin by assuming an optimal TOA estimator and
study the performance given different adaptive antennas. Next, the optimum STAP
weights will be derived for an optimal TOA estimator. It will be shown that max
C/N beamforming is optimal in the sense that max C/N with an optimal TOA esti-
mator represents an optimal combined STAP/GNSS receiver for stationary Gaussian
interference. Consequently, it provides useful bounds for STAP/GNSS performance.
Finally, a non-ideal, implementation of TOA estimator will be discussed. Simulations
will show that its performance comes very close to the bound.
5.1 Optimum TOA Estimation
Unfortunately, actual TOA estimators can vary widely in implementation. For
instance, there are many different tracking loop designs used in GNSS receivers.
Thus, it is difficult, at first, to understand how one can model TOA variance in a
53
general, yet applicable, way so that a STAP-based receiver could be designed for it.
As a first step toward addressing this, the approach of this study will be to assume an
optimum TOA estimator and design optimal STAP weights for it. An optimal TOA
estimator is one that satisfies the Cramer-Rao bound for estimation variance given
particular input signals (in this case, from the output of the adaptive antenna). This
paper will focus on the generalized correlation method for TOA estimation [13]. An
optimal TOA estimator was shown in [9] to be a prefiltered coherent peak picking
correlator (CPPC). This estimator performs cross-correlation of the received signal
with an identically filtered reference signal which is assumed to be in phase. The
cross-correlation is evaluated at all delays, and the peak location is estimated as
the maximum; hence, this is a theoretical TOA estimator. The optimal prefilter is
shown to be the inverse of the noise power spectral density. This TOA estimator
is asymptotically maximum-likelihood and achieves the CRLB (provided sufficient
SNR) for the given signal spectrums. One can consider that this is what is meant
by an optimal TOA estimator, which this chapter optimizes STAP weights for. Also,
when this study refers to TOA estimates, it refers to the unsmoothed or unprocessed
estimate as discussed in Chapter 2. A real receiver would process this estimate in
the context of previous estimates and other sources of information before determining
navigation data.
The Cramer-Rao lower bound for estimating the time-of-arrival (TOA) of this
known signal (and known phase) in Gaussian noise is given by [9] as
CRLB−11 = 8π2T
∫f 2Sd(f)
Su(f)df, (5.1)
where Sd is the power spectrum of the SOI and Su is the power spectrum of the
undesired noise. In most GNSS literature, Sd is clean and undistorted and Su is the
sum of incident interference and white noise. However, in a STAP/GNSS receiver,
this signal would have been filtered by the STAP processor, and both Sd and Su
will be functions of the STAP weights, w. Therefore, this CRLB (which will be
denoted CRLB1) is dependent on the STAP weights. The next logical question
is, which weights produce noise and signal spectra which minimize this CRLB? To
answer this, a second CRLB, CRLB2, is defined, which represents the lower bound on
TOA estimation variance given K signals, one from each antenna element. Figure 5.1
54
Figure 5.1: STAP/GNSS system model with two Cramer-Rao lower bounds for TOAestimation in non-white Gaussian noise.
depicts the difference between the two bounds. CRLB1 is minimum TOA estimation
variance given the output signal of the STAP processor. CRLB2 is the minimum
TOA estimation variance given K signals from the antenna array. CRLB2 represents
a bound on the entire system, including STAP, and is thus independent of a particular
STAP implementation. Now, a new sense of ”optimum” beamforming can be defined;
namely, the weights which satisfy the condition
CRLB1 = CRLB2 (5.2)
can be said to be optimum with respect to TOA variance (when paired with an
optimum TOA estimator). It will be shown that as the number of taps increase,
max C/N beamforming converges to satisfy this condition. Alternatively, CRLB1 by
itself is a convenient means to compare the relative performance of different STAP
methods without having to be tied to a specific receiver implementation.
55
5.2 Derivation of the CRLB for TOA and Phase
As stated earlier, CRLB1 is given by Equation 5.1. This chapter will follow similar
steps to those in [9] to find CRLB2, which represents the Cramer-Rao lower bound
for TOA estimation given the reception of a signal from the K antenna elements of
the array.
Consider the signal, xk(t), at the output of the kth front-end channel of which
there are 2N samples available. This signal contains information about the SOI
including its phase ψ and delay τ0 relative to the receiver. The TOA estimator has
access to T seconds of received signal. In the frequency domain, the received data
which contains both the desired signal and noise is
Qk(ζ) =
∫ T
0
xk(t)e−jζω0tdt (5.3)
where ω0 = 2π/T and ζω0 are the discrete frequency points. Its mean contains simply
the desired component,
Yk(ζ) =
∫ T
0
xdk(t− τ0)ejψe−jζω0tdt. (5.4)
It follows that
Qk(ζ) = Yk(ζ) + Zk(ζ), (5.5)
where Zk is the undesired Gaussian noise and interference. From these samples,
vectors of the desired signal and received data for each element are defined as
qk = [Qk(−N) . . . Qk(N)]T (5.6)
yk = [Yk(−N) . . . Yk(N)]T (5.7)
zk = [Zk(−N) . . . Zk(N)]T (5.8)
where there are 2N + 1 points and the vectors for each element are composed into
combined vectors
q =
q1...qk
y =
y1...yk
z =
z1...zk
. (5.9)
The mutual power spectral matrix is given by
Hu = var {q|ψ, τ0}
= E{z∗zT
}(5.10)
56
and each submatrix of Hu is denoted[Hu
]i,j
and is given by
[Hu
]i,j
= E{z∗i z
Tj
}, (5.11)
It is assumed that the interference is independent between frequencies and[Hu
]i,j
is diagonal. Since Hu is composed of diagonal submatrices, it follows that H−1u will
also have diagonal submatrices. The diagonal component will be specially denoted as
samples of a function Hvi,j(f),
[H−1u
]i,j
= diag
{1
T[ Hv
i,j(−N) . . . Hvi,j(N) ]
}(5.12)
The conditional probability distribution of the received signal is
p(Q| ψ, τ0) =1
det [πHu]e−(q−y)HH
−1u (q−y), (5.13)
and it follows that the log-likelihood function is
LQ(ψ, τ0) = −{ln det [πHu] + (q − y)∗H−1
u (q − y)}
(5.14)
= K0 −K∑
i=1
K∑
j=1
(qi − yi)∗
[H−1u
]i,j
(qj − yj) (5.15)
= K0 −K∑
i=1
K∑
j=1
Li,j(ψ, τ0). (5.16)
taking the mean of the second derivative with respect to the delay produces
E
{∂2Li,j(ψ, τ0)
∂τ 20
}= E
{ N∑
ζ=−N
−(ζω0
)2[Y ∗
i (ζ)Qj(ζ)
+Yj(ζ)Q∗
i (ζ)]T−1Hv
i,j(ζ)
}(5.17)
=N∑
ζ=−N
− 2
T
(ζω0
)2Ha∗i (f)Ha
j (f)CdGd(f)Hvi,j(f) (5.18)
Under the assumption of large T, then the number of samples is large and the sum-
mation approaches an integral
E
{∂2Li,j(ψ, τ0)
∂τ 20
}= −8π2T
∫f 2Ha∗
i (f)Haj (f)CdGd(f)Hv
i,j(f)df (5.19)
Using a connection to Equation 4.53, we define
Hd = HdHs (5.20)
57
the the elements of Hd are
Hdi,j(f) = Ha∗
i (f)Haj (f)CdGd(f)e−2πfτ0 (5.21)
and it follows that, using a small relative delay approximation (τ0 ≈ 0),
E
{∂2LQ(ψ, τ0)
∂τ 20
}=
K∑
i=1
K∑
j=1
E
{∂2Li,j(ψ, τ0)
∂τ 20
}(5.22)
=K∑
i=1
K∑
j=1
−8π2T
∫f 2Hd
i,j(f)Hui,j(f)df (5.23)
which is identical to the results in [9], only modified to account for the responses
of multiple antenna elements. We note that it is independent of both the unknown
phase and delay. Continuing, the CRLB has the form
CRLB2 = −E{∂2LQ(θ, τ0)
∂u2k
}−1
(5.24)
=
[8π2T
K∑
i=1
K∑
j=1
∫f 2Hd
i,j(f)Hvi,j(f)df
]−1
(5.25)
and, using Equation 4.41,
CRLB−12 = 8π2T
⟨f 2H∗
d,H−1u
⟩(5.26)
= 8π2T
⟨f 2H∗
d,1
THsH
−1u
⟩(5.27)
= 8π2⟨f 2H∗
d, H−1u
⟩. (5.28)
This bares a close resemblance to the bound for post-correlation SINR of the con-
verged max C/N beamformer. In fact, it can be shown that the max C/N beamformer
converges to the condition CRLB1 = CRLB2. We begin with the equation for the
STAP-dependent CRLB,
CRLB−11 = 8π2T
∫f 2Sd(f)
Su(f)df (5.29)
The desired power spectral density is found by inserting Hsys from Equation 4.67 into
Equation 2.3 to produce
Sd(f) = CdGd(f) |Hsys(θd, φd, f)|2 (5.30)
= Su(f)√CdGd(f)Hsys(θd, φd, f)e−j2πfτ0 (5.31)
= Su(f)
[ K∑
i=1
K∑
j=1
Hdi,j(f)Hv
i,j(f)
]. (5.32)
58
This yields,
CRLB−11 = 8π2T
K∑
i=1
K∑
j=1
∫f 2Hd
i,j(f)Hvi,j(f)df (5.33)
= CRLB−12 . (5.34)
Therefore, the max C/N adaptive beamformer converges to the optimal pre-filter for
TOA estimation, as defined in Section 5.1.
The max C/N STAP algorithm is also optimal for phase estimation. If the deriva-
tive with respect to the unknown phase, ψ, is taken instead in Equation 5.17, it is
clear that the results of the previous section are identical except that there is no
(2πf)2 term, and it follows from Equation 5.26 that the minimum phase estimation
variance is
CRLB−1ψ,1 = 2T
∫Sd(f)
Su(f)df (5.35)
for the finite tap case and
CRLB−1ψ,2 = 2
⟨Hd, H
−1u
⟩(5.36)
for the infinite case. This is proportional to the post-correlation SINR for the con-
verged max C/N beamformer. Therefore, it is sufficient to state that optimizing
post-correlation SINR is equivalent to optimizing phase estimation variance. How-
ever, this considers the TOA and phase estimation as independent and is therefore
the most optimistic choice for the minimum variance.
5.3 Simulation Results
The section provides simulation results comparing the TOA estimation perfor-
mance of different STAP algorithms. The results will focus on noise errors in the
estimates. Bias errors introduced by the adaptive antenna will not be dealt with.
Details of the simulation setup is as described in Chapter 2. The antenna array of 7
isotropic elements was simulated in the presence of a single SOI. The SOI was a weak
( 30dB below the noise floor) and had a sinc-squared power spectrum with a 20MHz
wide main lobe. Its angle-of-incidence was φ = 0◦ and its elevation was swept from
θ = −90◦ to 90◦. The SINR performance was measured for each of these angles using
59
the the 4 different STAP algorithms previously discussed. All have 7 taps, and the
system bandwidth is 24 MHz.
Figure 5.2 shows the standard deviation of the unsmoothed TOA estimate after a
10ms correlation. These results correspond to the pre-correlation and post-correlation
SINR shown in Figure 4.1. These TOA results are the CRLB, which is the minimum
achievable standard deviation for the spectrums produced by the respective STAP
methods. All methods have the same performance except for power minimization,
whose performance is limited since it does not provide beam steering. First, this
provides a convenient way to get pseudorange estimates in units of distance instead
of just power levels; however, it must be remembered that these estimates do not
correspond directly to navigation accuracy. On the other hand, it is interesting to
note that aside from the power inversion method, all STAP methods have identical
performance. Therefore, all methods have the same TOA estimation variance when
paired with an optimal TOA estimator.
Figure 5.3 shows the same data as in Figure 5.2 only in a different form. In this
case, the inverse of the variance is plotted in dB. Though the absolute values on the
graph are not important, this allows a more direct comparison to the SINR plots in
Fig 4.1. In fact, there is a relative performance discrepancy when comparing TOA
variance to post-correlation SINR, much like there was comparing post-correlation to
pre-correlation SINR. However, this is the simplest case since there is no interference.
Figure 5.4 shows the case with an interferer present. The interference is wideband
(20 Mhz) and strong (40dB above the noise floor) with a flat power spectral density.
It is incident from φ = 0◦ and θ = 80◦. Now the STAP filter is responding to the
interference, and there are performance differences between the different STAP meth-
ods. Again, Figure 5.5 shows the inverse of the variance plotted in dB. It is clear that
the max C/N and simple beam steering methods still have the optimal performance.
MMSE is very close to the optimal performance; however, the performance of the max
SINR method is diminishing. It is interesting because, again, there is a discrepancy
between the relative post-correlation SINR performance and the relative TOA vari-
ance. The results confirm that the max C/N algorithm produces the optimal TOA
performance.
60
−80 −60 −40 −20 0 20 40 60 800
20
40
60
80
100
120
140
160
180
200
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
40
45
50
55
60
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.2: Minimum achievable TOA standard deviation for different STAP algo-rithms corresponding to the scenario in Figure 4.1.
61
−80 −60 −40 −20 0 20 40 60 80−70
−65
−60
−55
−50
−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) Inverse TOA Variance (dB)
−70 −60 −50 −40 −30 −20−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) Inverse TOA Variance (dB), zoomed
Figure 5.3: The results of Figure 5.2 but represented as variance and inverted. Thisform is directly comparable to the post-correlation SINR in Figure 4.1 (no interfer-ence).
62
−80 −60 −40 −20 0 20 40 60 800
20
40
60
80
100
120
140
160
180
200
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
40
45
50
55
60
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.4: Minimum achievable TOA standard deviation for different STAP algo-rithms corresponding to the scenario in Figure 4.3.
63
−80 −60 −40 −20 0 20 40 60 80−70
−65
−60
−55
−50
−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) Inverse TOA Variance (dB)
−70 −60 −50 −40 −30 −20−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) Inverse TOA Variance (dB), zoomed
Figure 5.5: The results of Figure 5.4 but represented as variance and inverted. Thisform is directly comparable to the post-correlation SINR in Figure 4.3 (one interferer).
64
5.4 Non-Ideal TOA Estimator
The previous sections dealt with the prefiltered CPPC which is a theoretical TOA
estimator with optimal performance. Unfortunately, there are some properties of it
that are not practical. Most notably, real discriminators only have access to cross-
correlation function estimates at a finite number of delays. One commonly discussed
discriminator is the non-coherent early-minus-late processor (NELP) [9]. The NELP
uses only two cross-correlation functions estimates in order to estimate the TOA. The
discriminator is given by
d =|Ryd(τ0 + Td)| − |Ryd(τ0 − Td)|
Gd
(5.37)
where Td is the correlator tap spacing and τ0 is the previous estimate of the relative
delay. Gd is the discriminator gain and is typically adjusted to make the TOA estima-
tor unbiased. The NELP is qualitatively different from the CPPC in two important
ways. First, since it is a function of estimates from more than one delay, its noise
behavior is dependent not only on the noise variance but also its covariance between
delays. Second, distortion of the desired signal spectrum will alter the discriminator
gain. However, as shown in [9], this discriminator has the potential to achieve near
optimal performance depending on the noise spectrum.
To demonstrate how STAP processing can distort the cross-correlation function,
Figure 5.6 shows the cross-correlation functions for the corresponding system re-
sponses in Figure 3.5. As observed in Chapter 3, some STAP algorithms form matched
responses in the SOI direction. The effect is to change the cross-correlation main lobe
shape. Sometimes the distortion causes shifting or skewing in the main lobe which
can lead to bias errors. However, it is clear from the figure that the primary distortion
is widening of the main lobe. Although this widening of the main lobe will not create
bias errors (the peak location does not change), it will affect noise errors. When
the main lobe is widened the discriminator decreases since the denominator must get
smaller in order to keep the TOA estimator unbiased. This will effectively lower the
discriminator gain. Therefore, it will be advantageous to avoid STAP filters which
form matched filters.
Simulations were performed in order to measure the TOA variance using the
NELP. Although analytic treatments of this discriminator exist [9], this work will
65
−150 −100 −50 0 50 100 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ (ns)
|Ryd
| (no
rmal
ized
)
MMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−40 −30 −20 −10 0 10 20 30 40
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
τ (ns)
|Ryd
| (no
rmal
ized
)
MMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.6: Cross-correlation functions for different STAP algorithms, correspondingto the system responses in Figure 3.5.
utilize a numerical approach. However, if the entire STAP system were to be applied
numerically, considerable computational resources would be spent in the generation
66
of signals, the application of the STAP filter, and the cross-correlation in the receiver.
Thus, it is advantegous to skip these steps by generating samples from the analytical
representation of the cross-correlation function in Chapter 3. Then, these samples
can still be processed numerically by the discriminator. One can follow a common
method for generating jointly-normal random variables in order to create instances
of Ryd(τ).
Figure 5.7 shows the standard deviation of the unsmoothed TOA estimate after a
10ms correlation. These results correspond to the pre-correlation and post-correlation
SINR shown in Figure 4.1. These results were found using the NELP discriminator.
Unlike the CPPC, the NELP produces performance differences for different STAP
algorithms even in the abscence of interference. For the cases where the STAP filter
is forming a matched response, the main lobe is being distorted and the discriminator
gain is being effected. The max SINR method performs poorly and the MMSE method
is less than optimal. On the other hand, the max C/N and simple beam steering
methods perform the best, and their performance with the NELP is close to that of
the CPPC.
Figure 5.8 shows the same standard deviation of the unsmoothed TOA estimate
in the presence of interference. Again, the conclusions are similar to the case without
interference. Both near and far from the interferer direction, the max C/N method
performs the best. The simple beam steering method also performs very well. The
MMSE and Max SINR methods, however, suffer in performance. It is interesting to
note that it were these two methods performed best in Chapter 3 where pre-correlation
SINR was used as the metric. By observing post-correlation SINR and TOA variance,
the true relative performance of the different algorithms is revealed.
67
−80 −60 −40 −20 0 20 40 60 800
20
40
60
80
100
120
140
160
180
200
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
40
45
50
55
60
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.7: TOA standard deviation of an NELP for different STAP algorithms cor-responding to the scenario in Figure 4.1.
68
−80 −60 −40 −20 0 20 40 60 80−70
−65
−60
−55
−50
−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −20−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.8: TOA inverse variance of an NELP for different STAP algorithms corre-sponding to the scenario in Figure 4.1.
69
−80 −60 −40 −20 0 20 40 60 800
20
40
60
80
100
120
140
160
180
200
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −2010
15
20
25
30
35
40
45
50
55
60
TO
A S
td. D
ev. (
cm)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.9: TOA standard deviation of an NELP for different STAP algorithms cor-responding to the scenario in Figure 4.3.
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−80 −60 −40 −20 0 20 40 60 80−70
−65
−60
−55
−50
−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(a) TOA Std. Dev. (cm)
−70 −60 −50 −40 −30 −20−45
−40
−35
−30
−25
−20
Inve
rse
TO
A V
aria
nce
(dB
)
Elevation (deg)
PIBSMMSEMax SINRMax C/N
(b) TOA Std. Dev. (cm), zoomed
Figure 5.10: TOA inverse variance of an NELP for different STAP algorithms corre-sponding to the scenario in Figure 4.3.
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5.5 Summary
The previous chapters have dealt with progressively deeper considerations between
the STAP processor and the GNSS receiver. This study began, as much previous work
had, by measuring performance as the SINR at the output of the array. Looking fur-
ther in the system, the weights were optimized for the C/N, which is essentially
the noise performance after correlation. In this chapter, the optimum weights for
time-of-arrival (TOA) estimation variance were discussed. TOA estimates are the
basis of the pseudorange estimates from which GNSS navigation processors find posi-
tion. Performance bounds for TOA and phase estimations of a STAP/GNSS receiver
were defined. A theoretically optimal STAP/GNSS system was also identified. This
granted perspective on existing STAP implementations and showed their relation to
optimal performance.
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CHAPTER 6
Conclusions
Adaptive antennas are used to provide interference suppression for GNSS receivers.
These adaptive antennas are composed of an antenna array with STAP-based antenna
electronics, and there has been increased interest to pushing the capabilities of these
systems. Toward this end, this work has contributed to numerous aspects, including
the identification of performance bounds and the discussion of optimal theoretical
systems which achieve these bounds. Furthermore, common implementations were
simulated and compared to these bounds. It was revealed that there are imple-
mentable systems which come very close to the performance bounds. Additionally,
it was shown that there can be discrepancies in the relative performance of different
STAP algorithms if one uses different performance metrics. The following section will
provide a more detailed explanation of all of these aspects for each chapter.
6.1 Overview
Chapter 3 introduced the differences between pre-correlation SINR and post-
correlation SINR performance metrics. Simulations showed that the noise at the
output of the STAP processor is significantly non-white for some STAP algorithms,
which causes the processing gain of the receiver to be detrimentally affected. As
a result, post-correlation SINR (or C/N) is not necessarily directly proportional to
pre-correlation SINR to the extent that there can be significant relative performance
differences based on which metric is observed. In this regard, STAP methods which
effectively created matched filters for the desired signal are not optimum for spread-
spectrum applications like GNSS. It was demonstrated in simulations that effects of
system bandwidth on STAP are also more clear when observing the post-correlation
73
SINR. The chapter provided equations relating pre and post-correlation SINR, in-
cluding a novel form vector form which provided useful in the following chapter. In
this regard, STAP methods which effectively created matched filters for the desired
signal are not optimum for spread-spectrum applications like GNSS. It was demon-
strated in simulations that effects of system bandwidth on STAP are also more clear
when observing the post-correlation SINR. The chapter provided equations relating
pre and post-correlation SINR, including a novel form vector form which provided
useful in the following chapter.
Chapter 4 established the performance bound for post-correlation SINR for an
adaptive antenna. The weight algorithm which achieves this bound was also derived
and was called max C/N adaptive beamforming. The new technique was derived by
utilizing a slight modification to existing work on maximizing pre-correlation SINR.
Although the nature of the max C/N beamforming algorithm makes it impractical
to implement directly, there are many ways to approximate it suitably. Simulations
showed that existing methods come very close to its performance. The maximum
post-correlation was defined for both a finite and infinite number of STAP filter taps.
This allows the bound to apply more generally and for the convergent properties of
the max C/N filter to be studied. It was useful to define the convergent behavior of
the max C/N filter as the number of taps became large since it related directly to
optimal delay and phase estimation in the following chapter.
Chapter 5 extended the performance analysis of the STAP/GNSS system to the
estimation of delay and phase of the received signal. In the first part of the chapter,
the CRLB for TOA and phase estimation was derived. It was shown that max C/N
converges to this bound as the number of taps increase. Simulations also showed
plots of the TOA estimation variance for specific STAP implementations when they
are paired with an optimal TOA estimator. Next, a non-ideal TOA estimator was
implemented and the estimation variance was compared between different STAP al-
gorithms. It was shown that there exist implementable STAP/GNSS systems which
get very close to the optimum performance.
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6.2 Future Work
GNSS errors are commonly categorized into two types: noise errors and bias
errors. Noise errors represent the variance around the mean measurement and have
been dealt with exclusively in this paper. On the other hand, bias errors, which
represent the error in the mean, have not been extensively dealt with. There has
long been concern that STAP can introduce bias errors into the SOI in the process of
suppressing interference. The optimal bias error is simply zero, which one can achieve
via prediction and correction of the biases as in [10]. Again, this requires complete
knowledge of the antenna and GNSS receiver. For other situations, when prediction
and correction are not possible, a deeper exploration of bias errors is needed.
The optimal STAP algorithm discussed in this work requires complete knowledge
of the antenna; however, one never has perfect knowledge of the antenna response.
Measurement errors and manufacturing errors may cause deviations from the ideal
response in an implementable STAP system. Future work should include a sensitivity
analysis to such errors. Furthermore, it would be advantageous to study what rev-
elations can be gained about improving antenna design based on the equations and
behavior of optimum STAP processing given in this work.
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