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

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−35

−30

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−20

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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

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−10

Pre

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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

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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.

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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

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−65

−60

−55

−50

−45

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−30

−25

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rse

TO

A V

aria

nce

(dB

)

Elevation (deg)

PIBSMMSEMax SINRMax C/N

(a) TOA Std. Dev. (cm)

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−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.

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100

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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

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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

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Inve

rse

TO

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)

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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

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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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