UNIVERSITY OF CINCINNATI
______, 20 _____
I,______, hereby submit this as part of the requirements for the degree of:
______in: ______It is entitled: ______
Approved by: ______
Digital Direction Finding System Design and Analysis
A thesis submitted to the
Division of Graduate Studies and Research of the University of Cincinnati
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE (M.S.)
in the Department of Electrical & Computer Engineering and Computer Science of the College of Engineering
2003
by
Huazhou Liu
B.E., Xi’an Jiaotong University P. R. China, 2000
Committee Chair: Professor Howard Fan
ABSTRACT
Direction Finding (DF) system is used in many military and civilian operations such as surveillance, reconnaissance, and rescue, etc. In the past years, direction finding system is implemented usually using analog RF techniques such as Butler matrix and analog beamforming. Analog direction finding systems have drawbacks inherent from their analog properties such as expensive implementation, inflexibility to adjust or change functionality, intensive calibration procedures and etc. The digital technique relies on reconfigurable logic implementations. Thus it is more flexible and less expensive compared with its analog counterpart. In a digital direction finding system, all the received signals by array elements are sampled and digitized into digital format.
They are processed by a high throughput digital processor. The whole system is much more reliable and accurate. Digital implementation of the direction finding system becomes practically exploitable in recently years.
In this research work, we design a high throughput digital direction finding (DDF) system. It implements the digital Butler matrix to accomplish the direction finding task. Through theoretical timing error analysis, we then estimate the performance that the digital direction finding system can achieve. We also analyze how to choose the geometry of the antenna array, array size and so forth, based on theoretical and practical considerations.
ACKNOWLEDGMENT
I would like to express my sincerest gratitude to my advisor Dr. Howard Fan for his outstanding guidance, constant encouragement and patience. I wish to thank Dr.
James Caffery, Shu Wang and Huiqin Yan for their valuable discussions and hard work in this project. Without their help, this thesis would have not been possible. I am thankful to Dr. Ali Minai for his time and helpful comments. I also wish to thank all of my friends and colleague in my lab for their help. Finally, I would like thank my parents for their love and supports over the years.
And this work was funded in part by Nova Engineering, Cincinnati, OH, under U.S.
Department of Defense Small Business Innovative Research (SBIR) contract.
CONTENTS
CONTENTS...... I
LIST OF FIGURES ...... III
LIST OF TABLES ...... V
1 INTRODUCTION ...... 1
1.1 ANTENNA ARRAY...... 1
1.2 FUNDAMENTAL PARAMETERS OF ANTENNA ARRAY ...... 1
1.2.1 Radiation Pattern...... 2
1.2.2 Array Factor ...... 2
1.2.3 Main Lobe...... 3
1.2.4 Side Lobes...... 3
1.2.5 Half -Power Beamwidth...... 3
1.3 ARRAY GEOMETRY ...... 3
1.3.1 Linear Array ...... 4
1.3.2 Circular Array ...... 5
1.3.3 Hexagonal Array...... 6
2 BEAMFORMING...... 8
2.1 ANALOG BEAMFORMING ...... 9
2.1.1 Analog Butler Matrix...... 9
2.2 DIGITAL BEAMFORMING...... 10
2.2.1 Description of DBF...... 10
2.2.2 Time-domain Beamforming ...... 12
2.2.3 Frequency-domain Beamforming ...... 21
3 SYSTEM DESCRIPTION ...... 23
3.1 A/D SAMPLING AND SYNCHRONIZATION...... 23
3.2 DIGITAL BUTLER MATRIX ...... 24
- i - 3.2.1 Narrow Band Butler Matrix...... 24
3.2.2 Wideband Butler Matrix ...... 25
3.2.3 Shading ...... 27
3.2.4 Direction Finding Strategy ...... 28
4 ARRAY PARAMETER ANALYSIS ...... 31
4.1 ANTENNA ARRAY SIZE CONSIDERATION...... 31
4.1.1 Equal-spaced Linear Array...... 31
4.1.2 Uniform Circular Array...... 37
4.2 UNIFORM CIRCULAR ARRAY WITH A CENTER ELEMENT ...... 48
4.3 HEXAGONAL ARRAY ...... 50
5 TIMING ERROR ANAYSIS...... 53
5.1 EQUAL-SPACED LINEAR ARRAY ...... 57
5.2 UNIFORM CIRCULAR ARRAY ...... 61
5.2.1 Four Element Uniform Circular Array Plus a Center Element ...... 62
5.2.2 Four Element Uniform Circular Array...... 64
5.3 SIMULATION RESULTS...... 66
5.3.1 Equal-spaced Linear Array...... 67
5.3.2 Four Element Uniform Circular Array Plus a Center Element ...... 69
5.3.3 Four Element Uniform Circular Array...... 74
5.3.4 Overall System Simulations ...... 77
6 CONCLUSION ...... 80
7 BIBLIOGRAPHY...... 81
- ii - LIST OF FIGURES
FIGURE 1.1 RECTANGULAR RADIATION PATTERN...... 2
FIGURE 1.2 UNIFORMLY SPACED LINEAR ARRAY ...... 4
FIGURE 1.3 A CIRCULAR ARRAY WITH EQUALLY SPACED 4 ELEMENTS...... 5
FIGURE 1.4 A HEXAGONAL ARRAY...... 6
FIGURE 2.1 A 4X4 BUTLER BEAMFORMER...... 9
FIGURE 2.2 NARROW-BAND BEAMFORMER STRUCTURE...... 12
FIGURE 2.3 WIDEBAND BEAMFORMER STRUCTURE...... 13
FIGURE 2.4 WIDEBAND FREQUENCY DOMAIN BEAMFORMER ...... 22
FIGURE 3.1 BLOCK DIAGRAM OF DIGITAL DIRECTION FINDING SYSTEM...... 30
FIGURE 4.1 SIN(NX)/N/SIN(X)...... 34
FIGURE 4.2 BEAM PATTERN OF 4-ELEMENT CIRCULAR ARRAY WITH DIFFERNENT RADIUS ...... 41
FIGURE 4.3 BEAM PATTERN OF CIRCULAR ARRAY WITH DIFFERNENT ANTENNA NUMBER AND FIXED R..42
FIGURE 4.4 BEAM PATTERN OF CIRCULAR ARRAY WITH FIXED INTER-SPACE ...... 43
FIGURE 4.5 AZIMUTH ANGLE BEAMWIDTH WITH DIFFERENT ELEVATION ANGLE...... 45
FIGURE 4.6 AZIMUTH ANGLE BEAMWIDTH WITH DIFFERENT AZIMUTH ANGLE ...... 46
FIGURE 4.7 ELEVATION ANGLE BEAMWIDTH WITH DIFFERENT ELEVATION ANGLE ...... 46
FIGURE 4.8 ELEVATION ANGLE BEAMWIDTH WITH DIFFERENT AZIMUTH ANGLE...... 47
FIGURE 4.9 BEAM PATTERN COMPARE W/O CENTER ELEMENT ...... 50
FIGURE 4.10 COMPARISON OF BEAM PATTERN BETWEEN HEXAGONAL ARRAY AND CIRCULAR ARRAY ..51
FIGURE 5.1 VARIANCE OF DDF ERROR VS. DOA...... 68
FIGURE 5.2 VARIANCE AND MAX OF DDF ERROR VS. SAMPLE TIMING ERROR...... 68
λ FIGURE 5.3 MEAN DDF ERROR WITH f = 32MHz AND R = (WITH CENTER ELEMENT)...... 70 2
λ FIGURE 5.4 MAX DDF ERROR WITH f = 32MHz AND R = (WITH CENTER ELEMENT) ...... 70 2
λ FIGURE 5.5 MEAN DDF ERROR WITH f = 86MHz AND R = (WITH CENTER ELEMENT) ...... 71 2
- iii - λ FIGURE 5.6 MAX DDF ERROR WITH f = 86MHz AND R = (WITH CENTER ELEMENT) ...... 71 2
λ FIGURE 5.7 MEAN DDF ERROR WITH f = 32MHz AND R = (WITH CENTER ELEMENT)...... 72 π
λ FIGURE 5.8 MAX DDF ERROR WITH f = 32MHz AND R = (WITH CENTER ELEMENT)...... 72 π
λ FIGURE 5.9 MEAN DDF ERROR WITH f = 86MHz AND R = (WITH CENTER ELEMENT)...... 73 π
λ FIGURE 5.10 MAX DDF ERROR WITH f = 86MHz AND R = (WITH CENTER ELEMENT) ...... 74 π
λ FIGURE 5.11 MEAN DDF ERROR WITH f = 32MHz AND R = (WITHOUT CENTER ELEMENT) ...... 75 2
λ FIGURE 5.12 MAX DDF ERROR WITH f = 32MHz AND R = (WITHOUT CENTER ELEMENT) ...... 75 2
λ FIGURE 5.13 MEAN DDF ERROR WITH f = 86MHz AND R = (WITHOUT CENTER ELEMENT)...... 76 2
λ FIGURE 5.14 MAX DDF ERROR WITH f = 86MHz AND R = (WITHOUT CENTER ELEMENT)...... 76 2
λ FIGURE 5.15 PERFORMANCE OF 4-ELEMENT UCA WITH R = ...... 78 2
λ FIGURE 5.16 PERFORMANCE OF 4-ELEMENT UCA WITH CENTER ELEMENT WITH R = ...... 78 2
λ FIGURE 5.17 PERFORMANCE OF 4-ELEMENT UCA WITH R = ...... 79 π
λ FIGURE 5.18 PERFORMANCE OF 4-ELEMENT UCA WITH CENTER ELEMENT WITH R = ...... 79 π
- iv - LIST OF TABLES
TABLE 4-1 POSITION OF THE FIRST SIDE LOBE IN LINEAR ARRAY ...... 37
TABLE 4-2 LEVEL OF THE FIRST SIDE LOBE IN LINEAR ARRAY...... 37
TABLE 4-3 LOCATION OF THE FIRST SIDE LOBE (AZIMUTH ANGLE) OF A 4-ELEMENT UCA...... 48
TABLE 4-4 LEVEL (DB) OF THE FIRST SIDE LOBE OF A 4-ELEMENT UCA...... 48
TABLE 4-5 SIDE LOBE LEVEL COMPARISON BETWEEN HEXAGONAL AND CIRCULAR ARRAY ...... 51
TABLE 4-6 SIDE LOBE POSITION COMPARISON BETWEEN HEXAGONAL AND CIRCULAR ARRAY...... 51
TABLE 4-7 MAIN LOBE BEAMWIDTH OF THE HEXAGONAL ARRAY AND CIRCULAR ARRAY ...... 52
TABLE 5-1 NOTATIONS FOR DDF ERROR ANALYSIS...... 54
- v - 1 INTRODUCTION
1.1 Antenna Array
An antenna array consists of multiple stationary antenna elements, which are often fed coherently. The received signals obtained by antenna array elements, which are located at different points in space in the field of interest, are manipulated by array processing techniques to extract useful characteristics of the received signal field (e.g., its signature, direction, speed of propagation) [1]. Beamforming and DoA estimation using antenna arrays have been explored for many decades. Application of antenna array and array processing has also been suggested in recent years for mobile communications systems to improve the systems performance by increasing channel capacity and spectrum efficiency, extending coverage range, tailoring beam shape, steering multiple beams to track many mobiles and compensating aperture distortion electronically.
1.2 Fundamental Parameters of Antenna Array
To describe the performance of an antenna array, definitions of various parameters are necessary. Here we provide some of the parameters and definitions that are particularly relevant to this research.
- 1 - Radiation Intensity
Main Lobe
HPBW Side Lobe
− π π θ
Figure 1.1 Rectangular radiation pattern
1.2.1 Radiation Pattern
The radiation pattern of an antenna is commonly defined as the spatial distribution of a quantity that characterizes the electromagnetic field generated by antenna. Usually, it is described in terms of its principal E-plane (The plane containing the electric-field vector and the direction of maximum radiation) patterns [3].
1.2.2 Array Factor
The array factor represents the far-field radiation pattern of an array of isotropically radiating elements. In general, the far-field pattern of any array is given by the multiplication pattern of the field of the single element positioned at the origin of the array and the array factor. In this research, the array factor will be denoted by AF()ϕ,θ , where θ represents the elevation angle and ϕ represents the azimuth angle (if applicable) in space.
- 2 - 1.2.3 Main Lobe
A main lobe is defined as “the radiation lobe containing the direction of maximum radiation” [2].
1.2.4 Side Lobes
Side lobes are defined as “the any other lobes other than the main lobe”.
1.2.5 Half -Power Beamwidth
The half-power beamwidth (HPBW) is defined as: “In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half the maximum value of the beam” [3].
1.3 Array Geometry
Array geometrical configuration is a key factor of an antenna array. Geometry controls the radiation pattern and almost all other factors of an antenna array.
Antenna arrays with different geometries have different properties, which make an antenna array with certain geometry be used only for a particular purpose or in particular environment conditions. Here, we briefly introduce three simple and important array geometries, which will be used in this research work.
- 3 - 1.3.1 Linear Array
ϕ 0 ϕ n 0 si d
dd
Figure 1.2 Uniformly spaced linear array
An array of identical elements with signals all of identical magnitude and each with a progressive phase is referred to as a uniform array [3]. Figure 1.2 is a drawing of a uniformly spaced linear array with N identical isotropic elements. If the incident
signal impinges upon the array at an angleϕ 0 , the phase difference between two
adjacent elements is d sinϕ 0 . Set the signal at the origin as the reference. The phase
of incident signal at element i relative to that of reference element is iκd sin ϕ 0 ,
ω where κ = . So the array factor is: c
∗ ∗ ∗ ∗ jκdsinϕ0 j2κdsinϕ0 j(N−1)κdsinϕ0 (1.1) AF(ϕ0) = w0 + w1 e + w2 e +L+ wN−1 e
which can be expressed in terms of vector inner product:
Η AF(ϕ0 ) = w v (1.2)
where
Τ (1.3) w = [w0 w1 L wN−1]
- 4 - is the complex weighting vector and
jκdsinϕ j(N−1)κdsinϕ Τ v = []1 e 0 L e 0 (1.4)
is the array propagation vector that contains the information on the edge of arrival of the signal.
1.3.2 Circular Array
Incident Signal Z
Y
D ith antenna
θ0
ϕi O ϕ0 X
Figure 1.3 A circular array with equally spaced 4 elements
The circular array, in which the isotropic antenna elements are placed in a circular ring, is one array configuration of very practical interest. Referring to Figure 1.3, for equally spaced elements in the circular ring, the azimuth angle of the ith antenna is
2π ϕ = (i −1), i = 1, , N . We let the center of the circle be the reference point. It is i N L
easy to prove that the length of OD is OD = R cos(ϕ 0 −ϕ i )sinθ 0 . Then the phase shift
- 5 - between signals received at center and received at the ith antenna is
R cos(ϕ −ϕ )sinθ ω 0 i 0 . So the array factor of a circular array is given as c
N R cos( ϕ 0 − ϕ i ) sin θ 0 ∗ jω c Η (1.5) AF (ϕ 0 ,θ 0 ) = ∑ w i e = w v i =1
Following the same steps, we can get the array propagation vector for a circular array:
R cos( ϕ −ϕ ) sin θ R cos( ϕ −ϕ ) sin θ R cos( ϕ −ϕ ) sin θ Τ ⎡ jω 0 1 0 jω 0 2 0 jω 0 N 0 ⎤ v = e c e c e c ⎢ L ⎥ (1.6) ⎣ ⎦
1.3.3 Hexagonal Array
R1
R2
o
Figure 1.4 A hexagonal array
The hexagonal array can be treated as consisting of a number of concentric six- element circular arrays of different radius. Figure 1.4 give a simple example of hexagonal array with 12 elements, 6 of which are located at the vertices of the hexagon and the other 6 elements are placed at the middle of each line of the hexagon,
- 6 - respectively. As shown in Figure 1.4, we can think of the hexagon array consisting of
two 6-element circular arrays with radii R1 and R2 , respectively. Using the results in the previous section, it is easy to get the array factor for hexagonal array. We assume the hexagonal array consisting of N concentric circular arrays. Based on equation
(1.5), we get the array factor of a hexagon array as:
R cos( ϕ − ϕ ) sin θ N −1 6 jω i 0 i ,k 0 ∗ c AF (ϕ 0 ,θ 0 ) = ∑∑wi , k e ik==0 1 (1.7)
π π R(2N − i) Where ϕ = i + (k −1) and R = , which stands for the azimuth i,k 3 3N i π 2N cos( i) 3N angle of each antenna and the radius of the concentric circular array, respectively.
- 7 - 2 BEAMFORMING
The quintessential goal of antenna array signal processing is the estimation of parameters by fusing temporal and spatial information, captured via sampling a wave signal with a set of judiciously placed antenna sensors. The signal is assumed to be generated by a finite number of emitters, and contains information about signal parameters characterizing the emitters. The first approach to carry out space-time processing of data sampled at an array of sensors was spatial filtering or beamforming.
The temporal and spatial characteristics combined with the laws of physics allow us to determine source’s location. Thus, in addition to temporal filtering, array processing techniques can enhance an array’s spatial filtering properties. The term beamforming is to distinguish between the spatial properties of signal and noise. The system used to do the beamforming is called a beamformer [9]. The term, beamformer, is derived from the fact that early spatial filters were designed to form pencil beams in order to receive a signal radiating from a specific location and attenuate signals from other locations [4].
From delay-and-sum beamforming, the oldest and simplest array signal processing algorithm, to adaptive beamformings, many algorithms were provided in order to get better performance under different realistic scenarios. As a result of recently major advances in areas of integrated circuit design technology and digital signal processing technology, digital beamforming is provided, based on well-established antenna technology and digital technology. In the following of this chapter, we will briefly introduce basic concepts and important beamforming algorithms.
- 8 - 2.1 Analog Beamforming
2.1.1 Analog Butler Matrix
−180° −180°
45° 45°
0° 0°
(a) 4× 4 ButlerMatrix (b) hybrid in (a)
Figure 2.1 A 4x4 Butler beamformer
The Butler matrices [30] have been used in many applications to generate a number of staring beams. A Butler matrix, shown in Figure 2.1, consists of fixed phase shifts interconnected to hybrids and yields orthogonal beams. Butler matrix is the analog implementation of the Fast Fourier Transform (FFT). But it is developed before the
FFT. There is an important difference between FFT and Butler matrix, i.e., a Butler matrix processes signals in the analog domain, whereas the FFT processes signals in the digital domain. Butler matrices have been implemented in different media, including waveguides for high-power use [31], microstrip [32] and integrated optic
- 9 - form [33]. A 64x64 Butler matrix seem to be the largest possible when the relative permittivity microstrip technology is used [34].
2.2 Digital Beamforming
Analog beamformer has its drawbacks inherent from its analog properties. For example, it is impossible to change its function after the hardware is built. The calibration procedure is labor intensive. Some signal processing algorithms are hard or impossible to be implemented in an analog system. As the recently advance in integrated circuit technology and digital signal processing technology, digital beamforming (DBF) technology is becoming practically exploitable.
2.2.1 Description of DBF
In a DBF system, the RF signal is detected and digitized at the antenna. After sampling and quantization by analog-to-digital converters (ADC) at the antenna elements, all the amplitude and phase information of the signals are preserved in the digital form. The beamforming is carried out by adjusting the amplitudes and phase of these digital signals such that they form the desired beams.
Because the whole procedure after ADCs is carried out by a digital signal processor,
DBF has many attractive features beyond the capabilities of its analog counterpart.
First of all, all of the received RF signal information is captured at the antennas and is available for processing in a digital signal processor, so DBF has much enhanced flexibility without degradation in signal-to-noise ratio (SNR). Secondly, because the digital processor is driven by software, the configuration of the system can be easily
- 10 - changed and adjusted. That makes the whole system flexible to operate for different purposes without any hardware change. We can even implement several functions at the same time. It also can do real time calibration when operating in different environments without the labor intensive calibration procedure needed by an analog beamformer. Thirdly, adaptive beamforming can be easily implemented to improve the system capacity by suppressing co-channel interference [2].
Though DBF has many attractive features, some problems need to be addressed and solved to improve its performance. For example, the beams are formed by summing the appropriate saved samples. The sampling interval and the number of samples determine the number of discrete directions where a beam can be pointed exactly. So the practical requirement of an adequate set of directions where simultaneous beams need to be pointed implies that the array signals be sampled at much higher rates than required by the Nyquist criterion to reconstruct the waveform back from the samples
[23]. To solve this problem, we can use high-speed digital devices, large storage and large bandwidth cables to get high sampling rate. Or the high sampling rate can be obtained by interpolating between samples. This beamformer is called digital- interpolation beamformer [24]. Other issues about DBF can be found in [2] and [6].
- 11 - 2.2.2 Time-domain Beamforming
x1 (t) ∗ w1
x (t) 2 ∗ w 2 y(t) + M x (t) N ∗ wN
Figure 2.2 Narrow-band beamformer structure
Consider a narrow-band beamformer first, shown in Figure 2.2. The output of the beamformer at time t , y(t) is given by a linear combination of the data at the N sensors at time t:
N ∗ y (t ) = ∑ w i x i (t ) (2.1) i =1
It is conventional to multiply the data by conjugates of the weights to simplify notation. We assume throughout that the data and weights are complex since in many applications a quadrature receiver is used at each sensor to generate in phase and quadrature (I and Q) data.
In the wideband case, the performance of the beamformer, shown in Figure 2.2, deteriorates. In order to process broadband signals, a Time Delay Line (TDL) structure is provided, shown in Figure 2.3. The output in this case can be expressed as
- 12 - x1 (t) T T
∗ ∗∗ ∗∗ ww1,,00 ww11,0,1 ww1,1K,0−1
x (t) 2 T T
w∗ w∗ w∗∗ w21,,00 w12,0,1 w21,K,0−1 y(t) + M
xN (t) TT
∗∗ ∗∗ ∗∗ wwN1,0,0 wwN1,0,1 wwN1,0K−1
Figure 2.3 Wideband beamformer structure
N K −1 ∗ y (t ) = ∑∑w i , p x i (t − pT ) (2.2) i =1 p = 0
where K −1 is the number of delays in each of the N sensor channels and T is the delay interval of time delay line.
We can simultaneously formulate the two beamformers in one equation by appropriately defining a weight vector w and data vector x(t) . We let w and x(t) be a N dimension column vector referring to (2.1) and NK dimension vector referring to
(2.2). We also ignore the time index for notation convenience. So that equation (2.1) and (2.2) can be written as
Η y = w x (2.3)
And if the components of x(t ) can be modeled as zero mean stationary processes, then for a given weight vector w , the mean output power of the processor is given by
- 13 - p ( w ) = Ε [y y ∗ ] (2.4) Η = w Rw
Where Ε[]• denotes the expectation operator and R is the array correlation matrix defined by
Η R = Ε [x ( t ) x ( t ) ]
2.2.2.1 Conventional Beamformer
The conventional (or Bartlett) beamformer is a natural extension of classical Fourier- based spectral analysis to sensor array data. For an array of arbitrary geometry, we choose the “steering vector” α(ω θ ϕ) as the weights of the conventional beamformer,
α(ω θ ϕ) w = N (2.5)
where α(ω θ ϕ) has the same form as propagation vector v but with ϕ replacing
ϕ 0 and θ replacing θ 0 , if applicable. The mean output power of the beamformer is
α Η (ω θ ϕ )R α (ω θ ϕ ) p ()α ()ω θ ϕ = N 2 (2.6)
When ω = ω0 , θ = θ 0 and ϕ = ϕ0 , that is, we steer the array to point at the direction of incident signal, then the mean output of the conventional beamformer is equal to the power of incident signal.
- 14 - The steering vector compensates the phase delay between different antenna elements.
So the beamformer is similar to steering the array mechanically in the look direction except that it is done electrically by adjusting the phase. This beamformer maximizes the signal to noise ratio (SNR) under the condition that only uncorrelated noise and no directional interferences exist [6].
Though the conventional beamformer provides maximum output SNR, its performance deteriorates as some intentional or unintentional directional interferences exist. And also as revealed in equation (2.6), the output power is a periodogram in the spatial dimension. We can treat this periodogram in the spatial dimension in the same way as we treat classical periodogram in temporal time series analysis. From [35], we know that the classical periodogram has resolution limitation, which means that, in our case, if we work with the spatial dimension alone, it cannot distinguish two spatial frequency components that are separated by less than 2π / N . In other words, no matter how the available spatial data quality is, it cannot resolve sources whose incident angles are closer than 2π / N . For example, two signals with incident angles separated less than 12o cannot be resolved by a 10-element ULA of half-wavelength inter-element spacing. But beamforming also uses temporal dimension, in which case, multiple spatial beams will be formed with a much finer resolution than the spatial periodogram alone can achieve.
2.2.2.2 Optimal Beamformer
There is one set of popular beamforming methods, which, to some extent, alleviates the limitations of the conventional beamformer, such as unable to cancel effects of the
- 15 - interference signals. In the statistically optimal beamformer, the weights are chosen based on the statistics of data received at the array [4], in order to optimize the beamformer response so that the effect of interference signals and noise is minimized.
Here we briefly introduce one typical type of optimal beamformer. These beamformers use linear constraints to control the response of the beamformer. The basic idea behind linearly constrained minimum variance (LCMV) beamforming is to constrain the response of the beamformer so that signals from the direction of interest are passed with specified gain and phase. The weights are chosen to minimize output variance or power subject to the response constraint [4]. Here we just give one example beamformer in this class, which is proposed by Capon [10]. Because it is to pass the signal from steered angle without distortion, it is called minimum variance distortionless response (MVDR) beamformer. The optimization problem was posed as
Η min p ()w subject to w α (ω θ ϕ ) = 1 w (2.7)
wherep(w ) is as defined in equation (2.4). The optimal solution w can be found using the method of Lagrange multipliers:
R − 1α (ω θ ϕ ) w = Η − 1 α ()()ω θ ϕ R α ω θ ϕ (2.8)
Capon’s method significantly outperforms the conventional method. Through the power minimization procedure (2.7), the capability to suppress effects from interference signals is achieved at the cost of reduced noise suppression capability.
- 16 - The Capon’s method uses only a single constraint (2.7). As we know, the beamformer has degree of freedom N-1, if it has N antenna elements. So we can at most add N-1 constraints to the optimization problem to get much better beam pattern as we expected. Then the single constraint optimization problem in (2.7) becomes:
Η min p ()w subject to C w = g w (2.9)
The matrix C is termed the constraint matrix, and the vector g , termed the gain vector, has constant design constraints. The beamformer described herein is referred to as a generalized side lobe canceller (GSC). Reference [9] has a good introduction analysis of GSC. We can design different constraints and choose the gain vector under different realistic considerations. For example, the point constraints [11] fix the beamformer response at points of spatial direction and temporal frequency. The derivative constraints [12] force the derivatives of the beamformer response at some point of direction and frequency to be zero. The eigenvector constrains [13] are provided base on a least squares approximation to the desired response and used to control beamformer response over direction or/and frequency.
There are other types of optimal beamformer, such as the multiple side lobe canceller
[4], optimization using reference signals [4], [14], [15], [16]. There are two main drawbacks of the optimal beamformer. Firstly, the optimum beamformer requires knowledge of some signal characteristics, such as statistics of signals or/and direction, etc. If we cannot get enough information, we cannot use the optimal beamformer. Or if the information is inaccurate, the performance of the optimal beamformer deteriorates. Secondly, the optimal beamformer can only minimize the effect of uncorrelated interference signals. If there is correlated interference, such as
- 17 - interference due to multipath, the beamformer will cancel a portion of the desired signals.
2.2.2.3 Adaptive Algorithms
Knowledge of statistics (usually second-order statistics) is required to determine the optimum beamformer weight. However, the statistics of the data received at the array are generally unknown and change over time. To overcome these difficulties, the weights are usually determined by adaptive algorithms. An adaptive algorithm means any algorithm whose characteristics depend on the data it receives or refers to only those algorithms that update themselves as each observation is acquired [17]. It is possible to perform satisfactorily in an environment where complete knowledge of the relevant signal characteristics is not available. Some well-known adaptive algorithms are provided and well-studied in adaptive beamforming. Here we briefly describe characteristics of some algorithms.
2.2.2.3.1 LMS algorithm
The least mean square (LMS) algorithm is an important member of the family of stochastic gradient algorithms [9]. A significant feature of the LMS algorithm is its simplicity. It does not require matrix inversion and measurements of the pertinent correlation functions. The LMS adaptive algorithm is described by
w ( n + 1) = w ( n ) + µ g ( w ( n )) (2.10)
- 18 - g (• ) is an unbiased estimate of the gradient of the mean squared error,
MSE ( w ( n )) , which is defined as
⎡ Η 2 ⎤ MSE (w ()n )= E d ()n − x ()n w ()n ⎣⎢ ⎦⎥
where d ( n ) is the reference signal. And µ is the step size for iteration. In reality, an exact measurement of the gradient vector is not possible. The usual strategy is to use instantaneous estimates of the gradient vector to update the weights, i.e.:
∗ Η w ( n + 1) = w ( n ) + µ x ( n )[ d ( n ) − x ( n ) w ( n )] (2.11)
The convergence of the LMS algorithm depends upon the eigenvalues of R. In an environment yielding a large eigenvalue spread of R, the algorithm converges with a slow speed.
2.2.2.3.2 RLS algorithm
The recursive least square (RLS) algorithm is an extension of the method of lease squares. The fundamental difference between the RLS algorithm and the LMS algorithm is to replace the gradient step size µ with a gain matrix R −1 (n ) at the nth iteration. This modification makes the rate of convergence of the RLS algorithm typically an order of magnitude faster than that of the LMS algorithm. The RLS adaptive algorithm is given as
− 1 w ( n + 1) = w ( n ) + R ( n ) g ( w ( n )) (2.12)
where R(n) is given by
- 19 - N − 1 1 Η R ( n ) = ∑ x ( n ) x ( n ) (2.13) N n = 0
And the estimate will be updated when the new samples arrive using
Η R ( n ) = δ R ( n − 1) + x ( n ) x ( n ) (2.14)
δ is a real scalar smaller than but close to one
2.2.2.3.3 CMA algorithm
The constant modulus algorithm (CMA) is one of the most popular blind algorithms for beamforming. It utilizes the property of a constant envelope of the modulated signal. It updates the weight by minimizing the cost function
Η 2 2 ε ( n ) = E [( w ( n ) x ( n ) − σ ) ]
We can use the LMS algorithm to update the weights base on this cost function. The advantage of the CMA is that it is easy to implement because it does not require a reference signal as in the LMS and the RLS algorithms. On the other hand, the modulation type of signals for the CMA is limited because it requires that the envelope of the desired signal be constant. [19] and [20] have further detailed discussion.
There are some other algorithms, like SCORE algorithm, conjugate gradient method, neural network approach and etc, which are not discussed here. Good reviews can be found in [6], [4] and [18].
- 20 - 2.2.3 Frequency-domain Beamforming
The beamformers discussed so far are in the time domain by delaying and summing either in the analog format or in the digital format. Of course the time domain view of beamforming has obvious frequency domain counterparts: Delay corresponds to linear phase shift. Frequency domain beamforming implements the calculations entirely in the frequency domain by Fourier transforming the inputs, applying the phase shifts, and summing and inverse transforming the result [17].
The wideband time domain beamformer described before also has its counterpart in the frequency domain. Usually, wideband beamformer is implemented with a narrowband decomposition structure like Figure 2.4. The wideband signal from each antenna element is transformed into the frequency domain using the FFT algorithm. If the number of frequency bins is large enough, each frequency bin can be treated as a narrowband signal. A simple method is to process the different bin separately using the standard narrowband method. Then the outputs of narrowband beamformers need to be combined in some appropriate way. When the signals in different frequency bins are independent, the performance of the time and frequency domain beamformers are the same. Otherwise, the frequency domain methods can only get suboptimal performance. The tradeoffs and comparisons of time and frequency domain beamformers can be found in [25]. An optimal method is to exploit and combine the information from different frequency bins [26], [27].
- 21 - k x1 (t) ∗ FFT w1,k
y1 ( f ) x (t) k y ( f ) y(t) 2 FFT ∗ 2 IFFT w2,k +
y ( f ) M N
xN (t) ∗ FFT wN,k
Figure 2.4 Wideband frequency domain beamformer
- 22 - 3 SYSTEM DESCRIPTION
In this research, we provide a novel direction finding system. That is a transformation of the analog Butler Matrix, which forms a set of beams in fixed directions, into a digital implementation.
3.1 A/D Sampling and Synchronization
The digital system block diagram is shown in Figure 3.1. Each antenna output is immediately amplified by the front end low noise amplifier (LNA), and then sampled at the A/D sampling modules, located locally directly after the antenna and LNA.
These A/D modules are synchronized by a system-wide master clock, which is the
A/D sampling frequency f s . Because the master clock is located at a central location apart from the A/D sampling modules, due to factors such as non-uniform cable lengths, synchronization errors in A/D exist. There are several approaches to compensate for the sampling offsets, such as Time Delay Line (TDL), Frequency
Domain (FD) phase compensation. The details of synchronization error calibration are not covered in this research. We just assume some kind of appropriate calibration procedure is done to reduce the sampling error. Under this assumption, due to the periodic nature of the master clock signal, the A/D modules take samples within one sample period of each other regardless of the length of the cables connecting the
master clock to each A/D modules. Thus, for chosen sampling frequency f s , the
1 maximum sampling error between any two A/D modules is . Clearly, for higher f s sampling frequencies, the maximum sampling error may be further reduced to a
- 23 - certain degree. At the A/Ds, hardware to trigger sampling at the zero crossings (for positive or negative slope of the sinusoid) can be employed. The sinusoidal clock will maintain its shape over the cables and experience only a phase shift representing transmission delay and cable effects. This fixed phase shift can be compensated with calibration.
The sampled received antenna signals are put into an FFT block to be transformed from time domain into the frequency domain. We use a fixed sampling frequency for
different bands in the VHF band. The only requirement for sampling frequency f s is that it should be much higher than the Nyquist frequency of the high end of the VHF band. For the low end of the VHF band, the RF sampling is simply more over sampled.
3.2 Digital Butler Matrix
Digital Butler matrix is a transformation of the analog Butler matrix, which defines a set of fixed beams in pre-defined directions, into a digital implementation. As we discussed previously, digital antenna array beamforming is carried out by weighting digital signals, received and sampled from antennas, thereby adjusting their amplitudes and phases such that when added together they form the desired beams [2].
3.2.1 Narrow Band Butler Matrix
Firstly, we discuss narrow band fixed beamformer and then extend it to wideband case.
- 24 - Assuming the incident narrow band signal (frequencyω0 ) from a known direction
[]θ 0 ϕ0 (they stand for elevation angle and azimuth angle respectively). Our objective is to choose the beamformer weight vector to minimize the squared error between the actual response and desired response. The choice for the beamformer
weight vector is the array steering vectorα(ωi θ 0 ϕ 0 ) . The resulting array and beamformer is termed a phased array since the output of each sensor is phase shifted prior to summation. In an environment consisting of only uncorrelated noise and no directional interferences, this beamformer provides maximum SNR [6]. And the actual response is characterized by a main lobe and side lobes. We can further adjust the magnitude of each element of steering vector, using tapering and windowing method, to trade off main lobe width against side lobe levels to form the response into
a desired shape. If we want to create K beams for center frequencyω0 , we can define a steering matrix as
A = [α(ω 0 θ 1 ϕ 1 ) L α(ω 0 θ K ϕ K )] (3.1)
Where the ith column of A is the steering vector of direction []θi ϕi and
frequencyω0 . In the narrow band case, this steering matrix can form any pre-defined beams.
3.2.2 Wideband Butler Matrix
Because the steering matrix can only produce beams for one central frequency, we
need more different steering vectors to produce beams in the same direction[]θi ϕi for different central frequencies to handle the wideband case. But first of all, we need to explore the relationship between a wideband signal and a narrow band one. A
- 25 - general strategy of frequency domain beamformer for wideband signals is to translate a wideband signal into the frequency domain using the FFT, and each frequency bin is processed by a narrow-band beamformer [6]. The weighted signals from all antenna elements are summed to produce an output at each FFT bin. The weights are selected independently for different frequency bin. In other words, the strategy is to divide the bandwidth of a wideband signal into several frequency “bin”s (sub-bands), each of which has much narrower bandwidth. So the frequency content of each sub-band is assumed to be constant. We then can treat each sub-band separately in parallel as in the narrow band case.
In our system, we translate the sampled received antenna signals into the frequency domain by an FFT algorithm. If we choose the size of FFT, L, large enough, each FFT output bin can be treated like one narrowband signal. We then can apply one column in the steering matrix, corresponding to that frequency, to one bin and get the beam patterns for this frequency of interest. As a consequence, to form a beam in a given
direction[]θ 0 ϕ0 for a wideband signal, L steering vectors are required for each of the L narrowband divisions.
j Let {X i } i = 1,2,L, L be the output of an L-point FFT at the jth antenna element.
The L frequency components are equally spaced across the bandwidth of the wideband signal. The wideband problem then becomes L narrowband ones. Thus,
j consider the ith frequency bin, or equivalently, X i j = 1,2,L, N , across all array
elements. The steering vector α(ωi θ 0 ϕ 0 ) will be designed for []θ 0 ϕ0 corresponding to the ith frequency. Obviously for a different frequency bin the
- 26 - steering vector would be different even for the same incident angle[]θ 0 ϕ0 . The beamformer output for this frequency bin is simply given by
Η 1 2 N Τ Y i = α ()ω i θ 0 ϕ 0 X i where X i = [X i X i L X i ] (3.2)
Since i ranges from 1 to L, L weight vectors are needed to produce L frequency bins of the wideband beamformer output. The total energy of the output of the beamformer is given by
L 2 E = ∑ Y i i =1 (3.3)
Actually, since the data keeps coming in blocks of length L, Xi is also a function of time (in number of blocks). So E also needs to be averaged over the time blocks. The dimension of Butler matrix for one direction is then L × N .
3.2.3 Shading
As we just mentioned before, some kind of tapering and windowing method needs to be implemented to get a desired beam pattern in a beamforming system. This is also called shading. It is obvious that, in order to get good accuracy for direction finding and robustness against interference, we need a narrow main lobe and low side lodes in the beam pattern. Antenna synthesis provides us many methods to design exactly or approximately an acceptable beam pattern [3]. One of the most well-known methods is Dolph-Tschebyscheff method. It is optimum in that for a given tolerable side lobe level R, the narrowest main lobe width will be achieved by the Dolph-Tschebyscheff shading, whose side lobes are all equal to R [7].
- 27 - For a given frequencyωi and incident angle[θ 0 ϕ0 ], we have Dolph-Tschebyscheff
weight, w i , and the steering vectorα(ωi θ 0 ϕ 0 ) . So finally the ith weight vector in
the digital Butler matrix is element multiplication of w i andα(ωi θ 0 ϕ 0 ) , i.e.:
Wi =α(ωi θ0 ϕ0 )⊗wi (3.4)
Τ Τ Where, for two vectors a =[a1 a2 L aN ] and b =[b1 b2 L bN ] , we
Τ define: a⊗b =[]a1b1 a2b2 L aNbN .
The details of how to design Dolph-Tschebyscheff weight is beyond this work. [3] and [28] are good tutorials for design Dolph-Tschebyscheff weight for linear array. A good method is provided in [29].
3.2.4 Direction Finding Strategy
Direction finding based on the digital Butler matrix operates in the following two steps:
1. Coarse estimation of DoA. Set a fixed number of beams in the entire possible
DoA area. The separation between two adjacent look angles is (much) larger
than the desired accuracy. The look direction of the beam with the highest
output energy is a coarse estimate of the direction from which the signal is
arriving.
2. Fine estimation of DoA. Narrow the search area around the look direction
obtained from step 1. And set a fixed number of beams to refine the look
direction. The separation between two adjacent look angles is (much) smaller
- 28 - than that in step 1, and can be equal to the required resolution. The look
direction of the beam with the highest output energy is the refined estimate of
the direction from which the signal is arriving.
3. Repeat Step 2, if necessary, till certain DoA resolution is achieved.
- 29 - LNA
A/ D
Master Clock
S / P
FFT
e jwLt4 e jwLt3 e jwLt2 jw t e jw1t1 e L 1 w L,4 w L,3 w L,2
w 1,1 wL ,1
Figure 3.1 Block Diagram of Digital Direction Finding System
- 30 - 4 ARRAY PARAMETER ANALYSIS
In this chapter, we will consider one important issue in designing a direction finding system using antenna array. That is how to choose the antenna array geometry and how to set array parameters, such as the distance between two adjacent elements, based on the system performance requirements.
4.1 Antenna Array Size Consideration
Firstly, we discuss how to choose the antenna array size to get best performance. In order to simplify the discussion, we only focus our discussion on arrays, which is excited with a single frequency signal, with steering vector pointing to DoA of incident signal as weights.
4.1.1 Equal-spaced Linear Array
We choose the array weight w = α (ω ϕ ), which makes the array point at the
direction ϕ . If the incident signal impinges upon the array at an angleϕ 0 , so we get the array factor from (1.1) as
jκd(sinϕ −sinϕ) j2κd(sinϕ −sinϕ) j(N−1)κd(sinϕ −sinϕ) AF(ϕ) =1+e 0 +e 0 +L+e 0
jNκd(sinϕ0−sinϕ) 1−e = jκd(sinϕ −sinϕ) 1−e 0 (4.1)
Let
β ()ϕ = κd(sinϕ0 − sinϕ) (4.2)
- 31 -
In order to simplify the formulations, we use β instead of β (ϕ ) in the following formulations. And insert (4.2) to (4.1), we get:
jNβ − jNβ 2 1− e 1− e AF(ϕ) = × 1− e jβ 1− e− jβ 2 − e jNβ + e− jNβ = () 2 − ()e jβ + e− jβ
⎛ Nβ ⎞ sin2 ⎜ ⎟ 2 = ⎝ ⎠ ⎛ β ⎞ sin2 ⎜ ⎟ ⎝ 2 ⎠
Then we get:
⎛ Nβ ⎞ sin⎜ ⎟ 2 AF()β ()ϕ = ⎝ ⎠ ⎛ β ⎞ sin⎜ ⎟ ⎝ 2 ⎠ (4.3)
It is obvious that the magnitude of linear array factor is a periodic function of β .
From (4.2) and properties of sinusoid function, we can get the range of β
κd ()sin ϕ 0 − 1 ≤ β ≤ κd (1 + sin ϕ 0 ) (4.4)
Based on the previous formulations, we will derive several important aspects of an equal-spaced linear antenna array below.
- 32 - 4.1.1.1 Ambiguity of Direction
The visible range of a linear antenna array is determined by (4.4). In order to avoid ambiguity of direction, β must be in one period of the periodic magnitude function
(4.3), i.e. 0 ≤ β ≤ 2π . And because − 1 ≤ sin ϕ 0 ≤ 1 , we have
0 ≤ β ≤ 2κd ≤ 2π ⇒ κd ≤ π
π c λ ⇒ d ≤ = = κ 2 f 2 (4.5)
(4.5) give the condition of distance between two adjacent elements, which makes the antenna array able to detect the angle of incident signal without ambiguity. The following discussion is based on the condition that the array element distance obeys
(4.5).
4.1.1.2 Beamwidth of Main Lobe
If the distance between two adjacent elements obeys (4.5), there is only one main lobe in the visible range. The beamwidth of the main lobe is the angular space between the half-power points on each side of the main lobe. Following this definition, the
position of the half-power point ϕ h is determined by
2 2 1 2 N AF(ϕh ) = AFmax = 2 2 (4.6)
Then we can get:
- 33 - ⎛ Nβ (ϕ )⎞ sin ⎜ h ⎟ 2 2 ⎝ ⎠ = ⎛ β ()ϕ ⎞ 2 N sin ⎜ h ⎟ ⎝ 2 ⎠ (4.7)
We can solve β ()ϕ h from (4.7) and insert it into (4.2). Then we can find the beamwidth of the main lobe
⎛ β (ϕ h )⎞ 2 ϕ h − ϕ o = 2 arcsin ⎜ sin ϕ 0 − ⎟ − ϕ 0 ⎝ κd ⎠ (4.8)
From (4.8), it is obvious that the beamwidth of main lobe is determined by the
β ()ϕ h by the solution of (4.7). β (ϕ ) is the position of half-power of κd h sin ()Nx sin (Nx ) . Here we plot the function below with different N. N sin x N sin x
sin(Nx)/N/sin(x) 1 N=5 N=10 0.9 N=20
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x (π)
Figure 4.1 Sin(Nx)/N/sin(x)
- 34 - sin ()Nx From Figure 4.1, we can see, as N increases, the main lobe of function N sin x
becomes narrower. So β ()ϕ h becomes smaller as N increases. We then can conclude that, in order to get narrower main lobe, we need to add more elements to the array. At the same time, we can also increase the distance between two adjacent
λ β (ϕ h ) elements in the range 0 ≤ d ≤ to let smaller, so as to get narrower 2 κd beamwidth.
4.1.1.3 Side Lobes
The position and the magnitude of the first side lobe are another important performance factor. We can get numerical solutions of the location of each side lobe by setting the derivative of (4.3) to zero.
Nβ ⎛ Nβ β Nβ β ⎞ 2 sin ⎜ N cos sin − sin cos ⎟ d AF ()β 2 ⎝ 2 2 2 2 ⎠ = = 0 β dβ sin 3 2 (4.9)
The solutions of (4.9) are
⎧ 2mπ β = m = ±1,±2, ⎪ N L ⎨ and β ≠ 2mπ β Nβ ⎪ N tan = tan ⎩⎪ 2 2
2mπ The solutions β = m = ±1,±2, are actually locations of the nulls. So N L the locations of the side lobes are
β Nβ N tan = tan 2 2 (4.10)
- 35 -
When N is a large number, we can get approximate solution by assuming that side lobes are roughly at the middle of two adjacent nulls. We already obtained the null locations from solutions of (4.9), i.e.
β 2mπ mλ sin ϕ − sin ϕ = = = m = ±1,2, 0 m κd Nκd Nd L
So the position of the first side lobe is at:
⎛ 2 sin ϕ 0 − sin ϕ ±1 − sin ϕ ± 2 ⎞ ϕ s ≅ arcsin ⎜ ⎟ ⎝ 2 ⎠
⎛ 3λ ⎞ = arcsin ⎜ ± + sin ϕ 0 ⎟ ⎝ 2 Nd ⎠
The position of the first side lobe is determined by the aperture of the linear array. As the aperture increases, the first side lobe is nearer to the main lobe. And the magnitude of the first side lobe is:
1 2 N AF ()ϕ ≅ ≅ s ⎛ 3π ⎞ 3π sin ⎜ ⎟ ⎝ 2 N ⎠
When it is compared with magnitude of the main lobe
AF ()ϕ 2 s ≅ i.e. − 13 .5 dB AF ()ϕ 0 3π (4.11)
(4.11) gives us the level of the first side lobe when N approaches to a very large value.
For example, in our system, we use 5-element equal-spaced linear array, which points
6 at 0°, with the following settings: ωc = 2πf c = 2π (86×10 ). The position and level of the first side lobe are obtained numerically. The results are listed below:
N=5 N=10 N=20 N=1000
- 36 - λ d = 35.4810° 16.6803° 8.2301° 1.6393° 2 λ d = -- 35.0343° 16.6366 ° 3.2799° 4 Table 4-1 Position of the first side lobe in linear array
N=5 N=10 N=20 N=1000 λ d = -12.0412 dB -12.9662 dB -13.1882 dB -13.2585 dB 2 λ d = -- -12.9662 dB -13.1882 dB -13.2585 dB 4 Table 4-2 Level of the first side lobe in linear array
From Table 4-1 and Table 4-2, we can conclude that as the number of antenna elements becomes larger, the first side lode moves further near to main lobe. The level of the first side lobe becomes lower and converges to around -13.5dB. And as the distance between two adjacent elements becomes smaller, the position of the first side lobe moves further away from the main lobe.
4.1.2 Uniform Circular Array
We choose the array weight w = α (ω θ ϕ ), which makes the array point at the
direction []θ ϕ . If the incident signal impinges upon the array at an angle[]θ 0 ϕ0 , so we get the array factor from (1.5) as
ωR N j ()cos(ϕ −ϕ ) sin θ −cos(ϕ −ϕ ) sin θ c 0 i 0 i AF (ϕ,θ ) = ∑ e i=1 (4.12)
In order to simplify (4.12), we introduce two auxiliary variables, which are defined below:
1 2 2 2 µ = R[]()()sin θ cos ϕ − sin θ 0 cos ϕ 0 + sin θ sin ϕ − sin θ 0 sin ϕ 0 (4.13) - 37 - And
R ()sin θ cos ϕ − sin θ cos ϕ cos ζ = 0 0 µ (4.14)
Using (4.13) and (4.14), we can rewrite the exponential in (4.12) as:
ω R ()cos( ϕ − ϕ ) sin θ − cos( ϕ − ϕ ) sin θ c 0 i 0 i ω R = (cos ϕ cos ϕ sin θ + sin ϕ sin ϕ sin θ c 0 i 0 0 i 0
− cos ϕ cos ϕ i sin θ − sin ϕ sin ϕ i sin θ ) ω R = (()cos ϕ sin θ − cos ϕ sin θ cos ϕ c 0 0 i
+ ()sin ϕ 0 sin θ 0 − sin ϕ sin θ sin ϕ i ) ωµ R ()cos ϕ sin θ − cos ϕ sin θ = ( 0 0 cos ϕ c µ i R ()sin ϕ sin θ − sin ϕ sin θ + 0 0 sin ϕ ) µ i ωµ = ()cos ς cos ϕ + sin ς sin ϕ c i i
Thus we can rewrite (4.12) as follow
ω N −1 j µ cos ()ζ −ϕ c i AF (ϕ ,θ ) = ∑ e i = 0 (4.15)
ϕ We can calculate the array factor using (4.13-4.15), when R, N, i , θ 0 and ϕ0 are given. But it is a very time consuming task even for a moderately large value of N.
We can use Jacobi-Anger Expansion
∞ jz cos ()x k jkx e = ∑ j J k ()z e k = −∞
- 38 - and the Fourier expansion of discrete delta function
2 π mk N − j 1 N δ ()k = N ∑ e and δ ()k = δ (k + N ) m = 1
where J k ()• is the Bessel function of the first kind, to rewrite (4.15) as:
ω N − 1 j µ cos ()ζ − ϕ AF (ϕ ,θ ) = ∑ e c i i = 0 ⎛ 2 π i ⎞ N − 1 ∞ jm ⎜ − ζ ⎟ m ⎛ ω ⎞ ⎝ N ⎠ = ∑∑ j J m ⎜ µ ⎟e i = 0 m = −∞ ⎝ c ⎠ ⎛ π ⎞ 2 π i ∞ jm ⎜ − ζ ⎟ N − 1 jm ⎛ ω ⎞ ⎝ 2 ⎠ N = ∑ J m ⎜ µ ⎟e ∑ e m = −∞ ⎝ c ⎠ i = 0 ⎛ π ⎞ ∞ jmN ⎜ − ς ⎟ ⎛ ω ⎞ ⎝ 2 ⎠ = N ∑ J mN ⎜ µ ⎟e m = −∞ ⎝ c ⎠ (4.16)
Based on properties of Bessel function and (4.16), we can conclude that there is only one main lobe, which can be pointed to any direction over the surface of a hemisphere.
The Bessel function is defined in the range [0 ∞] of its argument. But in (4.16),
ω J • is restricted to the range ⎡ ⎤ , where µ is the maximum value mN () ⎢0 µmax ⎥ max ⎣ c ⎦ that µ can achieve. We call this range “visible range”. And the number of side lobes
and positions are determined by J mN (•) in the visible range. Because the value of a
Bessel function of large order is very small, so the shape of AF ()ϕ,θ , when the number of antenna elements N is large, can approximated by
⎛ ω ⎞ AF (ϕ ,θ ) ≅ NJ 0 ⎜ µ ⎟ ⎝ c ⎠ (4.17)
Using (4.17), we can do array pattern analysis for large N case. But in this project, we only use several antenna elements. And there is no more simple formulation for the
- 39 - array factor for uniform circular array. So we choose the numerical analysis for circular array based on (4.12).
4.1.2.1 Ambiguity of Direction
Assuming that the elevation angle of the incident signal is 90°, then (4.12) is a periodic function of ϕ with period 2π radius.
ωR N j ()cos(ϕ −ϕ )−cos(ϕ −ϕ ) c 0 i i AF (ϕ ) = ∑ e i=1 (4.18)
And it can be analyzed in terms of a Fourier series and Jacobi-Anger Expansion. We just borrow the final results of analysis from [28] here: the circular array will not
2π radiate or receive spatial frequencies higher than about R. The antenna elements λ can be regarded as spatially sampling the signal having the highest spatial
2π frequency R. The spatial frequency is defined as [37]: λ
u =κ sinθ cosϕ
v =κ sinθ sinϕ which is the counterpart of frequency in temporal sampling. Then by applying the
Nyquist sampling theory, the circumferential sampling rate, i.e. the inter-element spacing, should correspond to at least twice the highest spatial frequency [36]. So
2π 2π λ 2× R ≤ N ⇒ R ≤ λ N 2 (4.19)
λ That is the distance between any two elements should be less than . In the 2 following analysis, we assume that the condition in (4.19) always be satisfied. - 40 - Radius = 2λ/π (θ=90° and 4 antenna) Different Radius (θ=90° and 4 antenna) 90 1 90 1 120 60 120 60
150 0.5 30 150 0.5 30
180 0 180 0
210 330 210 330
240 300 240 300 270 270
R = λ/π R = λ/2π R = λ/4π
Figure 4.2 Beam pattern of 4-element circular array with differnent radius
Based on (4.18), Figure 4.2 generates the beam patterns of a 4-element circular array,
whenθ =θ0 =90° and the array points atϕ0 =0° . When the distance of inter-elements
λ 2πR increases beyond , e.g. R = 2λ /π (inter-space is equal to , which is λ in 2 N this case) in the figure, there are three main lobes, which points at 0°, 95° and 265°,
ω respectively. And as R decreases, µ decreases. Thus the visible range, ⎡ ⎤, max ⎢0 µmax ⎥ ⎣ c ⎦ decreases. So the number of side lobes decreases. We can see from Figure 4.2,
λ when R = 2λ /π , there are 3 side lobes; when R = λ /π , inter-space= , there are 2 only 2 small side lobes left; in the remaining two cases, R = λ / 2π and R = λ / 4π , there are no side lobes at all. On the other hand, because the visible range decreases as the radius decrease, the width of the main lobe becomes larger, and vice versa.
- 41 - Different antenna number (θ=90° and R=λ/π) 90 1 N = 4 N = 8 120 60 N = 16 0.8 N = 32
0.6 150 30 0.4
0.2
180 0
210 330
240 300
270
Figure 4.3 Beam pattern of circular array with differnent antenna number and fixed R
Figure 4.3 gives a good explanation for (4.16), it shows that how the different number of the array elements changes the shape of the beam pattern. And it also explains the effects of the Bessel functions of high order to the beam pattern. From Figure 4.3, we can see that for different number of antenna elements on the array, the change of the shape of the main lobe is not obvious. But the positions and the levels of the side lobes change a lot for N changing from N=4 to N>4. From (4.16), the side lobe levels are determined by Bessel functions of high order. But they have little effects on the shape of the main lobe. So when N becomes large, we can use (4.17) to proximate
(4.16). In Figure 4.3, the pattern change little for N=8 and beyond.
- 42 - Fixed inter-space -- λ/2 (θ=90°) 90 1 N=4 N=8 120 60 N=16 0.8
0.6 150 30 0.4
0.2
180 0
210 330
240 300
270
Figure 4.4 Beam pattern of circular array with fixed Inter-space
Figure 4.4 shows the beam patterns of a circular array with fixed inter-space, but different number of antenna elements, which leads to different radius of the array. We can see from the figure that as N increases, as we expected, the number of the side lobes increases whereas the main beam width decreases.
4.1.2.2 Beamwidth of Main Lobe
In the previous section, we have already drawn conclusions on how the beamwidth changes with N and R. Here, we will first discuss how the beamwidth changes when the array points to a different azimuth angle and elevation angle. We will then calculate the beamwidth of the main lobe numerically. In this section, we will treat the beamwidth along the azimuth angle and the elevation angle separately.
- 43 - ⎛ ω ⎞ As mentioned before, the shape of the main lobe is determined by J 0 ⎜ µ ⎟ , ⎝ c ⎠
especially for a large N case. So J 0 (•) and µ determine the beamwidth of the main lobe. From (4.13), we can get:
⎧ θ + θ θ − θ 2R cos 0 sin 0 ; ϕ = ϕ ⎪ 2 2 0 µ = ⎨ ϕ − ϕ ⎪ ⎛ 0 ⎞ 2R sin θ 0 sin ⎜ ⎟ ; θ = θ 0 ⎩⎪ ⎝ 2 ⎠ (4.20)
Firstly, we can draw a conclusion from (4.20) that the main lobe beamwidth along both the elevation angle and the azimuth angle changes with different elevation angle and does not vary much with the azimuth angle.
In order to get a better shape of the beam pattern for the demonstration purpose, in the following simulations we use a 80-element uniform circular array with a radius equal to 4λ .
In order to show the results obtained from (4.20), we plot the beam patterns of the circular array. First, we show how the azimuth angle beamwidth changes with different azimuth angle and elevation angle. Figure 4.5 plots the beam patterns along
the azimuth angle at different elevation angles with ϕ0 = 0° . And Figure 4.6 plots the beam patterns along the azimuth angle with array pointing at different azimuth angles
when θ0 = 45° . Comparing these two figures, it is obvious that the azimuth angle beamwidth of the main lobe changes significantly with different elevation angle. And the azimuth angle has almost no effect on the azimuth angle beamwidth.
- 44 -
Following the similar way, we show how the elevation angle beamwidth changes with different azimuth angle and elevation angle in the same way. Figure 4.7 plots the
beam patterns along the elevation angle at different elevation angles with ϕ0 = 0° .
And Figure 4.8 plots the beam patterns along the elevation angle with array pointing
at different azimuth angles when θ0 = 45° . From Figure 4.7, it is obvious that the elevation angle beamwidth of the main lobe becomes wider as the elevation angle becomes larger. From Figure 4.8, we see again that the azimuth angle has almost no effect on the beamwidth along the elevation angle either. The shapes of the main lobe at different azimuth angles are almost identical.
Azimuth angle beamwidth at different elevation angle ( =0 ) φ0 ° 0 10
-1 10
-2 10 Array output (dB)
-3 10
θ1 = 30° = 45 θ2 ° = 90 θ3 ° -4 10 -50 -40 -30 -20 -10 0 10 20 30 40 50 Azimuth angle φ
Figure 4.5 Azimuth angle beamwidth with different elevation angle
- 45 - Azimuth angle beamwidth at different azimuth angle ( =45 ) θ0 ° 0 10
-1 10
-2 10 Array output (dB)
-3 10
= 30 φ1 ° = 45 φ2 ° -4 10 0 10 20 30 40 50 60 70 80 90 Azimuth angle φ
Figure 4.6 Azimuth angle beamwidth with different azimuth angle
Elevation angle beamwidth at different elevation angle ( =0 ) φ0 ° 0 10
-1 10
-2 10
-3
Array output(dB) 10
-4 10
= 45 θ1 ° = 60 θ2 ° -5 10 0 10 20 30 40 50 60 70 80 90 Elevation angle θ
Figure 4.7 Elevation angle beamwidth with different elevation angle
- 46 - Elevation angle beamwidth at different azimuth angle ( =45 ) θ0 ° 0 10
-1 10 Array output (dB)
= 0 φ1 ° = 30 φ2 ° = 45 φ3 ° = 90 φ4 ° -2 10 0 10 20 30 40 50 60 70 80 90 Elevation angle θ
Figure 4.8 Elevation angle beamwidth with different azimuth angle
4.1.2.3 Side Lobes
As we stated before, the side lobe position and level are determined by J mN (• ) in the visible range. When N is large enough, we can ignore the effects of Bessel functions of higher orders and analyze the position and level of side lobes by
exploring properties of J 0 ()• . In our project, we only use a small number of antenna elements. So we can only find the position and level of side lobes numerically. Here we calculate the first side lobe position and level under several conditions which may be considered in our project.
Firstly, in our system, we calculate a 4-element uniform circular array, which points at
6 the azimuth angle ϕ 0 = 0° with ωc = 2πf c = 2π (86×10 ). The position and level of
- 47 - the first side lobe are obtained numerically. The results show that the first side lobe is always at θ = 90° . So in Table 4-3, we only list the azimuth angle (degree) of the first side lobe. The results are listed below:
θ 0 0° 30° 45° 60° 90° λ R = 45 116.6332 106.2814 97.6131 90 2 λ R = -- 233.1658 239.7236 245.2764 250.2261 3 λ R = -- -- 227.9246 232.3116 236.2814 4 Table 4-3 Location of the first side lobe (azimuth angle) of a 4-element UCA
Table 4-4 gives us the level of the first side lobe. The unit of the value in the table is dB.
θ 0 0° 30° 45° 60° 90° λ R = -4.3549 -0.3014 -0.0379 -0.0017 -4.1214e-6 2 λ R = -- -8.2496 -5.5279 -4.2563 -3.5258 3 λ R = -- -- -21.6932 -14.9032 -11.9648 4 Table 4-4 Level (dB) of the first side lobe of a 4-element UCA
4.2 Uniform Circular Array with a center element
The (4.16) reveals that the array pattern of a uniform circular array is composed by
Bessel functions with different order. For large N, the main lobe beamwidth and the
first side lobe level, where J 0 (3.8) = 0.4026 (The position and level of the first side lobe
of J 0 ()• ), are determined numerically by Bessel function with the first order. Because it is only -7.9dB below the main lobe, so a reduction of the side lobe level is therefore essential for practical application. There are several techniques to reduce the side lobe
- 48 - level. One is attained by an appropriate phase and amplitude taper. Here we introduce another method: to add another element at the center of the ring. So the array pattern is modified as:
⎛ ω ⎞ AF (ϕ ,θ ) = I + NJ 0 ⎜ µ ⎟ ⎝ c ⎠ (4.21)
Because the center element is treated as the reference, its output is just a DC value, I.
After adding a DC value to J 0 (•), the whole function is shifted up by I. If I is large relative to N, we can reduce the first side lobe level, even eliminate the first side lobe level at the cost of increased second side lobe level. We can further reduce the side lobe level or change the other aspects of the beam pattern by using concentric rings, such as hexagonal array, which may produce more desirable results.
In Figure 4.9, we compare the beam patterns of a circular array with a center element and that of which without the center element. We can point out from the plot that, after adding one center element, the level of the first side lobe decreases at the cost of increased level of the second side lobe.
- 49 - Array pattern w/o center element 0 10
-1 10 Array output (dB)
-2 10
Without center element With center element -3 10 0 10 20 30 40 50 60 70 80 90 Elevation angle θ
Figure 4.9 Beam pattern compare w/o center element
4.3 Hexagonal Array
Here we give a brief discussion and numerical analysis for the hexagonal array. In the left subplot of Figure 4.10, we plot the beam pattern of a 12-element hexagonal array
2λ using the following parameters: R = , ω = 2πf = 2π (86×106 ) , θ = 45° and 1 3 c c 0
ϕ0 = 0°. For comparison purpose, we also plot the beam pattern of a 12-element UCA
2λ with the same parameters, except R = , in the right subplot of the figure. 3
- 50 -
Figure 4.10 Comparison of Beam pattern between hexagonal array and circular array
From Figure 4.10, we can see that the second side lobe level of the UCA is much lower than the first side lobe level. After change the geometry of the UCA to form the hexagonal array, by keeping six elements at the original position and shrinking the radius of the circle which the other six elements, the first side lobe level is decreased.
And the second side lobe level increases. Below we give the numerical results, from which it is seen that the UCA has lower side lobe levels than the hexagonal array.
First side lobe Level Second side lobe level Circular array -7.8989 dB -10.2956 dB Hexagonal array -6.2174 dB -6.2174 dB Table 4-5 Side lobe level comparison between hexagonal and circular array
First side lobe position Second side lobe position Elevation Azimuth Elevation Azimuth Circular array 17.15° 126.79° 76.93° 180° Hexagonal array 35.21° 180° 68.94° 92.35° Table 4-6 Side lobe position comparison between hexagonal and circular array
- 51 - From Table 4-5 and Table 4-6, we can see that even though the first and second side lobe levels of hexagonal array are little higher than those of circular array. But the first side lobe of hexagonal array is much further away from the main lobe, compared with circular array. From Table 4-6, the first side lobe of hexagonal array points at
180° (azimuth angle). The main lobe and the first side lobe point at opposite directions. But the first side lobe of circular array points only at 126.79°.
Table 4-7 lists the beamwidth of the main lobe for the hexagonal array and the circular array. We calculate the beamwidth along azimuth angle and elevation angle, separately. From Table 4-7, we can see that the main lobe beamwidth of hexagonal array is wider than that of the circular array.
Azimuth angle Elevation angle Circular array 62.4° 39.7107° Hexagonal array 66.8768° 46.8919° Table 4-7 Main lobe beamwidth of the hexagonal array and circular array
- 52 - 5 TIMING ERROR ANAYSIS
In this chapter, we compute the DDF angle error that results from given timing errors at each antenna, introduced by cable lengths and other electrical length differences, for a given array geometry. The purpose of such analysis is that for those timing errors that cannot be compensated by calibration, we can estimate the performance, such as resolution, that the direction finding system can achieve. And for those timing errors that can be compensated by calibration, we can also analyze how the cable lengths and other electrical length differences affect the performance of the overall
DDF system, so that we can determinate the maximum cable length differences allowed for the system to meet the given direction finding resolution requirement.
We consider the following three array geometries in our research:
• Equal-spaced linear array
• 4 element uniform circular array
• 4 element uniform circular array with one more element in the center of the
circle
The following notations are used for the analysis in this chapter:
The center frequency of the k th bin in incident
ωk single
L FFT length at each antenna
The number of frequency bins in each sub-band, M which are the interested frequency bins in all L
- 53 - bins. So apparently M ≤ L
N The number antenna array elements
R Radius of the circular array (circular array only)
Distance between two antenna elements (linear d array only)
ϕ0 Azimuth angle of incident signal
θ 0 Elevation angle of incident signal (circular array)
T Sample time interval
Weight for k th bin of FFT (one row of FFT
matrix) Fk k k ( L −1) ⎡ − j 2π − j 2π ⎤ 0 L L F k = ⎢e e L e ⎥ ⎣ ⎦
Steering vector for frequency ωk , incident
α(ωk θ ϕ) elevation angle θ and azimuth angle ϕ
Phase delay of k th frequency bin at i th antenna id sin(ϕ ) φ = ω 0 k,i k c element (linear array only)
Phase delay of k th bin at i th antenna element.
ωk R φk,i = cos(ϕ0 −ϕi )sin(θ 0 ) 2π C (circular array) ϕi = (i −1) i = 1,2,..., N −1 N
Sampling time error. τ i uniformly distributed in
T T τ (i = 0,1, , N −1) ⎡ 0 0 ⎤ i L ⎢− ⎥ (T0 is determinated by the cable ⎣ 2 2 ⎦
length effects)
c The speed of light
Table 5-1 Notations for DDF error analysis
- 54 - We set the first antenna, the one at one end of a linear antenna array or the one at the center of a circular array, as the reference one, which has no sampling time error, i.e.
τ 0 = 0 . The sampled incident signal at the antenna array is represented as the sum of
jω t M narrowband signals, each of which represents a narrowband signal e k and the combination of which forms the wideband signal. In addition, each narrowband signal must have its own steering vector. Consequently, samples of the received signal across the array can be arranged as:
0 j(−φk ,1+ωkτ1) j(−φk ,N 1+ωkτ N 1) ⎡ e e L e − − ⎤ ⎢ j(ω T−φ +ω τ ) j(ω T−φ +ω τ ) ⎥ e jωkT e k k ,1 k 1 e k k ,N −1 k N −1 S = a ⎢ L ⎥ ∑ k ⎢ ⎥ k ⎢ M M O M ⎥ jω (L−1)T j(ωk (L−1)T−φk ,1+ωkτ1) j(ωk (L−1)T−φk ,N 1+ωkτ N 1) ⎣⎢e k e L e − − ⎦⎥ 0 ⎡ e ⎤ ⎢ ⎥ e jωkT ⎢ ⎥ 0 j(−φk ,1+ωkτ1) j(−φk ,N −1+ωkτ N −1) = ∑ak []e e L e k ⎢ ⎥ ⎢ M ⎥ jωk (L−1)T ⎣⎢e ⎦⎥ (5.1)
where each column of S contains samples of one array element and each row contains samples of the signals received by different antenna elements in one snapshot. The
φk ,i + ω kτ i term stands for the overall phase delay introduced by antenna elements’ position difference and timing errors. The summation is overall frequency bins in a sub-band. We then transform the time domain digital signal to the frequency domain using the well-known FFT algorithm and then select the corresponding frequency bins of the band of interest. The index k in (5.1) spans those frequency bins of interest which are input to the Digital Butler matrix. Define the steering vector for the kth bin is:
- 55 - T 0 jφ k ,1 jφ k , N −1 α (ω k θ ϕ ) = [e e L e ] (5.2) where
id sinϕ Linear Array φ = ω k,i k c ω R Circular Array φ = k cos(ϕ −ϕ )sinθ k,i C i
The output of the Butler matrix for the kth bin is given as:
2 H Jk = Fk Sα(ωk θ ϕ)
H H H =α (ωk θ ϕ)S FkFk Sα(ωk θ ϕ) ⎡ e0 ⎤ ⎢ j(φ −ω τ ) ⎥ e k,1 k 1 H ⎢ ⎥ 0 j(−φk,1+ωkτ1) j(−φk,N−1+ωkτN−1) = Akα (ωk θ ϕ) []e e L e α(ωk θ ϕ) ⎢ M ⎥ ⎢ j(φ −ω τ ) ⎥ ⎣⎢e k,N−1 k N−1 ⎦⎥
N−1 2 j(φk,i −φk,i −ωkτi ) = Ak 1+∑e i=1 (5.3) where
⎡ e 0 ⎤
⎢ jω iT ⎥ 2 e 0 − jω iT − jω i ( L−1)T H ⎢ ⎥ Ak = ∑ ai []e e L e Fk Fk i ⎢ ⎥ ⎢ M ⎥ jω i ( L−1)T ⎣⎢e ⎦⎥ (5.4)
In a desired frequency domain transformation, the output of one frequency bin
contains only energy of corresponding frequency. The FFT algorithm has a leakage
phenomenon that results in errors between the computed and the desired frequency
domain transformation. It occurs when a frequency component of the original signal
is not an integer multiple of the sampling period (T). This is seen in (5.4). If
i ω = 2π , then the inner products in (5.4) become zero if i ≠ k , and L if i LT
- 56 - 2 i = k , yielding Ak = La k . Otherwise, the energy of the frequency appears to leak into other frequency. As the sampling rate and length of FFT increase, this phenomenon relieves and the energy leaks from other frequency is very small. So we
2 can assume Ak ≈ La k .
The output of the beamformer is the summation of all of the frequency bins in a given sub-band as
J = ∑ J k k
N −1 2 j(φk ,i −φ k ,i −ωkτ i ) = ∑∑Ak 1+ e k i=1 N −1 N −1 2 2 = ∑∑Ak [(1+ ∑ cos(φk ,i −φ k ,i −ω kτ i )) + ( sin(φk ,i −φ k ,i −ω kτ i )) ] k i=1 i=1 (5.5)
Now we are prepared to discuss the three different array geometries separately.
5.1 Equal-spaced Linear Array
The linear array has very good linear property inherited in its geometry, such as phase difference of signals received by two elements is a linear function of the distance between two antenna elements. We use the linear property in the following work to simplify the formulation deduction.
id sin ϕ id sin ϕ φ − φ = ω 0 − ω k ,i k ,i k c k c ω d (sin ϕ − sin ϕ ) = i k 0 c
= i∆φ k ()ϕ (5.6) Where
- 57 - d(sinϕ0 − sinϕ) ∆φk ()ϕ = φk,i −φ k,i = ωk c (5.7)
Inserting (5.6) and (5.7) into (5.5), we get
N −1 N −1 2 2 J ()ϕ = ∑∑Ak {[1+ ∑ cos(i∆φ k (ϕ )−ω kτ i )] + [ sin(i∆φ k (ϕ )−ω kτ i )] } k i=1 i=1 (5.8)
Note that ∆φk (ϕ ) is a linear function of ωk through its dependence on the bin center frequency. The estimated DoA angle is
ϕ = arg max{J(ϕ)} ϕ (5.9)
Because J (ϕ ) is a continuous function of ϕ , in order to solve the optimization problem formulated in (5.9), we need to find the derivative of J (ϕ) with respect to ϕ
and set the derivative equal to zero. In order to simply the formulation we use ∆φk
and J instead of ∆φk (ϕ) and J (ϕ) in the following formulations.
N−1 N−1 dJ ⎛ ⎞⎛ ⎞ωk d cosϕ = 2∑∑Ak ⎜1+ ∑cos(i∆φk −ωkτ i )⎟⎜ isin(i∆φk −ωkτ i )⎟ dϕ k ⎝ i=1 ⎠⎝ i=1 ⎠ c N−1 N−1 ⎛ ⎞⎛ ⎞ωk d cosϕ − 2∑∑Ak ⎜∑sin(i∆φk −ωkτ i )⎟⎜ icos(i∆φk −ωkτ i )⎟ k ⎝ i=1 ⎠⎝ i=1 ⎠ c 2d cosϕ N−1 = ∑∑iAkωk sin(i∆φk −ωkτ i ) c k i=1 = 0 (5.10)
Because in a realistic system, τ i , which is the time delay error after calibration , is
very small, and i∆φk is to compensate the error and is small, too. So it is reasonable
to assume that i∆φk − ωkτ i is very small. We apply the approximation sin(x) ≈ x which simplifies (5.10) to
- 58 - dJ 2d cos ϕ N −1 = ∑∑iAkω k sin( i∆φ k −ω kτ i ) dϕ c k i=1 2d cos ϕ N −1 = ∑∑iAkω k (i∆φ k −ω kτ i ) c k i=1 N −1 2d cos ϕ 2 2 = ∑∑(i Akω k ∆φ k − iAkω k τ i ) c k i=1 = 0
In a linear array, we cannot detect signals from angles near 90o , So we can ignore the possible solution for the previous equation, cosϕ = 0 . We discuss the other solution,
N −1 2 2 ∑∑(i Akω k ∆φ k − iAkω k τ i ) = 0 . k i=1
Solving the last expression results in the equation
N −1 N −1 2 2 ∑∑i Akω k ∆φk = ∑∑iAkω k τ i k i=1 k i=1
Put (4.7) into the previous equation and simplify as follows
N −1 A ω 2 iτ d(sinϕ − sinϕ) ∑∑k k i A ω 2 0 = k i=1 ∑ k k c N −1 k ∑i 2 i=1 N −1 iτ d sinϕ − sinϕ ∑ i 0 = i=1 c N −1 ∑i 2 i=1 (5.11)
We define the DDF angle error as ∆ϕ = ϕ 0 − ϕ , i.e. the angle difference between the incident angle and the estimated angle. As we stated before ∆ϕ is very small. We
- 59 - apply the approximation sin(∆ϕ) ≈ ∆ϕ and cos(∆ϕ) ≈ 1 to obtain the following identity:
sin ϕ 0 − sin ϕ = sin ϕ 0 − sin (ϕ 0 − ∆ϕ ) = sin ϕ − sin ϕ cos ∆ϕ + cos ϕ sin ∆ϕ 0 0 0 ≈ sin ϕ − sin ϕ + ∆ϕ cos ϕ 0 0 0 = ∆ϕ cos ϕ 0 (5.12)
Now substitute (5.12) into (5.11) we obtain
N −1 ∑iτ i sinϕ − sinϕ = i=1 0 d N −1 ∑i 2 c i=1 N −1 ∑iτ i ∆ϕ cosϕ = i=1 0 d N −1 ∑i 2 c i=1 N −1 ∑iτ i ∆ϕ = i=1 d cosϕ N −1 0 ∑i 2 c i=1 (5.13)
⎡ T0 T0 ⎤ As we assume that the τ i is independent and uniformly distributed in ⎢− ⎥ , ⎣ 2 2 ⎦
we can get the mean and variance of τ i as:
Mean(τ i ) = E[τ i ] = 0 (5.14) and
T 2 Var(τ ) = E[τ 2 ] = 0 i i 12 (4.15)
Using the independence of τ i and (5.14) and (5.15), we can compute the mean value of ∆ϕ
- 60 - Mean(∆ϕ) = E[∆ϕ] N −1 ∑iE[]τ i = i=1 d cosϕ N −1 0 ∑i 2 c i=1 = 0 5.16)
and its variance is
Var(∆ϕ) = E[(∆ϕ) 2 ] N −1 2 ∑i var()τ i i=1 = 2 ⎛ d cosϕ N −1 ⎞ ⎜ 0 ∑i 2 ⎟ ⎝ c i=1 ⎠ 2 ()cT0 = 2 2N(N −1)(2N −1)()d cosϕ 0 (5.17)
Finally, the maximum value that ∆ϕ can achieve in (5.13) is obtained when all the
T T τ s equal to the maximum value they can achieve at the same time, i.e. 0 or − 0 . i 2 2
T N −1 0 i 2 ∑ 3cT max( ∆ϕ ) = i=1 = 0 d cosϕ N −1 d cosϕ (4N − 2) 0 ∑i 2 0 (5.18) c i=1
5.2 Uniform Circular Array
In the circular array, we lost linear property among antenna elements, which is inherent in the linear array. So we cannot get the closed form for the DDF error of the circular array. Numerical solution for DDF error becomes our tool to analyze circular array case. Our strategy is to use a gradient ascend algorithm to find the maximum value of J, the corresponding azimuth and elevation angles are the estimated angle of incident signal. The DDF output J is a function of two real variables. That is,
- 61 - 2 J : ℜ → ℜ . The gradient of J at [θ 0 ϕ0 ] is ∇J ([θ 0 ϕ 0 ]) . The direction of maximum rate of increase of a real-valued differentiable function at a point is orthogonal to the tangent of the level set of the function through that point. Thus the
direction of ∇J ([]θ 0 ϕ 0 ) is the direction of maximum rate of increase of J
at[]θ 0 ϕ0 . Then the gradient ascend algorithm can be formulated as follows: Given a point []θ ϕ (k ) , find the next point [θ ϕ](k+1) by moving by the amount µ∇J ([]θ ϕ (k ) ) from[θ ϕ](k ) , where the positive scalar µ is called step size
[8]. The mathematical formulation of the gradient descent algorithm is:
(k+1) (k) (k) []θ ϕ = [θ ϕ] + µ∇J([θ ϕ] )
After convergence, we obtain the estimated direction numerically, which are the direction finding system output. Compared with the DoA of the incident signal, we can do numerical analysis of DDF error. To make the mathematical formulation simple and elegant, in the following analysis for the circular array, we consider only two cases: 4-element uniform circular array (UCA) and 4 UCA plus a center element.
We can analyze the circular array with any other number of elements in the same way.
We consider the 4 UCA plus a center element case first and then the 4 element case, since the 4 element case is a subset of the former.
5.2.1 Four Element Uniform Circular Array Plus a Center Element
When N = 5 , the four antenna elements on the circular ring are at the intersection of
⎧ π 3π ⎫ the ring with the x-axis and the y-axis, i.e. ϕi ∈ ⎨0 π ⎬ . These particular ⎩ 2 2 ⎭
- 62 - ω R angles simplify the formulation of φ = k cos(ϕ −ϕ )sin(θ ) and let us obtain the k,i c 0 i 0
derivatives of βk,i = φk,i −φ k,i −ωkτ i directly as
∂βk,1 ω R ∂β k,1 ωk R = − k cosϕ cosθ = sinϕ sinθ (5.19) ∂θ c ∂ϕ c
∂β k,2 ω R ∂β k ,2 ωk R = k sinϕ cosθ = cosϕ sinθ (5.20) ∂θ c ∂ϕ c
∂β k,3 ∂β k,1 ∂β k,3 ∂β k ,1 = − = − (5.21) ∂θ ∂θ ∂ϕ ∂ϕ
∂β ∂β ∂β k,3 ∂β k,1 k ,4 = − k,2 = − (5.22) ∂θ ∂θ ∂ϕ ∂ϕ Thus, we can get derivative of J, equation (5.5), with respect to the estimated azimuth angle (ϕ ) and elevation angle (θ ) at every point. So based on (5.5) and (5.19 – 5.22), we can get
∂J 2R 4 = ∑∑Akωk {(1+ cosβ k,i )[()sin β k,1 − sin β k,3 cosϕ cosθ ∂θ c ki=1
− (sin β k,2 − sin β k,4 )sinϕ cosθ] 4 − (∑sin β k,i )[()cosβ k,1 − cosβ k,3 cosϕ cosθ i=1
− ()cosβ k,2 − cosβ k,4 sinϕ cosθ]}
2R β k,1 − β k,3 β k,1 + β k,3 β k,1 − β k,3 = ∑ Akωk {[2sin (cos + 2cos ) c k 2 2 2 β − β β − β β + β − β − β + 4cos k,2 k,4 sin k,1 k,3 cos k,2 k,4 k,1 k,3 ]cosϕ cosθ 2 2 2 β − β β + β β − β −[2sin k,2 k,4 (cos k,2 k,4 + 2cos k,2 k,4 ) 2 2 2 β − β β − β β + β − β − β + 4cos k,1 k,3 sin k,2 k,4 cos k,1 k,3 k,2 k,4 ]sinϕ cosθ} 2 2 2 (5.23)
Following the same steps, we get
- 63 - ∂J − 2R β k,1 − β k,3 β k,1 + β k,3 β k,1 − β k,3 = ∑ Akωk {[2sin (cos + 2cos ) ∂ϕ c k 2 2 2 β − β β − β β + β − β − β + 4cos k,2 k,4 sin k,1 k,3 cos k,2 k,4 k,1 k,3 ]sinϕ sinθ 2 2 2
β − β β + β β − β +[2sin k,2 k,4 (cos k,2 k,4 + 2cos k,2 k,4 ) 2 2 2 β − β β − β β + β − β − β + 4cos k,1 k,3 sin k,2 k,4 cos k,1 k,3 k,2 k,4 ]cosϕ sinθ} (5.24) 2 2 2
With the derivative of J with respected ϕ and θ , we use the gradient ascend algorithm to obtain a numerical solution of [ϕ θ ], which maximizes the J. In the
gradient ascend algorithm, it is reasonable to set [ϕ 0 θ 0 ] as the initial point for the algorithm with µ as the step size, and continue the iteration until the magnitude of the gradient vector becomes less than a certain value, for example 10-5.
⎡ ∂J ⎤ ⎢ ⎥ ⎡ϕ k ⎤ ⎡ϕ k 1 ⎤ ∂ϕ = − + µ ⎢ k ⎥ ⎢ ⎥ ⎢ ⎥ ∂J ⎣θ k ⎦ ⎣θ k −1 ⎦ ⎢ ⎥ (5.25) ⎣⎢ ∂θ k ⎦⎥
After the gradient method converges, we can calculate the DDF error as:
⎡∆ ϕ ⎤ ⎡ϕ k ⎤ ⎡ϕ 0 ⎤ ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ (5.26) ⎣ ∆ θ ⎦ ⎣θ k ⎦ ⎣θ 0 ⎦
5.2.2 Four Element Uniform Circular Array
We follow the same steps to form a gradient-based numerical solution for the four element uniform circular array. Other than the fact that the center element is missing, all other formulation and notations are the same as in the five antenna case.
The sampled received signal at the antenna elements is similar to (5.1):
- 64 - j(−φ +ω τ ) j(−φ +ω τ ) j(−φ +ω τ ) ⎡ e k ,1 k 1 e k ,2 k 2 L e k ,4 k 4 ⎤ ⎢ ⎥ e j(ωkT−φk ,1+ωkτ1) e j(ωkT−φk ,2 +ωkτ 2 ) e j(ωkT−φk ,4 +ωkτ 4 ) S = A ⎢ L ⎥ ∑ k ⎢ ⎥ k ⎢ M M O M ⎥ j(ωk (L−1)T−φk ,1+ωkτ1) j(ωk (L−1)T−φk ,2 +ωkτ 2 ) j(ωk (L−1)T−φk ,4 +ωkτ 4 ) ⎣⎢e e L e ⎦⎥ ⎡ e0 ⎤ ⎢ ⎥ e jωkT ⎢ ⎥ j(−φk ,1+ωkτ1) j(−φk ,2 +ωkτ2 ) j(−φk ,4 +ωkτ 4 ) = ∑Ak []e e L e k ⎢ ⎥ ⎢ M ⎥ jωk (L−1)T ⎣⎢e ⎦⎥ (5.27)
and (4.4) becomes
Τ j φ j φ j φ α (ω θ ϕ ) = e k , 1 e k , 2 e k , 4 k [ L ] (5.28)
Following similar steps to the five element case, we express J as
J = ∑ J k k
4 2 j(φk ,i −φ k ,i −ωkτ i ) = ∑∑Ak e ki=1 4 4 2 2 = ∑∑Ak [(∑cos(φk,i −φ k,i −ω kτ i )) + ( sin(φk,i −φ k,i −ω kτ i )) ] k i=1 i=1 4 4 2 2 (5.29) = ∑∑Ak [(∑cos β k,i ) + ( sin β k,i ) ] kii=1 =1
and the derivatives become
- 65 - ∂J 2R 4 = ∑∑Akωk {( cos β k,i )[()sin β k,1 − sin β k,3 cosϕ cosθ ∂θ c ki=1
− (sin β k,2 − sin β k,4 )sinϕ cosθ ] 4 − (∑sin β k,i )[()cos β k,1 − cos β k,3 cosϕ cosθ i=1
− ()cos β k,2 − cos β k,4 sinϕ cosθ ]}
8R β k,1 − β k,3 β k,1 − β k,3 = ∑ Akωk {[sin cos + C k 2 2 β − β β − β β + β − β − β cos k,2 k,4 sin k,1 k,3 cos k,2 k,4 k,1 k,3 ]cosϕ cosθ 2 2 2 β − β β − β −[sin k,2 k,4 cos k,2 k,4 + 2 2 β − β β − β β + β − β − β cos k,1 k,3 sin k,2 k,4 cos k,1 k,3 k,2 k,4 ]sinϕ cosθ} 2 2 2 (5.30) and
∂J −8R β k,1 − β k,3 β k,1 − β k,3 = ∑ Akωk {[sin cos + ∂ϕ c k 2 2 β − β β − β β + β − β − β cos k,2 k,4 sin k,1 k,3 cos k,2 k,4 k,1 k,3 ]sinϕ sinθ 2 2 2
β − β β − β +[sin k,2 k,4 cos k,2 k,4 + 2 2 β − β β − β β + β − β − β 4cos k,1 k,3 sin k,2 k,4 cos k,1 k,3 k,2 k,4 ]cosϕ sinθ} 2 2 2 (5.31)
which can be used in the gradient ascend algorithm in (5.25) and to calculate the DDF error using (5.26).
5.3 Simulation Results
In our simulation, one more point needs to be emphasized. The center point of a circular array is treated as the reference point in our analysis. Since time delay error
τ i is only relative among all antenna elements in an array, we set no time delay error
- 66 - at the reference point. We need to be careful in setting time delay errors to the generated signals of each antenna when we do simulations at Matlab. In the five antenna case, there is an antenna at the center. It is obvious to set no time delay error to the center antenna and to set random time delay errors to the rest four antennas according to the probability distribution of the time delay error. But in the four antenna case, there is no center antenna and all four antennas have the time delay
error τ i in our analysis. In the simulation, however, it is reasonable to treat an
arbitrary one of the four antennas, the k th antenna, without time delay (τ k = 0 ), as the reference in terms of the time delay errors. And set random time delay errors to the rest three antenna elements according to the error probability distribution.
5.3.1 Equal-spaced Linear Array
As an example of linear array, we use 5-element array with following settings: N = 5 ,
λ ω = 2πf = 2π (86×106 ), L = 4096 , T = 5ns , d = and Bandwidth = 5MHz. c c 2
Firstly, we plot (5.17), whenT0 = 0.5ns , for variousϕ0 .
- 67 - -4 x 10 Variance of DDF Error 8
6
4 Variance of Error
2
0 0 10 20 30 40 50 60 70 80 ( ) φ0 °
Figure 5.1 Variance of DDF error vs. DOA
From Figure 5.1, it is seen that the error increases exponentially as incident
angleϕ0 increases.
Variance of DDF Error ( =45 ) Maximum of DDF Error ( =45 ) φ0 ° φ0 ° 0.018 0.35
0.016 0.3
0.014
0.25 0.012 ) ° 0.2 0.01
0.008 0.15 Variance of Error Maximum of Error ( 0.006 0.1
0.004
0.05 0.002
0 0 -2 -1 0 1 -2 -1 0 1 10 10 10 10 10 10 10 10 T (ns) T (ns) 0 0
Figure 5.2 Variance and max of DDF error vs. sample timing error - 68 - Figure 5.2 plots variance and maximum DDF error, which we got at (5.17) and (5.18)
versus T0 forϕ0 = 45° . We can figure out the maximum allowed timing error for certain DDF error requirement from previous plot. For example, if the maximum allowed DDF error is 0.2o, then maximum tolerable timing error is 5 ns.
5.3.2 Four Element Uniform Circular Array Plus a Center Element
In the simulations for circular array in this section and following section, settings
listed below apply: L = 4096 , T = 5ns , ϕ0 = 135° , θ 0 = 45° and Bandwidth = 5MHz.
The timing error term T0 was set to range between 10 ps and 10 ns. From our whole
DDF system simulations, we find that the energy variations do not contribute much to the beam patterns, so it should has little effect on the DDF error. We just assume the energy of different frequency be equal in our simulations.
λ We mainly consider UCA radii R = case under center frequency frequencies 2
λ f = 32MHz and f = 86MHz . We also plot result of UCA radii R = for π comparison purpose.
- 69 - Mean DDF Error 3 10 Azimuth φ Elevation θ
2 10
1
) 10 °
0
Mean of Error ( 10
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.3 Mean DDF error with f = 32MHz and R = (with center element) 2
Maximum DDF Error 3 10 Azimuth φ Elevation θ
2 10
) 1 ° 10
0 10 Maximum of Error ( Error of Maximum
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.4 Max DDF Error with f = 32MHz and R = (with center element) 2
- 70 - Mean DDF Error 2 10 Azimuth φ Elevation θ
1 10 ) °
0 10 Mean of Error (
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.5 Mean DDF error with f = 86MHz and R = (with center element) 2
Maximum DDF Error 3 10 Azimuth φ Elevation θ
2 10
) 1 ° 10
0 10 Maximum of Error ( Error of Maximum
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.6 Max DDF error with f = 86MHz and R = (with center element) 2
- 71 - Mean of DoA Error 3 10
2 10
1 10 ) o
0 Mean of error( 10
-1 10 Azimuth φ Elevation θ
-2 10 -11 -10 -9 -8 10 10 10 10 Sample time error(s)
λ Figure 5.7 Mean DDF error with f = 32MHz and R = (with center element) π
Maximum DoA error 3 10
2 10 ) o
1 10 Max of error(
0 10
Azimuth φ Elevation θ
-1 10 -11 -10 -9 -8 10 10 10 10 Sample timing error(s)
λ Figure 5.8 Max DDF error with f = 32MHz and R = (with center element) π
- 72 - Mean of DoA Error 3 10
2 10
1 10 ) o
0 Mean of error( 10
-1 10 Azimuth φ Elevation θ
-2 10 -11 -10 -9 -8 10 10 10 10 Sample time error(s)
λ Figure 5.9 Mean DDF error with f = 86MHz and R = (with center element) π
- 73 - Maximum DoA error 3 10
2 10 ) o
1 10 Max of error(
0 10
Azimuth φ Elevation θ
-1 10 -11 -10 -9 -8 10 10 10 10 Sample timing error(s)
λ Figure 5.10 Max DDF error with f = 86MHz and R = (with center element) π
Compared with from Figure 5.3 to Figure 5.10 as the radii of UCA increase, the mean and maximum value of DDF error decreased. This result is expected based on our analysis in previous chapter.
5.3.3 Four Element Uniform Circular Array
In the following simulations, we only use 4-element UCA. To compare the results with counterpart in previous section, we can conclude that the 4-element UCA with center element out performance the 4-element UCA. As we discussed before, the center element makes the side lobe lower. This function makes the performance of
DDF system better.
- 74 - Mean DDF Error 2 10 Azimuth φ Elevation θ
1 10 ) °
0 10 Mean of Error(
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.11 Mean DDF error with f = 32MHz and R = (without center element) 2
Maximum DDF Error 3 10 Azimuth φ Elevation θ
2 10
) 1 ° 10
0 10 Maximum of Error ( Error of Maximum
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.12 Max DDF error with f = 32MHz and R = (without center element) 2
- 75 - Mean DDF Error 3 10 Azimuth φ Elevation θ
2 10
1
) 10 °
0
Mean of Error ( 10
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.13 Mean DDF error with f = 86MHz and R = (without center element) 2
Maximum DDF Error 3 10 Azimuth φ Elevation θ
2 10
) 1 ° 10
0 10 Maximum of Error ( Error of Maximum
-1 10
-2 10 -11 -10 -9 -8 10 10 10 10
T (s) 0
λ Figure 5.14 Max DDF error with f = 86MHz and R = (without center element) 2
- 76 - 5.3.4 Overall System Simulations
Here we provide the results of the overall DDF system performance simulations. For the simulations, we used wideband (5 MHz) digital communication signals, which is composed of random amplitude raised cosine (RC) pulses such that length of one RC symbol is L = 300 samples. The center frequency of wideband RC signal is 86 MHz.
The sampling rate is 200 MHz. The simulated azimuth angle range is from –45o to
+45o and the elevation angle is fixed at 45o. Both a 4-element UCA and a 4-element
UCA with center element were simulated. We use circular array with radius R = λ 2.
The SNR at the array is fixed to be 15 dB. Random timing jitter is assumed to have a triangular distribution with a maximum value of 10 ps. Fixed time delays simulating various cable lengths are inserted, with the largest time delay difference between two antenna elements being about 3.8 ns, corresponding to maximum 1.14 m in length difference. A total of 180×60 = 10800 beams (180 beams for azimuth angle estimation and 60 beams for elevation angle estimation) are defined for the array for direction finding. An FFT length of 4096 is used for the digital beamforming.
The plots show the mean and maximum errors for azimuth and elevation angle estimation versus the azimuth of the incident signal. From the simulation results, we see that the elevation angle estimation errors are independent of the azimuth angles.
- 77 - 0.26
Mean of Azimuth θ mean Mean of Elevation φ 0.24 mean Max of Azimuth θ max Max of Elevation φ max 0.22
0.2 ) o
0.18
0.16
0.14
0.12 Error of DOA Estimates ( Estimates DOA of Error
0.1
0.08
0.06 -40 -30 -20 -10 0 10 20 30 40 Azimuth ( o )
λ Figure 5.15 Performance of 4-element UCA with R = 2
0.4
Mean of Azimuth θ mean Mean of Elevation φ mean 0.35 Max of Azimuth θ max Max of Elevation φ max
0.3 ) o
0.25
0.2
Error of DOA Estimates ( 0.15
0.1
0.05 -40 -30 -20 -10 0 10 20 30 40 Azimuth ( o )
λ Figure 5.16 Performance of 4-element UCA with center element with R = 2
Compared with the simulation results of two cases, the performance of UCA with center element is better.
- 78 - In the following simulations, We use circular array with radius R = λ π , which
satisfies (4.19).
0.5 Mean of Elevation θmean Mean of Azimuth φ 0.45 mean Max of Elevation θmax Max of Azimuth φmax 0.4 ) o 0.35
0.3
0.25
0.2 Error of DOA Estimates ( 0.15
0.1
0.05 -40 -30 -20 -10 0 10 20 30 40 Azimuth ( o )
λ Figure 5.17 Performance of 4-element UCA with R = π
0.5 Mean of Elevation θmean Mean of Azimuth φ 0.45 mean Max of Elevation θmax Max of Azimuth φmax 0.4 ) o 0.35
0.3
0.25
0.2 Error of DOA Estimates ( Estimates DOA of Error 0.15
0.1
0.05 -40 -30 -20 -10 0 10 20 30 40 Azimuth ( o )
λ Figure 5.18 Performance of 4-element UCA with center element with R = π
- 79 - 6 CONCLUSION
In this thesis, we design a high throughput digital direction finding (DDF) system. We design digital wideband fixed beamformer by transforming analog Butler matrix into a digital implementation. In order to estimate the performance that this direction finding system can achieve, and in order to determine the maximum cable length differences allowed for the system to meet the given direction finding resolution requirement, we did theoretical timing error analysis. We also analyzed how to choose the geometry of the antenna array, array size and so forth, based on theoretical and practical considerations. The resulting DDF system is more economic, robust for hostile environment, more flexible for using under different conditions, easy to calibrate, etc. It can be used in many military and civilian operations such as surveillance, reconnaissance and rescue.
The future work can focus on shading parameter design. And adaptive beamformer can be implemented in DDF system.
- 80 - 7 BIBLIOGRAPHY
[1] S. Haykin, Array Signal Processing, Prentice-Hall, Inc., 1985.
[2] John Litva and Titus Kwok-Yeung Lo, Digital Beamforming in Wireless
Communcations, Artech House Publishers, 1996.
[3] Constantine A. Balanis, Antenna Theory: Analysis and Design, John Wiley &
Sons, Inc., 1997.
[4] Barry D. Van Veen and Kevin M. Buckley, “Beamforming: A Versatile
Approach to Spatial Filtering”, IEEE ASSP Magazine, April 1998.
[5] Lal V. Godara, “Application of Antenna Arrays to Mobile Communications,
Part I Performance Improvement, Feasibility, and System Considerations”,
Proceedings of the IEEE, Vol. 85, NO. 7, July 1997.
[6] Lal V. Godara, “Application of Antenna Arrays to Mobile Communications,
Part II: Beam-Forming and Direction-of-Arrival Considerations”, Proceedings
of the IEEE, Vol. 85, NO. 8, August 1997.
[7] Sophocles J. Orfanidis, Electromagnetic Waves and Antennas,
http://www.ece.rutgers.edu/~orfanidi/ewa/.
[8] Edwin K.P. Chong and Staanislaw H. Zak, An Introduction to Optimization,
John Wiley & Sons, Inc., 2001.
[9] Simon Haykin, Adaptive Filter Theory, Prentice Hall, 2002.
[10] J. Capon. “High-Resolution Frequency-Wavenumber Spectrum Analysis”, Proc.
IEEE, 57(8):2408-1418, Aug. 1969.
[11] Kelly, Jr., E.L. and Levin, M.L., “Signal parameter estimation for seismometer
arrays”, MIT, Lincoln lab, Technical Report 339, Jan. 1964.
- 81 - [12] Owsley, N.L., “A recent trend in adaptive spatial processing for sensor arrays:
constrained adaptation”. In Signal Processing, ed., J.W.R. Griffiths, et al., pp.
591-604, Academic Press, 1973.
[13] Buckley, K.M., “Spatial/spectral filtering with linearly-constrained minimum
variance beamformers”, IEEE Trans. On ASSP, Vol. ASSP-35, pp. 249-266,
March 1987.
[14] S.P. Applebaum, “Adaptive arrays”, IEEE Tran. Antennas Propagation, vol.
AP-24, pp. 585-598, 1976.
[15] B. Windrow, P.E. Mantey, L.J. and B.B. Goode, “Adaptive antenna systems”,
Proc. IEEE, vol. 55, pp. 2143-2158, 1976.
[16] B. Widow, K.M. Duvall, R.P. Gooch and W.C. Newman, “Signal cancellation
phenomena in adaptive antennas: Causes and cures”, IEEE Tran. Antennas
Propagation, vol. AP-30, pp.469-478, 1982.
[17] Don H. Johnson and Dan E. Doudgeon, Array Signal Processing: Concepts and
techniques, Prentice Hall, 1993.
[18] Hiroyuki Tsuji and Mitsuhiko Mizuno, “Applications of Adaptive Array
Antennas in Mobile Communications”, Electronics and Communications in
Japan, Part 3, Vol. 83, No. 12, 2000.
[19] I. Chiba, W. Chujo, and M. Fujise, “Beamspace constant modulus algorithm
adaptive array antennas”, in Proc. IEE 8th Int. Conf. Antennas and Propagation,
pp. 975–978, Edinburgh, Scotland, 1993.
[20] K. Nishimori, N. Kikuma, and N. Inagaki, “The differential CMA adaptive
array antenna using an eigen-beamspace system”, IEICE Trans. Comm., vol.
E78-B, pp. 1480–1488, Nov. 1995.
- 82 - [21] L. J. Horowitz, H. Blatt, W. G. Brodsky, and K. D. Senne, “Controlling
adaptive antenna arrays with the sample matrix inversion algorithm”, IEEE
Trans. Aerosp. Electronic System, vol. AES-15, pp. 840–847, 1979.
[22] R. Schreiber, “Implementation of adaptive array algorithms”, IEEE Trans.
ASSP, vol. ASSP-34, pp. 1038–1045, 1986.
[23] A. Papoulis, “A new algorithm in spectral analysis and bandlimited
extrapolation”, IEEE Trans. Circuits Syst., vol. CAS-22, pp. 735–742, 1975.
[24] R. G. Pridham and R. A. Mucci, “A novel approach to digital beamforming”,
Journal of Acoustic Society of America., vol. 63, pp. 425–434, 1978.
[25] W. S. Hodgkiss, “Adaptive array processing: Time vs. frequency domain”,
ICASSP, Washington, D.C., 1979.
[26] D. Kraus and J.F. Böhme, “Maximum likelihood location estimation of
wideband sources using the EM algorithm”, In Adaptive System in Control and
Signal Processing, Proc. 1992.
[27] M. Wax and I. Ziskind, “on unique localization of multiple sources by passive
sensor arrays”, IEEE Trans. On ASSP, ASSP-37(7), pp. 996-1000, July 1989.
[28] Robert J. Mailloux, Phased Array Antenna Handbook, Artech House, 1994.
[29] Mattias Dahl, Ingvar Claesson, Sven Nordebo and Sven Nordholm,
“ Chebyshev Optimization of Circular Arrays”, Chapter of Optimization
Methods and Applications, kluwer Academic Publishers, 2001.
[30] J. L. Butler, “Digital, matrix, and intermediate frequency scanning”, in R.C.
Hansen, ed., Microwave Scanning Arrays, Academic Press, New York, 1966.
[31] Levy R.”A high power X-band Butler matrix”, Microwave, J. 27, p.153, April
1984.
- 83 - [32] Wallington J. R. “Analysis, design and performance of a microstrip Butler
matrix”, Proc. 6th Eur. Microwave Conf., Brussels, p. A14.3.1, 1973.
[33] Charczenco W. et al., “Integrated optical Butler matrix for beamforming in
phased array antennas”, Proc. SPIE-Optoelectron, Signal Processing Phased
Array Antennas II, p. 196, 1990.
[34] Hall P. S. and Vetterlein S. J., “Review of radio frequency beamforming
techniques for scanned and multiple beam antennas”, IEE Proc., Part H:
Microwaves, Opt. Antennas, 1990.
[35] John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing –
Principles, Algorithms and Applications, 3rd edit, Prentice Hall, 1996.
[36] Davies, D. E. N., “Circular Arrays”, Ch. 12 in The Handbook of Antenna
Design, Vol. 2, A. Rudge et al., eds., London: Peter Peregrinus, 1987.
[37] Prabhakar S. Naidu, Sensor Array Signal Processing, CRC Press, 2001.
- 84 -