Tài liệu Digital Signal Processing Handbook P30 docx

Farrell T-Netix/SpeakEZ 30.1 Introduction 30.2 Background Theory Wave Propagation •Spatial Sampling•Spatial Frequency 30.3 Narrowband Arrays Look-Direction Constraint•Pilot Signal Constr

Kevin R Farrell “Inverse Problems in Array Processing.”

2000 CRC Press LLC <http://www.engnetbase.com>.

Inverse Problems in Array

Processing

Kevin R Farrell

T-Netix/SpeakEZ

30.1 Introduction 30.2 Background Theory

Wave Propagation •Spatial Sampling•Spatial Frequency

30.3 Narrowband Arrays

Look-Direction Constraint•Pilot Signal Constraint

30.4 Broadband Arrays 30.5 Inverse Formulations for Array Processing

Narrowband Arrays •Broadband Arrays•Row-Action Projec-tion Method

30.6 Simulation Results

Narrowband Results•Broadband Results

30.7 Summary References

30.1 Introduction

Signal reception has numerous applications in communications, radar, sonar, and geoscience among others However, the adverse effects of noise in these applications limit their utility Hence, the quest for new and improved noise removal techniques is an ongoing research topic of great importance in

a vast number of applications of signal reception

When certain characteristics of noise are known, their effects can be compensated For example,

if the noise is known to have certain spectral characteristics, then a finite impulse response (FIR) or infinite impulse response (IIR) filter can be designed to suppress the noise frequencies Similarly, if the statistics of the noise are known, then a Weiner filter can be used to alleviate its effects Finally, if the noise is spatially separated from the desired signal, then multisensor arrays can be used for noise suppression This last case is discussed in this article

A multisensor array consists of a set of transducers, i.e., antennas, microphones, hydrophones, seismometers, geophones, etc that are arranged in a pattern which can take advantage of the spatial location of signals A two-element television antenna provides a good example To improve signal reception and/or mitigate the effects of a noise source, the antenna pattern is manually adjusted to steer a low gain component of the antenna pattern towards the noise source Multisensor arrays typically achieve this adjustment through the use of an array processing algorithm Most applica-tions of multisensor arrays involve a fixed pattern of transducers, such as a linear array Antenna pattern adjustments are made by applying weights to the outputs of each transducer If the noise arrives from a specific non-changing spatial location, then the weights will be fixed Otherwise,

if the noise arrives from random, changing locations then the weights must be adaptive So, in a military communications application where a communications channel is subject to jamming from random spatial locations, an adaptive array processing algorithm would be the appropriate solution Commercial applications of microphone arrays include teleconferencing [6] and hearing aids [9] There are several methods for obtaining the weight update equations in array processing Most

of these are derived from statistically based formulations The resulting optimal weight vector is then generally expressed in terms of the input autocorrelation matrix An alternative formulation

is to express the array processing problem as a linear system of equations to which iterative matrix inversion techniques can be applied The matrix inverse formulation will be the focus of this article The following section provides a background overview of wave propagation, spatial sampling, and spatial filtering Next, narrowband and broadband beamforming arrays are described along with the standard algorithms used for these implementations The narrowband and broadband algorithms are then reformulated in terms of an inverse problem and an iterative technique for solving this system of equations is provided Finally, several examples are given along with a summary

30.2 Background Theory

Array processing uses information regarding the spatial locations of signals to aid in interference suppression and signal enhancement The spatial locations of signals may be determined by the wavefronts that are emanated by the signal sources Some background theory regarding wave propa-gation and spatial frequency is necessary to fully understand the interference suppression techniques used within array processing The following subsections provide this background material

30.2.1 Wave Propagation

An adaptive array consists of a number of sensors typically configured in a linear pattern that utilizes the spatial characteristics of signals to improve the reception of a desired signal and/or cancellation

of undesired signals The analysis used in this chapter assumes that a linear array is being used, which corresponds to the sensors being configured along a line Signals may be spatially characterized

by their angle of arrival with respect to the array The angle of arrival of a signal is defined as the angle between the propagation path of the signal and the perpendicular of the array Consider the wavefront emanating from a point source as is illustrated in Fig.30.1 Here, the angle of arrival is shown asθ.

Note in Fig.30.1that wavefronts emanating from a point source may be characterized by plane waves (i.e., the locus of constant phase form straight lines) when originating from the far field or

Fraunhofer, region The far field approximation is valid for signals that satisfy the following condition:

s ≥ D2

wheres is the distance between the signal and the array, λ is the wavelength of the signal, and D is the

length of the array Wavefronts that originate closer thanD2/λ are considered to be from the near field

or Fresnel, region Wavefronts originating from the near field exhibit a convex shape when striking

the array sensors These wavefronts do not create linear phase shifts between consequetive sensors However, the curvature of the wavefront allows algorithms to determine point source location in addition to direction of arrival [1] The remainder of this article assumes that all wavefronts arrive from the far field region

FIGURE 30.1: Propagating wavefront.

30.2.2 Spatial Sampling

In Fig.30.1it can be seen that the signal waveform experiences a time delay between crossing each sensor, assuming that it does not arrive perpendicular to the array The time delay,τ, of the waveform

striking the first and then second sensors in Fig.30.1may be calculated as

τ = d

whered is the sensor spacing, c is the speed of propagation of the given waveform for a particular

medium (i.e., 3× 108m/s for electromagnetic waves through air, 1.5 × 103m/s for sound waves through water, etc.), andθ is the angle of arrival of the wavefront This time delay corresponds to a

shift in phase of the signal as observed by each sensor The phase shift,φ, or electrical angle observed

at each sensor due to the angle of arrival of the wavefront may be found as

φ = 2λ πd

Here,λ ois the wavelength of the signal at frequencyf oas defined by

λ o= f c

Hence, a signalx(k) that crosses the sensor array and exhibits a phase shift φ between uniformly

spaced, consequetive sensors can be characterized by the vector x(k), where:

x(k) = x(k)











1

e −jφ

e −2jφ

e −j (K−1)φ











Uniform sensor spacing is assumed throughout the remainder of this article

30.2.3 Spatial Frequency

The angle of arrival of a wavefront defines a quantity known as the spatial frequency Adaptive

arrays use information regarding the spatial frequency to suppress undesired signals that originate from different locations than that of the target signal The spatial frequency is determined from the periodicity that is observed across an array of sensors due to the phase shift of a signal arriving at some angle of arrival

Signals that arrive perpendicular to the array (known as boresight) create identical waveforms at each sensor The spatial frequency of such signals is zero Signals that do not arrive perpendicular to the array will not create waveforms that are identical at each sensor assuming that there is no spatial aliasing due to insufficiently spaced sensors In general, as the angle increases, so does the spatial frequency It can also be deduced that retaining signals having an angle of arrival equal to zero degrees while suppressing signals from other directions is equivalent to low pass filtering the spatial frequency This provides the motivation for conventional or fixed-weight beamforming techniques Here, the sensor values can be computed via a windowing technique, such as a rectangular, Hamming, etc

to yield a fixed suppression of non-boresight signals However, adaptive techniques can locate the specific spatial frequency of an interfering signal and position a null in that exact location to achieve greater suppression

There are two types of beamforming, namely conventional, or “fixed weight”, beamforming and adaptive beamforming A conventional beamformer can be designed using windowing and FIR filter theory They utilize fixed weights and are appropriate in applications where the spatial locations of noise sources are known and are not changing Adaptive beamformers make no such assumptions regarding the locations of the signal sources The weights are adapted to accommodate the changing signal environment

Arrays that have a visible region of−90◦to+90◦(i.e., the azimuth range for signal reception) require that the sensor spacing satisfy the relation

d ≤ λ

The above relation for sensor spacing is analogous to the Nyquist sampling rate for frequency domain analysis For example, consider a signal that exhibits exactly one period between consequetive sensors

In this case, the output of each sensor would be equivalent, giving the false impression that the signal arrives normal to the array In terms of the antenna pattern, insufficient sensor spacing results in grating lobes Grating lobes are lobes other than the main lobe that appear in the visible region and can amplify undesired directional signals

The spatial frequency characteristics of signals enable numerous enhancement opportunities via array processing algorithms Array processing algorithms are typically realized through the imple-mentation of narrowband or broadband arrays These two arrays are discussed in the following sections

30.3 Narrowband Arrays

Narrowband adaptive arrays are used in applications where signals can be characterized by a single frequency and thus occupy a relatively narrow bandwidth A signal whose envelope does not change during the time their wavefront is incident on the transducers is considered to be narrowband A narrowband adaptive array consists of an array of sensors followed by a set of adjustable gains, or weights The outputs of the weighted sensors are summed to produce the array output A narrowband array is shown in Fig.30.2

The input vector x(k) consists of the sum of the desired signal s(k) and noise n(k) vectors and is

FIGURE 30.2: Narrowband array.

defined as

wherek denotes the time instant of the input vector The noise vector n(k) will generally consist of

thermal noise and directional interference At each time instant, the input vector is multiplied with the weight vector to obtain the array output, which is given as

whereC K is the complex space of dimensionK The array output is then passed to the signal

processor which uses the previous value of the output and current values of the inputs to determine the adjustment to make to the weights The weights are then adjusted and multiplied with the new input vector to obtain the next output The output feedback loop allows the weights to be adjusted adaptively, thus accommodating nonstationary environments

In Eq (30.8), it is desired to find a weight vector that will allow the outputy to approximately

equal the true target signal For the derivation of the weight update equations, it is necessary to know

what a priori information is being assumed One form of a priori information could be the spatial

location of the target signal, also known as the “look-direction” For example, many array processing algorithms assume that the target signal arrives normal to the array, or else a steering vector is used

to make it appear as such Another form of a priori information is to use a signal at the receiving end

that is correlated with the input signal, i.e., a pilot signal Each of these criteria will be considered in the following subsections

30.3.1 Look-Direction Constraint

One of the first narrowband array algorithms was proposed by Applebaum [2] This algorithm is

known as the sidelobe canceler and assumes that the direction of the target signal is known The

algorithm does not attempt to maximize the signal gain, but instead adjusts the sidelobes so that interfering signals coincide with the nulls of the antenna pattern This concept is illustrated in Fig.30.3

Applebaum derived the weight update equation via maximization of the signal to interference plus thermal noise ratio (SINR) As derived in [2], this optimization results in the optimal weight vector

as given by Eq (30.9):

FIGURE 30.3: Sidelobe canceling.

In Eq (30.9),R xx is the covariance matrix of the input,µ is a constant related to the signal gain,

and t is a steering vector that corresponds to the angle of arrival of the desired signal This steering

vector is equivalent to the phase shift vector of Eq (30.5) Note that if the angle of arrival of the

desired signal is zero, then the t vector will simply contain ones.

A discretized implementation of the Applebaum algorithm appears as follows:

w(j+1)= w(j) + αwq− w(j)− βx(k)y(k) (30.10)

In Eq (30.10), wqrepresents the quiescent weight vector (i.e., when no interference is present), the superscriptj refers to the iteration, α is a gain parameter for the steering vector, and β is a gain

parameter controlling the adaptation rate and variance about the steady state solution

30.3.2 Pilot Signal Constraint

Another form of a priori information is to use a pilot signal that is correlated with the target signal.

This results in a beamforming algorithm that will concentrate on maintaining a beam directed towards the target signal, as opposed to, or in addition to, positioning the nulls as in the case of the sidelobe canceler One such adaptive beamforming algorithm was proposed by Widrow [20,21] The resulting weight update equation is based on minimizing the quantity(y(k) − p(k))2wherep(k) is the pilot

signal The resulting weight update equation is

This corresponds to the least means square (LMS) algorithm, where is the current error, namely (y(k) − p(k)), and µ is a scaling factor.

30.4 Broadband Arrays

Narrowband arrays rely on the assumption that wavefronts normal to the array will create identical waveforms at each sensor and wavefronts arriving at angles not normal to the array will create a linear phase shift at each sensor Signals that occupy a large bandwidth and do not arrive normal to the array violate this assumption since the phase shift is a function off oand varying frequency will cause a varying phase shift Broadband signals that arrive normal to the array will not be subject to frequency dependent phase shifts at each sensor as will broadband signals that do not arrive normal

to the array This is attributed to the coherent summation of the target signal at each sensor where the phase shift will be a uniform random variable with zero mean A modified array structure,

however, is necessary to compensate the interference waveform inconsistencies that are caused by variations about the center frequency This can be achieved by having the weight for a sensor being

a function of frequency, i.e., a FIR filter, instead of just being a scalar constant as in the narrowband case Broadband adaptive arrays consist of an array of sensors followed by tapped delay lines, which

is the major implementation difference between a broadband and narrowband array A broadband array is shown in Fig.30.4

FIGURE 30.4: Broadband array

Consider the transfer functions for a given sensor of the narrowband and broadband arrays, shown by

and

Hbroad(w) = w1+ w2e −jwT + w3e −2jwT + + w J e −j (J −1)wT (30.13) The narrowband transfer function has only a single weight that is constant with frequency How-ever, the broadband transfer function, which is actually a Fourier series expansion, is frequency dependent and allows for choosing a weight vector that may compensate phase variations due to signal bandwidth This property of tapped delay lines provides the necessary flexibility for process-ing broadband signals Note that typically four or five taps will be sufficient to compensate most bandwidth variances [14]

The broadband array shown in Fig.30.4obtains values at each sensor and then propagates these values through the array at each time interval Therefore, if the valuesx1throughx Kare input at time instant one, then at time instant two,x K+1throughx2Kwill have the values previously held by

x1throughx K,x2K+1throughx3Kwill have the values previously held byx K+1throughx2K, etc Also, at each time instant, a scalar valuey will be calculated as the inner product of the input vector

x and the weight vector w This array output is calculated as

whereC J K is the complex space of dimensionJ K.

Although not shown in Fig.30.4, a signal processor exists as in the narrowband array, which uses the previous output and current inputs to determine the adjustments to make to the weight vector

w The output signaly will approach the value of the desired signal as the interfering signals are

canceled until it converges to the desired signal in the least squares sense

Broadband arrays have been analyzed by Widrow [20], Griffiths [10,12], and Frost [7] Widrow [20] proposed a LMS algorithm that minimizes the square of the difference between the observed output and the expected output, which was estimated with a pilot signal This approach assumes that the

angle of arrival and a pilot signal are available a priori Griffiths [10] proposed a LMS algorithm that assumes knowledge of the cross-correlation matrix between the input and output data instead of the pilot signal This method assumes that the angle of arrival and second order signal statistics are known

a priori The methods proposed by Widrow and Griffiths are forms of unconstrained optimization.

Frost [7] proposed a LMS algorithm that assumes a priori knowledge of the angle of arrival and the frequency band of interest The Frost algorithm utilizes a constrained optimization technique, which Griffiths later derived an unconstrained formulation that utilizes the same constraints [12] The Frost algorithm will be the focus of this section

The Frost algorithm implements the look-direction and frequency response constraints as follows For the broadband array shown in Fig.30.4, a target signal waveform propagating normal to the array, or steered to appear as such, will create identical waveforms at each sensor Since the taps in each column, i.e.,w1throughw K, see the same signal, this array may be collapsed to a single sensor FIR filter Hence, to constrain the frequency range of the target signal, one just has to constrain the sum of the taps for each column to be equal to the corresponding tap in a FIR filter havingJ taps

and the desired frequency response for the target signal

These look-direction and frequency response constraints can be implemented by the following optimization problem:

where Rxxis the covariance matrix of the received signals, h is the vector of FIR filter coefficients defining the desired frequency response, and CT is the constraint matrix given by







11 1 00 0 00 0

00 0 11 1 00 0

00 0 00 0 11 1





 .

The number of rows in CT is equal to the number of taps of the array and the number of ones in

each row is equal to the number of sensors The optimal weight vector woptwill minimize the output power of the noise sources subject to the constraint that the sum of each column vector of weights is equal to a coefficient of a FIR filter defining the desired impulse response of the array

The Frost algorithm [7] is a constrained LMS method derived by solving Eqs (30.15) and (30.16) via Lagrange Multipliers to obtain an expression for the optimum weight vector, Frost [7] derived the constrained LMS algorithm for broadband array processing using Lagrange multipliers The function to be minimized may be defined as

H (w) = 1

whereλ is a Lagrange multiplier and F is a vector representative of the desired frequency response.

Minimizing the functionH (w) with respect to w will obtain the following optimal weight vector:



xxC

−1

An iterative implementation of this algorithm was implemented via the following equations:

w(j+1)= Phw(j) − µR xxw(j)i

+ CCTC−1

whereµ is a step size parameter and

P = I − CCTC−1

w(0) = CCTC−1

h

where I is the identity matrix and

h= h1 h2 h J 

.

30.5 Inverse Formulations for Array Processing

The array processing algorithms discussed thus far have all been derived through statistical analysis and/or adaptive filtering techniques An alternative approach is to view the constraints as equations that can be expressed in a matrix-vector format This allows for a simple formulation of array processing algorithms to which additional constraints can be easily incorporated Additionally, this formulation allows for efficient iterative matrix inversion techniques that can be used to adapt the weights in real time

30.5.1 Narrowband Arrays

Two algorithms were discussed for narrowband arrays, namely, the sidelobe canceler and pilot signal algorithms We will consider the sidelobe canceler algorithm here The derivation of the sidelobe canceler is based on the optimization of the SINR and yields an expression for the optimum weight vector as a function of the input autocorrelation matrix We will use the same constraints as the sidelobe canceler to yield a set of linear equations that can be put in a matrix vector format Consider the narrowband array description provided in Section30.3 In Eq (30.7), s(k) is the

vector representing the desired signal whose wavefront is normal to the array and n(k) is the sum of

the interfering signals arriving from different directions A weight vector is desired that will allow the

signal vector s(k) to pass through the array undistorted while nulling any contribution of the noise

vector n(k) An optimal weight vector woptthat satisfies these conditions is represented by:

and

wheres(k) is the scalar value of the desired signal Since the sidelobe canceler does not have access

tos(k), an alternative approach must be taken to implement the condition of Eq (30.20) One method for finding this constraint is to minimize the expectation of the output power [7] This expectation can be approximated by the quantityy2, wherey = x T (k)w Minimizing y2subject to the look-direction constraint will tend to cancel the noise vector while maintaining the signal vector This criteria can be represented by the linear equation: