Kevin R. Farrell. “Inverse Problems in Array Processing.” 2000 CRC Press LLC. <http://www.engnetbase.com>. InverseProblemsinArray Processing KevinR.Farrell T-Netix/SpeakEZ 30.1Introduction 30.2BackgroundTheory WavePropagation • SpatialSampling • SpatialFrequency 30.3NarrowbandArrays Look-DirectionConstraint • PilotSignalConstraint 30.4BroadbandArrays 30.5InverseFormulationsforArrayProcessing NarrowbandArrays • BroadbandArrays • Row-ActionProjec- tionMethod 30.6SimulationResults NarrowbandResults • BroadbandResults 30.7Summary References 30.1 Introduction Signalreceptionhasnumerousapplicationsincommunications,radar,sonar,andgeoscienceamong others.However,theadverseeffectsofnoiseintheseapplicationslimittheirutility.Hence,thequest fornewandimprovednoiseremovaltechniquesisanongoingresearchtopicofgreatimportancein avastnumberofapplicationsofsignalreception. Whencertaincharacteristicsofnoiseareknown,theireffectscanbecompensated.Forexample, ifthenoiseisknowntohavecertainspectralcharacteristics,thenafiniteimpulseresponse(FIR)or infiniteimpulseresponse(IIR)filtercanbedesignedtosuppressthenoisefrequencies.Similarly,if thestatisticsofthenoiseareknown,thenaWeinerfiltercanbeusedtoalleviateitseffects.Finally,if thenoiseisspatiallyseparatedfromthedesiredsignal,thenmultisensorarrayscanbeusedfornoise suppression.Thislastcaseisdiscussedinthisarticle. Amultisensorarrayconsistsofasetoftransducers,i.e.,antennas,microphones,hydrophones, seismometers,geophones,etc.thatarearrangedinapatternwhichcantakeadvantageofthespatial locationofsignals.Atwo-elementtelevisionantennaprovidesagoodexample.Toimprovesignal receptionand/ormitigatetheeffectsofanoisesource,theantennapatternismanuallyadjustedto steeralowgaincomponentoftheantennapatterntowardsthenoisesource.Multisensorarrays typicallyachievethisadjustmentthroughtheuseofanarrayprocessingalgorithm.Mostapplica- tionsofmultisensorarraysinvolveafixedpatternoftransducers,suchasalineararray.Antenna patternadjustmentsaremadebyapplyingweightstotheoutputsofeachtransducer.Ifthenoise arrivesfromaspecificnon-changingspatiallocation,thentheweightswillbefixed.Otherwise, c  1999byCRCPressLLC 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 as θ. Note in Fig. 30.1 that 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 farfield approximation isvalid forsignals thatsatisfy thefollowing condition: s ≥ D 2 λ (30.1) where s is the distance between the signal and the array, λ is the wavelength of the signal, and D is the lengthof thearray. Wavefronts thatoriginate closerthan D 2 /λare consideredto befromthe nearfield 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. c  1999 by CRC Press LLC FIGURE 30.1: Propagating wavefront. 30.2.2 Spatial Sampling In Fig. 30.1 it can be seen that the signal waveform experiences a time delay between crossing each sensor, assuming that it doesnot arrive perpendicular to the array. The time delay, τ, of the waveform striking the first and then second sensors in Fig. 30.1 may be calculated as τ = d c sin θ (30.2) where d is the sensor spacing, c is the speed of propagation of the given waveform for a particular medium (i.e., 3 × 10 8 m/s for electromagnetic waves through air, 1.5 × 10 3 m/s for sound waves through water, etc.), and θ 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 = 2πd λ o sin θ = ω o d c sin θ. (30.3) Here, λ o is the wavelength of the signal at frequency f o as defined by λ o = c f o . (30.4) Hence, a signal x(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        . (30.5) Uniform sensor spacing is assumed throughout the remainder of this article. c  1999 by CRC Press LLC 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 alsobe deduced thatretaining signals having anangle of arrival equal to zero degrees while suppressingsignals fromother directionsis equivalentto lowpass filtering the spatialfrequency. 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 2 . (30.6) The above relation for sensor spacing is analogous to the Nyquist sampling rate for frequency domain analysis. Forexample, consider a signalthat exhibitsexactlyoneperiod betweenconsequetivesensors. 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 oftheweighted sensorsaresummedtoproducethearray output. Anarrowband 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 c  1999 by CRC Press LLC FIGURE 30.2: Narrowband array. defined as x(k) = s(k) + n(k) (30.7) where k 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 y(k) = x T (k)w, x, w ∈ C K , (30.8) where C K is the complex space of dimension K. 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 output y 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): w opt = µR −1 xx t . (30.9) c  1999 by CRC Press LLC 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) + α  w q − w (j)  − βx(k)y(k) . (30.10) In Eq. (30.10), w q represents the quiescent weight vector (i.e., when no interference is present), the superscript j 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. Thisresultsina beamforming algorithmthat will concentrateonmaintaining abeamdirectedtowards the target signal, as opposed to, or in addition to, positioning the nulls as in the case of the sidelobe canceler. One suchadaptive beamformingalgorithm wasproposedby Widrow [20,21]. The resulting weight update equation is based on minimizing the quantity (y(k) − p(k)) 2 where p(k) is the pilot signal. The resulting weight update equation is w (j+1) = w (j) + µ(k)x(k) . (30.11) 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 of f o and 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, c  1999 by CRC Press LLC 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 broadbandarrays, shown by H narrow (w) = w 1 (30.12) and H broad (w) = w 1 + w 2 e −jwT + w 3 e −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. 30.4 obtains values at each sensor and then propagates these values through the array at each time interval. Therefore, if the values x 1 through x K are input at time instant one, then at time instant two, x K+1 through x 2K will have the values previously held by x 1 through x K , x 2K+1 through x 3K will have the values previously held by x K+1 through x 2K ,etc. Also, at each time instant, a scalar value y will be calculated as the inner product of the input vector x and the weight vector w. This array output is calculated as y(k) = x T (k)w, x, w ∈ C JK , (30.14) where C JK is the complex space of dimension JK. 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 c  1999 by CRC Press LLC w. The output signal y 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. Broadbandarrays havebeenanalyzedbyWidrow[20], Griffiths[10,12],andFrost[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 pilotsignal. Thismethod assumesthat theangle ofarrival andsecondordersignal statisticsare 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., w 1 through w 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 having J 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: minimize : w T R xx w (30.15) subject to : C T w = h (30.16) where R xx is the covariance matrix of the received signals, h is the vector of FIR filter coefficients defining the desired frequency response, and C T is the constraint matrix given by C T =      11 . 100 . 0 . 00 . 0 00 . 011 . 1 . 00 . 0 . . . 00 . 000 . 0 . 11 . 1      . The number of rows in C T 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 w opt will 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 2 w T R xx w + λ T  C T w − h  (30.17) where λ is a Lagrange multiplier and F is a vector representative of the desired frequency response. Minimizing the function H(w) with respect to w will obtain the following optimal weight vector: w opt = R −1 xx C  C T R −1 xx C  −1 h . (30.18) c  1999 by CRC Press LLC An iterative implementation of this algorithm was implemented via the following equations: w (j+1) = P  w (j) − µR xx w (j)  + C  C T C  −1 h (30.19) where µ is a step size parameter and P = I − C  C T C  −1 C T w(0) = C  C T C  −1 h where I is the identity matrix and h =  h 1 h 2 . 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 Section 30.3. 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 w opt that satisfies these conditions is represented by: w T opt s(k) = s(k) (30.20) and w T opt n(k) = 0 (30.21) where s(k) is the scalar value of the desired signal. Since the sidelobe canceler does not have access to s(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 quantity y 2 ,wherey = x T (k)w. Minimizing y 2 subject 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: x T (k)w = 0 . (30.22) c  1999 by CRC Press LLC [...]... suppression 30. 7 Summary This article has formulated the array processing problem as an inverse problem Inverse formulations for both narrowband and broadband arrays were discussed Specifically, the sidelobe canceler c 1999 by CRC Press LLC algorithm for narrowband array processing and Frost algorithm for broadband array processing were analyzed The inverse formulations provide a flexible, intuitive implementation... CRC Press LLC FIGURE 30. 6: Narrowband input spectrum FIGURE 30. 7: Output spectrum for inverse formulation c 1999 by CRC Press LLC FIGURE 30. 8: Output spectrum for Applebaum array FIGURE 30. 9: Broadband input spectrum c 1999 by CRC Press LLC FIGURE 30. 10: Output spectrum for inverse array FIGURE 30. 11: Output spectrum for Frost array c 1999 by CRC Press LLC TABLE 30. 2 Experiment Input Scenario for Broadband... formulation derived in the previous section Once again, the minimization of the cost function in Eq (30. 15) can be achieved by Eq (30. 22), assuming that the target signal arrives normal to the array The constraint for the desired frequency response in the look direction can also be implemented in a similar fashion to that of the narrowband array in Eq (30. 23) Instead of constraining the sum of the weights... the array at uniformly spaced angles ranging from −90◦ to +90◦ in unit increments The interference power is 2.6 dB greater than the desired signal The resulting interference suppression observed in the array output is illustrated in Fig 30. 12 The maximum interference suppression (i.e., for interference arriving at ±90◦ ) is 11.0 dB for the RAP method and 11.2 dB for the Frost method FIGURE 30. 12: Interference... different input subspaces, which lies in the target signal subspace Since the RAP method consists of only row operations, it is convenient for parallel implementations This technique, described by Eqs (30. 24), (30. 29), and (30. 30), comprises the RAP method for array processing 30. 6 Simulation Results Several simulations were performed to compare the inverse formulation of the array processing problem... solving a system of linear equations The RAP method has found numerous applications in digital signal processing [16] and is applied here to adaptive beamforming The RAP method for iteratively solving the system in Eq (30. 24) is given by the update equation: w (j +1) = w (j ) + µ where i i ai T ai ai (30. 29) is the error term for the ith row defined as: i = bi − ai w (k) (30. 30) In Eqs (30. 29) and (30. 30),... signal Interference 1 Interference 2 0 27 41 3.0 1.5 4.0 gain value µ = 0.05 The h vector specifies a low pass frequency response with a passband up to 4 KHz The input and output signal spectrums are shown in Figs 30. 9 through 30. 11 The inverse formulation and Frost algorithms again demonstrate similar performance The broadband array processing algorithms are also evaluated for a microphone array application... for the inverse formulation and Applebaum algorithm are shown in Figs 30. 6 through 30. 8 The inverse formulation and Applebaum algorithms demonstrate similar performance for this example TABLE 30. 1 Experiment Input Scenario for Narrowband Signal Frequency (KHz) Target signal Interference 1 Interference 2 Interference 3 30. 6.2 Angle (deg) 0 28 41 72 2.0 3.0 1.0 4.0 Broadband Results The broadband array. .. Another solution can be obtained by using the Moore-Penrose generalized inverse, or pseudo -inverse, of A via (30. 28) w † = A† b where A† and w † represent the pseudo -inverse of A and the pseudo -inverse solution for w, respectively These methods all provide an immediate solution for the weight vector, w, however, at the expense of requiring a matrix inversion along with any instabilities that may be apparent... Eq (30. 22) implies that the weight vector be orthogonal to the composite input vector as opposed to just the noise component However, the look-direction constraint imposed by the following equation will maintain the desired signal 1 1 1 w =1 (30. 23) This equation satisfies the look-direction constraint that a signal arriving perpendicular to the array will have unity gain in the output The constraints . Kevin R. Farrell. Inverse Problems in Array Processing. ” 2000 CRC Press LLC. <http://www.engnetbase.com>. InverseProblemsinArray Processing KevinR.Farrell. PilotSignalConstraint 30. 4BroadbandArrays 30. 5InverseFormulationsforArrayProcessing NarrowbandArrays • BroadbandArrays • Row-ActionProjec- tionMethod 30. 6SimulationResults