Wideband Beamforming for Multipath Signals Based on Frequency Invariant Transformation

Transcription

1 International Journal of Automation and Computing 9(4), August 2012, DOI: /s z Wideband Beamforming for Multipath Signals Based on Frequency Invariant Transformation Wei Liu Communications Research Group, Department of Electronic & Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, UK Abstract: It is well known that the performance of conventional adaptive beamformers degrades severely due to the presence of coherent or correlated interferences (multipath propagation) and various techniques have been developed to improve the performance of the beamformer. However, most of the work in the past has been focused on the narrowband case. In this paper, the wideband beamforming problem in the presence of multipath signals is addressed, with a novel approach proposed by employing a pre-processing stage based on the frequency invariant beamforming (FIB) technique. In this approach, the received wideband array signals are first processed by an FIB network, and then a traditional narrowband adaptive beamformer or an appropriate instantaneous blind source separation (BSS) algorithm can be applied to the network outputs. It is shown that with the proposed structure, cancellation of the desired signal is reduced, leading to a significantly improved output signal to interference plus noise ratio (SINR). Keywords: Wideband beamforming, multipath signals, frequency invariant beamforming, blind beamforming, convolutive mixtures. 1 Introduction Beamforming has been studied extensively due to its applications in various areas ranging from sonar and radar to wireless communications [1 5]. It is well known that the performance of traditional adaptive beamformers degrades severely due to the presence of coherent or correlated interferences (multipath propagation) and various techniques have been developed to improve the performance of the beamformer, such as the spatial smoothing technique [6, 7]. However, most of the efforts have been focused on the narrowband case. For wideband signals, this problem has not been thoroughly investigated. The spatial smoothing technique was extended to the wideband multipath case in [8], with a limited success as also shown in the simulations part of this work. Based on various coherent wideband direction of arrival (DOA) estimation techniques [9 11], a focusing processing method based on the main DOA of the desired signal is proposed for wideband beamforming in a multipath environment [12, 13]. However, this focusing method has actually reduced the whole array system into an equivalent narrowband beamformer. This can be seen by considering a widely used beamforming scenario when the main direction of the desired signal comes from the broadside: in this case, no focusing processing is needed since the desired signal has the same steering vector for all frequencies; then the proposed structure is simplified to a direct combination of the frequency bin outputs by a single common weight vector, which gives a final output equivalent to an instantaneous linear combination of the received time-domain wideband array signals, i.e., a narrowband beamforming structure. As a result, it can only work effectively when the interfering signals have a relatively narrow bandwidth. A major difference between a narrowband signal and a wideband signal for the multipath case in the context of adaptive beamforming is that, for a narrowband signal of interest, its multipath delayed version is often assumed to be a scaled version of the original one with only a simple magnitude and phase change and the signal of interest could be completely canceled by the multipath signals after adaptive beamforming. However, if the signal of interest is wideband, in general any delayed version of the signal will only be partially correlated with it; as a result, after adaptive beamforming, the desired signal will not be completely canceled. In our proposed approach, this property will be exploited based on the frequency invariant beamforming (FIB) technique. In the new structure, the received array signals are first passed through an FIB network [14 16], which transforms the wideband beamforming problem into a narrowband one and then a traditional narrowband adaptive beamformer can be applied to the network outputs. It is shown that after this frequency invariant transformation (FIT), cancellation of the desired signal due to its multipath signals is reduced significantly with a much improved output signal to interference plus noise ratio (SINR). On the other hand, the beamforming problem can also be solved by considering each of the network outputs as an instantaneous mixture of the impinging source signals and an instantaneous blind source separation (BSS) algorithm can be applied instead to extract the signal of interest [17, 18], which leads to the so-called blind beamformer [19 24]. The advantage of this second method is that we do not need to know the main DOA of the signal of interest as in the previous case. It will be shown that although the network outputs are still convolutive mixtures of the impinging source signals without multipath, an instantaneous BSS algorithm will be able to recover the desired signal effectively. This paper is organised as follows. A discussion of the wideband multipath problem and a detailed analysis of the focusing method are given in Section 2. Two solutions based on the FIB technique are proposed in 3. Simulation results are provided in Section 4, and conclusions are drawn in Section 5. Manuscript received March 2, 2011; revised December 23, 2011

2 W. Liu / Wideband Beamforming for Multipath Signals Based on Frequency Invariant Transformation The multipath problem with wideband signals A general array structure with M sensors is shown in Fig. 1, where a plane-wave signal s l (t) impinges from a DOA angle θ l. We assume that there are in total L impinging signals and at the 0-th sensor, they are denoted by s l (t), l = 0,, L 1. Then, the signal received at the m-th sensor will be x m(t) = L 1 l=0 s l(t τ m,l ), where τ m,l is the delay from the 0-th to the m-th sensor. For narrowband signals, this delay can be expressed as a complex number and x m(t) becomes a weighted sum of the signals s l (t), l = 0,, L 1, which represents an instantaneous mixing problem. For wideband signals, this delay cannot be expressed as a single complex coefficient and the convolution of δ(t τ m,l ) with s l (t) represents a convolutive mixing problem. In a compact form, we have x(t) = A s(t) (1) where the symbol { } denotes the convolution operation and s(t) = [s 0(t) s 1(t) s L 1(t)] T x(t) = [x 0(t) x 1(t) x M 1(t)] T [A] m,l = a m,l = δ(t τ m,l ), m = 0,, M 1, l = 0,, L 1. (2) where the coefficients vector and the input samples vector are defined as [ T w = w T 0 w T 1... wj 1] T (5) w j = [w 0,j w 1,j... w M 1,j] T (6) [ T x = x T [n] x T [n 1]... x T [n J +1]] (7) x[n j] = [x 0[n j] x 1[n j]... x M 1[n j]] T. (8) Fig. 2 A wideband beamforming structure based on tapped delay-lines Assume that s 0(t) is the desired signal and s 1(t) is a multipath version of s 0(t) given by s 1(t) = αs 0(t δt ), where α is a scalar and δt is the corresponding delay. Note that the discussion and the proposed scheme can be extended to the case with more than one multipath signal in a straightforward way. All of the remaining L 2 signals are uncorrelated interferences. For adaptive beamforming, as an example, we consider the linearly constrained minimum variance (LCMV) beamformer [28], for which the main direction θ 0 of the signal of interest is assumed to be known. The LCMV problem can be formulated as min w wt R xxw subject to C T w = f (9) Fig. 1 An array structure with M sensors, where a plane wave s l(t) impinges from a DOA angle θ l For convenience, we consider the discrete form of the signals. Then we have s l [n] = s l (nt ), l = 0,, L 1, and x m[n] = x m(nt ), m = 0,, M 1, where T is the sampling period. Now the elements a m,l of A become discretetime filters. The filter h m,l [n] at the position a m,l A is given by h m,l [n] = sin((n τ m,l )π) T (n τ m,l )π. (3) T Wideband beamforming can be achieved by employing either the traditional tapped delay-line (TDL) structure or the most recently proposed sensor delay-line structure [25 27]. Without loss of generality, we only consider the TDL-based structure here, as shown in Fig. 2, where each of the received array signals is processed by a TDL and then added together to form the beamformer output y[n], given by y[n] = w T x (4) where R xx is the correlation matrix of the observed array data x, given by R xx = E{xx T }, C is the constraint matrix and f is the response vector. The constraint equation C T w = f guarantees that the resultant beamformer has the desired response specified in it. Its solution w opt can be obtained by the Lagrange multipliers method, given by w opt = R 1 xx C(C T R 1 xx C) 1 f. (10) By minimizing the output variance of the beamformer, the uncorrelated interferences will be suppressed effectively. However, for the correlated interference s 1(t), it will try to cancel at least part of the desired signal in the output to reduce the overall output signal variance. Suppose the impulse response of the beamformer to the desired signal s 0(t) from the direction θ 0 is simply a delay T 1 and to s 1(t) from θ 1 is given by h 1(t), i.e., at the beamformer output, the component corresponding to s 0(t) is s 0(t T 1) and that corresponding to s 1(t) is αs 0(t δt ) h 1(t). In the worst case, when h 1(t) has a magnitude response 1/α and a phase response (T 1 δt )ω, the desired signal will be canceled completely by its multipath version at the beamformer output.

3 422 International Journal of Automation and Computing 9(4), August 2012 To avoid cancellation of the desired signal, a focusing processing method based on the main DOA of the desired signal was proposed for wideband beamforming in a multipath environment [13, 29]. In this structure, as shown in Fig. 3, each of the received array signals x m[n], m = 0, 1,, M 1 is first transformed into the frequency domain and then a focusing transformation is applied to each set of frequency bin outputs with the same frequency so that we have the same steering vector for different frequencies. h m,j, m = 0, 1,, M 1, j = 0, 1,, J 1 are the coefficients of this focusing transformation. After this focusing processing, we apply a single set of coefficients w m, m = 0, 1,, M 1 to obtain the J frequency domain outputs, where J is the length of the discrete Fourier transform (DFT) and also the length of the original TDLs. An inverse DFT (IDFT) is then performed leading to the time domain beamformer output signal y[n]. The values of w m can be decided based on different optimisation criteria and constraints. Since w m is common for different frequencies, this structure can be transformed into an equivalent form shown in Fig. 4. broadside. Then, we can choose the focusing coefficients as h m,j = e jω j (τ M τ m(θ 0 )), where τ M is a proper delay chosen by the design and ω j is the frequency corresponding to the j-th frequency bin of the DFT. Each set of the focusing coefficients for a fixed m is actually implementing a simple delay in the time domain together with the associated DFT and IDFT operations. Therefore, the structure in Fig. 3 can be further transformed into the one shown in Fig. 5. Clearly, this is the well-known narrowband beamforming structure (after exchanging the positions of the coefficients w m and the delays τ M τ m(θ 0)), and the delays after w m are simply used to steer the beamformer to the main direction θ 0 of the signal of interest. This idea is similar to the one proposed in [12], where the focusing coefficients are chosen as h m,j = e j(ˆω 0 ω j )τ m(θ 0 ) with ˆω 0 being the chosen focusing frequency. In this case, it is implementing a time advancement in addition to a fixed phase shift independent of frequency. Fig. 5 The equivalent narrowband beamformer structure to the ones shown in Figs. 3 and 4 Fig. 3 The wideband beamforming structure for the focusing processing based method With these analyses, we can see that the previously proposed focusing based wideband beamformer will only work effectively when the signal bandwidth is relatively narrow since it is essentially based on a narrowband beamforming structure. To overcome the limitation of the focusing based method, we need to adopt a proper wideband beamforming structure, such as the TDL-based or SDL-based ones. In the next we will introduce the proposed scheme based on an FIB network. 3 The proposed approach based on frequency invariant beamforming Fig. 4 An equivalent beamforming structure to the one shown in Fig. 3 The steering vector a(ω, θ) of the array as a function of signal frequency ω and DOA angle θ is given by [ ] a(ω, θ) = e jωτ0(θ) e jωτ 1(θ) e jωτ M 1(θ) (11) where τ m(θ) is the delay from the 0-th to the m-th sensor. As a result, we have τ 0(θ) = 0. Assume the main direction of the signal of interest is θ 0 and we want to focus the steering vectors for θ = θ 0 at different frequencies to the Since the proposed scheme involves the design of frequency invariant beamformers, in the next, we will give a brief review to it first; based on the FIB design, we then introduce the narrowband beamforming based scheme and the instantaneous BSS based scheme, respectively, followed by discussions. 3.1 Frequency invariant beamforming Frequency invariant beamforming is a technique for wideband array design to form a response only as a function of the DOA angle of the impinging signals, independent of the signal frequency [14, 15, 30 39]. Here, we use the design method proposed in [15] for uniform linear arrays (ULAs) as an example. Note that the two proposed schemes are not limited to ULAs and can be applied to any array geometries as long as a frequency invariant beamformer can be

4 W. Liu / Wideband Beamforming for Multipath Signals Based on Frequency Invariant Transformation 423 designed based on it, such as planar arrays, circular arrays, etc. For the wideband beamformer shown in Fig. 2, its beam response P (ω, θ) is given by P (ω, θ) = M 1 m=0 k=0 J 1 w m,k e jmω τ e jkωt (12) where τ = d sin θ with d being the adjacent sensor spacing c and c the wave propagation speed. With the normalized angular frequency Ω = ωt, (12) can be rewritten as P (Ω, θ) = M 1 m=0 k=0 J 1 w m,k e jmµω sin θ e jkω (13) with µ = d. Substituting Ω1 = µω sin θ and Ω2 = Ω, (13) ct yields P (Ω 1, Ω 2) = M 1 m=0 k=0 J 1 w m,k e jmω1 e jkω 2. (14) As the spatio-temporal spectrum of the impinging signal lies on the line defined by Ω 1 = µω 2 sin θ, we can replace sin θ by Ω 2 µω 1 in a given desired frequency invariant beam response P (sin θ), and apply a 2-D inverse Fourier transform to the resultant P (Ω 1, Ω 2) to obtain the desired coefficients w m,k, m = 0,, M 1 and k = 0,, J 1. For more details, please refer to [15]. As in general, the sampling period is half that of the highest signal frequency and d is half the corresponding wavelength, we have d = 1 c (2T ) = ct and µ = 1. 2 Therefore, without loss of generality, we will only consider the case with µ = 1 in the design and simulations. Fig. 6 shows a design result with M = 17 sensors and a tapped delay-line length of J = 80. The frequency invariant property is clearly visible over the band Ω [0.40π, π]. The proposed structure is shown in Fig. 7, where x[n] is the signal vector at time n and each block labeled as FIB i, i = 0, 1,, N 1 represents a frequency invariant beamformer with a response P i(θ). The output of block FIB i is denoted by b i[n]. FIB 0 is the main beam pointing to θ 0 and the remaining N 1 beams are auxiliary beams pointing to the remaining directions. All the auxiliary beams have a zero response at θ 0 so that there is no component corresponding to s 0(t) at their outputs. The final beamformer output is given by y[n], a weighted sum of the FIB network outputs y[n] = b 0[n] ŵw T b[n] (15) where b[n] = [b 1[n] b 2[n] b N 1[n]] T ŵw = [w 1 w 2 w N 1] T. (16) The optimum ŵw opt can be obtained by minimizing E{ y[n] 2 } ŵw opt = R 1 bb p bb (17) where R bb = E{b[n]b[n] T } and p bb = E{b[n]b 0[n]}. Alternatively, a classical adaptive algorithm such as the normalized least mean square (LMS) algorithm, can be employed to update the weight vector iteratively ŵw[n + 1] = ŵw[n] + where µ is the stepsize. µ y[n]b[n] (18) b[n] T b[n] Fig. 7 The proposed FIT-based beamforming structure employing a narrowband beamforming scheme Fig. 6 An FIB design example with coefficients for a uniform linear array with a broadside main beam 3.2 The proposed scheme using a narrowband beamformer Assuming the main direction θ 0 of the signal of interest is know, a solution can be obtained based on the traditional beamforming idea by employing the FIB technique. The proposed structure is actually the same as the previously proposed beamspace adaptive wideband arrays [14, 16, 40]. However, the traditional beamspace structure is proposed for working in a non-multipath environment and it is the first time to realize that such a structure can actually work effectively in a multipath environment, which will become clear after the following detailed analysis. Note the responses P i(θ), i = 0, 1,, N 1 of the frequency invariant beamformers are independent of frequency. Then, each of their outputs can be expressed as a weighted sum of the impinging signals with b i[n] = p i s[n] (19) s[n] = [s 0[n] s 1[n] s L 1[n]] T p i = [P i(θ 0) P i(θ 1) P i(θ L 1)]. (20)

5 424 International Journal of Automation and Computing 9(4), August 2012 Then, the component b i,l corresponding to s l [n] in the output b i[n] will be b i,l = P i(θ l )s l [n]. For the desired signal s 0[n] from θ 0, since all of the other FIB i, i = 1, 2,, N 1 have a zero response at θ 0, i.e., P i(θ 0) = 0 for i = 1, 2,, N 1, the component b 0,0 corresponding to s 0[n] in the output b 0[n] becomes b 0,0 = P 0(θ 0)s 0[n] = P 0(θ 0)s 0(nT ) (21) and the signal component y 0 corresponding to s 0[n] in the final beamformer output y[n] will be the same as in b 0[n], i.e. y 0 = b 0,0 = P 0(θ 0)s 0(nT ). (22) For the component y 1 in y[n] corresponding to s 1[n], the scaled delayed version (multipath) of s 0[n], it is N 1 y 1 = (P 0(θ 1) P i(θ 1)w i)s 1[n] = i=1 N 1 α(p 0(θ 1) P i(θ 1)w i)s 0(nT δt ). (23) i=1 Since s 1[n] = s 0(nT δt ) is only partially correlated with s 0[n] = s 0(nT ) for the zero lag, no matter how to adjust the coefficients of ŵw, the desired signal will not be completely canceled and the total output y s corresponding to the signal of interest in the beamformer output y[n] is given by y s = y 0 + y 1 = P 0(θ 0)s 0(nT ) + N 1 α(p 0(θ 1) P i(θ 1)w i)s 0(nT δt ). (24) i=1 Suppose the uncorrelated interferences have been suppressed sufficiently at the output, then y[n] will include a filtered version of the desired signal s 0(nT ) plus some noise components. Now the task of beamforming in terms of interference suppression has been completed. At the next stage, to recover the original signal from its filtered version, a deconvolution operation is needed where methods such as those proposed in [41, 42] can be adopted. Note that the DOAs of the impinging signals can be estimated based on the FIB network using some narrowband DOA estimation methods [30], and we can then use this DOA information to design some beamformer for interference suppression. Actually, all of the beamforming problems can be solved in this way to different degrees if the complexity and difficulty involved in DOA estimation is not a big issue in the implementation. However, in many of the traditional beamforming problems, we only know the main direction of the signal of interest, we do not know the directions of the correlated or uncorrelated interferences, and we want to avoid the additional DOA estimation process if possible, especially for online applications. The advantage of the proposed scheme is that we do not need to do any DOA estimation and a simple narrowband beamformer will be able to solve the problem effectively. In the next proposed scheme, it is a completely blind scenario since even the main direction of the signal of interest is not required. 3.3 The proposed scheme using an instantaneous BSS algorithm In the first proposed scheme using a narrowband beamformer, the main DOA angle of the signal of interest is assumed to be known. As mentioned in Section 2, the received wideband array signals can be considered as convolutive mixtures of the impinging sources, so that convolutive BSS algorithms can be applied to x[n] directly without any knowledge of the DOA angles of the signals. However, convolutive BSS algorithms have an extremely high computational complexity and it is extremely difficult if it is not prohibitive to implement them online when a large number of sensors are involved. As studied in [43], if there are not multipath signals, the outputs of the FIB network can be considered as instantaneous mixtures of the wideband source signals and we can employ a simple instantaneous BSS algorithm to recover the signal of interest, leading to a blind wideband beamformer. This idea can be extended to the multipath directly. From (19), we can see that each b i[n] is a weighted sum or an instantaneous mixture of the original L source signals. Applying an instantaneous BSS algorithm to them to extract the desired signal, we obtain the structure shown in Fig. 8, with its output y[n] given by where y[n] = N 1 i=0 w ib i[n] = w T b[n] (25) b[n] = [b 0[n] b 1[n] b N 1[n]] T w = [w 0 w 1 w N 1]. (26) Fig. 8 The FIT-based structure using an instantaneous BSS algorithm This is actually a simplified BSS structure, called blind source extraction (BSE) [17, 18], since we only extract one signal from the mixtures, while normally we can recover all of the source signals simultaneously using a general BSS algorithm. Depending on the statistical properties of the source signals, different BSS/BSE algorithms can be employed. As an example, if the source signals are independent of each other with at most one being Gaussian, we can recover them based on the criterion that the separated signals are as independent of each other as possible. If no multipath signals are present and all source signals are independent of each other, it is easy to see that we can recover the signal of interest by applying an instantaneous BSS/BSE algorithm [43].

6 W. Liu / Wideband Beamforming for Multipath Signals Based on Frequency Invariant Transformation 425 However, for the multipath case when the source signal s 1[n] is a delayed version of s 0[n], we will not be able to separate them from each other since they are not independent of each other. Fortunately, both of them are independent of the remaining signals, and as long as the number of mixtures N is not smaller than the source signal number L, we will be able to extract s 0[n] and s 1[n] together from the mixtures b[n] by finding a linear combination w of b[n]. Then, in the ideal case, only a mixture of s 0[n] and s 1[n] (similar to the form of y s[n]) plus some noise is left in the final output y[n]. In the following simulations, the desired signal is sub- Gaussian and an instantaneous BSE algorithm based on minimizing the normalized kurtosis value of the output y[n] [18, 44] can be employed where with w[n + 1] = w[n] µφ(y[n]) b[n] (27) φ(y[n]) = m2(y)[n] m 4(y)[n] y3 [n] y[n] (28) m q(y)[n] = (1 λ)m q(y)[n 1] + λ y[n] q, q = 2, 4 (29) where λ is the forgetting factor for estimating m 1(y)[n]. 3.4 Discussions One important issue about any beamforming method is its robustness. The robustness of the proposed scheme is dependent on how a frequency invariant beamformer can be designed robustly (against array model errors) and how the following narrowband beamformer can perform robustly against steering vector errors (as long as the model errors have been absorbed successfully by the design of frequency invariant transformations). The design of robust frequency invariant beamformers is an ongoing research topic; for the following narrowband beamformer, any robust beamformers can be used and the key contribution of this work is to show that a combination of the FIB network and the traditional narrowband beamformer or instantaneous BSS/BSE algorithms can solve the multipath problem effectively. For the FIB network, it is not limited to any specific designs, while for the following narrowband beamformer or the instantaneous BSS/BSE algorithms, they are not limited to any specific solution either. We can try different frequency invariant beamformer designs, different narrowband beamformers (either robust or not) and different instantaneous BSS/BSE algorithms depending on the application scenarios. Moreover, from (24), we can see that the output signal of the proposed beamformer is a simple filtered version of the original source signal and the number of taps for this filtering operation is the same as the number of different directions for the signal of interest. To recover the original source signal completely, we need a second step, i.e. deconvolution. This is a different area from beamforming and its success will depend on the statistical properties of the original source signal itself and is not closely related to the array structure anymore. From the beamforming point of view, the task of recovering the component corresponding to the signal of interest is basically completed, since the other interfering signals have been suppressed successfully without attenuating the desired signal component. Interestingly, this result coincides with the general perception about BSS for convolutive mixtures, i.e. for convolutive BSS itself, we can only recover the sources subject to an arbitrary filtering operation. As a further study to this topic in the future, we can try to combine the deconvolution process and the beamforming process together (completely integrated, not divided into two separate steps) in some way to recover the original signal directly. 4 Simulations Our simulations are based on a ULA with M = 17 sensors. For the proposed schemes, 5 frequency invariant beams are designed based on the desired response shown in Fig. 9, each with dimensions Their performances are compared with both the original and the spatial-smoothing based LCMV beamformers in terms of the output SINR for different input signal to noise ratios (SNRs). For the original LCMV beamformer, its dimensions are 17 80, the same as the frequency invariant beams. For the spatialsmoothing based beamformer, the sub-array size is 6 and in total there are = 12 sub-arrays. The smoothed correlation matrix is obtained by averaging the 12 individual correlation matrices, which represents an equivalent beamformer with 6 sensors and a TDL length of 80. The desired signal comes from the broadside θ 0 = 0 and both LCMV beamformers have a simple broadside constraint in order to receive the desired signal with a unity response. Two uncorrelated interfering signals with a signal to interference ratio (SIR) of 0 db arrive from the directions 60 and 50, respectively. All of the source signals are bandlimited to the normalized frequency range Ω [0.45π, π]. In order to use the BSE algorithm in Equation (27), the desired signal has been chosen to be sub-gaussian with a normalized kurtosis value of about 0.7. All of the interfering signals are Gaussian. Fig. 9 The desired responses of the five frequency invariant beams In the first set of simulations, there is one multipath signal, which is a delayed version of the desired signal by 20 samples and scaled by 0.5 in magnitude, and arrives from θ 1 = 30. We change the input noise level from SNR = 0 db to SNR = 20 db and then draw the output SINR curve for all of the four beamformers: the original

7 426 International Journal of Automation and Computing 9(4), August 2012 LCMV beamformer, the spatial-smoothing based LCMV beamformer, the FIB-based beamformer in Fig. 7 and the FIB-based beamformer in Fig. 8. The result is shown in Fig. 10 with the four beamformers denoted by original LCMV, spatial-smoothing, proposed #1 and proposed #2, respectively. Moreover, the two nulls at the interfering signal directions (60 and 50 ) are clearly visible. Fig. 11 Output SINR of the four beamformers in the second set of simulations Fig. 10 Output SINR of the original LCMV beamformer ( original LCMV ), the spatial-smoothing based LCMV beamformer ( spatial smoothing ), the FIB-based beamformer in Fig. 7 ( proposed #1 ) and the FIB-based beamformer in Fig. 8 ( proposed #2 ), for different values of input SNR in the first set of simulations It can be seen that the proposed schemes can deal with the multipath problem effectively and the output SINR for both of the proposed beamformers increases steadily with respect to a reducing input noise level. On the other hand, the original LCMV beamformer has failed to pass through the signal of interest at its output and its output SINR stays at about 8 db with a tiny fluctuation with respect to the input SNR level. The spatial-smoothing method can improve the output SINR of the traditional LCMV beamformer, but the improvement is not as significant as in the proposed schemes. In the second set of simulations, we add one more multipath signal to the first setting and it is a delayed version of the desired signal by 10 samples and scaled by 0.5 in magnitude, arriving from θ 2 = 70. The results are shown in Fig. 11, which are very similar to those in Fig. 10. Interestingly, an additional multipath signal has not degraded the two proposed beamformers performance in a notable way and for some of the input SNR values, the output SINR with two multipath signals is even a little bit higher than the one multipath signal case. A possible explanation for this phenomenon may be that the second multipath signal can add either constructively or destructively to the first multipath signal, and their overall effect to the main desired signal will not change linearly with respect to the number of multipath signals. This is a complicated issue and further study is needed in the future. At last, a resultant beampattern for the second set of simulations using the first proposed scheme ( proposed #1 ) is shown in Fig. 12. Since each of the frequency invariant beamformers in the FIB network has a frequency invariant response, as a linear combination of them, the resultant beampattern also exhibits a frequency invariance property. Fig. 12 A resultant beampattern for the second set of simulations using the first proposed scheme ( proposed #1 ) 5 Conclusions A novel beamforming approach for multipath wideband signals based on the frequency invariant beamforming technique has been proposed. In this approach, the received array signals are first processed by an FIB network and then a traditional narrowband adaptive beamformer or an appropriate instantaneous BSS/BSE algorithm can be applied to its output to recover the desired signal, depending on whether the main direction of arrival of the desired signal is available or not. The desired signal recovered by this approach will be subjected to an arbitrary filtering effect and further deconvolution operation is needed to remove this effect. Simulation results show that an improved output SINR has been achieved by the proposed scheme compared to the traditional beamforming methods and it can deal with the correlated wideband interference problem effectively.

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