Complex CORDIC-like Algorithms for Linearly Constrained MVDR Beamforming

Transcription

1 Complex CORDIC-like Algorithm for Linearly Contrained MVDR Beamforming Mariu Otte Information Proceing Lab Univerity of Dortmund Otto Hahn Str Dortmund, Germany Martin Bücker (martin.bucker@reearch.nokia.com) Nokia Reearch Center Meemanntr Bochum, Germany Jürgen Götze (goetze@dt.e-technik.uni-dortmund.de) Information Proceing Lab Univerity of Dortmund Otto Hahn Str Dortmund, Germany Abtract. In thi paper we invetigate the ue of CORDIC like approximate rotation for complex valued ignal proceing application. In particular we deign a proceor array for MVDR beamforming with multiple contraint entirely baed on linear (Gau) and circular (Given) complex tranformation. The required tranformation are expreed a factorized tranformation uch that they can be repreented a et of real CORDIC module. A factorized rotation cheme i applied during the entire algorithm. Each real CORDIC module i replaced by CORDIC like approximate rotation. The performance of the preented linearly contraint MVDR beamformer i invetigated by comparing the bit error rate (auming 4 QAM modulated ignal) and the beampattern. Keyword: CORDIC, factorized rotation cheme, approximation, parallel implementation, beamforming 1. Introduction In recent year ignal proceing for communication ha become one of the main area of reearch and development. Epecially, the growing number of real time application in the area of wirele communication require the development of parallel ignal proceing algorithm and architecture. Uually, mot of the algorithm and architecture preented in the literature aume real input data. Thi i particularly the cae when it come to the VLSI implementation of the architecture. The complex cae i uually covered by tandard mean (e. g. four real multiplication for the execution of a complex multiplication). In many c 2006 Kluwer Academic Publiher. Printed in the Netherland. mvdr.tex; 7/09/2006; 12:36; p.1

2 2 practical application, however, complex data do occur, e.g., adaptive beamforming [13], multi uer detection [26]. The proceor array for MVDR adaptive beamforming [19, 18] eentially implement the olution of a complex valued leat quare problem, which i obtained by incorporating (one or multiple) contraint in the given minimization problem. Furthermore, a direct computation of the output ignal i implemented by a final multiplication of the output ignal of the right hand ide by the quare root of the converion factor. The computation for incorporating the contraint are not implemented in thee proceor array but referred to a preproceing tep. The proceor array are entirely compoed of proceor cell which evaluate (diagonal cell) or apply (off-diagonal cell) circular rotation. Therefore, the complexity of the parallel implementation i mainly determined by the complexity of the evaluation and application of the circular (Given) rotation. Conequently, different trategie, which have been preented for modifying the circular rotation, can be conidered in virtue of an efficient implementation: M1 approximate rotation [21, 2] M2 factorized rotation [6, 23, 11, 12] M3 CORDIC [3, 5, 14] M4 normalized rotation [20] M5 Combination of the modification: M1+M2 factorized approximate rotation [8] M1+M3 CORDIC like approximate rotation [7, 10, 9] The implementation of two ided complex rotation baed on CORDIC ha alo been dicued in [16, 15, 25]. In thi paper we preent an extended proceor array for MVDR beamforming with multiple linear contraint. Uing the Schur complement the incorporation of the linear contraint into the minimization problem can be formulated a a partial Gauian elimination. The reulting leat quare problem i olved by the QR decompoition, a uual. Even the final multiplication for the direct computation of the output ignal can be formulated a a linear tranformation by uing the Schur complement. Thereby, we achieve an implementation which i entirely baed on linear Gau tranformation and circular Given rotation and can be implemented on an upper triangular array of proceor. mvdr.tex; 7/09/2006; 12:36; p.2

3 In order to implement the complex valued tranformation they are formulated in a factorized form. Thi factorization i compoed of a real valued linear and circular tranformation, repectively, and phae hift of the compleumber involved in the tranformation. Thee phae hift can alo be referred to circular tranformation in the complex plane. Therefore, the complex tranformation can be referred to a number of real tranformation (jut like a complex multiplication i compoed of four real multiplication). Exploiting thee factorization in detail, however, it i poible to formulate a factorized rotation cheme for a complex tranformation, i.e. one of the phae hift i accumulated in a diagonal matrix which accompanie the involved matrice during the algorithm. Thi i imilar to the idea of factorized rotation [6, 23, 12] where the caling factor are wapped out into a diagonal matrix. The diagonal matrix i compenated at the end of the computation. Thereby, the number of required real rotation i reduced and two of the real rotation are jut ued to annihilate the imaginary part of the involved compleumber. A mentioned above the entire upper triangular proceor array for MVDR beamforming with multiple contraint can be implemented baed on proceor cell executing linear and circular complex 2 2 tranformation. Thee tranformation are referred to real linear and circular 2 2 tranformation. Each real tranformation i implemented uing a linear and circular CORDIC proceor, repectively. The phae factor accumulated in the diagonal matrix are compenated by the final linear tranformation for directly computing the output ignal. We apply CORDIC-like approximate rotation to the real CORDIC module repreenting a complex tranformation. At firt the ue of approximate (linear and circular) CORDIC like rotation for the complex tranformation i invetigated. Fortunately, all the real tranformation, of which a complex tranformation i compoed of, require the ame accuracy of the approximation, i. e. all the real CORDIC like module can be implemented uing the ame number of µ rotation. Then, CORDIC like approximate rotation are ued for all real CORDIC module building the (linear and circular) complex tranformation in the preented MVDR beamforming proceor array. The preented MVDR beamformer with multiple contraint i applied to a modulated binary ignal (auming 4 QAM). The beampattern and the BER (bit error rate) of the preented MVDR beamforming algorithm and architecture are analyzed in order to obtain a performance profile [22] for the approximation of the CORDIC module. Different approache for incorporating the contraint are invetigated. Simple method for finding the hift value (the pecific µ rotation) of the CORDIC like approximate rotation are invetigated. Even the 3 mvdr.tex; 7/09/2006; 12:36; p.3

4 4 ignal 2(t) ignal 1(t) interferer enor element (t) 3 Figure 1. Scenario with four enor element, two information bearing ignal 1(t) and 2(t) from known direction and one interferer 3(t) from an unknown direction. ue of CORDIC like module without caling factor compenation i dicued. The imulation how, that the required computational effort (approximation accuracy, caling) trongly depend on the pecific condition of the application (SNR, modulation cheme). The paper i organized a follow. In ection 2 we explain the underlying ignal model and review the MVDR beamforming algorithm. An efficient parallel implementation entirely baed on complex tranformation cell i preented. In ection 3 we briefly explain CORDIC and CORDIC-like technique. The factorized tranformation preented in ection 4 allow the contruction of complex CORDIC-like proceor cell in term of real CORDIC cell. Thi lead to a very efficient hardware architecture. Reult from computer imulation are preented in ection 5 to illutrate the performance of the decribed implementation. 2. MVDR Algorithm In thi ection, an adaptive beamformer with minimum variance ditortionle repone i dicued. Particularly, we focu on it parallel implementation Signal Model Conider a cenario with M omnidirectional enor element located in a plane at poition m i. The antenna array receive a mixture of deired ignal which hould be decoded, undeired interferer from unknown direction and background noie equal to all direction. The cenario i depicted in Figure 1. Note, that for the receiving ignal 1 (t) the econd ignal 2 (t) alo repreent an interferer with known direction of arrival. Wherea 3 (t) i an interferer with unknown direction of arrival. For implicity, we aume that the direction of propagation i equal at each enor and the waveform are planar. Thu, the farmvdr.tex; 7/09/2006; 12:36; p.4

5 field aumption hold. The individual enor output x i (t) hould be in baeband form, taking into account the received ignal uually are ome kind of modulated ignal tranformed into equivalent lowpa ignal. Therefore, we aume a narrow-band approximation, i. e., every antenna element receive the ame ignal, but delayed in time. The equivalent lowpa ignal at enor i can be written a x i (t) = x(t)exp( j2πf c τ i ) + n i (t), (1) where x(t) i the complex baeband ignal at a virtual reference enor element placed in the origin of the antenna coordinate ytem given in Figure 1, τ i i the time delay of the ignal at enor i relative to the reference enor, f c i the carrier frequency and n i (t) i white Gauian noie. Note that equation (1) only hold for exactly one direction. Thu, a complete decription of Figure 1 require two equation. Combining the exp-term belonging to every antenna and to every known direction of propagation in a N M matrix C, with N the number of known ignal direction, will allow a more compact notation. The matrix i defined a follow: e jφ 1,1 e jφ 2,1 e jφ M,1 e jφ 1,2 e jφ 2,2 e jφ M,2 C =......, (2) e jφ 1,N e jφ 2,N e jφ M,N where φ i,k = 2πf c τ i,k. The index k correpond to the kth direction of arrival. For a given array geometry and given direction of arrival, the time delay τ i,k can be eaily calculated by projection conideration. The dicrete ignal x i (n) are conidered a output of analog frontend with one front-end per enor. Detail of the front-end tructure are beyond the cope of thi paper. The ampled ignal are arranged in a n M matrix X, with x 1 (1) x 2 (1) x M (1) x 1 (2) x 2 (2) x M (2) X =..... (3). x 1 (n) x 2 (n) x M (n) where n i the number of ample taken at each antenna. Now, weighting the enor output with complex factor w m, the ummation of thee product reult in a patial filter, a o-called beamformer [13]. Due to the fact, that every weight vector w i = [w 1,...,w M ] T correpond to one deired output ignal e i, we define a ignal matrix E = [e 1 e 2 e L ], where L N, containing the filter output. 5 mvdr.tex; 7/09/2006; 12:36; p.5

6 6 The mot traightforward technique for adjuting the weight i to demand that the deired ignal hould be emphaized, wherea noie and interfering ignal propagating from other direction hould be uppreed [17]. Thi lead to the following leat quare repreentation, min w i e i = Xw i 2 2 for i [1, L] ubject to CW = B, (4) where L i the number of deired output ignal and B denote the gain matrix. Motly the element in N K matrix B are taken from the et {0, 1}; 0 for interference uppreion, 1 for unity gain of the information ignal. In turn, the weight matrix W i compoed of vector w i, uch that W = [w 1 w 2 w L ]. So the leat-quare criterion trie to minimize the average output of the beamformer and imultaneouly fulfilling the contraint. For the cenario depicted in Figure 1 we would obtain N = 2 contraint (two known ignal direction 1 (t), 2 (t)), uch that C : 2 M, B = I 2 in order to obtain E = [e 1 e 2 ] Solving the Contrained Optimization Problem One approach to olve the contrained optimization problem i to reformulate the problem a a leat quare problem without contraint [13]. With C = [C 1 C 2 ], W T = [W T 1 WT 2 ] and X = [X 1 X 2 ], where = [ 1 2 ] denote the partitioning of matrix in two ubmatice, whereby the firt one ha N column, the contraint equation can be written a C 1 W 1 + C 2 W 2 = B (5) Solving for matrix W 1, we get Thu, Therefore, with and W 1 = C 1 1 (B C 2W 2). (6) XW = X 1 W 1 + X 2 W 2 = (X 2 X 1 C 1 1 C 2)W 2 ( X 1 C 1 1 B) we have to olve L leat quare problem X 2 = X 2 X 1 C 1 1 C 2 (7) B 2 = X 1 C 1 1 B (8) min X 2 W2 B (9) W 2 mvdr.tex; 7/09/2006; 12:36; p.6

7 in order to get W 2 and then calculate W 1 from (6). In a further tep we have to calculate the deired output ignal by E = XW. (10) It i beneficial, particularly with regard to a parallel hardware implementation, to incorporate the tep involved in the olution of the contrained leat quare problem (i. e. equation (6), (7), (8), (9), (10)) into one matrix triangularization proce. Let the (n + N) (M + K) matrix M be defined a M = C 1 C 2 B X 1 X (11) Applying a equence of Gauian tranformation G pq () to the matrix M, where G pq () annihilate the element m pq uch that C 1 become upper triangular and X 1 i annihilated entirely, reult in M = p,q G pq () M = R 1 C 2 B 1 0 X 2 B 2, (12) where R 1 i an upper triangular matrix and the n (M N + K) matrix [X 2 B 2 ] i the Schur complement of M [24]. Thi i actually jut a partial Gauian elimination of M. Note, that we can chooe the tranformation uch that the diagonal element of R 1 become real. The leat quare problem incorporated in the lower left-hand block can be olved by QR decompoition of X 2 in uch a way that X 2 = [ ] [ ] R Q2 Q 2, where [Q 0 2 Q ] i unitary. Again the required unitary tranformation i compoed of Given rotation G pq (θ). Defining the matrix [ P H 2 P H ] H [ ] = B 2 Q2 Q and applying a equence of Given rotation to M we ee that the partial triangularization proce of (12) i continued by unitary tranformation: R 1 C 2 B 1 G pq (θ) 0 X 2 B 2 p,q = R 1 C 2 B 1 R 2 P P S (13) A before, the Given rotation can be choen uch that the diagonal element of R 2 become real. mvdr.tex; 7/09/2006; 12:36; p.7

8 8 By uing the equation above, we actually can compute the output ignal without computing W: E = XW = X 2 W 2 B 2 (14) = Q 2 R 2 W 2 B 2 Since R 2 W 2 = P 2 and B 2 = Q 2 P 2 + Q P S hold, one obtain E = Q P S (15) 2.3. Proceor Array Since the entire algorithm i formulated a a triangularization of the matrix M, it can be implemented on an upper triangular proceor array. In Figure 2 uch an array i depicted. The two proceor row at the top (haded dark) perform the partial Gauian elimination, while the lighter haded cell perform the QR-decompoition. In thi example we conider a five element enor array with two ignal from known direction impinging on it, hence we have two contraint (B equal the 2 2 identity matrix). The boundary cell calculate the linear and unitary tranformation, repectively. The internal cell carry out the tranformation. Figure 3 illutrate the cell function. The two multiplier at the bottom of the array carry out the multiplication that follow from equation (15). We call it γ-factor compenation. Due to the fact that they detroy the uniform array tructure it would be deirable to embed them in the array. Of coure, we can ue the lightly modificated cell of Figure 3 (b) to do the multiplication. Thi trivial fact will become more important in our later implementation. Since the ignal are complex valued, the proceor cell have to handle complex data. A we ee from Figure 3, the cell have to carry out mainly multiplication and ummation. However, the diagonal cell have to calculate quare root and/or diviion, repectively. Due to it computational complexity, thee computation take more time than the multiplication. To overcome thi problem, we will now concentrate on the internal tructure of the proceor cell. In virtue of an efficient VLSI implementation, CORDIC proceor have been ued to implement linear and circular tranformation entirely baed on hift-and-add operation. In the following we look at the implementation of the complex valued tranform baed on CORDIC proceor and the ue of CORDIC-like approximate rotation. mvdr.tex; 7/09/2006; 12:36; p.8

9 9 0 X 2 B X 1 C2 C 1 linear Gauian tranformation 1 unitary tranformation Figure 2. Proceor Array. e 1 e 2 3. CORDIC-like Algorithm The CORDIC (COordinate Rotation on a DIgital Computer) algorithm make it poible to carry out vector rotation (and hence to calculate trigonometric function) only by uing hifter and adder, which i very attractive from a hardware point of view. The general CORDIC iteration i given by [ ] xk+1 y k+1 = [ 1 md k 2 σ k d k 2 σ k 1 z k+1 = z k + d k α k ][ xk y k ] (16a) (16b) Dependent on the choice of m and σ k we can ditinguih between different type of CORDIC. By chooing m = 1 and σ k = k we obtain mvdr.tex; 7/09/2006; 12:36; p.9

10 10 (a) r xn r r (initialization) (b) r +1 r + r (initialization) +1 (c) γ n r c γ n+1 c r n r r rc + γ n+1 γ n c (d) Figure 3. Proceor cell. c r c +1 r rc + +1 r + c the claic one originally developed by Volder [27] which can be treated a a caled rotation of vector [x k y k ] T. That i, if T correpond to the exact circular/linear/hyperbolic rotation, KT, K R, denote a caled rotation. Different value for m and σ k yield to caled hyperbolic rotation and linear rotation [28]. The different mode are ummarized in Table I. Table I. CORDIC mode Mode m σ k α k circular 1 k arctan(2 σ k ) linear 0 k 2 σ k hyperbolic 1 1,..., 4, 4, 5,..., 12, 13, 13, 14,... arctanh(2 σ k ) Moreover the et of poible value for d k allow u to elect either non-redundant (d k { 1; 1}) or redundant (d k { 1; 0; 1}) CORDIC. To perform a complete circular, linear or hyperbolic tranformation we have to pa through the iteration in equation (16) until the remaining error fulfill the requirement in ome ene. While claical CORDIC compute w micro-rotation (a µ-rotation i one recurion of the full CORDIC equence, i. e. the execution of equation (16a) with pecific m, σ k, d k, w i the word length), an approximate CORDIC rotation i defined a one µ-rotation. Note that in approximate ignal proceing [22] it i often ufficient to break off the iteration after a mvdr.tex; 7/09/2006; 12:36; p.10

11 11 v v θ ab(v) final poition v θ v final poition v Figure 4. Evaluation (top) and application (bottom) mode of CORDIC. Dependent on the mode (final poition are from left to right: hyperbolic, linear, circular) an exact rotation move the pointer along one of the dahed line. few tep, thu only to apply few µ-rotation. The correponding exact vector tranformation then i given by v = K [ c(θ) r (θ) l (θ) c(θ) ] v (17) where c(θ) denote co(θ) in circular, coh(θ) in hyperbolic and 1 in linear mode. r (θ) = l (θ) = in(θ) in circular mode, r (θ) = l (θ) = inh(θ) in hyperbolic mode and r (θ) = 0, l (θ) = θ in linear mode. Furthermore, we can ubdivide CORDIC into two operation mode: vectoring (evaluation) and rotation (application) mode. Figure 4 illutrate the mode that have different input and output parameter if treated a black boxe. Now, aume the embedding of CORDIC cell into an array of interconnected cell. Uing the non-redundant CORDIC we can eaily realize the communication between the cell by tranmitting a vector of [d 1, d 2,...,d w ] from one cell to the other. In cae of redundant CORDIC, however, we have to tranmit the d k vector a well a the value of σ k = k, i,e. the correponding angle. The main drawback of CORDIC i that in circular and hyperbolic mode we have to carry out a final multiplication to achieve a caling factor compenation. Table II lit thee cale factor in different mode. For computing the cale factor it i advantageou to ue a fixed number of iteration o that the factor can be calculate in advance. Variou technique have been uggeted to peed up thi cale factor correction [4, 1]. mvdr.tex; 7/09/2006; 12:36; p.11

12 12 Table II. Scale factor K in different mode. Mode circular hyperbolic É w 1 É k=0 w 1 k=0 K Ô 1 + d 2 k 2 2k Ô 1 d 2 k 2 2σ k Obviouly, the maximum rotation angle of CORDIC i limited. The angle cannot exceed w k=0 α k. In vectoring mode thi i the region of convergence. To overcome thi retriction we have to perform a prerotation. The pre-rotation ha to be a imple a poible. Rotation through 180 degree (invert v) or through 90 degree (wap component of v and invert one) are commonly ued. In linear mode, rotation through 180 degree or imple caling of y-component are poible. In the equel, we ue a equence of CORDIC-like approximate rotation. That i, we have to contruct a et of µ-rotation {µ 0, µ 1,...}, where every µ k correpond to one tuple {d k, k k }. The crucial point for approximate rotation i to find the approximate angle d k α k. Here, we aume a floating point repreentation of x k and y k, repectively. That i, x k = m x,k 2 e x,k and y k = m y,k 2 e y,k. The direction of rotation d k follow from the ign of y k. The eaiet way to obtain σ k = k in linear and circular mode i to ubtract the exponent, i.e. to calculate σ k = e x,k e y,k. Other more difficult method that find the optimal value for σ k, i. e. the angle d k α k which i cloet to the exact rotation angle, can be found in [9]. For method that apply the caling of the CORDIC-like approximate rotation the reader i alo referred to [9]. 4. Complex CORDIC-like Algorithm In thi ection we implement complex CORDIC module in term of real CORDIC. In order to do o, we firt formulate complex rotation in a factorized form. It i hown, that the general complex module can be implified in cae of uing them in the MVDR beamformer Unitary Rotation Auming a unitary 2 2 matrix T u that i applied to a vector v = [r a] T, where both r and a repreent compleumber; i. e. r = R{r}+ mvdr.tex; 7/09/2006; 12:36; p.12

13 ji{r} and a = R{a} + ji{a}. Then T u i defined by [ c T u = ], (18) c where = a/ r 2 + a 2, c = r/ r 2 + a 2 and therefore T u v = [ r 2 + a 2 0] T. Let ϕ a and ϕ r the phae angle of a and r, repectively. Now, T u can be decompoed into four matrice: [ ] [ ] 1 0 co δ inδ T u = 0 e j(ϕa+ϕr) inδ co δ }{{}}{{} γ ϕ T 3z (δ) a r 2 + a [ ] e jϕa } {{ } T 2 (ϕ a) [ ] e jϕ r }{{} T 1 (ϕ r) 13 (19) r where inδ = and co δ = 2 r 2 + a 2. If we now rewrite the two element complex vector v a a four element vector ˆv with real element, where R{r} ˆv = I{r} R{a} (20) I{a} we can formulate the factorized complex tranformation in equation (19) in term of four real tranformation in the following fahion. The application of Given rotor T 1 (ϕ r ) = co ϕ r in ϕ r 0 0 inϕ r co ϕ r (21) and T 2 (ϕ a ) = co ϕ a in ϕ a 0 0 inϕ a co ϕ a (22) on vector ˆv correpond to the multiplication re jϕr and ae jϕa, repectively. Likewie, the application of matrix co δ 0 inδ 0 T 3z (δ) = 0 co δ 0 inδ inδ 0 co δ 0 (23) 0 inδ 0 co δ mvdr.tex; 7/09/2006; 12:36; p.13

14 14 on vector ˆv correpond to the application of T 3z (δ) on v. Up to now, we have ubtituted three of the four complex factor in equation (19). The remaining factor γ ϕ can alo be expreed a a real rotation. However, it i advantageou not to perform thi rotation in combination with the other, but to defer thi tep. Auming the phae hift γ ϕ are not performed in combination with the other tranformation. That i, there exit a remaining 2 2 diagonal matrix coniting of independent phae factor e jϕ 1 and e jϕ 1, repectively. To perform the next nulling tep in the Given rotation equence, we have to carry out the unitary tranformation given in equation (18). From thi it follow that we have to calculate [ c ] [ ] [ ] e jϕ 1 0 r c 0 e jϕ, (24) 2 a where = a ej(ϕa+ϕ 2) r 2 + a 2 and c = r ej(ϕr+ϕ 1) r 2 + a 2 (25) After a few algebraic manipulation we get [ ][ ][ ] [ 1 0 co δ in δ e jϕ r 0 r 0 e j(ϕa+ϕr+ϕ 1+ϕ 2 ) inδ co δ 0 e jϕa a ]. (26) The important point lie in noticing that we can accumulate the phae factor in the leading diagonal matrix. And o, the phae compenation can be performed in a final tranformation tep and need not to be computed in every proceor cell. Thi i imilar to the idea ued in factorized rotation [6, 23, 11, 12], where the caling of the rotation are accumulated in an accompanying diagonal matrix. Since T u according to (19) can be conidered a a factorized rotation, we can handle the firt diagonal matrix accordingly. In contrary to factorized rotation, a compenation of the diagonal matrice during the algorithm in order to avoid overflow i not required, ince the diagonal matrice γ ϕ only contain phae factor Linear Rotation Similar to unitary rotation, it i poible to write a complex linear rotation a a product of four matrice that are uitable for a real repreentation. In the linear cae, the 2 2 tranformation matrix T l that i applied to vector v = [r a] T to annihilate it econd component i given by T l = [ ] [ 1 0 a r 1 = 1 0 a e j(ϕa ϕr) 1 r ]. (27) mvdr.tex; 7/09/2006; 12:36; p.14

15 Like T u we define a decompoition of T l a [ ] [ ] e jϕ r T l = 0 e jϕa a r 1 }{{}}{{} γ ϕl T 3l ( ) [ ] e jϕa } {{ } T 2 (ϕ a) 15 [ ] e jϕ r 0. (28) 0 1 }{{} T 1 (ϕ r) Comparing the term in equation (28) with the term in equation (19) we found that the tranformation T 1 and T 2 appear in both cae. The unitary tranformation T 3z, however, ha to be replaced by a linear tranformation T 3l. Again, thi complex tranformation may be formulated a a 4 4 real tranformation matrix: T 3l ( ) = (29) A well a in equation (19) we need to perform a final multiplication with γ ϕl, actually a phae correction of the two element of T 3l T 2 T 1 v. Like the foregoing it i poible to incribe the phae hift in a diagonal matrix, hence to accumulate them in a eparate diagonal matrix Implementation Now, if we ubtitute all real tranformation for CORDIC block the complex CORDIC cell arie from the factorized repreentation. The complex block are diplayed in Figure 5. The elementary real CORDIC cell the complex cell baed on are lited in Table III. Recall, that the diagonal value of R 1 and R 2 are real valued, repectively. Hence, we can omit everal real CORDIC cell in the above tated complex CORDIC block. The implified block ued in the array are hown in Figure 6. The complete proceor array entirely baed on CORDIC cell i hown in Figure 8. Comparing with Figure 2 we have done a few modification. Firt, the partial Gauian tranformation to create matrix [ C B] = [R 1 C 2 B 1 ] out of [C 1 C 2 B] are carried out in a econd DOA-proceing block. That i, due to the high reolution capability of direction of arrival etimation, a better performance can be achieved by calculating the tranformation with an increaed accuracy. Since the regiter value in the upper part of the array need not to be updated in every ample tep (DOA vary relatively low with time), the execution time of the DOA etimation i not critical. The value c i,k and b i,k are then aigned to the regiter of the linear proceor cell. A a conequence, feeding of the contraint matrice into the array i dipenable. Thi method increae the accuracy of the mvdr.tex; 7/09/2006; 12:36; p.15

16 16 Table III. Real CORDIC module. Mode Symbol Input Output circular vector mode +1 y n Θ y n+1 +1 yn y n+1 = 0 Θ = arctan yn circular rotation mode +1 y n Θ y n+1 yn Θ +1 y n+1 linear vector mode y n +1 y n+1 +1 yn y n+1 = 0 = yn linear rotation mode +1 y n y n+1 yn +1 y n+1 contraint in the cae of CORDIC-baed approximate tranformation. Note, however, that ince the regiter contain c i,k and b i,k the proceor array work with a pecified accuracy for all CORDIC module (i. e. fixed number of µ-rotation per CORDIC module). Furthermore, the γ-factor calculation, that wa done in the diagonal cell in Figure 2 i now hifted to a new cell column inerted in the array tructure. The correponding cell are hown in Figure 7 (b), (c). Simultaneouly, in thi column the phae hift γ ϕ and γ ϕl can be carried out. A mentioned above, we need to replace the final multiplication in Figure 2 with block that have a imilar internal tructure a the ret of the array. Let b i and c arbitrary compleumber. [ By formal ] 1 [b1, b application of a Gauian tranformation on M = 2,...] c 0 T we annihilate c in order to compute the Schur complement of M, i. e. [ ] 1 [b1, b G 2,1 ()M = 2,...] (30) 0 c[b 1, b 2,...] Obviouly, the Schur complement equal the product c[b 1, b 2,...]. Therefore, the multiplication may be implemented a a tranformation evaluation and application. Fortunately, we can ue the ame cell for vectoring a we ued it in the upper part of the array (ubfigure (a) in mvdr.tex; 7/09/2006; 12:36; p.16 4

17 17 Θ 1 +1 Θ 1 +1 y n Θ y n+1 y n Θ 2 (a) (b) Θ 1 Θ 3 +1 Θ 1 Θ 3 +1 y n Θ 2 Θ y n y n+1 Θ 2 Θ 3 (c) Figure 5. Complex CORDIC for linear evaluation (a), linear application (b), circular evaluation (c) and circular application (d). (d) r r r r Θ 2 +1 Θ 2 (a) (b) Θ 1 Θ 3 r Θ 1 Θ 3 r (c) r (d) r Θ 3 +1 Figure 6. Simplified complex CORDIC for linear evaluation (a), linear application (b), circular evaluation (c) and circular application (d). Figure 6). Furthermore, the application cell (ubfigure (a) in Figure 7) ha a very imilar tructure a the hexagonal cell. mvdr.tex; 7/09/2006; 12:36; p.17

18 18 0 Θ 2 +1 Θ 1-1 Θ 2 +1 K 1 Θ (a) (b) (c) Figure 7. Subfigure (a) how a CORDIC-baed multiplication cell. The triangular cell (b) and (c) perform the γ-factor calculation and accumulate the phae hift. X 1 X c 11 c 12 c 13 c 14 c 15 b11 b12 c 22 c 23 c 24 c 25 b21 b22 γ 1 Figure 8. Complete proceor array. e 1 e 2 mvdr.tex; 7/09/2006; 12:36; p.18

19 19 5. Simulation In thi ection, computer imulation of the propoed array are preented. In the following imulation it i aumed that the ignal we want to decode are 4-QAM modulated ignal. The parameter have been choen to repreent a cenario with M = 5 antenna and 3 impinging ignal. The antenna are uniformly ditributed in a circle contellation, whereby the ditance between adjacent enor i et to half the wavelength. The firt imulation experiment i conducted with two equi-powered ignal impinging from the known direction 90 and 63.4, repectively. An interferer with unknown direction i impinging from The reulting bit error rate a a function of the ignal-to-noie ratio are hown in Figure 9. The proceor cell work with different number of CORDIC iteration. The required hift value are alway calculated by an exponent ubtraction (non-optimized angle calculation, ee ection 3). The reult are compared with a imulation trial where exact rotation were ued. Notice that already three µ-rotation yield to almot the ame bit error rate a the exact computation. Figure 10 how BER curve for the ame experiment a before, but with caled rotation. That i, all cale factor compenation are omitted. Even the multiplication cell carry out approximate multiplication. It i important to notice, that even in thi coare approximation the BER converge to the exact curve. In practice, however, we have to decide if the pecific problem can cope with caled rotation (computational performance/eae of implementation veru BER). We now turn our attention to the reulting beampattern. To determine the underlying beampattern of the array we have read out the regiter cell of the array after certain imulation tep. From thi, a dicued in ection 2.2, we can calculate the inherent weight vector. The reulting beampattern are hown in Figure 11. The ignal and interferer direction are indicated by dahed line. The SNR at each antenna i et to 8 db. Although the curve eem quite different, they meet one expectation. Both, the exact calculated beampattern a well a the beampattern of the approximated olution fulfill the contraint amplification equal one perfectly. Furthermore, the uppreion of the known interferer i very accurate. The uppreion of the unknown interferer increae with the number of µ-rotation. To characterize our approach we invetigate the quantification of the tradeoff between output quality (BER) and computational effort. For thi, Figure 12 how the performance profile [22] of the array uing CORDIC-like approximate rotation with cale factor compenmvdr.tex; 7/09/2006; 12:36; p.19

20 exact 1 micro rotation 2 micro rotation 3 micro rotation 10 1 BER SNR [db] Figure 9. Bit error rate uing CORDIC-like approximate rotation with cale factor compenation. Every real CORDIC proceor cell execute the ame number of CORDIC iteration (µ-rotation). Additionally, BER in cae of uing exact rotation i given exact 1 micro rotation 3 micro rotation 5 micro rotation 10 1 BER SNR [db] Figure 10. Bit error rate uing CORDIC-like approximate rotation without cale factor compenation. Every real CORDIC proceor cell execute the ame number of CORDIC iteration (µ-rotation). Additionally, BER in cae of uing exact rotation i given. ation. Note, that four µ-rotation uffice to achieve the full poible performance. 6. Concluion In thi paper a ytolic proceor array for MVDR beamforming with multiple contraint ha been preented. The propoed implementation i entirely baed on complex linear and unitary rotation. We have demvdr.tex; 7/09/2006; 12:36; p.20

21 attenuation [db] exact 1 micro rotation 2 micro rotation angle ξ [degree] Figure 11. Steered repone extracted from the array (with cale factor compenation). Three ignal plu noie are preent. Two ignal are uppreed, the deired ignal power gain i 0 db. The SNR at each antenna i et to 8 db. The amplitude function i given by a(ξ) = 20 log 10 [e jφ 1(ξ),..., e jφ M(ξ) ]w BER micro rotation Figure 12. Bit error rate veru number of µ-rotation (SNR = 8 db). The dahed line indicate BER in cae of exact rotation. igned pecial CORDIC-baed module for thee complex tranformation. To reduce the number of operation, ome of the involved phae hift were accumulated into a diagonal matrix and compenated in a final phae compenation multiplication. Furthermore, we employed a non-optimized angle calculation cheme for the real CORDIC-like approximate rotation which i very cheap to compute. mvdr.tex; 7/09/2006; 12:36; p.21

22 mvdr.tex; 7/09/2006; 12:36; p.22