Integral formulation of the measured equation of invariance

Complexity evaluation of MOMA: We observe that the proposed structure consists of a CSA tree, which requires D = rlogg/(log,3- 1)1 node levels, and a final adder/converter block. The CSA tree has < rpi31D12 nodes (i.e. the area of the triangle containing the tree). Each node consists of two CSAs of n bits and of an 8x n multiplexer. The final MAC has three CSAs and four carry lookahead adders (CLA) of n+3 bits at most and an 8 x n multiplexer. The response time is + kSA + tCLA(n+3) f tMUX(8xn) (1) Recalling that t,,(,, = 2+‘flog4h] gate delays & for operands of h bits [6], TA is 13(logg+log~log~m) and appears to be more attractive than results in the literature [l - 31. Application to the generation of IAm: It is well known that, given a positive integer X = cX,2 5 2-1, x, E {O,l}, the value of Idrn can be obtained as T-1 This computation can be easily performed using a multioperand modular adder, whose inputs consist of terms x, +Irn. Note that each of these terms can be yielded by rlog,ml AND gates between x, and the bits of 12Im. Therefore iAm can be evaluated by means of a multioperand modular adder as in Fig. 2, with r inputs, each n bits wide. As far as response time T,,,, is concerned, it can be derived from e n. 1 replacing p by the length r of X and giving h the value rl og,m 4 +3 (Fig. 3), i.e. + ~CSA + (4bg,(bg, ml + 3)l + tn/lux(axn)) (2) To evaluate the performance of this conversion approach in a specific case, we choose the value r = 128 and observe from Fig. 1 that tcsa = 6Ag and t,,,(,,) = 2&; then eqn. 2 becomes log, 128 - 1 (2 x 6 + 2) Tconv = [ log,3 - 1 1 + 6 + 2 + 4[log4( [log, ml + 3)1 + 2 6 0.58 - x 14 + 10 + 4riog,(pog, + 311 2 155 + 4 [iog, ( [log, + 311 Considering the worst case m = 2Iz8-1, it results that T,,,, = 155+4x4 = 171 Ag. In terms of complexity, m = O(2) and therefore Tc0,, = B(log,r). 0 IEE 1996 Electronics Letters Online No: 19960026 G. Alia (Dipartimento di Ingegneria dell’lnformazione, Universitd di Pisa, Via Diotisalvi 2, 56126 Pisa, Italy) E. Martinelli (Dipartimento di Ingegneria Elettricu ed Elettronica, Universita di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy) 24 October 1995 References 1 SKAVANTZOS, L.: ‘Design of multi-operand carry-save adders for arithmetic modulo (2”+1)’, Electron. Lett., 1989, 25, (17), pp. 1152- 1153 2 KOC, c.K., and HUNG, c.Y.: ‘Multioperand modula addition using carry save adders’, Electron. Lett., 1990, 26, (6), pp. 361-363 3 PIESTRAK, s.J.: ‘Design of residue generators and multioperand modular adders using carry-save adders’, IEEE Trans., 1994, C- 423, (l), pp. 69-77 applications to computer technology’ (McGraw-Hill, New York, 1967) 5 ALIA, G., and MARTINELLI, E.: ‘A VLSI structure for X (mod n) operations’, VLSI Signal Process., 1990, 1, pp. 257-264 6 SCOTT, N.R.: ‘Computer number systems and arithmetic’ (Prentice- Hall International Editions, Englewood Cliffs, NJ, 1985) 4 SZABO, N.S., and TANAKA, RJ.: ’Residue arithmetic and its ELECTRONICS LETTERS 4th January 1996 Vol. 32 Integral formulation of the measured equation of invariance J.M. Rius Indexing terms: Electromagnetic wave scattering, Finite d$ference methods, Integral equations A novel integral formulation of the measured equation of invariance is derived from the reciprocity theorem. This formulation leads to a sparse matrix equation for the induced surface current, resulting in great CPU the and memory savings over the conventional approaches. The algorithm has been implemented for two-dimensional perfectly conducting scatterers. Introduction: Recently, the measured equation of invariance (MEI) method has been introduced by Mei et al. [I] as a mesh truncation condition for the finite difference method. A later paper [2] analyses some theoretical and numerical aspects of the MEI. In this Letter we present the new concept of the ME1 as a surface integral equation method derived from the reciprocity theorem, which produces a sparse matrix with the same number of unknowns as the conventional boundary element - method of moments (BE-MOM) technique. This new formulation has an important advantage over the original fmite difference formulation: it requires a boundary discretisation instead of a volume discretisation, while maintaining the sparsity of the matrix. Derivation from the reciprocity theorem: Let C be the boundary of a perfectly conducting (PEC) scatterer and (7, a portion of this boundary. Introduce two sets of electric and magnetic sources as follows: 1. The induced currents on the surface of the I’EC scatterer J’, = iz x $Ic and 5?& = -iz x plc = 0, where i? and F? are the electric and magnetic fields actually existing in the presence of the scatterer. According to the equivalence theorem [4l], this set of sources radiates the scattered fields in I?s = I? - 31, $. = F? - free space where and F?l are the fields in the absence of the scatterer, or the incident fields. 2. A set (x, 2,) that exists only over C, arid radiate fields p,,, pn. With these sets of sources, the reciprocity theorem [4] for a linear and isotropic medium is where I?; and *, the components of the scattered fields tangential to the scatterer boundary, account for the fact that and 2, represent surface currents. eqn. 1 is the integral equation formulation of the ME1 @E-MEI). Fig. 1 Boundary meshing of scatterer for IE-MEI For simplicity, only the discretisation for 2-D problems will be presented. The boundary C and its portion C, are discretised into Nand M segments of equal length h, respectively. Let C, be centred at node n of C Fig. 1. The currents J’, and A?, are expanded along C,, into M pulse basis functions. If we define = hJ’, (ln e,) and +n nem = h&?n (I, No. 1 23 we obtain m= M1 xz [Zn,mm . z%e3m) - knem . m14 m=- M--l =pn.xdl (2) which is the discrete version of the IE-MEI. The addition of subscript indices is modulo N, so n 0 m assumes values from 0 to N - 1. The right-hand term in eqn. 2 is the residual R,, = I, i?n . s’, dl . This residual has previously been obtained in [2] for the fite difference ME1 using a different procedure. The Pn field corresponds to the ‘null field’ gnu,,, of [2]. In our experience, the residual R, can be made arbitrarily small for convex scatterer shapes with appropriate 7?n,nem and d n,nem coefficients. Assuming that R,, is zero, in two dimensions we have m= M--l 2 [an,nemEtS(lnem) - bn,nfBmG(~n@rn)l = 0 (3) ,=-M--l where for TM p2larisation gn,nem = - 7?,,nQm 2, 2; = -E:2, bnknOm = . 1, F?; = Wl, for TE olarisation a,,,,, = 7?n,,em ’ 1, 2; = E;l, b,,,mm = $Sn,nem . &, 2s = H,s& and 1 is the unit vector tangential to the scatterer boundary C. and b,,,em we use the same procedure as in the fmite difference formulation of the ME1 111: if in eqn. 2 the function is orthogonal to the induced current p3, the residual is zero for all possible incident fields (the postulate of invariance of the ME1 [l]). This fact suggests that if we know P 2 2M - 1 linearly independent scattered field solutions (called ‘measuring functions’) owing to P different incident fields, forcing eqn. 3 for each mesh node n we can obtain a set of an,nem and b,,,, coefficients. This set of coefficients will also satisfy eqn. 3 with the actual scattered field. The measuring functions are obtained as the field radiated by a set of functions op (37 defined on the scatterer boundary C, called ‘metrons’. Eqn. 3, which previously appeared in a conference presentation [3], is a linear relation between the tangential components of the electric and magnetic fields at M consecutive nodes in the portion C, of the scatterer boundary centred at node n, where n = O... N - 1. This relation can be expressed more concisely as where an,nam are the elements of global matrix [A], bn,nem are the elements of global matrix [B] and E; (Inem), H; (Znem) are the components of vectors [E;] and [H;], respectively. The matrices [A] and [B] are cyclic band diagonal with bandwidth M. Since the tangential component of the electric scattered field is known along the object boundary, [E;] = - [E,‘], we can obtain the electric induced current as [Js] 1 [fC] + [fft“] = [H,] - [B]-l[AIIEjl To compute the coefficients [AI[E,SI - [Bl[H,Sl = 0 (4) (5) Numerical considerations: The following set of metrons has been used to obtain the results presented in this Letter: P-1 2 0 5 I’ 5 L (6) IPI 5 - (l’) = e-j2TPl‘/L P = 2kRm,,~ + 1 These are harmonic functions along the scatterer boundary C to the order P = kR,,x, where &,, is the radius of the smallest cylinder enclosing the scatterer and x is a coefficient slightly larger than one. Typical values of x range from 1.05 to 1.30. In our experience, this set of metrons guarantees a small residual for twodimensional problems when the PEC scatterer is a convex cylinder. With regard to the number of nodes M in the linear relation eqn. 3, the best results are obtained with M 2 5, although the best trade-off between computation time and accuracy is achieved with M= 3. The memory requirement of our implementation of the IE-ME1 in MATLAB 4.2 language is -9.5 N complex numbers for a linear relation involving M = 3 boundary nodes. This means that for each megabyte of available memory the IE-ME1 method can allocate 6900 unknowns. For each unknown at the mesh truncation boundary, the number of operations is in principle proportional to P times the number of integration points at the boundary, Q. In our case, Q = N. However, it is noted that the field, owing to the metrons, as in eqn. 6 is a Fourier transform. For example, the electric field in the TM case is and similar relations exists for the TM magnetic field and for the TE case. Therefore, if the boundary samples are equally spaced, the FFT algorithm can be used and the total number of operations is proportional to NQ log, Q. In that case, it grows with frequency as fZ log,$ 10’ 10‘ 10’ a unknowns ,N b unknowns, N 101 io2 103 164112/ Fig. 2 Computer memory a With CPU time b Required by IE-ME1 compared to BEM-MOM Fig. 2 shows the computer memory and CPU time required by the IE-ME1 method compared with method of moments (MOM). For an honest comparison, the same discretisation and computer code has been used in both methods for computing the radiation operators. TM I I b=205h 0 20 40 60 80 \R aspect angle, 0 161113/ Fig. 3 Monostatic RCS of electrically large ogive IE-ME1 results are compared to GO + GTD high-frequency methods 0 GO+GTD MEIM=3 Results: Fig. 3 shows the monostatic RCS of an ogival cylinder of dimensions 768 h x 205 h and perimeter 1620 h. This result was computed in a 486 DX-66 PC using only 2.3 MB of memory. The results of the IE-ME1 are compared to high frequency approximations: geometrical optics (GO) and geometrical theory of diffraction (GTD). The agreement is excellent. Acknowledgments: The author is indebted to Prof. K. Michalski of Texas A&M University for the valuable discussions on the ME1 method. This work has been supported by the Spanish ‘Comisi6n Interministerial de Ciencia y Tecnologia’ (CICYT) under project TIC 93-0518. 0 IEE 1996 Electronics Letters Online No. 19960028 J.M. Rius (Department of DSC, Universitat Politecnica de Catalunya, Gran Capitun s/n, edifici 0-3, 08034 Barcelona, Spain) 25 August 1995 References 1 MEI, K.K., POUS, R., CHEN, z., LIU, Y.w., and PROUTY, M.D: ‘The measured equation of invariance: A new concept in field computations’, IEEE Trans. Antennas Propag., 1994, 42, (3) JETVIC, J.o., and LEE, R.: ‘A theoretical and numerical analysis of the measured equation of invariance’, ZEEE Trans. Antennas Propag., 1994, 42, (81, pp. 1097-1105 2 24 ELECTRONICS LETTERS 4th January 1996 Vol. 32 No. I RIUS, J.M., PARRON, J., POUS, R., and CADAMA, A.: 'The measured equation of invariance: A proof for the postulates and a new formulation as a sparse matrix integral equation method'. 1994 URSI Radio Science Meeting, Seattle, Washington, 19-24 June 1994 HARRINGTON, R.F.: 'Time-harmonic electromagnetic fields' (McGraw-Hill, 196 1) lyll Y2 JYq Y4 (i) Lowpassfilter: A lowpass filter is obtained by substituting the admittances specified in the 1st row of Table 1 in eqn. 1 Y5 Assuming C1 = C2 = 1F and comparing with the standard form of the second-order filter transfer function, lealds to the following simple design equations Highpass Bandpass Current mode filter structure based on dual output transconductance ampifiers B. Al-Hashimi v gl sC1 g2 sC2 g3 gl g2 sC1 sC2 g3 Indexing terms: Current-mode circuits, Active filters A new current-mode fiter structure based on the dual output operational transconductance amplifier is described. By appropriate choice of components, the structure provides lowpass, highpass and bandpass responses. Experimental results which confirm the theoretical analysis are given. Introduction: In recent years there has been considerable interest in two particular developments in active filter design. First, the use of current mode devices in filter configurations, examples include current conveyors [ 1,2], operational transconductance amplifiers [3,4] and current feedback amplifier [5,6] based filters. Secondly, the realisation of current rather than voltage-based network functions [7]. These developments bring the benefits of simple, low sensitivity fdter configurations which may be implemented at high frequencies. In the case of operational transconductance amplifier (OTA) based filters, two types of amplifier have been used: single output OTAs and dual output OTAs. It has been shown that dual output OTA based filters perform better than the single output OTAs [8 ~ lo], in particular in higher frequency of operation. Also, dual output operational transconductance amplifiers (DO-OTAs) usually simplify the development of current transfer functions owing to the multiple current outputs. In this Letter, a new current-mode filter structure based on the DO-OTA is described. Fig. 1 General filter structure ELECTRONICS LETTERS 4th January 1996 Vol. 32 (ii) Highpass filter: Substituting the admittances specified in the 2nd row of Table 1 into eqn. 1, and letting C1 = C2 = 1 F and gl = g2 = g3 = g, a highpass filter is obtained as shown 5- Gs2 I,, - - (39 + G)s2 + (4g2)s + g3 This filter has the following design equations: (iii) Bandpass filter: Substituting the admittancm specified in the last row of Table 1 into eqn. 1 and using C1 = C2 = 1 F and gl = g2 = g3 = g, a bandpass filter is obtained as shown Io - Gs I,, - - 2s2 + (59 + G)s + g2 This fdter has the following design equations: Experimental results: At the simplest level, the transconductance amplifier of Fig. 1 can be replaced by a single bipolar transistor, since the various filters (lowpass, highpass and bandpass) are obtained using only one of the differential inputs of the transconductance amplifier. The required G of the amlplifier is set up by the resistor RE, where G = l/(RE+re). Using this simplification, and setting up the required currents, the lowpass DO-OTA based filter is shown in Fig. 2. The transistors TR2 and TR3 form a current source to supply the required feedback and output currents, while TR4 and TR5 mirror the current in TR1 to provide an output. To simplify the testing of the filter, it is more convenient to work with voltages than currents and for this reason, transistor TR6 is used. The gain of this circuit is RJR,. IR4@----6 IR5 IO TRI RE V" IR3 o)-q IR2 hvo -vcc Fig. 2 Transistor implementation of lowpass DO-OTA basedfilter The circuit of Fig. 2 was constructed using TPQ6502 npnipnp transistor arrays, passive components with 1 YO tolerances and k12V supplies, to realise a second order lowpass with Q = 10 and Fo = 30kHz. Fig. 3 shows the practical results obtained, in which the measured Q and F, were found to be in error by only 1 and 0.5%, respectively. To investigate the high frequency performance of the proposed structure, a Butterworth lowpass fdter with a cutoff frequency of 1 MHz was implemented. The practical frequency response of the fdter is shown in Fig. 4, where the measured cutoff frequency was 0.997MHz. Finally, highpass and bandpass DOOTA based filters were also built and found lo perform well in practice. No. 1 25