Deviation is defined as the ratio of the nonlinearity producing term to the desired term. Hence computing the minimum deviation with respect to bias current yields ev; [.[OZV:. 112 + 2@ + OZyW2)b”+ (1 + 2@@”2 + 02y2@),] (4) Computing an average for I,,, over the entire input range yields a value of 16.1 pA for e = 560 pV and the other parameters as previously defined. This value is again close to the experimentally determined value of 17 pA. - Conclusion: A simple linear CMOS circuit to compute a ‘oneover’ function is analysed and fabricated. This circuit, which is implemented in a 2 pm n-well process, capitalises on the linear region of the MOS transistor. The main contribution to nonlinearity is found to be produced by mobility degradation and not by higher order terms of V,. Hence, the nonlinearity appears as a constant which is easily cancelled. This circuit does not require high degrees of component matching because any offsets can easily be cancelled. The circuit operates with 8 bit accuracy and is mostly limited by pick-up noise. It has been used in a tracking system where the target velocity is inversely proportional to an integrated pulse width. 0IEE 1993 12th July 1993 R. Etienne-Cummings, R. Hathaway and J. Van der Spiegel (Department of Electrical Engineering, Uniuersity of Pennsylvania, MO S. 33rd St., Philadelphia, PA 19104, USA) References GREGONAN, R., and TEMES, T. c.: ‘Analog MOS integrated circuits for signal processing’(Wiley and Sons, New York, 1986) M.: ‘Four-transistor continuous-time MOS transconductor’, Electron. Lett., 1987,23, pp. 1099-1100 3 KHACHAB, N. I., and ISMAIL, M.: ‘MOS multiplier/divider cell for analogue VLSI’, Electron. Lett., 1989,25,pp. 1550-1552 4 m ~ z o uc., , LIDGEY, F. I., and HAIGH, D. G . (Eds.): ‘Analogue IC design: the current-mode approach (Peter Pereginus Ltd., London, 1990) 1 2 ISMAIL, reader is referred to this article to see the flexibility of these new algorithms. We expect that in one form or another these algorithms will become the standard technique for decoding RS codes. The solution technique presented here (and henceforth called ‘the algorithm’) is similar to that in Reference 1 and ‘algorithm 11’, in Reference 2, but it is not identical to these. It is straightforward to implement, and the mathematical justification presented here is the most direct known to the authors. The algorithm finds a pair of polynomials P(x), Q(x) over some field F, satisfying the key equation yi Q(xJ = P(xi) for i = 1, ..., m with where m is a given positive integer, where the xi are given distinct elements in F, and where the yi are given elements in F. (Here [ k ] denotes the integer part of k.) Actually there is a slightly tighter restriction on the degrees of the polynomials, discussed in the final Section. Algorithm: The algorithm generates parallel sequences of polynomials P i x ) , Qxx), K(x), Fi’(x) so that P,(x) and Q.(x) solve eqn. 1. In the following, the notation ( P , Q), denotes a two-element vector with components P,(x), Q i x ) . Moreover J and i are integers, and d and c are elements of the underlying field F. A C-like syntax is used. J = 0; Po(x) = 0 ; Qo(x)= I ;Vo(x)= 1;W,(x)= 1 ;/*initialise */ for(i = 1; i s m ; i++){/*loop on i from 1 t o m * / if((d = y i Q i - l ( x i )- P i - , ( x i ) ) = =O)/*discrepancy at x i * / { ( V , W)i = ( X - x X V , mi- 1; ( P , Q)i = ( P , Q ) i - 1; J=J+I;} else {/* if d # 0 */ ( P , Q)i = ( x - xiXP, Q ) i c=CviW-i(xJ- K - i ( x i ) ) / d ; ( V ,w ) i = ( V , mi-1 - c(P, Q ) i - 1 ; else J = J - 1;} /* if (J= =0) swap ( V , W), and ( P , end d # 0 * / )/*end of loop on i */ ,; Expressions for the maximal degrees of the polynomials involved are given in the final Section; these may be useful in the implementation of this algorithm. ALGORITHM FOR SOLVING THE WELCH-BERLEKAMP KEY-EQUATION, WITH A SIMPLIFIED PROOF W. G. Chambers, R. E. Peile, K. Y.Tsie and N. Zein Indexinn terms: Decodina, Reed-Solomon codes An alternative technique due to Welch and Berlekamp for decoding Reed-Solomon codes has a key equation different in form from the key equation solved by the conventional BerlekampMassey algorithm or by the so-called Euclidean algorithm. The Letter presents an algorithm for solving the key equation which has a simple structure and which is readily shown to work. Introduction: The Welch-Berlekamp algorithm [I, 21 is an improved technique for decoding Reed-Solomon (RS) and related codes. It requires fewer computations than the classical BerlekampMassey algorithm, especially when the average error rate is kept well within the capabilities of the code; it employs a key equation of a significantly different form. The derivation of the key equation has recently been discussed in Reference 3 but not its algorithmic solution and proof. We fill these gaps in this Letter. Recently another algorithm has been published which appears to be equivalent to the WelchBerlekamp algorithm in how it can be used [4], and the 1620 Justification: We start with some definitions: For r 2 0 let R, den’ote a set of points with integer Cartesian co-ordinates (a, b) satisfying - 1 5 a s r l / [ r / 2 ] , - 1 I b $ [ r / 2 ] ; moreover (-1, -1) is excluded. We also define the ‘line sets’ L, as R, - R,- 1, that is the set of all points in R, not in R,- 1. We take Lo as R,. The sets Lo-L, are illustrated in Fig. 1, and their union is R, . The polynomial pair (P(x), Q(x)) is said to be at the point (deg P(x), deg Q(x)), where we take deg 0 as -1, and is said to lie in any set containing this point. It is called an r solution if it satisfies yiQ(xi) = P(xJ for i = 1 to r without both P(x) and Q(x) being identically zero. (For r = 0 any polynomial pair is an r solution.) It is called a minimal r solution either if P and Q are relatively prime (i.e. they have no polynomial common factors) or if removing any polynomial common factor from P and Q stops its being an r solution. A factor that is simply a field element is called a scalar factor. Next we need a uniqueness theorem: - Theorem 1 : (i) there is at most one minimal r solution in R, (ii) any other r solution in R, is a (polynomial) multiple of it Proof of theorem I : Suppose that there are two minimal r solutions ( P . Q ) and ( X , Y ) in R, which are not simply scalar multiples of each other; then the polynomial P Y - Q X has r roots, vanishing at x I , .. ., x,; however, its degree must be less than r, and so it vanishes. Therefore P Y equals Q X . Next let ELECTRONICS LETTERS 2nd September 1993 Vol. 29 No. 18 ~~ - A ( x ) be the HCF (highest common factor) of P and Q so that P = A P and Q = A Q with P and Q‘ relatively prime. Substituting into P Y = Q X and cancelling A we obtain P Y = Q X so that Q is a factor of P’Y and hence of Y ; it follows that Y = B Q , X = B P for some polynomial B(x). Let C(x) be the HCF of A and E ; then by the Euclid algorithm we can find polynomials F(x) and G(x) so that F A GB = C. It is then readily seen that (CP’, C Q ) = F . ( P , Q ) G . ( X , Y) is an r solution. Assume without loss of generality that deg A t deg E ; then C is a factor of A with deg C < deg A so that ( P , Q ) is not minimal, and we therefore have a contradiction. Thus apart from scalar multiples there is at most one minimal r solution in R,. This proves the first part of the theorem. The second part is a direct consequence: A second r solution in R, cannot be minimal and its reduction must give the unique minimal r solution in R,. We are now ready for the main argument, which proceeds by induction. The inductive hypothesis is as follows: For some value r 1 of i at the top of the loop on i we assume: + + + (i) ( P , Q), lies in L,(ii) ( P , Q), is an r solution (iii) (P, Q). is a minimal r solution (iv) ( V , W, lies in L+J+ verifying assumption (iii). The argument is by contradiction. If (P,Q),. = ( P . Q ) , + l is not minimal then by removing a poly- + nomial common factor we obtain an (r 1) solution ( X , Y ) lying in R,. Thus we have two r solutions in R,, that is (X,Y ) and (P, Q),. Thus by theorem 1 one must be a (polynomial) multiple of the other and so (P, and ( P , Q), are at points in Fig. 1 joined by a line of slope 1. However, the swap puts (P, on a line-set L,+ while ( P . Q), lies on L,, and as Fig. 1 shows there is no line of slope 1 joining a point in L, to one in L,,,. e),+ e),+ Final remarks: As far as the maxima of the degrees of the polynomials are concerned we note that because ( P , Q), lies in L,-J it also lies in R,-, so that deg P, < r - J - 1 - [(r - 4/21 and deg Q, s [(r - J)/2]. Similarly we find deg V, < r + J - [(r + J + 1)/2] and deg W, < [(r J 1)/2]. Moreover (P, Q), lies in R,_J which is a subset of R, so that deg P , Im - 1 - [m/2], deg Q , < [m/2], as claimed in eqn. 2. In decoding applications we know by other means [3] that provided the number of errors t satisfies 2t 5 m there is a minimal solution to eqn. 1 satisfying deg P < deg Q It. This is a minimal m solution lying in R, and so by theorem 1 it is the same as the solution found by the algorithm just described, apart from a scalar multiple. + + 0IEE 1993 (v) (V, W), is an r solution (not necessarily minimal). Evidently these assumptions are trivially obeyed for i = 1 when J = 0 and r = 0).At the bottom of the loop new polynomial pairs ( P , Q),.and (V, with r’ = r 1 and a new value J’ of J have been found. We must verify that the above assumptions are still true when r’ and J’ have replaced r and J. There are three mutually exclusive possibilities: wr. + + (1) The first is when d = 0; then J’ = J 1 and assumption (i) above is readily verified because ( P , Q),. = ( P , Q), and r ‘ - J ’ = r - J . Verifying the other assumptions is also straightforward. (2) The second case is when d # 0 with J > 0, so that J‘ = J - 1. The main difficulty is in checking assumption (iv). Updating (V,W) gives ( V , W),, = ( V , W). - c(P, where ( P , Q ) , lies in L. with U = r - J , and where (V, W), lies in L, with s = r + J + 1. Now s is odd if U is even and vice versa, and also s - U is not less than 3. (Typical examples of L, and L, are L, and L, in Fig. 1.) The outcome is decided by the degrees of the polynomials to be added. The degree of the sum of two polynomials with unequal degrees is the greater of these degrees, but if these degrees are equal then the degree of the sum can be less as leading terms may cancel. (3) The third case is when d # 0 with J = 0; then J’ equals 0 also and the ‘swap’ takes place. The main difficulty here is tb 28th July I993 of Electronic and Electrical Engineering, King’s College London, Strand, London WC2R ZLS, United Kingdom) R. E. Pede (Racal Research Ltd., Worton Drive, Worton Grange Industrial Estate, Reading, Berks RGZ OSB, United Kingdom) W.G. Chambers, K. Y. Tsie and N. Zein (Department References L., and BERLEKAMP, E. R.: ‘Error correction for algebraic block codes’. US Patent 4 633 470, September 1983 2 LIU, T. H.:‘A new decoding algorithm for Reed-Solomon codes’. PhD Thesis, University of Southern California, Los Angeles, CA, 1 WELCH, 1984 3 MORI], M., and UAHARA, M. : ‘Generalized key-equation of remainder decoding algorithm for Reed-Solomon codes’, IEEE Trans., 1992, IT-% pp. 1801-1807 4 SORGER, U. K.: ‘A new Reed-Solomon code decoding algorithm based on Newton’s interpolation’, IEEE Trans., 1993, IT-39, pp. 358-365 EFFECT OF ION IMPLANTED GERMANIUM PROFILE ON THE CHARACTERISTICS OF Si,-,Ge,/Si HETEROJUNCTION BIPOLAR TRANSISTORS K. Grahn, Z. Xia, P. Kuivalainen, M. Karlsteen and M. Willander Indexing terms: Transistors, Bipolar devices, Semiconductor devices and materials L1 L3 L5 An optimum profile for Ge ion implantation in SiGe/Si heterojunction bipolar transistors is determined by using a twodimensional simulator code for advanced semiconductor devices. The simulation code is based on a two-dimensional drift-diffusion model for heterostructure degenerate semiconductors with nonparabolicity included in the energy hand structure. The model allows accurate simulations of carrier transport in short base devices. The simulation results indicate that for high current gain the Ge profile maximum must be close to the base-collector junction, and that the unavoidable tail of the implanted germanium in the collector region does not deteriorate the gain. L7 F33/11 Fig. 1 Line sets L, of points in plane with integer Cartesian coordinates These co-ordinates are the degrees of polynomials in polynomial pairs; the set R, is readily visualised as the union of Lo, . .., L, Zntroduction: Pseudomorphic Si, - .GeJSi heterojunction bipolar transistors (HBTs) are of increasing interest due to their excellent high frequency and high current capabilities. ELECTRONICS LETTERS 2nd September 1993 Vol. 29 No. 18 1621

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