The measurement of the pulse duration is possibly affected by the propagation effect along the amplifier, therefore it becomes necessary to measure the chromatic dispersion of the amplifying fibre to determine, through computer simulation, the pulse shape and duration at the output of the ring. The mechanism that provides the build up of the bunch of ultrashort pulses is not clear yet; nevertheless it may concern the nonlinear polarisation evolution inside the cavity. Simultaneously to the pulse autocorrelation, a number of measurements were made to verify that the output state of polarisation effectively depends on the pump power. This observation suggests that the dynamical evolution of polarisation inside the ring depends on the power level of the circulating light. If the cavity presents a small anisotropic loss, and this is possible, the mechanism of selfadjustment of the light polarisation state induces a reduction of the laser threshold and pulse reshaping occurs [3, 41;then, in combination with the nonlinear propagation in the anomalous dispersion region of the fibre, the generated pulse may really shorten to hundreds of femtoseconds because of the soliton compression effect [7]. Conclusion: Passive and active modelocking in an erbium doped fibre single ring laser is reported. Whenever active modulation is used, the picosecond regime switches to the femtosecond regime by simply adjusting the polarisation state of the light circulating in the ring. This condition is independent of the modulation frequency as long as this is tuned to a harmonic of the cavity fundamental frequency. Bordoni and the Italian PT administration. N. Manfredini wishes to thank Soc. Cavi Pirelli for the study grant. 5th May 1992 F. Fontana and G. Grasso (Pirelli C a d s.p.a., Viale Sarca 202, 20146 Milano, Italy) N. Manfredini (Dip. Elettronica, Uniuersitd di Pavia, Via Abbiategrasso 209,27100 Pavia, Italy) M. Romagnoli and B. Daino (Fondazione Ugo Bordoni, Via B. Castiglione 59,00142, Rome, Italy) References 'Subpicosecond all-fibre erbium laser', Electron. DULING, I. N. 111: Lett., 1991.27, pp. 544-545 RICHARDSON, D. I., LAMING, R. I., PAYNE, D. N., PHILLIPS, M. w . , and MATSAS, v. I.: '3201s generation with passively mode-locked erbium fibre laser', Electron. Lett., 1991, 27, pp. 730-732 and MENYUK, c . R.: 'Soliton fiber ring laser', Opt. Lett., 1992.17, pp. 417-419 HOFER, M., FERMAN, M. E., HABERL, F., OBER, M. H., and SCHMIDT, A. 1.: 'Mode-locking with cross-phase and self-phase modulation', Opt. Lett., 1991,16, pp. 502-504 DAVEY, R. P., SMITH, K., and MCGUIRB A.: 'High speed, mode-locked, tunable, integrated erbium fibre laser', Electron. Lett., 1992, 28, pp. CHEN, c . J., WAI, P. K. A., 482-483 and PAY% D. N.: 'Energy quantisation in figure eight fibre laser', Electron. Lett., 1992, 28, pp. GRUDININ, A. B., RICHARDSON,D. I., 61-68 and WOOD, 0.: 'Trapping ofenergy into solitary waves in amplified nonlinear dispersive systems', Opt. Lett., 1987,12, pp. 1011-1013 BLOW, K. I., GORAN, N. I., Acknowledgment: This work has been carried out in the framework of the agreement between Fondazione Ugo GENERAL APPROACH TO EFFICIENT IMPLEMENTATION OF 2-D DENOMINATOR-SEPARABLE DIGITAL TRANSFER FUNCTIONS X.Nie, D. Raghuramireddy and R. Unbehauen Indexing terms: Digitalfilters, Filters A functional theoretical framework is reported for obtaining efficient 2-D digital filter realisations. A given 2-D transfer function having a separable denominator polynomial is developed into certain expansions which may be realised as two-stage structures. In such structures, only suitable 1-D substructures need to be searched for to obtain a complete realisation by connecting 1-D substructures using the corresponding weighting matrix. (HXz), H j z ) ) = 1/2Rj $I,I= H,(z)F:(l/z) dz/z where the coeffcients of Hf are the complex conjugates of those of H j x ) . The set of functions 1 {l,-,- 2 - a l 1 z - a z ,..,, z - a n builds a basis of the Hilbert space 2". By representing the basis functions as a vector g(z), H(z) can be expressed as H ( 4 = Cro,P1,....p.Mz) (2) If another basis of the Hilbert space is given as Yz),there is a nonsingular constant matrix T such that &) = TYZ) (3) The corresponding expansion of H ( z ) is Introduction: In 2-D signal processing, it is known that 2-D recursive digital filters having a denominator-separable transfer function (DSFT) possess a number of favourable properties with respect to the design and implementations [l-31. Circularly symmetric and quadrantal symmetric fan filters which are widely used in practice have denominator-separable transfer functions [SI. The purpose of this Letter is to explain a functional theoretical approach for this class of transfer function and point out a very general approach for generating a class of eficient realisations for DSTFs. First, we consider the transfer function of a 1-D stable digital filter H(z) having no poles in the region I z I 2 1. If the poles of H(z) are assumed to be simple for brevity and denoted by ai ( i = 1, 2, . .., n), we have H(z) = p o + -..k i=l (4) Let Yz) be an orthonormal basis of 2"as given in Reference 4; we then have the Parseval equality (5) where llH(z)1Iz denotes the 1, denotes the /,-norm of H(z), i.e. J(H(z), H W . 2-0 denominator-separable tranger functions : A denominatorseparable, stable, quarter-plane 2-D digital filter is described by its transfer function [l, 51 z - ai where n is the degree of the transfer function and p i ( i = 0, 1 , . .., n) are constants in the partial fraction expansion of H(z). NOW,we denote the set of all transfer functions whose poles are fixed at ai (i = 1, 2 , . .., n ) by R. It can be shown that the set R represents a Hilbert space under the inner product ELECTRONICS LETTERS 2nd July 1992 Vol. 28 No. 14 The order of the filter is (m, n) if D,(zl) and D,(z,) are polynomials of degree m and n, respectively. We assume that H(zl, z2) is analytic in J z l I z 1 and Jz21> 1 and continuous in 1293 Izl 12 1 and 1z212 1. Let the zeros of Dl(zl) and D2(z2) be simple and denoted by ai(i = 1, 2, ..., m) and /?,U = 1, 2,. .., n), respectively. With respect to each complex variable zI and z2, we can easily construct a set of basis functions. The set of basis functions formed by the variable z1 and poles ai is denoted by G I . Similarly, 9, is formed by the basis functions with the variable zz and the poles p p It can be shown that 6 , €3 6 , is exactly a basis for all possible functions H(zl, zz) with the same denominator, where 6 , Q !PZis defined as the tensor product of the two sets [6] 0 , and eZ.Given el as gl(z,) and GZas dz(z2),H(zl, zz) can be expressed by W I , (7) zz) = d1(z,)Cg,(z,) + + where C is an ( m 1) x (n 1) matrix of constants and the superscript ‘P indicates transpose. Eqn. 7 forms a general frame for various realisations of 2-D DSTFs with different 1-D basic building blocks. We next give a general realisation of the 2-D DSTF, and mention some important realisations in particular. Implication t o implementation: The general expression of eqn. 7 can be implemented as a two-stage realisation as follows. A + one-input-@ 1)-output network is synthesised which realises the vector gl(zl) as the fmt stage. In the second stage, g2(zz) should be realised as an (n 1)-input-one-output network. The complete realisation of H(zl, zz) will be completed by connecting the two stages through a weighting matrix C, as shown in Fig. 1. + input o--- matrix compared to the first-order 1-D structure used in the partial fraction expansion. It is predicted that it is equally true for the 2-D DSTF realisation based on eqn. 9. If other finite wordlength effects need to be accounted for, more general twostage structures have to he considered. Another interesting expansion is the orthonormal expansion [9]. In this expansion the vectors g,(z,) and g2(zz)are set to G 1673/11 Fs.1 General two-stage structure for realisationof 2-D DSTF The two-stage structure has first been explicitly introduced in Reference 5 for the partial fraction expansion of H(zl, z2) where 1 1 ,..., zl -a, z1 - a1 z1 - az and Because the basis functions form an orthonormal set [9], we have the Parseval equality where gij is the (i, j)th component of matrix C of eqn. 7. This structure has been shown to exhibit very low roundoff noise [lo]. It has been shown [lo, 111 that the digital lattice structure proposed by Gray and Markel [lZ 131 can easily be applied to realise another set of orthonormal basis functions and we will obtain a two-stage 2-D lattice structure. It is shown [lo, 111 that a two-stage 2-D lattice structure has slightly larger roundoff noise than the two-stage orthonormal structure based on eqn. 10. Furthermore, a two-stage structure can also be realised using wave digital filers as substructures for the realisation of gl(z,) and g&). In this way, we can obtain a class of efficient implementations for the 2-D DSTF in the general frame of the two-stage structure which can possess low roundoff nose and/or low sensitivity and be free from overflow oscillations. Finally, it may be mentioned that a complete comparison taking all aspacts of the implementation of the two-stage realisation of a 2-D DSTF using different substructures is yet to be made. At present, work on roundoff noise and sensitivity considerations is in progress. 9th April 1992 and X. Nie, D. Raghuramiddy and R. Unbehauen (Lehrstuhlfiir Allgemine und Theoretische Elektrotechnik, Cauerstrasse 7, D-8520 Erlangen, Germany) In References 5 and 7, the aspect of implementation expense has been discussed successfully. Another expansion of H(zl, az), which fits into the two-stage realisation, uses allpass firstorder sections [8]. The corresponding vectors gl(zl) andgz(zz) are set to 1 - afz, 1 - a:zl +-a1 zl-az -1 ,..., ~ ~ - a , , , and It may be noted that both realisations based on eqns. 8 and 9 yield parallel form structures for the vectors I & , ) and g2(zz) with first-order building blocks. It is established. that the first-order I-D allpass structure possesses a low sensitivity xre, x., RAGHUMMUEDDY, D., and UNBE~~AIJEN,R.: ‘A general approach for the realization of any IIR digital filters using first-order all-pas sections’, submitted to IEEE Trans on Circuits and Systems 1294 Refer1 TZAFFSTAS, s G. (Ed.): ‘Multidimensionalsystems, techniques and applications’(Marcel Dekker,Inc., New York, 1986) KAWAMATA, M., and HIGUCHI, T.: ‘Synthesis of 2-D separable denominator digital filters with minimum round-offn o k and no overflow oscillations’, IEEE Trans., April 1986, CAS-33,pp. 365372 3 KARIVARATHARAJAN, e., and SWAMY, M. N. s.: ‘Quadrantal symmetric associated with two-dimensional digital transfer functions’, IEEE Trans.,June 1978, CAS-25,pp. 340-343 4 WALSW, I. L: ‘Interpolation and approximation by rational functions in complex domain’ (American Mathematical Society Colloq. Publications, Vol. XX, 1960) 5 DABBAGH, M. Y., and ALEXANDER, w. E.: ’Multiprocessorimplementation of 2-D denominator separable digital filters for real-time processing’, IEEE Trans., June 1989,ASP-37, pp. 872-881 6 RICE, Y. R.: The approximation of functions II’ (Addison-Wiley Publishing Company, Reading, 1969) 7 RAGHUMMUEDDY, o., and UNBMAUW, R.: ‘Highly modular systolic structures for denominator-separable 2-D recursive filters’, IEEE Trans., Dcamber 1991, ASSP-39, pp. 2725-2728 8 RAGHURAMIREDDY, D., NI+ x., and UNBEHAUM, R.: ‘Minimaland low-msitivity implementations for a dass of two-dimensional digital filters’. To be presented at IEEE ISCAS-92, San Diego, USA 2 ELECTRONICS LETTERS 2nd July 1992 Vol. 28 No. 14 .,,, 9 ME, x., RAGHURAMIRBDOY, D., and UNBMAUEN, R.: 'Orthonormal minimal structure for n-dimensional denominator-separable recursive filters for multiprocessor implementation', Electron. Lett., November 1991,27, pp. 2206-2207 10 N E , x. : 'Beitrage zur Synthese zweidimensionaler Digitalfilter'. Doctorate Thesis to be submitted to Technische Fakultat der Universitat Erlangen-Niirnberg 11 BRANDENS", R.: 'Untersuchung einer Klasse von Strukturen zur Realisierung eines 2-D Digitalfilters'. Diplom-Engg.Thesis, Lehrstuhl fur Allgemeine und Theoretische Elektrotechnik Universitat Erlangen-Number& 1992 12 GRAY, A. H., JUN., and MARKEL,I. 0.:'Digital lattice and ladder filter synthesis', IEEE Trans. Audio Electroacoust., December 1973, AU-21, pp. 491-500 13 GRAY, A. H., JUN., and MARKEL, I. D.: 'A normalized digital filter structure', IEEE Trans., June 1975, ASP-23, pp. 268-277 assumed to be driven by a signal source having an equivalent output resistance of at least 50Q. -vc SILICON BIPOLAR LASER A N D LINE DRIVER I C WITH SYMMETRICAL OUTPUT PULSE SHAPE OPERATING U P TO l 2 G b i t l s H.-M. Rein, E. Bertagnolli, A. Felder and L. Schmidt Indexing terms: Integrated circults, Driver circuits, Lasers, ODtical communicafion A silicon bipolar laser and line driver IC with an outstanding symmetry of the (single-ended)output pulse shape is presented. The pulse shape can be optimally adjusted via only two external potentiometers, independent of operating speed and output current range. The circuit, fabricated in a 0.8pm selfaligned double-polysilicon technology, operates up to 12Gbitis. In high-speed digital communication systems and corresponding measurement equipment the shape of the generated pulses should be as symmetrical as possible with respect to the switching level. This can be easily achieved by operating the circuits in the differential mode (see, for example, Reference I). However, there are circuit applications which do not allow differential operation at the input and output terminals. For example, the laser driver at the transmitting end of an opticalfibre link must drive the laser diode by a single-ended output. Moreover, in measuring equipment the users often prefer single-ended signals at the input and output terminals to simplify the measuring setup. However, as a consequence of single-ended operation the demand on pulse symmetry has often not been sufficiently met by bipolar circuits (unless expensive measures using hybrid components were taken or a strong degradation of the pulse steepness could be tolerated). This is confirmed by the eye diagrams generated with the fastest bipolar laser driver ICs reported to date 12-43 with maximum data rates up to 11Gbit/s [2]. In this Letter a high-speed silicon bipolar driver IC, operating up to I2Gbit/s, is presented which does not have this disadvantage. The circuit principle used allows symmetrical single-ended pulses to be adjusted without degradation of the edge steepness.* Fig. 1 shows the diagram of the driver circuit which is mainly provided for driving laser diodes, e.g. via a matched 25R transmission line [l, 51. The circuit configuration is completely symmetrical (differential operation). It is built from two different stages, each consisting of three emitter follower pairs (E) followed by a limiting differential emitter-coupled stage (DS). The currents of both the DS (and thus the voltage swings) and the currents through the emitter follower pair E6 are off-chip adjustable. This also holds for the input offset voltages Vos, and Vos2 of both differential stages. The reasons for these options are given below. The circuit inputs, which are matched by on-chip 50R resistors (note R , % s o n ) , are * For a more detailed description of the theoretical background, see Reference 5 ELECTRONICS LETTERS 2nd July 1992 11- Vol. 28 No. 14 " m Fig. 1 Circuit diagram ojdriver IC The only off-chipcomponents are the potentiometers and external load ( R L ) For application as a laser driver, an external loading of R , = 25 R is assumed [I, 51 and the on-chip output resistor is chosen to be RQ = 75R to halve double reflections. Thus, a source current through the output stage DS2 of I, = 40mA results in an output current swing of AIQ = 30mA. By using the external voltage V,, I, and the AIQ can easily be controlled as required for laser-driver applications. A corresponding deterioration of the eye diagram can be avoided if simultaneously with I, the current through the driving emitter-follower stage (E6) is changed (see Fig. 1). As confirmed by measurements, no speed degradation occurs within the intended control range of 20mA II, I53mA (15mA 5 AI2 5 40mA). The circuit was fabricated with the bipolar part of a selfaligned double-polysilicon BiCMOS technology characterised by 0.8pm lithography which results in a minimum effective emitter width of 0.4pm and a transit frequency of the transistors of -25GHz (VcE= IV) [SI. However, it should be noted that the minimum emitter width is not necessarily the optimum; e.g. for output transistors TQ and T,.a n d for the transistor current sources, the effective emitter width was doubled (i.e. 0.8 pm). The total power consumption at a single supply voltage of 5 V is 1.2 W (including the external load). For completeness, some measurement results are presented before the measures used for achieving symmetrical pulse shapes are discussed. In all cases laser driver application was assumed ( R L = 25R, R, = 750) with I, = 40mA. The chip was mounted unpacked on a low-dielectric substrate and driven by a pulse pattern generator. Fig. 2 shows three output eye diagrams of the circuit in Fig. 1 measured at IOGbit/s. In Fig. 2a the differential output voltage up,, is given. The corresponding single-ended output voltage up is shown in Fig. 2b which demonstrates the asymmetry of the pulse shape. Using simple adjustment of two external potentiometers as discussed below, a symmetrical pulse shape of up is obtained without degradation of the edge steepeness, as demonstrated by Fig. 2c. It should be noted that in Fig. 2a and b the bases of the transistors Tn and T Q of the output stage are driven by symmetrical differential signals. Therefore, the asymmetry in Fig. 2b is mainly caused by the transistor junction capacitances loading the common emitter node. As a consequence, the sum of collector currents i, and i;. can differ from the constant source current I,. Two main reasons for the asymmetrical pulse shape can be observed (see Fig. 26) [SI: (1) At the end of the turn-on transient of TQ the parasitic capacitance at the common emitter node allows i, to exceed I, resulting in an undershoot of the falling edge of U,. In contrast, no overshoot in the rising edge of U, is observed, because the collector current cannot change its sign. (2) The turn-on delay of Tp (falling edge of up) is longer than its turn-off delay. This effect, which is well known from tran1295

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