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where the state vector x(k) represents the output signals at the are free from zero-input limit cycles and only the highpass delays. It is proved4 that the constant-input limit cycles in this section is free from both zero and constant-input limit cycles. filter can be eliminated by employing the technique of We show now that lowpass and bandpass structures of Refer- controlled-rounding if ence 2 can also be stabilised against constant-input limit cycles by making some simple modifications. The modified p = c p , p z ... ~ . - J ~ = ( I - A ) - ~ B (2) lowpass and bandpass structures are shown in Figs. 2 and 3, respectively, and their A, B and P matrices are given in eqns. 4 is machine representable. The structures given in References and 5, respectively: 1-3 are free of zero-input limit cycles, and hence for elimi- nating constant-input limit cycles they can be modified so that they satisfy eqn. 2. During this process, the recursive part of the filter is not altered so that the passband sensitivity pro- perty of the modified filter remains the same as that of the (4) original filter. Fig. 1 shows the modified structure of the one (5) The transfer functions H&) of Fig. 2 and H,,(z) of Fig. 3 are as follows: + mlXz + 1)’ -(1 HLA4 = 2’ + (m, + (1 + rn, + m2) - m2)z (1 + m,X1 - zz) H,‘z) = zz + (m, - mz)z + (1 + m , + m2) rQ\ Based on passive gyrator configurations, Kwan3 has proposed -1 “1 a2 several digital biquads which are free of zero-input limit cycles. In fact the structure of Reference 1 is also one of these. m These structures can also be modified in a similar way to Fig. 1 Modified version of structure of Reference I make them free from constant-input limit cycles. proposed in Reference 1. For this modified structure, the A, B Y. V. RAMANA RA0 21st March 1989 and P matrices are given below: C. ESWARAN Department of Electrical Engineering Indian Institute of Technology Madras 600 036. India Since P is machine representable, this structure can be stabil- References ised against constant-input limit cycles also using the tech- MEERKOTTER, K., and WEGENER, w.: ‘A new second-order digital nique of controlled-rounding. The transfer function H ( z ) filter without parasitic oscillations’,Arch. Elek. Ubertragung, 1975, realised by this structure is given below: 29, (7/8),pp. 312-314 A., ANTONIOU, and REZK, G.: ‘Digital filter synthesis using M. zz - (az - k , - k,)z ( k , + k,a,) - concept of generalised-immittance convertor’, IEE J. Electron. Cir- H ( z ) = -2 z2 - (a, + az)z + (1 + a1 - a z ) cuits & Syst., 1977, 1, pp. 207-216 KWAN, H : ‘Design of passive second-order digital filters’. Proc. . IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 1982, Based on the GIC analogue configuration, Antoniou and pp. 286289 Rezk’ have proposed five second-order digital structures DINIZ, P. s. R., and ANTONIOU, A.:‘On the elimination of constant- which realise lowpass, highpass, bandpass, notch and allpass input limit cycles in digital filters’, IEEE Trans. Circuits Syst., transfer functions. It is known5v6 that all these five structures 1984, CAS-31, (7), pp. 67M71 FSWARAN, c., and GANAPATHY, v.: ‘On the stability of digital filters mt designed using the concept of generalised-immittance convertor’, IEEE Trans. Circuits & Syst., 1981, CAS-28, pp. 745-747 --e PRASAD, K. s., and ESWARAN, c.: ‘Constant-input stability and multiple-output realization of GIC digital filters’, IEEE Trans. A -I Fig. 2 Modified GIC lowpass structure 1 Circuits Syst., to be published ON ARRIVAL OF PATHS IN FADING MULTIPATH INDOOR RADIO CHANNELS Indexing terms : Radiowave propagation, Radiocommunication, 1 Radio links, Data transmission Arrival of the paths in fading multipath channels obtained from several manufacturing floors and college campus labor- atories at 910 MHz are studied. The discrepancies between the empirical distribution of the arriving paths and Poisson arrivals are discovered. The modified Poisson process is shown to fit the arriving paths closely. Introduction: Recently, wideband meas~rementsl-~ and sta- 1708131 tistical modelling4 of indoor radio channels has attracted tre- Fig. 3 Modified GIC bandpass structure mendous attention for applications in oflices, manufacturing ELECTRONICS LETTERS 8th June 1989 Vol. 25 No. 12 763 floors, warehouses, campuses and hospitals.' All the reported Results and discussion: For the path arrival distribution, the wideband measurements and modellings are concerned with arrival time of the paths in each profile is divided into 5ns the statistics of the multipath spread. The only attempt in bins. The empirical curves were obtained by counting the modelling the arrival of the paths in a building, is reported by number of paths in the first N bins of each measured profile. Saleh and V a l e n ~ u e l a . ~ The probability of occupancy in each bin was then plotted Saleh and Valenzuela4 use the mathematical model orig- against the arriving delay (path index) and these points were inally suggested by Turin for urban radio data communica- connected by curves for clarity. The procedure is repeated for tions.' In this model, the complex envelope of the channel arriving paths in the first 5, 10, 15 and 20 bins. After careful impulse response is represented by visual inspection of many profiles, the threshold level L employed to detect a genuine path in any bin was fixed at h(t) = 1 Clk 6(t k= 1 - Tk)e'" 30dB below the highest peak in the profile. The transmitted impulse d(t) is received as the sum of L paths. The path k has amplitude akrdelay 7k and carrier phase shift Ok. Based on the measurements in a research laboratory, the arrival of the paths is modelled as a Poisson process. The paths are presumed to have independent uniform phases, and independent Rayleigh amplitudes. This letter reports the discrepancies following the Poisson arrival presumption and, based on empirical data collected from different manufacturing floors and college campus areas, presents a more accurate modified Poisson process to fit the arrival of the paths. Measurements: The measurement set-up for the multipath propagation experiments, involved modulation of a 910 MHz signal by a train of 3 ns (3 dB width) pulses with 500 ns repeti- 0 2 4 6 8 / 1 0 1 2 14 tion period.' The stationary receiver included a digital storage path index 15oo1,l scope and a personal computer. The transmitter and the recei- ver used vertically polarised quarter-wave dipole antennas Fig. 3 Comparison of empirical and theoretical Poisson distribution for placed about 1.5m above floor level. The transmitter was college laboratory areas moved to various locations in the site, and the received multi- Figs. 2 and 3 show a comparison between the empirical path profiles were stored in the computer. The distance path number distributions and the theoretical Poisson dis- between the transmitter and the receiver varied between 1 and tributions, for various values of N , for the manufacturing floor 50m. A total of about 480 profiles were collected from mea- areas and the college campus areas, respectively. The theoreti- surements made in five areas on three different manufacturing cal curves governed by the Poisson hypothesis use a mean floors' and two areas on college laboratory floors. Each profile was an average over time of 64 profiles collected over path arrival rate A for any value of N . The Poisson path 15-20s. A typical multipath profile, obtained from one of the number distribution is then given by locations is shown in Fig. 1. AK P , ( K ) = - e-' 52 K! This is plotted as a continuous curve though it has values for only integer path numbers. We observe considerable discrep- ancies between the empirical and the Poisson distributions for all values of N . This discrepancy reflects a tendency of the paths to arrive in clusters, rather than in a random manner. To explain similar discrepancies a modified Poisson model was proposed by Suzuki for urban radio channel modelling.6 According to the modified Poisson model, whenever there is a path in a bin, the mean arrival rate for the next bin is changed by a factor of K . The process becomes a standard Poisson sequence for K = 1. Depending on whether K is 200 250 300 350 400 450 greater or less than 1, the probability that there will be time , n s another path within the next bin increases or decreases, respectively. The underlying path occupancy rate Ai,is calcu- Fig. 1 Multipath profle obtained from location on manufacturingfloor 0 2 4 6 8 10 1 2 14 0 2 4 6 8 1 0 1 2 1 4 path index path index PT 1580141 Fig. 2 Comparison of empirical and theoretical Poisson distribution for Fig. 4 Comparison of empirical and modified Poisson distribution for manufacturing floor areas manufacturing floor areas 764 ELECTRONICS LETTERS 8th June 1989 Vol. 25 No. 12 lated from the empirical path occupancy rate r i by the follow- BULTITUDE, R. J. c., MAHMOUD, s. A.,and SULLIVAN, w. A.: ‘A com- ing relation : parison of indoor radio propagation characteristics at 910 MHz and 1.75GHz’. IEEE JSAC, Jan. 1988,pp. 2Cb-30 R. A.: SALEH, A. A. M.,and VALENZUELA, ‘A statistical model for rn indoor multipath propagation’. IEEE JSAC, Feb. 1987, pp. I.” = nZ1 (K - l)rn-l +1 128-137 PAHLAVAN, K.: ‘Wireless intra-office networks’, A C M Trans. O f . where 1, = r I . The path number distribution is then calcu- Inf Syst., 1988,(2) lated successively, using the recursive formulas given for this SUZUKI, H.: ‘A statistical model for urban radio propagation’, model in Reference 6. Figs. 4 and 5 show a comparison IEEE Trans. Commun., 1977, COM-25 . O NT N . TURIN, G L., CLAPP, F. D., J H S O , T L., FINE, S. B., and LAVRY, D.: between the empirical path number distributions and the ‘A statistical model of urban multipath propagation’, IEEE Trans. modified Poisson distributions for the manufacturing floors Veh. Technol., 1972, ( I ) , pp. 1-9 and college campuses, respectively. The curve fittings show considerable improvement over those of the Poisson model. 0.5,- 1 0 M H z CMOS OTA-C VOLTAGE-CONTROLLED QUADRATURE OSCl LLATOR Indexing terms: Circuit theory and design, Oscillators, Inte- grated circuits, Nonlinear networks A quadrature-type voltage-controlled oscillator with oper- ational transconductance amplifiers and capacitors (OTA-C) is presented. A monolithic integrated CMOS test circuit is introduced to verify theoretical results. The attainable frequency range of oscillation of the chip test circuit is path index 3-10.34 MHz. The total harmonic distortion (THD) is 0.2& 1.87% for corresponding peak-to-peak amplitude voltages Fig. 5 Comparison of empirical and modified Poisson distribution for between lOOmV and 1V. This amplitude can be controlled college laboratory areas either by using a diode connection of two MOS transistors or a proposed nonlinear resistor. This improvement is because the modified Poisson model uti- lises the empirical probability of occupancy for each bin, whereas the Poisson model just uses the sum of the probabil- Introduction: The quadrature mode of operation is a well- ities of occupancy for all bins. The optimum values of K known mode of waveform generation in the field of sinewave calculated for the manufacturing floors and the campus oscillator^.'-^ Two sinewaves in quadrature, that is with 90” environments are shown in Table 1. phase difference, are generated. This is accomplished with a two-integrator loop. Besides this, some form of regeneration is Table 1 OPTIMUM VALUES OF K usually included to ensure that the oscillation is created. The Number Number of bins N poles are initially located in the right half-plane (RHP) of the of complex frequency plane and then pulled back by a nonlinear Area locations 5 10 15 20 amplitude limiter. ~~ To make a quadrature oscillator into a voltage-controlled Manufacturing 288 1.184 0.904 1.03 2.46 oscillator (VCO), the integrators can be implemented using floors OTAs. The output current of a CMOS-OTA is controlled by the differential input voltage and its transconductance gain 9,. College 196 0.892 0.64 0.61 0.75 Furthermore, g, can be varied over several octaves by adjust- laboratories ing an external DC amplifier bias current IOk ,4 i.e. Conclusions: Fading multipath profiles obtained from several different manufacturing floors and some college laboratories at 910 MHz, have been used to model the arrival of the paths. where h, is a process-dependent parameter. The empirical distribution was compared with the Poisson In this letter an integrated CMOS OTA-C quadrature VCO and the modified Poisson distributions and the modified using a 3 pm double metal technology (processed by MOSIS, Poisson model is shown to provide a better fit. This observa- Marina del Rey, CA) is presented. The frequency of oscillation tion indicates that the paths tend to arrive in clusters rather can be externally adjusted by nearly two octaves with a THD than in a random manner. less than 1.9%. The real part of the poles can be adjusted, in practice, independently of the frequency of oscillation. A c k n o w l e d g m e n t ; This work was supported in part by the National Science Foundation under contract NCR-8703435. I 1 R. GANESH 8th March 1989 K. PAHLAVAN Department of Electrical Engineering Worcester Polytechnic Institute Worcester, M A 01609, U S A A -I-& - - L L - - 1 - m Fig. 1 Proposed O T A - C quadrature oscillator structure References PAHLAVAN, K.,GANFSH, R., and HOTALING, T.: ‘Multipath propaga- T h e o r y : The proposed quadrature VCO architecture’ is 1 tion measurements on manufacturing floors at 910 MHz’, Electron. shown in Fig. 1. It has one inverting (gml,C,) and one nonin- Lett., 1989, 25, pp. 225-227 verting integrator ( g m 2 , C2). R , is a nonlinear resistor whose 2 DEVASIRVATHAM, D. M. I.: ‘Time delay spread measurements of i / v characteristics are depicted in Fig. 2a. R , provides the 850 MHz radio waves in building environments’. GLOBECOM nonlinear amplitude limiter function. Implementations are dis- 1985 Conf. Rec., vol. 2, Dec. 1985, pp. 970-973 cussed later. The OTAs associated with gm3 and gm4 allow ELECTRONICS LETTERS 8th June 1989 Vol. 25 No. 12 765