A BASEBAND PULSE SHAPING FILTER FOR GAUSSIAN MINIMUM SHIFT KEYING 1 2 3 3 N. Krishnapura , S. Pavan , C. Mathiazhagan ,B.Ramamurthi 1 Department of Electrical Engineering, Columbia University, New York, NY 10027, USA 2 Texas Instruments, Edison, NJ 08837, USA 3 Department of Electrical Engineering, Indian Institute of Technology, Chennai, 600036, India Email: [email protected] measurement results. ABSTRACT A quadrature mo dulation scheme to realize the Gaussian pulse shaping is used in digital commu- same function as Fig. 1 can be derived. In this pap er, nication systems like DECT, GSM, WLAN to min- we consider only the scheme shown in Fig. 1. imize the out of band sp ectral energy. The base- band rectangular pulse stream is passed through a 2. GAUSSIAN FREQUENCY SHIFT lter with a Gaussian impulse resp onse b efore fre- KEYING GFSK quency mo dulating the carrier. Traditionally this The output of the system shown in Fig. 1 can b e describ ed is done by storing the values of the pulse shap e by in a ROM and converting it to an analog wave- Z t form with a DAC followed by a smo othing lter. g d 1 y t = cos 2f t +2k c f This pap er explores a fully analog implementation 1 of an integrated Gaussian pulse shap er, which can where f is the unmo dulated carrier frequency, k is the c f result in a reduced power consumption and chip mo dulating index k =0:25 for Gaussian Minimum Shift f area. Keying|GMSK[1] and g denotes the convolution of the rectangular bit stream bt with values in f1; 1g 1. INTRODUCTION with the impulse resp onse of the shaping lter unity gain at dc ht. The characterizing quantity of this system is Gaussian pulse shaping of the baseband pulse stream is the BT pro duct, where B is the -3 dB bandwidth of the resorted to in digital communication systems like DECT b Gaussian lter and T is the bit p erio d of the input stream. and GSM in order to limit the sp ectral energy outside b A smaller value of BT implies a longer resp onse relative the transmission band. Gaussian Frequency Shift Key- b to T of the Gaussian lter to a unit pulse. Typical num- ing GFSK [1] is a scheme where a Gaussian ltered pulse b b ers for BT are in the range 0.3, 0.8. For BT =0:5, stream is used to frequency mo dulate the carrier. Concep- b b the commonly used duration b eyond which the unit pulse tually, this is done by passing the baseband rectangular resp onse is truncated is 3T 1:5T from the p eak. pulse stream with values in f1; 1g through a lter with b b a Gaussian impulse resp onse. The same can b e thought 3. DIGITAL PULSE SHAPING of as adding or subtracting delayed versions of the unit pulse one bit wide pulse of unit amplitude resp onse of The blo ck diagram of a digital pulse shaping system is the Gaussian lter dep ending on the sign of the input bit. shown in Fig. 2. The input bit stream is fed into a shift The resulting \smo othed" stream is fed to a voltage con- register, whose length is equal to the number of bit du- trolled oscillator VCO as shown in Fig. 1. rations over which the unit pulse resp onse extends for Traditionally, this pulse shaping is done digitally. BT = 0:5, this is 3. The ROM contains the samples b In order to implement the system, the ideal unit pulse re- of the resp onse of the Gaussian lter for every p ossible sp onse which extends from 1 to +1 has to b e truncated combination of the bit sequence in the shift register. If to some nite duration. The samples of the truncated step the Gaussian is sampled m times in every p erio d and the k resp onse are stored in a ROM. A DAC converts this dig- pulse resp onse extends over k bits, the ROM has 2 m ital data to a staircase waveform which is smo othed bya values stored in it. continuous time lter and fed to the VCO Fig. 2. The output of the ROM feeds a DAC. The staircase In this pap er, we describ e a more direct realization waveform at the output of the DAC is ltered using a of Fig. 1, using a continuous time lter whose impulse re- continuous time lter b efore b eing fed to the VCO. sp onse is approximately Gaussian. We present the design In commonly used communication systems, 6{8 of this lter and show that in many cases, the complexity bits are required in the DACtokeep the spurious emission of this lter is the same as that of the smo othing lter due to the quantization error b elow the sp eci ed limit [2 ] used in the digital approach. E ectively, the ROM and [3 ] [4] [7 ]. For a given bit rate, a larger sampling rate re- the DAC can be eliminated from Fig. 2, resulting in a laxes the requirements on the smo othing lter, but this is reduction in the p ower consumption and chip area. dicult to achieve with high bit rates due to constraints on the p ower dissipation. For mo derate sampling rates, In the next section, we brie y discuss the GFSK the lter should have a large attenuation outside the band. system of Fig. 1. In section 3., we consider the design In addition, it should havea linear phase in order not to issues in a digital pulse shaping scheme. In section 4., distort the pulse shap e. To satisfy b oth these criteria, a we derive the continuous time Gaussian pulse shaping l- high order lter is necessary. To get a feel for the numb ers, ter and present some simulation results demonstrating its we note that in order to satisfy the DECT sp eci cations, suitability. Section 5. deals with the implementation of with BT = 0:5, 6 samples per bit duration and a 7{8 the pulse shap er in a 2 m n{well CMOS pro cess and the b 0-7803-4455-3/98/$10.00 (c) 1998 IEEE th in order to have sucient ro om for the tuning voltage at bit DAC,a7 order b essel lter is required for smo oth- the gate of the MOS resistors. The bu er used is a folded ing [5 ]. With a low sampling rate, the lter also requires casco ded di erential pair in unity gain feedback Fig. 6. some automatic tuning scheme to set the cuto frequency Two bu ers are used for each di erential biquad. accurately. The control voltage for the MOS resistors is derived 4. ANALOG PULSE SHAPING from the master{slave tuning scheme shown in Fig. 7 see [6 ] for details of op eration. The feedback lo op forces the The hinttowards obtaining a continuous time lter with a resonant frequency of the master lter another tunable Gaussian impulse resp onse comes from the \Central Limit Sallen{Key biquad to b e equal to the reference frequency. Theorem" CLT in statistics. This states that the proba- th By virtue of on chip comp onent matching, the i biquad bility density function p df of the sum of a large number has a resonant frequency of using the relations in Fig. 5 of indep endent identically distributed random variables r approaches a Gaussian. Since the p df of the sum of ran- C C 1;r ef 2;r ef dom variables is obtained by convolving the individual 2 ! =2f p;i ref pdf s, we infer that the cascade of a large numb er of iden- C C 1;i 2;i tical lters with p ositive impulse resp onses has a Gaus- which, b eing a function of capacitor ratios, can be set sian impulse resp onse. This can b e veri ed by cascading accurately on a chip. a large numb er of bu ered rst order RC sections. The reference frequency is chosen to be 2 MHz. As the name implies, CLT is a limiting result, and a The master lter is designed for ! = 2 2 MHz and p realization with as small a numb er of impulse resp onses a Q =1. TheQiskept low in order to avoid large voltage lter with as small an order as p ossible is desirable in swings across the MOSFETs whichwould cause tuning er- practice. To this end, we make the following generaliza- rors due to nonlinearity and also to avoid large capacitor tions. ratios see the expressions in Fig. 5. A cascade of large numb er of lters has a Gaussian Casco ded di erential pairs, op erating in op en lo op, impulse resp onse. are used for the comparators in Fig. 7. The XOR gate drives a charge pump to realize the integrator in the lo op. The impulse resp onse of a lter with Gaussian mag- The details are not shown due to lack of space. nitude resp onse and linear phase is Gaussian. Fig. 8 shows the chip photograph. Fig. 9 shows Bessel lter is optimized for linear phase. the magnitude resp onse for di erent reference frequencies. The step resp onse is shown in Fig. 10. The impulse re- With these facts in mind and with the aid of simulations, sp onse is approximated by a resp onse to a narrow pulse we conclude that a high order b essel lter results in a and is shown in Fig.